Les schémas de Codage Vidéo Linéaire (CVL) inspirés de SoftCast ont émergé dans la dernière décennie comme une alternative aux schémas de codage vidéo classiques. Ces schémas de codage source-canal conjoint exploitent des résultats théoriques montrant qu'une transmission (quasi-) analogique est plus performante dans des situations de multicast que des schémas numériques lorsque les rapports signal-à-bruit des canaux (C-SNR) dièrent d'un récepteur à l'autre. Dans ce contexte, les schémas de CVL permettent d'obtenir une qualité de vidéo décodée proportionnelle au C-SNR du récepteur.

Une première contribution de cette thèse concerne l'optimisation de la matrice de précodage de canal pour une transmission de type OFDM de ux générés par un CVL lorsque les contraintes de puissance dièrent d'un sous-canal à l'autre. Ce type de contrainte apparait en sur des canaux DSL, ou dans des dispositifs de transmission sur courant porteur en ligne (CPL).

Cette thèse propose une solution optimale à ce problème de type multi-level water lling et nécessitant la solution d'un problème de type Structured Hermitian Inverse Eigenvalue. Trois algorithmes sous-optimaux de complexité réduite sont également proposés. Des nombreux résultats de simulation montrent que les algorithmes sous-optimaux ont des performances très proches de l'optimum et réduisent signicativement le temps de codage. Le calcul de la matrice de précodage dans une situation de multicast est également abordé.

Une seconde contribution principale consiste en la réduction de l'impact du bruit impulsif dans les CVL. Le problème de correction du bruit impulsif est formulé comme un problème d'estimation d'un vecteur creux. Un algorithme de type Fast Bayesian Matching Pursuit (FBMP) est adapté au contexte CVL. Cette approche nécessite de réserver des sous-canaux pour la correction du bruit impulsif, entrainant une diminution de la qualité vidéo en l'absence de bruit impulsif. Un modèle phénoménologique (MP) est proposé pour décrire l'erreur résiduelle après correction du bruit impulsif. Ce modèle permet de d'optimiser le nombre de sous-canaux à réserver en fonction des caractéristiques du bruit impulsif. Les résultats de simulation montrent que le schéma proposé améliore considérablement les performances lorsque le ux CVL est transmis sur un canal sujet à du bruit impulsif.

In the last decades, a huge research eort has been devoted to design video coding and transmission systems to get the best received video quality for a given amount of channel resources. This is of paramount importance for cellular broadcasting, where channel conditions may be varying with time and among receivers, for multimedia transmission in wireless networks, but is also for communication over wired channels, such as Digital Subscriber Line (DSL) or Power Line Telecommunication (PLT) channels. The rst case is illustrated in Figure 1.1, where a multimedia transmission is performed via DVB-T (Digital Video Broadcasting -Terrestrial) to dierent users experiencing channels with dierent characteristics. The second case is depicted in Figure 1.2, where a multimedia server transmits data along the power line using HomePlug AV2 [YAA + 13], to receivers which are at dierent locations.

Traditional solutions for these use-cases consist in using a non-scalable or a scalable video encoder [START_REF] Schwarz | Overview of the scalable video coding extension of the H.264/AVC standard[END_REF]. Nevertheless, the source encoding is performed without knowing the actual channel characteristics, and this may cause channel underused or digital cli. This problem is illustrated in Figure 1.3. A single layer MPEG4 codec selects a bit rate for video compression equal to the channel transmission rate, depending on the modulation and channel FEC (Forward Error Correction) scheme. Assume that the chosen modulation and coding scheme (MCS) is 16 QAM and rate 1/2 convolutionnal code. If the channel quality (C-SNR) is less (for example at 10 dB) than the C-SNR for which the MCS is adapted, a cli eect will appear. On the other hand, if the channel quality is improved, for example C-SNR at 15dB, but the bit rate does not increase accordingly, then there is saturation problem. Hence in broadcast, choosing the bit rate that ts the receivers with the worst channel penalizes users with better channel conditions. Even though scalable coding facilitates transmission rate adaptation compared to a non-scalable scheme (such as H.264/AVC or HEVC [START_REF] Wiegand | Overview of the H.264/AVC video coding standard[END_REF][START_REF] Sullivan | Overview of the high eciency video coding (HEVC) standard[END_REF]) at similar coding rate, the global coding eciency of scalable schemes decreases with the number of scalability layers [START_REF] Wien | Performance analysis of SVC[END_REF].

Joint source-channel video coding (JSCVC) has the potential of dramatically improving the quality of the received video in such challenging conditions, as demonstrated by the breakingthrough SoftCast video coding and transmission system [START_REF] Jakubczak | Softcast: Clean-slate scalable wireless video[END_REF]. SoftCast is a JSCVC scheme that encodes the video content with linear-only operators (such as a full-frame DCT and scaling). For this reason, SoftCast-inspired schemes maybe referred to as Linear Video Coding networks show very clearly the advantages of SoftCast [START_REF] Jakubczak | Softcast: Clean-slate scalable wireless video[END_REF]. The details of SoftCast will be presented in Section 2.1.

Since SoftCast and, more generally, LVC does not use entropy coding and temporal prediction, the eciency in terms of pure source coding is less than that of classical video coders.

Nevertheless, signicant work has been done recently to improve the eciency of LVC. A chunk shape and size optimization is proposed in [XWF + 13] Nevertheless, all of the previously mentioned papers consider a single constraint on the total transmission power when evaluating the optimal scaling factors for the chunks. For some channels, such as DSL or PLT [YAA + 13], OFDM is employed and the power constraint depends on the subchannel, see for example, the constraint on the Power Spectrum Density (PSD) in PLT shown in Figure 1.4. Similarly, for multi-antenna transmission, each antenna may have its own power constraint [START_REF] Yu | Transmitter optimization for the multi-antenna downlink with per-antenna power constraints[END_REF]. In such situations, new power allocation schemes are needed. This is one of our contributions in this thesis. In Chapter (3), we propose an optimal precoding and decoding matrix design method for channels with per-subchannels power constraints. This methods involves multi-level water-lling [START_REF] Palomar | Optimum linear joint transmitreceive processing for MIMO channels with qos constraints[END_REF] and the solution of an inverse eigenvalue problem [START_REF] Zha | A note on constructing a symmetric matrix with specied diagonal entries and eigenvalues[END_REF]. Then in Chapter (4), suboptimal power allocation methods will be presented, reduce signicantly the execution time and have negligible performance loss compared to the optimal design techniques of Chapter (3).

Another important issue for video transmission is the mitigation of impulse noise. Several communication channels may be prone to impulse noise, such as the Digital Subscriber Line (DSL) [START_REF] Nedev | Analysis of the impact of impulse noise in digital subscriber line systems[END_REF] and the Power Line Telecommunications (PLT) channels [START_REF] Zimmermann | Analysis and modeling of impulsive noise in broad-band powerline communications[END_REF]. Impulse noise has a high magnitude (its power may be 50dB above that of the background noise), and when it is bursty, may corrupt the channel for more than 1 ms [START_REF] Zimmermann | Analysis and modeling of impulsive noise in broad-band powerline communications[END_REF]. If impulses are not corrected, the communication performance may be signicantly degraded [START_REF] Tareq | Impulse noise estimation and removal for ofdm systems[END_REF][START_REF] Lin | Impulsive noise mitigation in powerline communications using sparse bayesian learning[END_REF], even if LVC schemes are more robust than classical video coding scheme to noise and channel mismatch [START_REF]Softcast: one-size-ts-all wireless video[END_REF]. In Chapter 5, we have addressed this mitigation of impulse noise for SoftCast-based video transmission problem.

Contributions

In my thesis, we address two problems related to SoftCast-based video coding and transmission systems. The rst is the power allocation under per-subchannel power constraints; the second is the optimal subchannel provisioning for impulse noise correction.

The rst original contribution of this thesis is to optimize the power allocation for SoftCastbased video coding and transmission systems when the channel is made up of several parallel subchannels with dierent power constraints. The optimization consists in minimizing the receiver mean square error (MSE), and to do this, one has to nd a precoding matrix that transforms the chunks' coecients, modeled as independent Gaussian sources with dierent variances, so that they match the individual subchannel power constraints. One has also to determine an optimal decoding matrix at receiver. The optimization problem may be solved considering Karush-Kuhn-Tucker (KKT) conditions [START_REF] Boyd | Convex optimization[END_REF]. Nevertheless, this method leads to a system of nonlinear equations, which is dicult to solve directly as the number of subchannels increases.

A similar problem has been addressed in [START_REF] Yu | Transmitter optimization for the multi-antenna downlink with per-antenna power constraints[END_REF] in the context of downlink beamforming with per-antenna power constraints. Strong duality is used to transform this problem to an uplink beamforming problem (with signal to interference plus noise constraints) with uncertain noise. That solution is not directly amenable to our problem, since the downlink channel characteristics considered in [START_REF] Yu | Transmitter optimization for the multi-antenna downlink with per-antenna power constraints[END_REF] are independent of the beamforming vector to optimize. In our case the optimal decoding matrix depends on the precoding matrix.

In our work, the design of optimal precoding and decoding matrices with per-subchannel power constraints, after reformulation, will lead to an inverse eigenvalue problem. Such problem is found in several application contexts, see [START_REF] Chu | Inverse eigenvalue problems[END_REF] and the reference therein. One focuses on the specic class of Structured Hermitian Inverse Eigenvalue (SHIE) problem. This problem has been considered in [START_REF] Lee | Optimal linear coding for vector channels[END_REF] and encountered later in the context of CDMA [START_REF] Viswanath | Optimal sequences and sum capacity of synchronous CDMA systems[END_REF] and MIMO communication [START_REF] Palomar | Optimum linear joint transmitreceive processing for MIMO channels with qos constraints[END_REF][START_REF] Palomar | MIMO transceiver design via majorization theory[END_REF]. The design proposed in [START_REF] Lee | Optimal linear coding for vector channels[END_REF] is optimal when a set of sucient conditions on some problem parameters (in our case, the chunk variances, the subchannel power constraints, and the noise variances) are satised. When they are not satised, an heuristic approach has been proposed. Nevertheless, it is suboptimal and indeed no proof of optimality is provided in [START_REF] Lee | Optimal linear coding for vector channels[END_REF]. The multi-level water-lling approach proposed in [START_REF] Palomar | Optimum linear joint transmitreceive processing for MIMO channels with qos constraints[END_REF][START_REF] Palomar | MIMO transceiver design via majorization theory[END_REF], which aims to minimize the total transmission power with per-subchannel MSE constraints, allows one to nd iteratively the optimal solution, but with a large computing cost.

The main contributions of the rst part of our work consist in addressing the design of optimal precoding and decoding matrices with total and per-subchannel power constraints in the context of LVC. We provide an optimal solution and three lower-complexity alternative suboptimal solutions. For the optimal solution, the derivations of [START_REF] Lee | Optimal linear coding for vector channels[END_REF] are adapted, considering the majorization techniques used in the MIMO context by [START_REF] Palomar | Joint Tx-Rx beamforming design for multicarrier MIMO channels: A unied framework for convex optimization[END_REF] and the multi-level water-lling approach proposed in [START_REF] Palomar | Optimum linear joint transmitreceive processing for MIMO channels with qos constraints[END_REF][START_REF] Palomar | MIMO transceiver design via majorization theory[END_REF]. Inspired by the optimal approach, lowercomplexity suboptimal design methods are proposed which are able to reduce signicantly the design complexity. Moreover simulation results show that they have a very small performance degradation for most of the considered video sequences and are better than [START_REF] Lee | Optimal linear coding for vector channels[END_REF].

Moreover, we consider also the use case of point-to-multipoint video communication, which is a typical application of LVC schemes. In this case, the channel experienced by dierent users have dierent characteristics, in particular dierent noise levels. In such cases, the transmitter can only implement power allocation with respect to some target noise level, e.g., the average noise level among users [START_REF] Yu | Wireless scalable video coding using a hybrid digitalanalog scheme[END_REF][START_REF] Fan | Wavecast: Wavelet based wireless video broadcast using lossy transmission[END_REF], introducing thus a mismatch between the actual channels' noise and the one used to design power allocation. We analyze the robustness of the proposed schemes to mismatched channel characteristics in Sections 3.5, 3.6.5, and 4.6.3. This is very important, since these results show the applicability of the optimal precoding matrix design to the point-to-multipoint case.

The second contribution of our work is considering the problem of impulse noise mitigation when video is encoded using an LVC scheme and transmitted using an Orthogonal Frequency-Division Multiplexing (OFDM) scheme for multi-carrier modulation over a wideband channel prone to impulse noise. In the time domain, the impulse noise is modeled as independent and identically distributed Bernoulli-Gaussian variables. A Fast Bayesian Matching Pursuit (FBMP) algorithm [START_REF] Schniter | Fast bayesian matching pursuit, Information Theory and Applications Workshop[END_REF] is employed for impulse noise mitigation. This approach requires the provisioning of some OFDM subchannels to estimate the impulse noise locations and amplitudes. Provisioned subchannels cannot be used to transmit data and lead to a decrease of the video quality at receivers in absence of impulse noise. Using a phenomenological model (PM) of the residual noise variance after impulse mitigation in the subchannels, we have proposed an algorithm namely LVC with Optimal Subchannel Provisioning for Impulse noise Correction (LVC-OSP-IC), which is able to evaluate the optimal amount of subchannel to provision which minimizes the mean-square error of the decoded video at receivers. Simulation results show that the PM can accurately predict the number of subchannels to provision and that impulse noise mitigation can signicantly improve the decoded video quality compared to a situation where all subchannels are used for data transmission.

Organization of thesis

The rest of the thesis is organized as follows.

Joint source-channel coding schemes, SoftCast and its developments are presented in Chapter 2. Our results on optimal precoding and decoding matrix design are presented in Chapter 3.

We start in Section 3.3 by presenting the optimal power allocation when a total transmission power constraint is considered, which goes beyond the solution proposed in the original Soft-Cast, where a simplied precoding matrix design is considered. The design of the optimal precoding matrix under per-subchannel power constraints is presented in Section 3. are considered and show the advantage of optimal and suboptimal precoding matrix design approaches. The robustness of the proposed schemes to variations of the channel characteristics is analyzed in Sections 3.6.5, 3.5, and 4.6.3.

Chapter 5, presents the mitigation of impulse noise for SoftCast-based video transmission.

First, the application of FBMP for impulse noise mitigation is described in Section 5.4. Then Section 5.5 presents the method to compute the optimal number of subchannels to provision for impulse noise correction. Section 5.6 shows the simulation result, in which we can see that the mitigation of impulse noise for SoftCast-based video transmission has signicantly improve the performance. Chapter 6 concludes the thesis and presents some research perspectives.

Chapter 2

Related work and prior results

SoftCast [START_REF] Jakubczak | Softcast: Clean-slate scalable wireless video[END_REF] is a joint source-channel coding scheme that can resolve the unfairness problem encountered by conventional video codecs in the broadcast scenario. In the rst section of this Chapter, we present the principle of SoftCast which is an analog coding [START_REF] Jakubczak | Analog transmission of degradable content over wireless channels[END_REF][START_REF] Goblick | Theoretical limitations on the transmission of data from analog sources[END_REF] based joint source-channel video coding and video transmission scheme. In analog coding, the source component are directly mapped on the channel after multiplication with scaling factors.

The advantage compared to digital coding scheme will be shown. Then in Section 2.2, we illustrate the extensions of SoftCast that improve the global performance.

SoftCast: A joint source-channel coding scheme

Source-channel separation theorem tells us that in point-to-point communication we can perform source coding and channel coding separately. However, in the broadcast scenario, a joint source-channel coding scheme can be better than separate coding in some cases [START_REF] Gastpar | To code, or not to code: Lossy source-channel communication revisited[END_REF]. In Section 2.1.2 SoftCast [START_REF] Jakubczak | Softcast: Clean-slate scalable wireless video[END_REF] an analog coding based joint source-channel video coding and video transmission scheme is presented.

Information-theoretic foundations of SoftCast

In this section, we will present the information-theoretic ideas which support SoftCast. At rst the source-channel separation theorem is recalled. The denitions of rate R, channel capacity C and rate distortion function R (D) are given as in [START_REF] Thomas | Elements of information theory[END_REF].

Denition 1. Let X be a nite set of input channel symbols, Y be a nite set of output 2) an encoding function X n : {1, 2, . . . , M } → X n , yielding codewords x n (1) , x n (2) , . . . , x n (M ) .

The set of codwords is called codebook.

3) a decoding function

g : Y n → {1, 2, . . . , M } ,
which is deterministic rule that assigns a guess to each possible received vector.

The communication channel corresponding to these denitions is illustrated in Figure 2.1.

For a discrete memoryless channel without feedback one has p (y

n |x n ) = n i=1 p (y i |x i ). Denition 2. The rate R of an (M, n) code is R = logM n bits per transmission.
The choice of R will depend on the channel capacity, which is dened below.

Denition 3. The information channel capacity C of a discrete memoryless channel is

C = max p(x) I (X; Y ) ,
where I (X; Y ) is mutual information and the maximum is taken over all possible input distributions p (x) .

The capacity of a Gaussian channel with power constraint P and noise variance N is [CT06, Theorem 9.1.1]

C (P ) = 1 2 log 1 + P N bits per transmission. (2.1)
Denition 4. The maximum probability of error λ (n) for an (M, n) code is dened as

λ (n) = max i∈{1,2,...M } Pr (g (y n ) = i|X n = x n (i)) .
The channel coding theorem [CT06, Theorem 7.7.1] shows that for a discrete memoryless channel, for every rate R < C, there exists a sequence of 2 nR , n codes with maximum probability of error going to zero. Now let us introduce the rate-distortion code.

At rst, assumes a source sequence X 1 , X 2 , . . . , X n which are iid following p (x), x ∈ X and

X is a nite set. The source sequence X n is encoded to an index f n (X n ) ∈ 1, 2, . . . , 2 nR .
The decoder estimates X n of X n from this index and X n ∈ X n .

Denition 5. A 2 nR , n -rate distortion code consists of an encoding function

f n : X n → 1, 2, . . . , 2 nR , A decoding (reproduction) function, g n : 1, 2, . . . , 2 nR → X n .
The distortion D associated with the 2 nR , n code is dened as

D = Ed (X n , g n (f n (X n ))) where d (x n , x n ) = 1 n n i=1 d (x i , x i )
and d is a distortion measure and the expectation is with respect to the probability distribution on X D =

x n p (x n ) d (x n , g n (f n (x n ))) .
Denition 6. A rate distortion pair (R, D) is said to be achievable if there exists a sequence of 2 nR , n -rate distortion codes (f n , g n ) with lim n→∞ If the distortion D can be achieved for a suciently large n, we must have R (D) C. Source channel separation coding scheme works well as any joint source channel coding scheme in point-to-point communication provided the length of code n is innity. In multiusers scenario where the channel capacity of each receiver is dierent, the rates for each receiver are shown below.

x n p (x n ) d (x n , g n (f n (x n ))) D.
V n = g (Y n ). Let D = Ed V n , V n = 1 n n i=1 Ed V i , V i be
At rst, let us consider a simple case with one sender and two receivers [CT06, Example 15.6.6]. The sender emits a sequence of iid Gaussian variables with variance P . The channel between the sender and the receivers are assumed to be with additive white Gaussian noise (AWGN) but the variances of noise experienced by the receivers are dierent. For example, the rst receiver experiences a noise with a smaller variance σ 2 1 < σ 2 2 . In this case, the encoder could encode with a common rate R 2 (coarse version) to both receivers, the receiver whith the better channel can receive renement rate R 1 by using superposition coding [START_REF] Cover | Broadcast channels[END_REF]. Then the capacity region (R 1 , R 2 ) where the probability of error could go to zero is

R 1 < 1 2 log 1 + αP σ 2 1 (2.3) R 2 < 1 2 log 1 + (1 -α) P αP + σ 2 2 (2.4)
where 0 α 1.

To achieve this capacity region, the source has to use superposition coding. Receiver 2 which has the worst channel receives the coarse version of source symbol with distortion D (R 2 ).

Receiver 1 which has a better channel can receive the renement version. On the other hand, the coarse version can also be decoded by Receiver 1, hence the distortion of Receiver 1 is

D (R 1 + R 2 ) .
On the other hand, the condition of an optimal source-channel code [START_REF] Gastpar | To code, or not to code: Lossy source-channel communication revisited[END_REF] is

R (D) = C (P ) , (2.5)
where C (P ) represents the channel capacity which is a function of input cost (e.g. the power constraint P ). In [START_REF] Gastpar | To code, or not to code: Lossy source-channel communication revisited[END_REF], it has been shown that a joint source-channel coding scheme which sends directly this single iid Gaussian source with variance P over broadcast Gaussian channel performs well. This alternative solution is called as an uncoded scheme or it can be considerd as an analog coding scheme [START_REF] Goblick | Theoretical limitations on the transmission of data from analog sources[END_REF][START_REF] Jakubczak | Analog transmission of degradable content over wireless channels[END_REF]. At receiver side, the two receivers which respectively have Gaussian channel noise with zero mean and variance σ 2 1 and σ

R (D) = 1 2 log   P pσ 2 1 σ 2 1 +P   = 1 2 log 1 + P σ 2 1 = C (P ) .
(2.7)

Eq. (2.7) shows that the rate for Receiver 1 is equal to the corresponding channel capacity. It is also the case for Receiver 2. Therefore analog coding in this situation is an optimal sourcechannel code. On the other hand, the distortions of superposition coding for the Gaussian source and mean square error distortion measure is the distortion function of Gaussian source [START_REF] Gastpar | To code, or not to code: Lossy source-channel communication revisited[END_REF]. From (2.6), it can be deduced that the distortion rate function D (R) of Gaussian source is

D (R) = P 2 -2R . (2.8)
Therefore from (2.3), (2.4), and (2.8), the distortion region of superposition coding can be deduced. In [START_REF] Gastpar | To code, or not to code: Lossy source-channel communication revisited[END_REF], it has been shown that the analog joint source-channel coding scheme achieves a distortion pair point

P σ 2 1 σ 2 1 +P , P σ 2 2 σ 2 2 +P
which is strictly outside of distortion region achieved by superposition coding (see Figure 2.2). This is for the single Gaussian source.

The performance of this analog joint source channel coding with multi-variate Gaussian vector source is shown in below.

In [START_REF] Jakubczak | Analog transmission of degradable content over wireless channels[END_REF], the performance (the measure is mean square error) of analog coding based communication scheme and of digital communication is compared in point-to-point communication and in broadcast respectively with a multi-variate Gaussian vector source of dimension N .

The covariance of this source is a diagonal matrix with diagonal elements λ 1 , λ 2 , . . . , λ N , which are assumed in decreasing order. The source vector is transmitted over M AWGN channels with a specic SNR. In point-to-point communication, the optimum performance in terms of distortion of the analog coding based communication scheme and of the digital communication scheme are respectively D ana and D dig , which can be represented as and

D ana = k i=1 √ λ i 2 M SNR + K + N i=K+1 λ i , (2.9)
D dig = K K i=1 λ i (SNR + 1) M 1/K + N i=K+1 λ i ,
R (D) = N i=1 1 2 log λ i D i , (2.11)
where

D i =        γ if γ < λ i λ i if γ λ i , (2.12)
where γ is chosen such that N i=1 D i = D. Moreover, since the channel capacity of M AWGN channels (see (2.1)) is,

C (SNR) = M 2 log (1 + SNR) , (2.13)
one gets 2.10 by setting (2.11) equal to 2.13.

In [START_REF] Jakubczak | Analog transmission of degradable content over wireless channels[END_REF], the ratio

D dig
Dana is compared in dierent scenarios. It shows that in pointto-point communication, the analog scheme is better than digital system for a very low SNR.

In broadcast, analog scheme is nearly optimal as digital system when the dimension of source vector and dimension of channel is matched, otherwise the analog scheme is worse when the compression (N > M ) or expansion of bandwidth (N < M ). However the analog scheme is better when SNR of weak user is low. This advantage is helpful and has been used for video transmission in broadcast, in which the SNR is dierent among receivers. As we have seen in Figure 1.3, by using digital coding scheme for video compression in broadcast, it should decide the bit rate by considering the channel C-SNR of the worst channel and of the better channel. In this case, although the channel of one user has high C-SNR, the received video performance is not proportional to the channel quality. This is unfairness. Nevertheless an analog coding based video transmission scheme namely SoftCast [START_REF] Jakubczak | Softcast: Clean-slate scalable wireless video[END_REF], in which the received video performance is linear with the C-SNR. It is shown in Section 2.1.2.

SoftCast

The architectures of SoftCast is shown in Figure 2.3. The input video signal undergoes a linear 3D-DCT, consisting of a full-frame 2D-DCT followed by a temporal 1D-DCT on a Group of Pictures (GoP) of n F frames of n R × n C pixels. SoftCast works independently GoP by GoP.

After a GoP has been transformed, the resulting coecients are grouped into chunks. A chunk is a set of n r × n c spatial coecients belonging to the same temporal subband (assuming they follow a similar distribution). The n Ck chunks are sorted according to their variance λ i , where i = 1, . . . , n ck and only the rst of them may be sent, according to the bandwidth limitations and to the power constraint of the channel. More details of chunk selection will be given in Section 3.3. The map of the selected chunks is robustly transmitted (e.g. using a strong FEC) as metadata on the channel: since the number of chunks is relatively small, the rate overhead is not a big issue.

The selected chunks are scaled by power allocation for error protection in order to minimize the reconstruction MSE at the decoder. In SoftCast, only the total power constraint P T is considered. To compute the scaling factor g i for each chunk, in SoftCast, it is assumed that the channel SNR is high encough and the channel is AWGN. Under these hypotheses, it is possible to nd that

g i = λ -1 4 i P T n ck i √ λ i .
(2.14)

In order to increase the resilience to packet losses, SoftCast uses a Hadamard matrix to transform the chunks into equal-energy slices. The slices are then transmitted, e.g., via OFDM.

At the receiver, SoftCast uses the Linear Least Square Estimator (LLSE) to decode a sequence of received symbols. Thus, SoftCast uses linear transforms in compression, in error, and in loss protection. Combining with linear estimation at the receiver, all of these linear operations make that the quality of video in receiver scales linearly with the channel quality (C-SNR).

The performance of SoftCast is shown in Fig 2 .4. It shows that for conventional video coder, the video coder and channel coder should be been adjust when the SNR is changed, otherwise there is a cli eect when the SNR decreases or there is a saturation problem when the SNR increases. However for SoftCast, the video's quality increases linearly evaluated with channel quality. This is important in broadcast scenario, the receiver who has high SNR channel can receive a high PSNR video, the receiver who has low SNR channel receives low PSNR video.

It is unlike conventional video coder, which should choose an appropriate bit rate that could be transmitted over a low SNR channel, otherwise there is cli eect. This is unfair for the receiver who has high SNR channel. SoftCast has resolved this fairness problem.

However SoftCast does not use entropy coding and motion estimation to reduce the redundancy information in the video and no quantization for compression, which will in return decrease the performance of SoftCast [START_REF] Jakubczak | Analog transmission of degradable content over wireless channels[END_REF]. To overcome the limitation of pure analog communication scheme and also keep the benet of that, no cli or threshold eect. A general hybrid digital analog (HDA) source and channel coding version is proposed in [MP02, SPA02, SPA06],

which combines analog coding and digital coding. It provides a robust and graceful performance over a wide range channel SNR conditions. The one HDA architecture has been proposed in [START_REF] Polley | Hybrid channel coding for multiresolution HDTV terrestrial broadcasting[END_REF][START_REF] Schreiber | Advanced television systems for terrestrial broadcasting: Some problems and some proposed solutions[END_REF]. The SoftCast-based HDA will be presented in next section.

Improvements of SoftCast

Even though SoftCast oers a graceful video performance in broadcast scenario. However there is still a much room to improve SoftCast. For example, combines with digital coding scheme, e.g. quantization and motion estimation, to increase the performance; chunk size computation under power constraint and bandwidth constraint; the adaptation of SoftCast under more complex channel model and etc. In this section, we will present some important improvements of SoftCast.

Dcast

One important development of SoftCast is Dcast [START_REF] Fan | Distributed wireless visual communication with power distortion optimization[END_REF]. Dcast is a distributed video coding scheme [START_REF] Girod | Distributed video coding[END_REF] and it also can be considered as a HDA scheme [START_REF]Hybrid digitalanalog sourcechannel coding for bandwidth compression/expansion[END_REF]. The architecture of Dcast is shown in Figure 2.5. The key astute of Dcast is using side source information in encoding and decoding. The side information is computed by performing 2D-DCT over predicted frame, which is obtained by motion estimation and motion compsensation.

In encoding (Figure 2.5a), the side information is used for Coset. In decoding ( In the encoder of DCast (see Figure 2.5a), at rst there is coset coding. Let X be an original 2D-DCT transformed video frame in a vector form. X i is DCT coecient in ith subband. For each X i , Dcast has a uniform quantizer Q i () and get a residual value C i

C i = X i -Q i (X i ) .
(2.15) Then all C i are transmitted after power allocation under total power constraint P coset and modulation. In the receiver, C i is obtained after LMMSE decoding. The side information in receiver side is represented as S i s, where S i is the predicted DCT coecients of ith subband. In this case, the motion vectors which are estimated at encoder should be transmitted to receiver.

The motion vectors are transformed by DCT and then scaled under total motion vector power constraint P mv . Since the motion vector could not be perfectly transmitted at receiver, the side information S i s also should be computed at the encoder to nd the coset.

Next, the encoder designs Q i () with a specic quantization step such that

Q i (X i ) = Q i S i -C i (2.16)
with high probability. Therefore from (2.15) the reconstrcuted X i is

X i = C i + Q i S i -C i .
(2.17)

Moreover from (2.15),(2.16) and (2.17), it can deduce that the distortion D of X,

D = E X -X T X -X is close to the distortion D coset of C, D coset = E C -C T C -C .
In Dcast, it is shown that D coset is a function of P coset and P mv . Since the total power is P T = P coset + P mv , minimizing D then becomes an optimization problem min D coset s.t. P coset + P mv = P T .

Since in Dcast D coset is a convex function, the optimization problem can easily be solved by dierentiation of D coset with respect to variables P coset and P coset and set it to zero. Then the optimum power allocation pair (P coset , P mv ) is found and which will be used to scale the coset values and motion vector.

In Dcast, the side information is the predicted DCT coecients. In [SPX + 17], the side information is generated in a dierent way. A thumbnail of a image is decompressed and upsampled, and it is then used to retrieve correlated images from a database. Next a image is reconstructed by the retrieved images which will serve as a side information in decoding.

In decoding, using the local sparsity of residual image which is generated by subtracting the upsampled image from original image and exploiting the correlation between residual image and side information, the image is reconstructed.

WaveCast

Instead of using 1D-DCT to exploit the temporal correlation between frames, WaveCast [START_REF] Fan | Wavecast: Wavelet based wireless video broadcast using lossy transmission[END_REF] uses

g k = N p pa 2 σ k N P k=1 σ k . ( 2

.18)

At receiver side, it decodes the signal of base layer at rst, and then it subtracts it from the received stream to get the enhancement layer. In this case the components of enhancement layer can be considered as noise when the base layer is decoded. In order to achieve a bit error rate (BER) in base layer which is not larger than a targer P T E , an approach is shown in below.

In the I/Q modulation, the components of low variance PAUs are mapped on the I components and denoted as x a2 with average power P a2 , while the components of high variance PAUs are mapped on the Q components which are denoted as x a1 with average power P a1 . x a1 and x a2 compose x a . Then FEC coded and BPSK modulated base layer components x d are superposed with x a2 on I components. In this case, the transmitted signal is

x = x a + x d .

( ] and [START_REF] Lan | A practical hybrid digitalanalog scheme for wireless video transmission[END_REF], an expression of overall distortion of HDA scheme is given, then using this expression to choose the parameters of system, for example the quantization step.

Energy distribution Modeling

In SoftCast [START_REF] Jakubczak | Softcast: Clean-slate scalable wireless video[END_REF], after a GoP has been transformed by 3D-DCT, the resulting DCT coefcients are grouped into chunks. It is generally assumed that the coecients within a chunk follow the same distribution and have the same variance. In this way, we only need to compute the scaling factor for each chunk rather than for each DCT coecient. Moreover, only the variances of each chunk are transmitted as meta-data to receiver. Therefore the computation cost and overhead rate are reduced. However, the drawback is a reduced accuracy of the estimated variance of DCT components within each chunk, which can aect the overall performance

[XZW + 17a].
To improve this estimation, in [XWF + 13, XZW + 17b], it has been proposed an adaptive chunk division scheme and a piecewise log linear model of energy distribution instead of using rectangular equal size chunk (See Figure 2.9). In this way, the exprimental result shows that it improves SoftCast by 3 ∼ 5dB and reduces the meta-data at the same time.

At rst, the concept of transform gain [XWX + 16, XZW + 17a] is introduced. Let us consider a random vector x ∈ R N , which can be the vector of all pixels or of all DCT coecients in a frame. For each component x i with variance λ i , a scaling factor g i is computed from (2.14)

g i = λ -1 4 i P T N i=1 √ λ i .
(2.23)

Since in [XWX + 16, XZW + 17a] it is assumed that the receiver does not know the variance of channel noise n i , then the received compoent is simply decoded by inversing the scaling factor to get the reconstructed component

x i = 1 g i (g i x i + n i ) .
(2.24) Then the distortion of x i is

D i = E (x i -x i ) 2 = σ 2 n g 2 i , (2.25)
where σ 2 n is variance of channel noise.

One gets the total distortion D t as

D t = N i=1 D i = N i=1 σ 2 n g 2 i = σ 2 n P T N i=1 λ i 2 , ( 2
G (X|Γ ) = 1 N N i=1 λ X,i 1 N N i=1
λ F,i .

(2.28) Therefore if G (X|Γ ) is large, the tranform is helpful to increase PSNR.

On the other hand, we have mentioned before that in practice it is impossible to compute scaling factor for each component. Only scaling factors for chunks are computed, moreover with the assumption that elements in a chunk have same variance. Let the estimated variance of component x i be denoted as λ i , where i = 1, . . . , N. Then (2.26) becomes

D total = σ 2 n P T N i=1 λ i   N i=1 λ i λ i   .
(2.29) + 17a] shows that D total D total and that the equality holds if and only if

λ 1 λ 1 = λ 2 λ 2 = • • • = λ N λ N
, which means that the more accurate the variance estimations, the smaller the distortion.

From gure 2.9, it can be seen that the energy decreases along the distance ρ = √ u 2 + v 2 from the upper left corner which is low frequency part, while the energy are almost same along the angle θ = arctan u v . From these observations, two energy modeling scheme are proposed [XWF + 13, XZW + 17a]. In the rst, the chunk size is adapted along the distance, in the second it is piecewise log-linear modeling along the distance. These algorithms also has been used in HDA scheme, for example [CSY + 13, CXL + 15] .

ParCast+

ParCast+ [LHL + 14b] is a HDA scheme which considers the video transmission under fading channel in MIMO system. The encoding scheme is similar to WaveCast (Section 2.2.2), but the channel gains s 2 i of each subchannel which are fed back by Channel Side Information (CSI) are taken account into the scaling factor computation under total power constraint. In ParCast+, the optimal decoding matrix is also not considered in the precoding matrix design, which is as same as in SoftCast [START_REF] Jakubczak | Softcast: Clean-slate scalable wireless video[END_REF]. However under fading channel the scaling factors become [LHP + 12]

g i = (λ i s i ) -1 4 P T n ck i √ λ i s i .
Moreover, it proposes that a source component with high variance should be transmitted over a subchannel with higher channel gain, such that the reconstruction distortion can be reduced In SoftCast [START_REF] Jakubczak | Softcast: Clean-slate scalable wireless video[END_REF] the lower variance chunks should be discarded under bandwidth constraint.

In this case, eventhough the channel quality (C-SNR) is increased, the performance could not be increased proportionally or it is saturated. [START_REF] Cagnazzo | Shannon-Kotelnikov mappings for SoftCast-based joint source-channel video coding[END_REF] rst introduces Shannon-Kotel'nikov (SK) Mapping [START_REF] Hekland | Shannon-Kotelnikov mappings in joint source-channel coding[END_REF] in LVC.

SK mapping is helpful to reduce the number of discarded chunks under bandwidth constraint. For example, under 2 : 1 SK mapping (bandwidth reduction), two iid source symbols are mapped onto a point of a parametric curve (double Archimedes' spiral). Therefore under bandwith constraint, in order to reduce the number of discarded chunks, a pair of chunks could be combined to a SK mapped chunks. However the distortion of reconstruction by using 2 : 1 SK mapping has two contributions: one is the approximation of a couple of source points to one point of Archimedes' spiral; the other one is the channel noise which displaces the mapped point along the spiral arms. It is illustrated in Figure 2.11, in which ∆ is the distance between spiral arms. Moreover ∆ can be considered as quantization step and must be optimized given a channel state information (e.g. C-SNR).

The challenge of using SK mapping in LVC is the joint allocation of power and bandwidth to original chunks and SK mapped chunks. An illustration is shown in Figure 2.12. a iterative way to allocate the power and bandwidth respectively to original chunks and SK mapped chunks. Moreover SK-Cast [LLX + 17] is also a HDA scheme whose performance are better than those of WSVC (Section 2.2.3).

Conclusion

In this section, we have shown several important developments of SoftCast. [START_REF] He | Progressive pseudoanalog transmission for mobile video streaming[END_REF][START_REF] Fujihashi | Freecast: Graceful free-viewpoint video delivery[END_REF]. Convolutional neural network is also used at the decoder part of SoftCast to reduce the artifact [START_REF] Yin | Convolutional neural networks based soft video broadcast[END_REF]. Moreover in [START_REF]High-quality soft video delivery with gmrf-based overhead reduction[END_REF][START_REF] Fujihashi | Freecast: Graceful free-viewpoint video delivery[END_REF] the Gaussian Markov random eld (GMRF) is applied to reduce the metadata in SoftCast-based video transmission. All the papers that have mentioned here not only keeps the property of SoftCast that the video performance is linearly commensurate with C-SNR, but also improves the performance. However, only the total power transmission constraint and white Gaussian noise is considered in their problems. In other situations, the per-subchannel power constraint [YAA + 13] and impulse noise [ZD02, Ned03] will be encountered. In Chapter 3 and Chapter 4, we will show our work to resolve the per-subchannel power constraint for SoftCast-based video transmission. Next in Chapter 5, a proposed impulse noise correction scheme for SoftCast based video transmission scheme will be presented.

Chapter 3

Optimal Power Allocation

Introduction

This chapter presents the joint design of precoding and decoding matrices that minimize the MSE in a SoftCast-based LVC and video transmission system (Figure 2.3) under per-subchannel power constraints. This extends results in [START_REF]Softcast: one-size-ts-all wireless video[END_REF], where (i ) the optimal decoding matrix is not considered for the design of the optimal precoding matrix, (ii ) only a total power constraint is considered, (iii ) precoding matrix design for the multiusers case is not provided.

First, Section 3.2 presents the transmission model, then Section 3.3 describes the classical minimum MSE solution under a total power constraint, proposed in [START_REF] Lee | Optimal linear coding for vector channels[END_REF], with an alternative proof involving majorization techniques advocated by [PCL03, PLC04, PJ07], where the source components were assumed all with unit variance. In our LVC case, we extend this result to source components with dierent variances. The solution of this rst problem is then used in Section 3.4 to address the design of the precoding matrix minimizing the MSE under per-subchannel power constraints. In Section 3.6, the advantage of the proposed methods comparing respectively to [START_REF]Softcast: one-size-ts-all wireless video[END_REF] under total power constraint and to [START_REF] Lee | Optimal linear coding for vector channels[END_REF] under per-subchannel power constraint is shown. Moreover in Section 3.6.5 the robustness of the proposed scheme to mismatched channel characteristics un has also been analyzed. We consider a multi-user scenario, where the transmitter uses a common precoding matrix for the transmission to dierent users.

Precoding and decoding matrices

For the precoding and decoding matrix design, one assumes that at the output of the 3D-DCT, the coecients of similar variance are grouped into n Ck chunks of the same size n r × n c . Then a sequence of n r × n c vectors of dimension n Ck is formed by selecting one coecient per chunk for each vector, see Figure 3.1. These chunk vectors are assumed to be realizations of n r × n c independent and identically distributed zero-mean Gaussian random vectors t i , i = 1 . . . n r × n c with covariance matrix Λ = diag(λ 1 . . . λ n Ck ). The matrix Λ is assumed to be diagonal, since t i represents decorrelated 3D-DCT transformed pixels. In practice, the non-zero mean values of chunks are transmitted as metadata.

The chunk vectors t i have to be transmitted over n SC parallel AWGN subchannels with noise covariance matrix N = diag σ 2 1 , . . . , σ 2 n SC and individual power constraints p j , j = 1, . . . , n SC . One has to nd the optimal precoding and decoding matrices to minimise the MSE at receiver, while satisfying the per-subchannel power constraints. In what follows, the index i of t i is omitted, since all vectors t i have similar distribution and undergo the same processing. Moreover, without loss of generality, one assumes that the chunk indexing is such

that λ 1 • • • λ n Ck .
The vector t is multiplied by a precoding transform matrix G ∈ R n SC ×n Ck to get x = Gt.

(3.1)

The received vector is

y = Gt + v, (3.2) Symbol Set Represents n Ck N Dimension of source vector n SC N Nb of parallel subchannels G R n SC ×n Ck Channel precoding matrix H R n Ck ×n SC Decoding matrix n c × n r N Chunk size t R n Ck Chunk vector x R n SC Transmitted vector y R n SC Received vector v R n SC Noise vector σ 2 i R ++ variance noise of i-th subchannel N R n SC ×n SC ++ Channel noise vector covariance λ i R + Variance of a chunk Λ R n Ck ×n Ck + diagonal source covariance matrix p T R ++ Total power constraint ε R + mean-square reconstruction error γ R + Lagrange multiplier N Nb of transmitted components of chunk vectors p i R ++ Power constraint in ith subchannel s R n SC ++ Vector of SNR constraints s i R ++ SNR constraint in i-th subchannel S R n SC ×n SC SNR constraints matrix s eq R ++ Total SNR constraint in equivalent channel Z R n SC ×n SC Orthogonal transform matrix n SB N Nb of subblocks α and β R ++ Parameters for PAISP k N Nb of receivers Table 3.1: Main notations
where v is a vector of channel noise with E (v) = 0 and E vv T = N . To recover t, y is multiplied by a decoding matrix H ∈ R n Ck ×n SC to get t = Hy.

(3.

3)

The mean-square reconstruction error is

ε = tr E t -t t -t T = tr E (t -H (Gt + v)) (t -H (Gt + v)) T . (3.4)
Assuming that t and v are independent, E vt T = 0 and ε becomes ε = tr Λ -2HGΛ + HGΛG T H T + HN H T .

(3.5)

Total Power Constraint

Before considering individual per-subchannel power constraints, we addresse the MSE minimization problem under a total power constraint. Without loss of generality, the noise variance

indexing is such that σ 2 1 • • • σ 2 n SC .
Assuming that a total transmission power constraint p T = n SC i=1 p i has to be satised, the channel input vector x has to be such that (3.8)

E x T x = tr E xx T p T . ( 3
The Lagrangian function associated to (3.8) is

L T = tr Λ -2HGΛ + HGΛG T H T + HN H T +γ tr(GΛG T ) -p T , (3.9)
where γ 0 is a Lagrange multiplier.

Optimal decoding matrix

For a given precoding matrix G, the optimal decoding matrix H is obtained by setting to 0 the partial derivative of L T with respect to H. One gets

H = ΛG T GΛG T + N -1 .
HGΛG T H T + HN H T = ΛG T H T .
(3.12)

Now using (3.12) in (3.5) leads to

ε = tr Λ -2HGΛ + ΛG T H T . (3.13)
Using the properties of the trace and (3.10) in (3.13), one gets an expression of ε that depends

on G only ε = tr Λ -HGΛ = tr Λ -ΛG T GΛG T + N -1 GΛ . (3.14)
One may rewrite (3.14) as

ε = tr Λ 1 2 I -Λ 1 2 G T GΛG T + N -1 GΛ 1 2 Λ 1 2 (3.15)
The matrix inversion lemma leads to

I -GΛ 1 2 T GΛ 1 2 GΛ 1 2 T + N -1 GΛ 1 2 = I + GΛ 1 2 T N -1 GΛ 1 2 -1 and (3.15) becomes ε = tr Λ 1 2 I + GΛ 1 2 T N -1 GΛ 1 2 -1 Λ 1 2 = tr I + GΛ 1 2 T N -1 GΛ 1 2 -1 Λ (3.16) = f Λ diag Φ GΛ 1 2 (3.17)
where

Φ : A ∈ R n SC ×n Ck → I + A T N -1 A -1 ∈ R n Ck ×n Ck f Λ : u ∈ R n Ck → (λ 1 u 1 + • • • + λ n Ck u n Ck ) ∈ R + .
In (3.17), the argument of f Λ is the vector of the diagonal elements of Φ GΛ

1 2 . Introduce now the function f Λ = f Λ • Π u, Λ
, where Π u, Λ : R n Ck → R n Ck is the permutation that matches, for any u ∈ R n Ck , the smallest u i to the largest λ i , the second smallest u i to the second largest

λ i , etc. It has been shown in [PCL03, Appendix B] that given two vectors a ∈ R n and b ∈ R n ,
their scalar product is minimized when the elements of a are sorted in increasing order and those of b are sorted in decreasing order. As a consequence, ∀u ∈ R n Ck , f Λ (u) f Λ (u) , with equality if Π u, Λ is the identity, i.e., the values of u match those of Λ as described before.

The λ i s have been assumed sorted in decreasing order. Then, f Λ is a Schur-concave function [START_REF] Marshall | Inequalities: Theory of majorization and its applications[END_REF]3.A.4]. As a consequence, using [PCL03, Theorem 1], the matrix GΛ

1 2 that minimizes f Λ diag Φ GΛ 1 2
has the following structure

GΛ 1 2 = T    diag g 1 λ 1/2 1 . . . g λ 1/2 0 ×(n Ck -) 0 (n SC -)× 0 (n SC -)×(n Ck -)    , (3.18)
where min (n SC , n Ck ), and the g i s are scaling factors. In (3.18), T is the matrix whose columns are the eigenvectors of N -1 (sorted in decreasing order of their associated eigenvalues).

Here, as N -1 is diagonal and σ 2 1

• • • σ 2 n SC ,
T is simply the identity matrix.

If one introduces m i = g 2 i λ i , then m i is the power allocated to the components t i of chunk vector t and is the number of components actually transmitted. We show how to compute later on, depending on n SC and on the power constraint. If < n Ck , there are some null columns in G, meaning that some components cannot be transmitted. Likewise, if < n SC , there are null rows in G, which corresponds to the fact that the optimal solution does not use some subchannels. Now, using (3.18), (3.17) becomes

f Λ diag Φ GΛ 1 2 = f Λ diag I + Λ 1 2 G T N -1 GΛ 1 2 -1 (3.19) = n Ck i= +1 λ i + i=1 λ i 1 + m i /σ 2 i , (3.20)
The nal MSE consists of two contributions. The rst term n Ck i= +1 λ i represents the variances of the components of the chunk vector that have not been transmitted when < n Ck . These components are the n Ckwith the smallest variances since the λ i s are sorted in decreasing order. The second term depends on the variances λ i of the remaining components, the variances σ 2 i of the subchannel noise components, and the allocated powers given by m i .

Now the optimization problem (3.8) consists in nding an optimal power allocation vector, which can be formulated as

[m 1 . . . m ] = arg min [m 1 ...m ]∈R + i=1 λ i 1 + m i /σ 2 i + n Ck i= +1 λ i s.t. i=1 m i p T
To solve this convex optimization problem, we introduce the Lagrangian function

L T (m 1 . . . m , γ) = n Ck i= +1 λ i + i=1 λ i 1 + m i /σ 2 i + γ i=1 m i -p T , (3.21)
where γ is the Lagrange multiplier. Then, dierentiating (3.21) with respect to m i and setting to zero, one gets,

γ = λ i /σ 2 i (1 + m i /σ 2 i ) 2 , (3.22)
which can be written as

√ γ σ 2 i + m i = λ i σ 2 i .
(3.23) Summing (3.23) over i ∈ 1, . . . , and recalling that i=1 m i = p T , one gets

√ γ = i=1 λ i σ 2 i p T + i=1 σ 2 i . (3.24)
Finally, we can compute the power allocation for each component of the chunk vector from

(3.23), m i = λ i σ 2 i γ -σ 2 i (3.25)
where 1 i min (n SC , n Ck ). Since one should have m i > 0, one chooses as the largest integer less than min (n SC , n Ck ) that satises

λ i σ 2 i γ -σ 2 i > 0, i = 1, . . . , ,
where √ γ is given by (3.24).

From (3.20) one gets the minimum value of the distortion

f Λ diag Φ GΛ 1 2 = n Ck i= +1 λ i + √ γ i=1 λ i σ 2 i (3.26)
Finally, the non-zero diagonal elements of the precoding matrix G can be computed as

∀i ∈ 1, . . . , , g i =   λ i σ 2 i γ -σ 2 i   1/2 / λ i . (3.27)
Moreover, one has

Φ GΛ 1 2 = I + GΛ 1 2 T N -1 GΛ 1 2 -1 = diag 1 1 + m 1 /σ 2 1 , . . . , 1 1 + m /σ 2 , 1, . . . , 1 = diag   γσ 2 1 λ 1 , . . . , γσ 2 λ , 1, . . . , 1   .
The λ i s are decreasing and the σ 2 i s are increasing. The components of Φ GΛ

1 2
are thus sorted in increasing order and

Π diag Φ GΛ 1 2 ,Λ diag Φ GΛ 1 2 = diag Φ GΛ 1 2
.

As a consequence,

G =    diag (g 1 . . . g ) 0 ×(n Ck -) 0 (n SC -)× 0 (n SC -)×(n Ck -)    is also such that ε = f Λ diag Φ GΛ 1 2
is minimized.

In summary under total power constraint, the expression of the optimal precoding and decoding matrices are respectively

G =    diag (g 1 . . . g ) 0 ×(n Ck -) 0 (n SC -)× 0 (n SC -)×(n Ck -)    , (3.28)
and

H = ΛG T GΛG T + N -1
.

(3.29)

In (3.28), min (n SC , n Ck ) is the largest integer such that λ i σ 2 i γ -σ 2 i > 0, i = 1, . . . , (3.30)
with

√ γ = i=1 λ i σ 2 i p T + i=1 σ 2 i (3.31)
and

g i =   λ i σ 2 i γ -σ 2 i   1/2 / λ i , i = 1, . . . , . (3.32)
With this optimal precoding matrix G, the distortion 3.17 becomes,

ε = i=1 λ i σ 2 i 2 p T + i=1 σ 2 i + n C i= +1 λ i . (3.33)
Only the components of the chunk vector with the largest variances are transmitted on the subchannels with smallest noise variances, which is consistent with the results in [START_REF] Lee | Optimal linear coding for vector channels[END_REF]. In SoftCast original paper [START_REF] Jakubczak | Softcast: Clean-slate scalable wireless video[END_REF] the optimal decoding matrix is not considered in the precoding matrix design. Or it can be said in the other way that it assumes the SNR of channel (C-SNR)

p T n SC i=1 σ 2 i
is high enough such that the N could be approximated as zero in (3.29).

Chunk Index

A toy example

This example is adapted from [START_REF] Lee | Optimal linear coding for vector channels[END_REF]. We assume that there are total 10 independent channels and 10 chunks. The variance of each chunk is (10, 9, . . . , 1) and the covariance of channel noise is identity. The total power is p T = 5, 15, 25. The optimal power allocation for each chunk as shown in Figure 3.2,

We can see from Figure 3.2, when the total power allowed to be transmitted on the channel is not enough, for example p T = 5 , the two chunks which have smallest variances will not be transmitted even if the bandwidth is enough.

Per Subchannel Power Constraints

In this section, we present the optimal precoding matrix design under per-subchannel power constraint. In the following, we assume now, again without loss of generality, that the subchannels are indexed by decreasing SNR:

p 1 σ 2 1 p 2 σ 2 2 . . . pn SC σ 2 n SC
. The power used for transmission on subchannel i is n Ck j=1 g 2 ij λ j , which corresponds to the ith diagonal element of GΛG T . Therefore, the per-subchannel power constraints can be written as

∀ i ∈ {1, . . . , n SC }, GΛG T i,i p i . (3.34)
The function to be minimized is the same as Eq. (3.5). The Lagrangian of this constrained optimization problem is thus

L (G, H, γ) = tr Λ -2HGΛ + HGΛG T H T + HN H T + n SC i=1 γ i GΛG T i,i -p i ,
where γ = (γ 1 , . . . , γ n SC ) T is now a vector of Lagrange multipliers. For a given precoding matrix G, the optimum decoding matrix H is the same as in (3.29) and the objective function can again be expressed as

ε = tr I + GΛ 1 2 T N -1 GΛ 1 2 -1 Λ , (3.35) see (3.16). Now, introducing G = N -1 2 G, (3.36)
Eq. (3.35) becomes

ε = tr I + N 1 2 G Λ 1 2 T N -1 N 1 2 G Λ 1 2 -1 Λ = tr I + G Λ 1 2 T G Λ 1 2 -1 Λ , (3.37)
which has to be minimized with the constraints

∀ i ∈ {1, . . . , n SC }, N 1 2 G ΛG T N 1 2 i,i = p i . (3.38)
This constraint may be rewritten as

G ΛG T = S, (3.39) with S =          p 1 /σ 2 1 * * * * p 2 /σ 2 2 * * * * . . . * * * * p n SC /σ 2 n SC          . (3.40)
Thus, the per-subchannel power constraint enforces a structure to the matrix G ΛG T , namely it imposes that its diagonal elements are given by s i = p i σ 2 i , while the o-diagonal elements, represented as * can assume any real value.

Assume that some G minimizing (3.37) with the constraint expressed by (3.39) has been found, then the optimal precoding matrix is G = N 1 2 G and the corresponding H is found using (3.29). For this reason, one considers rst the problem of nding the optimal precoding matrix G with constraints on the signal-to-noise ratio (3.39) that minimizes (3.37). One can thus dene an equivalent channel [START_REF] Lee | Optimal linear coding for vector channels[END_REF] with per-subchannel power constraints corresponding to the SNRs of the original subchannels and uncorrelated unit-variance noise components.

An important property of the equivalent channel is shown below.

Lemma 1. [START_REF] Lee | Optimal linear coding for vector channels[END_REF] Consider a precoding matrix G leading to a given value ε of the distortion (3.37). For any n SC × n SC orthogonal matrix Z, the precoding matrix G = Z G leads to the same distortion ε.

As a consequence, one can consider the following approach (rst introduced in [LP76]) to minimize (3.37) with the constraint (3.39). First, one searches a precoding matrix G that satises the total equivalent channel power constraint dened as the sum of the SNRs of all subchannels. This can be solved using the results of Section 3.

m 1 ≥ • • • ≥ m n SC , s 1 ≥ • • • ≥ s n SC , satisfy k i=1 s i k i=1 m i (3.41) for all k = 1, 2, . . . , n SC -1 and n SC i=1 s i = n SC i=1 m i (3.42)
then there exists a Hermitian matrix with diagonal s and vector of eigenvalues m.

In practice, one nds G and evaluates m as shown in Section 3.4.1. If the sucient conditions of Theorem 2 are satised, there exists an orthogonal matrix Z such that the diagonal of

Z GΛ G T Z T is s. Several techniques are available to obtain Z in this case [ZZ95, VA99]. If
the sucient conditions are not satised, a suboptimal numerical method to obtain Z has been proposed in [START_REF] Lee | Optimal linear coding for vector channels[END_REF]. An optimal alternative approach is proposed in [START_REF] Palomar | Optimum linear joint transmitreceive processing for MIMO channels with qos constraints[END_REF] for the dual problem of power minimization under a per-channel MSE constraint and adapted in our context in Section 3.4.3.

Evaluation of m

To nd G and the related m, consider the minimization of (3.37) with the total SNR constraint

s eq = n SC i=1 p i /σ 2 i . (3.43)
From the result of Section 3.3, since the equivalent channel has uncorrelated unit-variance noise components, the solution of this problem is

G =    diag( g 1 , . . . , g ) 0 0 0    (3.44)
where G ∈ R n SC ×n Ck and min (n SC , n Ck ) is the largest integer satisfying

λ γ, √ γ = i=1 √ λ i s eq +
, and g i = λ i /γ -1 λ i .

(3.45)

As a consequence, GΛ G T is a diagonal matrix, with eigenvalues (and vector of diagonal elements) m with entries given by We describe the optimal power allocation procedure (called OptimalPrecoding)

m i = g 2 i λ i = λ i γ -1.

Z.

At the output of Algorithm 3.1, the optimal precoding matrix is block diagonal and consists 1 To simplify presentation, one assumes here that n Ck = n SC . If this is not the case, one may zero-pad the vector of subband variances (when n Ck < n SC ) or drop n Ck -n SC components of low variance (when n Ck > n SC ).

Algorithm 3.1 G = OptimalPrecoding(λ,s)

1 i = 1 % Initial number of subblocks 2 G = [] % Initialize G as an empty matrix 3 do 4 if i = 1 5 k (i) = 1, τ (i) = n SC 6 else 7 k (i) = τ (i-1) + 1 % Split position 8 τ (i) =
λ (i) = λ k (i) , . . . , λ τ (i) , s (i) = s k (i) , . . . , s τ (i)
14

G (i) , m (i) = OptTotalPower λ (i) , s (i) 15 v, τ (i) = CheckSuCond( m (i) ,s (i) ) 16
while v is false 17

Z (i) = SHIE m (i) , s (i) 18
G (i) = Z (i) G (i) 19 i = i + 1 % Increase number of subblocks 20 while τ (i-1) < n SC 21 n SB = i -1 % Final number of subblocks of n SB submatrices G =             Z (1) G (1) 0 • • • • • • 0 0 . . . . . . . . . . . . . . . Z (i) G (i) . . . . . . . . . . . . . . . 0 0 • • • • • • 0 Z (n SB ) G (n SB )             . (3.48)
In the loop 10-16, OptimalPrecoding tries to nd the largest subvectors λ (i) and s (i) such that the sucient conditions (3.41) and (3.42) are satised with the matrix G (i) designed in such a way that the total power constraint on these subvectors is satised. The transform matrix Z (i) is then evaluated.

OptimalPrecoding is a multi-level water-lling algorithm, in which the inverse γ -1 of the In the proposed scheme, the optimal scaling matrix replaces the power allocation and the Hadamard transform performed by SoftCast. Once the design has been performed, and since in most of the cases, the optimal precoding matrix G is block diagonal, the overhead related to the multiplication by G of each chunk vector is comparable to that of a scaling followed by an Hadamard transform and remains limited. The optimal precoding and decoding matrix design requires the knowledge of chunk variances, which are available at transmitter, and need to be sent to receivers as metadata, see Section 3.6.2. The characteristics of each subchannel need also to be known at transmitter. This information may be fed back by the receivers. In case of transmission to several receivers, as is the typically case in LVC schemes, the precoding matrix design in transmitter will be shown in Section 3.5. The precoding matrix will then be mismatched with the channels of most receivers. Nevertheless, as will be shown in Section 3.6.5, provided that each receiver adopts the decoding matrix adapted to the precoding matrix and to its actual channel conditions (which can be estimated e.g. using the pilot carriers of the OFDM scheme), the performance loss compared to a perfectly matched situation is rather small.

Precoding Matrix Design in multicast scenarii

In this section, we consider the precoding matrix design problem for a multiuser scenario, where a single SoftCast encoded stream is transmitted to k dierent users, each of which experiences dierent channel conditions. This type of problem has been considered, e.g., in [START_REF] Muhammad | Precoding design for mimo relay multicasting[END_REF] in the context of relay-assisted multicast. A min-max problem formulation is considered, where the aim is to design the precoding and decoding matrices so as to minimize the worst MSE among receivers. Here, our aim is to minimize the average MSE among receivers.

More precisely, the transmitter sends some SoftCast encoded stream in n SC subchannels.

For receiver i, the covariance matrix of the noise is referred to as N i . As in the single-user case, N i is assumed to be diagonal for i ∈ {1, . . . , k}.

N i = α i N ref (3.49)
where N ref is a diagonal matrix with diagonal elements σ 2 ref,1 , . . . , σ 2 ref,n SC and the α i s are positive coecients. In this case, the noise variance of the j-th subchannel of user i is α i σ 2 ref,j .

The precoding matrix G is the same for all users. Assuming that each receiver knows G, Λ,

and N i , it may use the optimal decoding matrix obtained from (3.10), Hi = ΛG T GΛG T + N i -1 .

Then, from (3.16), one gets the average distortion among receivers as

ε T = 1 k k i=1 ε i = 1 k k i=1 tr I + GΛ 1 2 T N -1 i GΛ 1 2 -1 Λ = 1 k tr k i=1 I + GΛ 1 2 T N -1 i GΛ 1 2 -1 Λ . (3.50)
The problem considered now is to design G so as to minimize (3.50), assuming that all N i are known at transmitter side (they may be fed back by the receivers, when k is not too large).

First, the precoding matrix design for linearly degraded multicast channels is considered in Section 3.5.1. Then the case of general multicast channels is considered in Section 3.5.1. 2. In what follows, the chunk indexing is such that λ 1 • • • λ n Ck .

Multicast scenario with linearly degraded multicast channels

In this section, we address the precoding matrix design problem under total power constraint and per-subchannel power constraint in the case of the linearly degraded multicast channels.

Total Power Constraint

Our problem is to nd a matrix G that minimizes

ε T = f Λ diag Φ T GΛ 1 2 (3.51)
where

Φ T : A ∈ R n SC ×n Ck → k i=1 I + A T N -1 i A -1 ∈ R n Ck ×n Ck (3.52) f Λ : u ∈ R n Ck → 1 k (λ 1 u 1 + • • • + λ n Ck u n Ck ) ∈ R + .
with the total power constraint or with the per-subchannel power constraint.

As in the single-user case, since the λ i s are assumed in decreasing order, f Λ is minimized when the components of its argument u, which are the diagonal elements of Φ T GΛ

f Λ (t) f Λ diag Φ T GΛ 1 2 = ε T ,
where the lower bound can be achieved if the argument of f Λ is the vector of the eigenvalues

of Φ T GΛ 1 2 , or if Φ T GΛ 1 2
is a diagonal matrix and with diagonal elements in increasing order. Since

Φ T GΛ 1 2 = K i=1 I + 1 α i GΛ 1 2 T N -1 ref GΛ 1 2 -1 , a sucient condition for Φ T GΛ 1 2
to be diagonal with elements in increasing order is that

GΛ 1 2 T N -1 ref GΛ 1 2
is a diagonal matrix with diagonal elements are in decreasing order.

In this case, from [PCL03, lemma12], under total power constraint (3.7), one deduces that the optimal structure of G is

G =    diag (g 1 . . . g ) 0 ×(n Ck -) 0 (n SC -)× 0 (n SC -)×(n Ck -)    . (3.53)
where min (n SC , n Ck ). Now, by using (3.53), (3.50) can be written as

1 k tr k i=1 I + GΛ 1 2 T N -1 i GΛ 1 2 -1 Λ = n Ck j= +1 λ j + 1 k k i=1 j=1 λ j 1 + m j σ -2 i,j . (3.54)
where σ 2 i,j represents the variance of noise at jth subchannel of ith receiver, m j = g 2 j λ j which represents the allocated power in the jth chunk and j = 1, . . . , .

Accounting for the total power constraint j=1 m j p T , one may introduce the Lagrangian associated to (3.51)

L T = n Ck j= +1 λ i + 1 k k i=1 j=1 λ j 1 + m j σ -2 i,j + γ j=1 m j -p T , (3.55)
where γ 0 is the Lagrange multiplier.

Let us dierentiate (3.54) respect to m j and set it equal to zero, we get

1 k k i=1 λ j σ -2 i,j 1 + m j σ -2 i,j 2 = γ. (3.56)
For some receivers, m j σ -2 i,j > 1 (SNR larger than one for the considered sub-channel) and for some others m j σ -2 i,j < 1 (SNR smaller than one). Let K + j and K - j the set of receivers for which m j σ -2 i,j > 1 and m j σ -2 i,j < 1, respectively. Then one may rewrite (3.56) approximately as

1 k    λ j m 2 j i∈K + j σ 2 i,j + i∈K - j λ j σ -2 i,j    = γ, (3.57)
from which one deduces

λ j m 2 j i∈K + j σ 2 i,j = kγ - i∈K - j λ j σ -2 i,j .
For all subchannel indexes j such that kγ -i∈K - i λ j σ -2 i,j > 0, one gets

m j = λ j i∈K + j σ 2 i,j kγ -i∈K - j λ j σ -2 i,j (3.58)
for the others, one should take m j = 0. When there are too many receivers for which the jth subchannel is poor, no power is allocated to that subchannel. Finally, γ is chosen such that j=1 m j = p T .

One obtains a relatively complex water-lling problem where and γ have to be adjusted so as to minimize ε T and kγ -i∈K - i λ j σ -2 i,j > 0. A simplied solution is shown below.

Assuming high SNR after power allocation for all subchannels and receiver, i.e.,

m j σ 2 i,j 1, (3.56) becomes 1 k λ j m 2 j k i=1 σ 2 i,j = γ,
and one gets

m j = λ j 1 k k i=1 σ 2 i,j
√ γ .

√ γ = j=1 λ j 1 k k i=1 σ 2 i,j p T , (3.60)
and

m j = λ j 1 k k i=1 σ 2 i,j j=1 λ j 1 k k i=1 σ 2 i,j p T ,
and the scaling factor g j in (3.53) can be computed as

g j = m j λ j . (3.61)
This analytical solution requires that the SNR of all subchannels is high enough after power allocation, which in turns requires a large p T . From (3.59), we can see that the variances of the noise to be considered in the precoding matrix design is the average of the variances of the noise for each receivers. This is no more the case when p T is not large enough, and one has to resort to a numerical solution of the water-lling problem.

Per-subchannel power constraint

The procedure to compute the precoding matrix in that case is similar to that in Section 3.4.

We assume that the subchannels are indexed such that p 1 /σ 2 ref,1

• • • p n SC /σ 2 ref,n SC .
One assumes rst that the eigenvalues of

GΛ 1 2 T N -1 i GΛ 1 2
are larger than one for i = 1, . . . , k. Then 3.50 becomes

ε T = 1 k k i=1 tr I + GΛ 1 2 T N -1 i GΛ 1 2 -1 Λ ≈ 1 k tr k i=1 GΛ 1 2 T N -1 i GΛ 1 2 -1 Λ = 1 k tr K i=1 1 α i GΛ 1 2 T N -1 ref GΛ 1 2 -1 Λ = 1 k tr K i=1 α i GΛ 1 2 T N -1 ref GΛ 1 2 -1 Λ = 1 k tr     GΛ 1 2 T K i=1 α i N ref -1 GΛ 1 2   -1 Λ   = 1 k tr     GΛ 1 2 T K i=1 N i -1 GΛ 1 2   -1 Λ   . (3.62) Now, introducing N o = K i=1 N i /k, (3.62) becomes ε T = tr GΛ 1 2 T N -1 o GΛ 1 2 -1 Λ . (3.63)
Here again, one has to consider the average channel among users. Then introducing the equivalent channel as in Section 3.4,

G o = N -1 2 o G, (3.63) becomes ε T = tr G o Λ 1 2 T G o Λ 1 2 -1 Λ ,
and the per-subchannel power constraint (3.34) becomes

∀ i ∈ {1, . . . , n SC }, N 1 2 o G o ΛG T o N 1 2 o i,i = p i . (3.64)
This constraint may be rewritten as

G o ΛG T o = S o , (3.65)
with

S o =          p 1 /σ 2 o,1 * * * * p 2 /σ 2 o,2 * * * * . . . * * * * p n SC /σ 2 o,n SC          . (3.66)
The following computation procedure to nd G is the same as that in Section 3.4. At rst the precoding matrix G o under total power constraint which is n Sc i=1 p i /σ 2 o,n SC and unit variance noise is computed using (3.61). Then, an orthogonal matrix Z o has to be found that satises the per-subchannel power constraint (3.66). At the end G = N

1 2 o Z o G o .
The second situation is when the eigenvalues of GΛ

1 2 T N -1 i GΛ 1 2
are smaller than one for i = 1, . . . , k. Then 3.50 becomes

ε T = 1 k k i=1 tr I + GΛ 1 2 T N -1 i GΛ 1 2 -1 Λ ≈ 1 k tr k i=1 I -GΛ 1 2 T N -1 i GΛ 1 2 Λ = 1 k tr kI -GΛ 1 2 T k i=1 N -1 i GΛ 1 2 Λ = tr I -GΛ 1 2 T N -1 L GΛ 1 2 Λ , (3.67)
where In what follows, the procedure to compute G is same as in the rst situation. At rst we compute a precoding matrix

N -1 L = 1 k k i=1
G L = N -1 2
L G in equivalent channel, at which subchannels are indexed such that p 1 σ -2

L,1

• • • p n SC σ -2
L,n SC . For the equivalent channel, ε T becomes

ε T = tr I -G L Λ 1 2 T G L Λ 1 2
Λ .

(3.68)

As in Section 3.5.1.1, the optimal structure of G L is

G L =    diag ( g 1 . . . g ) 0 ×(n Ck -) 0 (n SC -)× 0 (n SC -)×(n Ck -)    . ( 3
ε T = n Ck i=1 λ i - i=1 λ i m i . ( 3
G L = N 1 2 L Z L G L .

General multicast channels

In this section, the precoding matrix design for general multicast channels is considered. There are two situations considered. The rst one is when the eigenvalues of

GΛ 1 2 T N -1 i GΛ 1 2
are larger than one. Then other one is that the eigenvalues of GΛ

1 2 T N -1 i GΛ 1 2
are small.

Eigenvalues of GΛ

1 2 T N -1 i GΛ 1 2

larger than one

In this situation, under total power constraint, if GΛ 1 2 is an invertible matrix, one gets (3.62).

Then G is computed from (3.61), but in this case, the subchannels have to be indexed in such a way that σ 2 o,1

• • • σ 2 o,n SC .
Next, under per-subchannel power constraints, when n SC < n Ck , the last n Ck -n SC chunks are discarded. Moreover with the assumption that the eigenvalues of GΛ

1 2 T N -1 i GΛ 1 2
are larger than one, from (3.61), no chunk will be discarded, therefore we can assume GΛ 1 2 as invertible and one gets 3.62. The following computation of G under per-subchannel power constraint is the same as in Section 3.5.1.2 and subchannels are indexed

such that p 1 /σ 2 o,1 • • • p n SC /σ 2 o,n SC . 3.5.2.2 Eigenvalues of GΛ 1 2 T N -1 i GΛ 1 2
are small

In this situation, we also can get (3.67). Then the computation of G under per-subchannel power constraint is the same as in Section 3.5.1.2. Under total power constraint, subchannels are indexed in such a way that σ -2

L,1

• • • σ -2
L,n SC . Then, from the computation in equivalent channel at Section 3.5.1.2, one can deduce that the objective function to be minimized in here

is min m i ε T = n Ck i=1 λ i -i=1 λ i σ -2 L,i m i , s.t. i=1 m i p T where m i = g 2 i λ i .
3.6 Simulations 3.6.1 Simulation conditions

r SC = 2f SC 1 + β r , (3.71)
which is here equal to r SC = 37.56 × 10 3 real-valued symbols per second.

We consider the luminance component of a video source emitting r F frames per second. The size of each frame is n C × n R . To determine the way the chunks should be transmitted on the subchannel, one has to consider a given chunk size n c × n r . With this choice, the video source chunk rate is

r Ck = r F n R n C n r n c , ( 3
v Ck = n F r F r SC n r n c . (3.74)
For the typical values of the parameters considered in these simulations, v Ck > 1, i.e., several chunks may be transmitted on the same subchannel for the duration of a GoP.

To apply the precoding and decoding matrix design techniques, two approaches may be considered. The rst is to consider v Ck replicas of each subchannel, each replica (virtual subchannel)

only being able to transmit a single chunk. With this approach, during the transmission of a GoP, one has thus

n VSC = η SC v Ck = η SC n F r F r SC n r n c
virtual subchannels available for the transmission of

n Ck = n F n R n C n r n c
chunks. This approach is optimal, but leads to huge precoding and decoding matrices of n VSC × n Ck components. The alternative approach, adopted here, is to partition the n Ck chunks Their characteristics are given in Table 3.3. For the sake of simplicity, only the luminance component of these sequences has been considered. The precoding and decoding matrix design methods could be extended to color sequences using a proper weighting of the distortion of the chrominance components.

For each video sequence, the chunk size is chosen in such a way that n

Metadata

Metadata have to be transmitted without errors to the receiver so that it is able to decode the noisy precoded chunk vectors. A transmission of the precoding and decoding matrices should clearly be avoided, due to their size. Both matrices should be re-estimated at the receiver from metadata sent by the transmitter.

Considering total or per subchannel power constraints, the overhead due to metadata is similar to that of SoftCast, except for the information related to the channel characteristics.

Considering a GoP of n F frames containing n Ck = n F n R n C nrnc chunks, apart from the GoP size, frame size, and chunk size, a vector of n Ck bits has to be sent rst to indicate the transmitted chunks. Then, at most n Ck chunk mean values and variances have to be sent. The channel characteristics have to be known at the transmitter and at the receiver. In a point-to-point communication scenario, when they are fed back by the receiver to the transmitter, their retransmission as metadata is not required. In a multicast scenario, the transmitter may consider average channel characteristics from various receivers. These average characteristics have then to be sent as metadata to the receivers. This requires the transmission of n SC noise variances. Per subchannel power constraints are usually xed and are transmitted at most once during initialization of the communication.

Assume that the chunk mean values and variances as well as the channel characteristics are represented on 8 bits and that the metadata are channel coded with a rate 1/2 channel code.

For Kimono1 with GoPs of 8 frames and the chunk characteristics in

Total Power Constraint

A rst set of simulation is performed considering only a total power constraint. For each video, the organization of the chunk transmission is that described in Table 3.3. A unit variance noise is considered on each subchannel, while the total transmission power has been adjusted in such a way that the subchannel SNR is 15 dB (good transmission condition) or 5 dB (poor transmission condition).

The power allocation method described in Section 3.3 is compared to that presented in [START_REF]Softcast: one-size-ts-all wireless video[END_REF]. Simulation results in terms of average PSNR of the received sequences are reported in Table 3.4.

On the good channel, the two allocation methods perform similarly. For the poor channel the proposed allocation method clearly outperforms that considered in [START_REF]Softcast: one-size-ts-all wireless video[END_REF], as expected, since the latter assumes a relatively high channel SNR. This conclusion is conrmed by Figure

Kimonol

Proposed SoftCast

Per Subchannel Power Constraints

Per-subchannel power constraints are now considered using the channel model described in Section 3.6.1. We compare the optimal allocation method proposed here with heuristic precoding matrix design approach in [START_REF] Lee | Optimal linear coding for vector channels[END_REF].

In Section 3.4, we have seen that an precoding matrix design with per-subchannel power constraints can be formulated as a design problem with an equivalent channel with per-subchannel SNR constraints and unit subchannel noise variances. As a consequence, for the simulations, one assumes again unit noise variance on all subchannels and adjust the transmission power of chunks on each subchannel to have subchannel SNR matching those described in Figure 3.3.

Figure 3.6 illustrates the PSNR of frames of video sequences Kimono1 and Fourpeople when considering the heuristic precoding matrix design approach proposed by Lee in [START_REF] Lee | Optimal linear coding for vector channels[END_REF] and optimal allocation method proposed here. An average gain of 0.12 dB and 0.06 dB are respectively obtained with the proposed optimal design. Similar gains are observed with the other sequences.

Mismatch

In this section, one considers the impact of a channel mismatch on the proposed design technique under total power constraint and per subchannel power constraints. This represents scenarii such as a transmission to receivers with dierent channel characteristics or a precoding matrix design with outdated information on the channel characteristics.

To illustrate the eect of channel mismatch, one assumes that the total power constraint or per-subchannel power constraints are xed (provided by the standard, e.g. PLT channel, Figure 3.7), but that the channel noise or subchannel noises used for the precoding matrix design are not equal to actual channel noise or subchannel noises of receiver. Let N D be the diagonal noise covariance matrix used by the transmitter for the precoding matrix design and let N A be the covariance matrix of the actual noise aecting the subchannels. In case of channel mismatch, one has N D = N A . Both N D and N A are assumed to be perfectly known by the receiver, but the transmitter is only assumed to know N D . This is realistic in a point-to-point scenario when the receiver feeds back channel state information to the transmitter. In a point-to-multipoint scenario, dierent N A s are experienced by each receiver, and the transmitter has to select some average or worst-case channel characteristic N D , which has to be transmitted to the receivers as meta-information.

Denote as G D the optimal precoding matrix evaluated using (3.47) when considering N D .

If the noise covariance matrix is N A , one may still use the mismatched decoding matrix from

(3.29) H D = ΛG T D G D ΛG T D + N D -1 to get ε 1 = tr Λ -2H D G D Λ + H D G D ΛG T D H T D + H D N A H T D = tr Λ -H D G D Λ + H D (N A -N D ) H T D = tr I + G D Λ 1 2 T N -1 D G D Λ 1 2 -1 Λ + tr H T D H D (N A -N D ) .
(3.75)

The rst term in (3.75) corresponds to the MSE obtained without mismatch.

The term tr H T D H D (N A -N D ) may be positive or negative. When N A is smaller than N D , i.e., the subchannels are less noisy than expected, the MSE ε 1 will be smaller than expected during the precoding matrix design. When N A is larger than N D , the channel is worse than expected, and the MSE is larger than expected. In both cases, the MSE variation is commensurate with the dierence between N A and N D .

Alternatively, one may consider the decoding matrix adapted to N A which expression is deduced from (3.10) as

H A = ΛG T D G D ΛG T D + N A -1 (3.76) to get ε 2 = tr I + G D Λ 1 2 T N -1 A G D Λ 1 2 -1
Λ .

(3.77)

The decoding matrix H A is designed to minimize the reconstruction MSE considering that the precoding matrix is G D and the channel noise covariance matrix is N A . As a consequence, one has ε 2 ε 1 .

In the following simulations, the performance of video transmission under total power constraint and under per-subchannel power constraint for two receivers who have dierent channel conditions are tested. The channel for receiver1 consists in the n gCk subchannels with the largest SNR of bad SISO link (see Section 3.6.1). The variance of noise in subchannel can be deduced from the power constraint and the SNR. The power constraint is the integral of the Power Spectrum Density (PSD) over the bandwidth (here is 24.414kHz in PLT). The PSD is -50 dBm/Hz from 1.8 to 30 MHz and -80 dBm/Hz from 30 MHz up to 100 MHz, which is shown in Figure 3.7. From [YAA + 13], only carriers from 1.8 to 86.13 MHz are supported for 2 where the variance of noise in each subchannel is 5 times larger than that of receiver1. The general multicast channels is considered in Section 3.6.5.2.

Total Power Constraint

In this section, we simulate video transmission under total power constraint for 2 receivers in linearly degraded multicast channels. The transmission of video sequence Kimonol and BQMall are tested. The total power constraint is the sum of per-subchannel power constraints. The precoding matrix design approach presented in Section 3.5.1.1 is applied at transmitter. For both receivers, the optimal decoding matrix (

Per-subchannel Power Constraint

In this section, at rst, the linearly degraded multicast channels is considered. The proposed method in Section 3.5.1.2 is considered here. The transmssion of the video sequences Kimonol and BQMall are considered. Then the PSNR results of simulation are compared to those under point-to-point communication as shown in Tab 3.6. We can see the PSNR loss in mismatch is negligible.

Then we test the robustness of the proposed optimal method under general multicast channels. we consider the video sequence BQMall. In multicast scenario, we assume receiver 1 has the channel as shown in Figure 3.8a and refereed to channel 1 and receiver 2 has channel 2, which is generated by ipping randomly the subchannels of channel 1. Several channels can be generated by ipping a fraction of the subchannel, see Figure 3.8b for an example.

We consider three methods to compute the precoding matrix. In the rst one, the method proposed in Section 3.5.2.1 is considered, at which the per-subchannel power constraint in equivalent channel is p i /σ 2 o,i . In the second one, we compute the average per-subchannel SNRs of channel1 and of channel2. Then the subchannels are ordered in decreasing order of the average SNRs. Next, by using the method proposed in Section 3.4, we compute the precoding matrix for the equivalent channel with average per-subchannel SNR constraints. At end, the precoding matrix of equivalent channel is multiplied by an average covariance of noise obtained as division of the per-subchannel power constraint by the average SNR to adapt the per-subchannel power constraint. The third scheme consist in using the SNR constraints of channel 1 as a reference to compute precoding matrix for equivalent channel and by using the method proposed in Section 3.4. For receivers, the optimal decoding matrix is applied at decoder. The simulation results of PSNR for dierent receivers under dierent random ipping probability are shown in Table . 4.5.

We can see the method proposed in Section 3.5.2.1 is better than the others, the performance only decreases 0.3dB each time the probability of ipping increases by 5%. Figure 3.9 shows that in the case of probability of ipping is 10% for receiver2's channel, the reconstructed rst frames of BQMall for receiver 2 in multicast by considering dierent persubchannel power constraints as a reference in equivalent channel. Full sequences are available at https://drive.google.com/drive/folders/1umL5qeN35kT54JQhcu0-1SpANm8go5UE?usp=sharing.

Conclusions

In the context of LVC, this chapter addresses the problem of optimal precoding and decoding matrix design when the video has to be transmitted over parallel additive white Gaussian noise (AWGN) channel with dierent characteristics. One has considered rst that a total transmission power budget has to be allocated between the subchannels. Then, additional per-subchannel power constraints have been considered to address transmission contexts such as PLT channels or multi-antenna systems. At last, the transmission for several receivers who have dierent channel conditions is considered.

In the rst case, small gains compared to the reference SoftCast allocation have been observed essentially at low channel SNRs. In the second case, we have considered the same problem as in [START_REF] Lee | Optimal linear coding for vector channels[END_REF], but we provide an optimal solution which is adapted from the proposed Multi-level water-lling approach to resolve total transmission power minimization with persubchannel MSE constraints problem in [START_REF] Palomar | Optimum linear joint transmitreceive processing for MIMO channels with qos constraints[END_REF][START_REF] Palomar | MIMO transceiver design via majorization theory[END_REF]. In case of mismatch of the precoding matrix with the actual channel characteristics, the benets of the LVC paradigm are also preserved in our proposed solution.

Chapter 4

Sub-Optimal Power Allocation Under

Per-subchannel Power Constraint

In Chapter 3, an optimal power allocation approach under per-subchannel power constraint has been proposed, however it has a high complexity O (n 3 ). In this chapter four alternative suboptimal precoding are considered in the case of per-subchannel power constraints. In all cases, the resulting precoding matrix G still satises (3.34) but may lead to a larger MSE. In Section 4.1 a simple power allocation method is provided. Next, in Section 4.2, by inferring the split positions, the computation cost may be signicantly reduced with respect to optimal algorithm. Nevertheless, some parameters have to be suitably set up. Then the other two suboptimal power allocation schemes, which do not require any parameter tuning, are presented in Section 4.3 and 4.4. In Section 4.5, the limits of suboptimal methods will be shown. Finally in Section 4.6, the performance comparison and complexity comparison of proposed suboptimal methods with optimal method (Section 3.4) is shown.

Simple Chunk Scaling

In the rst method called Simple Chunk Scaling (SCS), the chunk of largest variance is transmitted over the subchannel with the best SNR, the chunk with the second largest variance is sent over the subchannel with the second best SNR, etc., Figure 4.1. To t the per-subchannel power constraints, the coecients of the i-th chunk are multiplied by g SCS,i = p i /λ i , i = 1 . . . n SC .

This allocation is clearly suboptimal but can be easily evaluated. It can be considered as the

Power Allocation with Inferred Split Position (PAISP)

In this suboptimal approach, one tries to infer the split positions of Algorithm 3.1. For that purpose, one rst analyzes the change of SNR constraints along the subchannels and the change of variance along the chunks. Consider the subvectors λ (i) = λ k (i) , . . . , λ τ (i) and

s (i) = s k (i) , . . . , s τ (i) of λ and s of length µ (i) = τ (i) -k (i) + 1 introduced
λ j γ -1 k j=k (i) s j k j=k (i) λ j τ (i) j=k (i) s j + k (i) + -1 j=k (i) λ j k j=k (i) (s j + 1) or a k b k (4.1) with a k = k j=k (i) λ j k (i) + -1 j=k (i) λ j and b k = k j=k (i) (s j +1) τ (i) j=k (i) s j + (4.2)
for k = k (i) , . . . , k (i) + -2, where µ (i) is the largest integer satisfying (3.45) 1 . Condition (3.42) corresponds to the total power constraint. It is satised by the design of G (s) .

1 The multi-level waterlling is such that there may be components in the last subblock (lowest chunk variances) that are not transmitted and in that case µ(n SB ). For intermediate subblocks in the loop 10-16 of Algorithm 3.1, = µ(i). This is consistent with the fact that on Figures 4.2a and 4.2b, the right part of the plot of the modied SNR constraint vector is less at than the corresponding part of the plot of the vector of standard deviations. The optimal power allocation algorithm will then check the conditions (4.1) for the subvectors λ 1 , . . . , λ τ (1) and s 1 , . . . , s τ (1) with τ (1) = n SC -1. Nevertheless, considering the plots in Figure 4.2, it is likely that the conditions (4.1) will again not be satised and are in fact likely to be satised for τ (1) closer to 293 than to 1382. To get shorter subvectors on which the conditions (4.1) are more likely to be satised, instead of using τ (i) obtained from CheckSuCond, the idea of PAISP is to consider subvectors (λ 1 , . . . , λ τ ) and (s 1 , . . . , s τ ) with τ τ (i) to avoid several iterations in the loop 10-16 of Algorithm 3.1. However, this may result in a value of τ smaller than the optimal split position. G, m = OptTotalPower(λ, s) To evaluate the complexity of PAISP to nd all subvectors. The worst case is when τ = µ-1

13 [v, τ ] = CheckSuCond ( m,s) 14 if v is true, a = c, c = a+b 2 , δ = |c -c | 15 else b = τ, c = a+b 2 , δ = |c -c | 16 if c = 1, then c = 1, δ = 0 17 end 18 λ (1) = (λ 1 . . . λ c ), s (1) = (s 1 . . . s c ) 19 λ (2) = (λ c+1 . . . λ µ ), s (2) = (s c+1 . . . s µ ) 20 G (1) =PAISP(λ (1) , s (1) ) 21 G (2) =PAISP(λ (2) , s (2)
m (0) i < τ i=1 s i (4.3) k i=1 m (0) i k i=1 s i (4.4) for k = τ + 1, . . . , n SC -1 and n SC i=1 m (0) i = n SC i=1 s i . (4.5)
From (4.3) and (4.5), one deduces that

n SC i=τ +1 m (0) i > n SC i=τ +1 s i ,
i.e., too much power has been allocated to the last n SC -τ components of λ. The total excess power is

∆ (0) = n SC i=τ +1 m (0) i - n SC i=τ +1 s i . (4.6)
The main idea of PALPA is to correct the values of m (0) n SC and (s τ +1 , . . . , s n SC ).

i , i = τ + 1, . . . ,
For that purpose, one evaluates rst

= max τ +1 n SC such that for i = τ + 1, . . . , , m (0) i - ∆ (0) -n SC j= +1 m (0) j -τ 0.
Since the m (0) i s are decreasing, one may consider only the constraint

m (0) - ∆ (0) -n SC j= +1 m (0) j -τ 0.
Then for i = τ + 1, . . . , n SC , the updated allocated powers are m

(1)

i =        m (0) i - ∆ (0) - n SC j= +1 m (0) j -τ if i 0 else. (4.7)
This correction corresponds to an increase of the water level, see Figure 4.4. It ensures that the source components with large variance are still allocated a larger power.

Proposition 1. The power allocation ajustement performed by PALPA using (4.7) is such that

for k = τ + 1, . . . , n SC -1, k i=τ +1 m (1) i k i=τ +1 s i and n SC i=τ +1 m (1) i = n SC i=τ +1 s i . (4.8)
Proof. One rst shows that the updated power allocation compensates the excess power, i.e.,

∆ (0) = n SC i=τ +1 m (0) i -m
(1) i

.

Using (4.7), one has

n SC i=τ +1 m (0) i -m (1) i = i=τ +1 m (0) i -m (1) i + n SC i= +1 m (0) i = i=τ +1   m (0) i -   m (0) i - ∆ (0) -n SC j= +1 m (0) j -τ     + n SC i= +1 m (0) i = i=τ +1 ∆ (0) -n SC j= +1 m (0) j -τ + n SC i= +1 m (0) i = ∆ (0) . Then to show Proposition 1, one has ∀k ∈ {τ + 1, . . . , n SC -1} τ i=1 m (0) i + k i=τ +1 m (0) i τ i=1 s i + k i=τ +1 s i τ i=1 m (0) i - τ i=1 s i k i=τ +1 s i - k i=τ +1 m (0) i k i=τ +1 m (0) i - k i=τ +1 s i ∆ (0) .
Now, one has

∆ (0) = n SC i=τ +1 m (0) i -m (1) i k i=τ +1 m (0) i -m (1) i , Algorithm 4.3 G = PALPA (λ, s) 1 µ = length(λ) % number of components of λ 2 G, m = OptTotalPower(λ, s) 3 [v, τ ]= CheckSuCond ( m, s) 4 if v is true % Conditions (3.41) and (3.42) satised 5 Z= SHIE( m,s) 6 G = Z G 7 else 8 ∆ (2) = µ i=τ +1 m i -µ i=τ +1 s i 9 λ (1) = (λ 1 . . . λ τ ), s (1) = (s 1 . . . s τ ) 10 m (0) (2) = ( m τ +1 . . . m µ ) 11 λ (2) = (λ τ +1 . . . λ µ ), s (2) = (s τ +1 . . . s µ ) 12 G (1) =PALPA(λ (1) , s (1) ) 13 G (2) =LPA( m (0) (2) , ∆ (2) , λ (2) , s (2) ) 14 end Algorithm 4.4 G (2) = LPA( m (0) ,∆ (0) , λ, s) 1 µ = length m (0) 2 for = µ down to 1 3 if m (0) - ∆ (0) -µ j= +1 m (0) j 0 break; 4 end 5 for i = 1 to µ 7 if i then m (1) i = m (0) i - ∆ (0) -µ j= +1 m (0) j 8 else m (1) i = 0 9 end 10 Z (2) = SHIE( m (1) ,s) 11 G (2) = Z (2) diag sqrt m (1) 1 /λ 1 , . . . , m (1)
µ /λ µ for all k ∈ {t + 1, . . . , n SC -1}. Thus k i=τ +1 m (0) i - k i=τ +1 s i k i=τ +1 m (0) i -m (1) i k i=τ +1 m (1) i - k i=τ +1 s i 0.
The proof of (4.8) follows the same lines.

Algorithm 4.3 corresponds to the PALPA algorithm that calls the Local Power Adjustment (LPA) method (4.7) described in Algorithm 4.4. The latter evaluates also the precoding matrix G (2) for the considered source subvector.

To evaluate the complexity of nding all the split positions with PALPA, the worst case is now obtained when τ = µ -1 at Line 3 of Algorithm 4.3. At each recursion, the complexity is again mainly due to OptTotalPower, which is linear in the length µ of the vector to be processed.

As a consequence, the total complexity is proportional to n SC µ=1 µ and is again O (n 2 SC ). From a practical point of view, it is better to modify Algorithm 4.3 in such a way to make it iterative rather than recursive. In our test, we use the iterative version of PALPA.

Limits of PAISP and PALPA

In this section, we consider two toy examples to show the drawbacks of PAISP (Section 4.2)

and PALPA (Section 4.4).

For PAISP, intuition tells us that the optimal values of α and β depend on the variation of SNRs. If the variation of SNRs is fast at the end and slow at beginning (with respect to the standard deviations), then this method may fail to infer the best splitting position. For example, we consider the vectors of standard deivations and the vector of SNRs shown in With these values of chunks standard deviations and SNR constraints, we ran the optimal power allocation algorithm and the two sub-optimal PAISP and PALPA. Then we compute the resulting MSE for the three of them. We foud that, while in this case PALPA will provide same mean square error as the optimal method, (namely, ε = 12.66), PAISP gives a higher distortion:

we found ε = 19.05 withα = 0.8 and β = 0.5 and ε = 24.71 withα = 0.5 and β = 0.3. The minimum MSE can be achieved with PAISP using α = 1, but this simply means that PAISP works exactly as the optimal method.

In the case of PALPA, the allocation may fail when there are chunks with very low variances compared to the neighbor chunks' variance, and when the SNR constraints are at around the split position. In this case, PALPA may allocate some power to chunks that would not be transmitted using the optimal allocation. An example is given in Fig. 4.6. As in the previous example, we ran the optimal and the suboptimal algorithms and computed the resulting distortion. We found that for the optimal method ε = 39.27, but for PALPA is ε = 46.60. We also observe that the optimal method would not allocate power to the last chunk, while PALPA does it. The power allocation m i for each component under optimal As a conclusion of this section, we observe that in some special cases PAISP and PALPA methods may underperform with respect to the optimal one. The toy example given here help in understanding when this may happen. However, the point here is to understand how often these "pathological" cases may happen with real videos and how much they inuence on video's quality. In order to answer to this question, we performed a comprehensive simulation campaign, which is detailed in the next section. However, we anticipate here the most important results: with real video signal of resolutions ranging from 416 × 240 to 1920 × 1080, PAISP and PALPA methods achieved distortions that are very close to the optimal one. Moreover, their execution times are sensibly lower and they admit parallel implementations that could further speed them up.

Simulation results

In this section, we compare the performance of optimal allocation method (Section 3.4) with the four proposed suboptimal ones in terms of average PSNR on the same video sequences under per subchannel power constraints. The simulation conditions and metadata transmission are same as in Section 3.6.1 and Section 3.6.2. We also compare the optimal allocation with PAISP, PAISP with Dichotomy and PALPA in terms of complexity. At the end, in Section 4.6.3, the robustness of proposed suboptimal method is also tested.

Comparison of the power allocation methods

In Section 3.4, we have seen that precoding matrix design with per-subchannel power constraints can be formulated as a design problem with an equivalent channel with per-subchannel SNR constraints and unit subchannel noise variances. As a consequence, for the simulations, one assumes again unit noise variance on all subchannels and adjust the transmission power of chunks on each subchannel to have subchannel SNR matching those described in Figure 3.3.

The results of the simulation are shown in terms of average PSNR of the received sequences in are also observed for Kimono1, due to the atness of the variance vector. For video sequences in Class D, since the size of the video is relative small, the subchannels to be used have higher SNR constraints and are relatively at compared to the vector of variances. In this case optimal precoding or PAISPs/PALPA can allocate power in a ecient way. For SlideShow in Class F, the high gains come from two aspects. First, many chunks have small variance, in this case the optimal and suboptimals designs can allocate power more eciently. Second, there are chunks in some GoPs with zero variance, hence in such case, some subchannels do not need to be used to transmit these chunks. With SCS these available subchannels cannot be used, contrary to the optimal precoding matrix design method, PAISPs, or PALPA, which increases signicantly the PSNR.

Finally, Figure 4.

Complexity comparison

The computation cost of the optimal power allocation, of PAISP, of PAISP with Dichotomy and of PALPA are compared on simulations performed using MatlabR2014b on an Intel(R)Xeon(R)CPU E5-1603 v3 @ 2.8GHz. Table 4.2 provides the speed-up factor (ratio of precoding matrix computation times) of PAISPs and PALPA compared to the optimal precoding matrix design method.

The parameters of PAISP have been taken as α = 0.75 and β = 0.5. For RaceHorses of class C, SlideShow of Class F, the speed-up of the suboptimal algorithms is close to one, since in most of the GoPs it is not necessary to perform vector splitting. For videos of Class D, there is no split within all GoPs, therefore the four methods again perform similarly. But for the video sequences of class B, class E, and the video BQMall of class C, the speed up is signicant, especially for PALPA and PAISP. The reasons is that the suboptimal algorithms can quickly nd the split positions. On the other hand, the complexity to obtain an orthogonal matrix

(Lemma 1) is O (n 2 ) [ZZ95]
where n is the length of subvector. Since the suboptimal algorithms may lead to more split positions than the optimal algorithm, the size of the subblocks is decreased and the computation costs related to the solution of the SHIE problem are also decreased. We also evaluated the speed-up factor of the heuristic approach in [START_REF] Lee | Optimal linear coding for vector channels[END_REF] 2 . We can see the proposed suboptimal methods are also faster than [START_REF] Lee | Optimal linear coding for vector channels[END_REF].

To further illustrate these results, consider the rst GoP of Kimonol sequence. The sorted variances of the chunk vector components and the per-subchannel SNR constraints are shown in Figure 4.2. Figure 4.12 represents the largest index τ at which condition (3.41) is violated at each iteration of the optimal allocation algorithm, of PAISP, and of PALPA. We also plot the test positions of PAISP Dichotomy (variable c in Algorithm 4.2) at each iteration. The optimal allocation algorithm requires 106438 iterations, whereas PAISP requires 640 iterations, PAISP Dichotomy 2171 iterations and PALPA 848 iterations. This explains the eciency of the proposed suboptimal methods for the Kimono1 sequence.

Figure 4.13 represents the same information as Figure 4.12 for a chunk vector in the 7th

GoP of BasketballDrill. The optimal allocation algorithm requires only 360 iterations, whereas PAISP requires 142 iterations, PAISP Dichotomy 61 iterations and PALPA 61 iterations. In this case the suboptimal algorithms do not reduce signicantly the time to nd the precoding 2 In [START_REF] Lee | Optimal linear coding for vector channels[END_REF], there is an another algorithm to nd the orthogonal matrix, but the complexity is higher than [START_REF] Zha | A note on constructing a symmetric matrix with specied diagonal entries and eigenvalues[END_REF]. In order to have a fair comparison, we adopte the method of [START_REF] Zha | A note on constructing a symmetric matrix with specied diagonal entries and eigenvalues[END_REF] in the implementation [START_REF] Lee | Optimal linear coding for vector channels[END_REF]. In conclusion, PAISP is faster than PAISP with Dichotomy, but it should be tuned using appropriate values of α and β. Table 4.3 illustrates the inuence of the values of α and β on the PSNR and on the speed-up factor for a subset of the considered video sequences. The values α = 0.75 and β = 0.5 provide a good compromise between PSNR degradation and speed-up.

Iteration index

Mismatch

In this section, we test the robustness of the proposed suboptimal methods in a multicast scenario. The simulation condition is the same as in Section 3.6.5. is 5 times larger than that of Receiver 1. The procedure to compute the precoding matrix is described in Section 3.5.1.2, except that the precoding matrix computation considers the equivalent channel and is done using a suboptimal method. Since the suboptimal methods have similar performance, we only do the simulation with PAISP. The results are shown in Table 4.4. We can see that the PSNR loss in this case of linear degraded channel is negligible.

Then results for a channel model for Receiver 2 obtained by ipping a fraction of subchannels of the channel of Receiver 1 are shown in Table ??. Likewise the experiments reported in Section 3.6.5.2, the design method proposed in Section 3.5.2.1 provides the best results, and the PSNR only decreases by 0.3dB each time the probability of ipping increases by 5%.

Conclusions

This chapter has presented four suboptimal precoding matrix design, when the video has to be transmitted over parallel additive white Gaussian noise (AWGN) channel, with dierent characteristics. Among these techniques, PAISP, PAISP with Dichotomy and PALPA may sig-nicantly reduce the matrix design complexity comparing to optimal approach, with a marginal degradation in terms of video PSNR. Comparing to SCS which can be considered as the most straightforward and natural extension of Parcast [LHL + 14b], the optimal method, the both versions of PAISP and PALPA have signicant average PSNR gains, ranging from 2.13 dB for class B videos to 11.55 dB for class F videos. From Table 4.1 and results shown in Section 3.6.4, we also can see that PAISPs and PALPA achieve better performance than [START_REF] Lee | Optimal linear coding for vector channels[END_REF]. Moreover, PAISPs and PALPA are faster than the heuristic method in [START_REF] Lee | Optimal linear coding for vector channels[END_REF] and than the optimal method. Moreover, the robustness of PAISP is also tested in the case of mismatch, , it is shown that the PSNR loss is rather small.

Chapter 5

Impulse error mitigation for LVC schemes

Introduction and main contributions

The characteristics of the transmission channel have been better taken into account into SoftCast-based video transmission. The rst papers considered wideband additive white Gaussian noise (AWGN) channels [START_REF] Jakubczak | Performance regimes of uncoded linear communications over AWGN channels[END_REF]. Fading channels and MIMO channels [HLL

+ 17, ZWW + 15, ZLMW17, LHL + 14a, LHL + 14b]
have then been considered. Optimal precoding schemes for per- subchannel power constrained channels have been designed in [ZAC + 16]. Nevertheless, all the above-mentioned papers consider mainly channels aected by Gaussian noise only. Several types of communication channels may be also prone to impulse noise, such as the Digital Subscriber Line (DSL) [START_REF] Nedev | Analysis of the impact of impulse noise in digital subscriber line systems[END_REF] and the Power Line Telecommunications (PLT) channels [START_REF] Zimmermann | Analysis and modeling of impulsive noise in broad-band powerline communications[END_REF]. Impulse noise has a high magnitude (its power may be 50dB above that of the background noise), and when it is bursty, may corrupt the channel for more than 1 ms [START_REF] Zimmermann | Analysis and modeling of impulsive noise in broad-band powerline communications[END_REF]. If impulses are not corrected, the communication performance may be signicantly degraded [ANQC14, LNE13],

even if LVC schemes are more robust than classical video coding scheme to noise and channel mismatch [START_REF]Softcast: one-size-ts-all wireless video[END_REF].

In this chapter, we address the problem of impulse noise mitigation when the LVC-encoded video is transmitted using an Orthogonal Frequency-Division Multiplexing (OFDM) scheme for multi-carrier modulation over a wideband channel prone to impulse noise. As in [START_REF] Tareq | Impulse noise estimation and removal for ofdm systems[END_REF], the impulse noise is modeled in the time domain by independent and identically distributed (iid)

Bernoulli-Gaussian variables. A Fast Bayesian Matching Pursuit (FBMP) [START_REF] Schniter | Fast bayesian matching pursuit, Information Theory and Applications Workshop[END_REF] algorithm, adapted to OFDM systems by [START_REF] Tareq | Impulse noise estimation and removal for ofdm systems[END_REF], is employed for impulse noise mitigation. This approach requires the provisioning of some OFDM subchannels to estimate the impulse noise locations and amplitudes. Since nothing can be transmitted on provisioned subchannels, this leads to a decrease of the number of chunks which may be transmitted and to a decrease of the video quality at receivers in absence of impulse noise.

To adresse this problem, we propose a phenomenological model (PM) structure to describe the residual noise in the OFDM subchannels after impulse noise estimation and removal. It amounts to a parametric model takes as input the channel and noise characteristics as well as the number of provisioned subchannels. We have combined the PM with a model of the evolution of the PSNR at the receiver in absence of impulse noise as a function of the channel and video characteristics and of the proportion of transmitted chunks. It is then possible to optimize the proportion of subchannels to provision. The parameters of the PM have been adjusted for dierent channel characteristics and one has observed that it leads to accurate estimates of the optimal proportion of subchannel to provision once the characteristics of the video are known. Simulation results show that impulse noise may then be eciently mitigated with a limited impact on the PSNR that may be obtained in absence of impulse noise.

The rest of the chapter is organized as follows. Related results are described in Section 5.2.

The SoftCast-based coding and transmission system is introduced in Section 5.3. The application of FBMP for impulse noise mitigation is described in Section 5.4. Section 5.5 presents the way the optimal number of subchannels to provision can be determined for impulse noise correction. Simulation results are described in Section 5.6 before drawing some conclusions in Section 5.7.

Table 5.1 summarizes the main notations used in this chapter.

Related work

One approach to mitigate the impulse noise is to use a clipping/blanking nonlinearity [START_REF]Analysis and comparison of several simple impulsive noise mitigation schemes for ofdm receivers[END_REF].

A threshold is chosen to decide whether there is a impulse noise in the received symbol. If the magnitude of the received symbol is above that threshold, this symbol is clipped or blanked. The improvement is little when the signal to impulse noise ratio (SINR) is large [START_REF] Sergey | Performance analysis and optimization of OFDM receiver with blanking nonlinearity in impulsive noise environment[END_REF]. Another approach is to use sparse vector recovery algorithms to estimate the impulse noise characteristics (position and amplitude) [CR08, Lam11, LNE13

2σ 2 i R ++ variance circular complex Gaussian noise of i-th subchannel N R n SC ×n SC ++ Ng = 2N = 2diag σ 2 1 , . . . , σ 2 n SC v I C m Impulse noise vector δ R Bernoulli variable w C
Impulse noise variable + 16] and the references there in). The quality of the recovery depends on the restricted isometry property of the conjugate transpose of precoding matrix [START_REF] Emmanuel | Highly robust error correction byconvex programming[END_REF]. In other papers, an a priori statistical information is exploited to estimate the sparse impulse noise samples, e.g., the sparse Bayesian learning approach [WR04, Tip01] is applied in [START_REF] Lampe | Bursty impulse noise detection by compressed sensing[END_REF][START_REF] Lin | Impulsive noise mitigation in powerline communications using sparse bayesian learning[END_REF][START_REF] Tareq | Impulse noise estimation and removal for ofdm systems[END_REF]. A adaptation of the FBMP algorithm [START_REF] Schniter | Fast bayesian matching pursuit, Information Theory and Applications Workshop[END_REF] for OFDM systems has been proposed in [START_REF] Tareq | Impulse noise estimation and removal for ofdm systems[END_REF]. This leads to a reduced run time by one order of magnitude with a performance similar to that of FBMP.

Compared to the state-of-the-art, our contributions are (i ) to adapt the FBMP-based impulse mitigation technique to LVC schemes in the context of OFDM for wideband channels prone to impulse noise; (ii ) to propose a PM of the residual noise after impulse noise mitigation; (iii ) to provide an algorithm for the selection of the number of subchannels to provision which minimizes the receiver Mean-Square Error (MSE). Simulation results illustrate the performance improvements provided by the impulse noise mitigation scheme once the optimal number of provisioned subchannels has been chosen.

Linear Video Coding and OFDM Transmission Scheme

The SoftCast-based LVC architecture [START_REF]Softcast: one-size-ts-all wireless video[END_REF] is rst briey recalled before focusing on the OFDM-based transmission scheme used to convey the LVC-processed video frames. The considered coding and transmission scheme is represented in Figure 5.1.

Joint source-channel coding

We focus on the luminance part of the video. The chrominance components undergo a similar processing. For the power allocation between luminance and chrominance components, the weighting approach adopted in MPEG may be employed [OSS

+ 12]. The input digital video signal is organized in Group of Pictures (GoP) of n F frames with n R × n C pixels each. Each
GoP is processed independently. First, a full-frame 2D-DCT is applied on each frame of the GoP, then a temporal 1D-DCT is applied on the transformed frames to perform temporal decorrelation. The transformed GoP coecients are grouped into chunks of n r × n c coecients from nearby spatio-temporal subbands. Each GoP contains thus

n Ck = n F n R n C nrnc chunks.
The elements of a given chunk are assumed iid and to follow the same zero-mean Gaussian (using, e.g., a strong FEC) as metadata on the channel. The resulting rate overhead may be typically neglected, provided that the number of chunks remains small compared to the number of pixels of each frame [START_REF]Softcast: one-size-ts-all wireless video[END_REF].

A scaling of the selected chunks is then performed to help minimizing the reconstruction MSE at receiver, assuming that each scaled chunk has been transmitted over a dierent subchannel. This optimization may be performed assuming a total power constraint as in [START_REF]Softcast: one-size-ts-all wireless video[END_REF] or considering a per-subchannel power constraint as in [ZAC + 16]. In the rst case, the re- silience to packet losses may be improved by a Hadamard matrix to transform the chunks into equal-energy slices.

Transmission

One considers an OFDM-based transmission scheme with n SC subchannels on which quasianalog signaling (64k-QAM as in [START_REF]Softcast: one-size-ts-all wireless video[END_REF]) or analog QAM is used. A total power p T is available for the transmission of each OFDM symbol.

To perform scaling and transmission, n r × n c chunk vectors t i , i = 1, . . . , n r × n c , each of dimension n Ck , are formed by selecting for each vector one coecient per chunk. The t i s can be seen as realizations of n r × n c iid zero-mean Gaussian vectors with covariance matrix Λ = diag(λ 1 . . . λ n Ck ). The chunk vectors are multiplied by a precoding matrix G ∈ R n SC ×n Ck designed in such a way that u i = Gt i satises a power constraint p T /2. Then n r × n c /2 vectors of complex symbols are formed by combining pairs of consecutive scaled chunk vectors

u i = u 2i-1 + ju 2i = G (t 2i-1 + jt 2i )
(5.1)

with u i ∼ CN 0, 2GΛG T , i = 1, . . . , n r × n c /2.
In what follows, the index i is omitted, since all vectors u i have similar distribution and undergo the same processing. The u i s are transformed into n r × n c /2 OFDM symbols, each satisfying the power constraint p T , using an inverse Discrete Fourier Transform (IDFT). A cyclic prex may be inserted and the symbols are then quadrature-mixed to passband.

One focuses on a transmission power and bandwidth constrained scenario where n SC n Ck .

The number of transmitted chunks is thus such that n SC .

Channel model

The transmitted signal is assumed to be corrupted by Gaussian noise and impulsive noise.

Using the model introduced in [ANQC14], the input vector y ∈ C n SC of the FFT at receiver may be represented as

y = F H u + v I + v g (5.2)
where F H is IDFT matrix, v g is a Gaussian noise vector and v I is an impulse noise vector.

After the DFT, F v g ∼ CN (0, N g ) can be modeled as a zero-mean complex circular Gaussian noise vector with N g = 2N and N = diag σ 2 1 , . . . , σ 2 n SC , and v I is an impulse noise vector. The components of v I are iid and such that v I,k = δ k w k , where δ k is the realization of a Bernoulli variable with parameter p I = Pr{δ k = 1} and w k ∼ CN (0, 2σ 2 I ) with σ 2 I > σ 2 i , i = 1, . . . , n SC .

Baseline receiver

At rst baseline receiver is introduced, without considering impulse noise. The components of the vector y go through an I/Q demodulator, a DFT to get

F y = u + F v g ,
which is fed to a demapper. An LMMSE estimate ti of the chunk vector t i is then evaluated using a decoding matrix H ∈ R n Ck ×n SC .

Finally, using the side information (i.e., the map of selected chunks, the chunk mean values, and their variances), the GoP is reconstructed by applying 3D-Inverse DCT (3D-IDCT) on the estimated chunk vectors.

Power allocation and chunk selection

Here, the precoding and decoding matrix design is briey recalled without accounting for the presence of the impulse noise. Without loss of generality, one assumes that the subchannel indexing is such that σ

G =    diag (g 1 . . . g ) 0 ×(n Ck -) 0 (n SC -)× 0 (n SC -)×(n Ck -)   
(5.4) and

H = ΛG T GΛG T + N -1 , (5.5)
where n SC is the largest integer such that

λ i σ 2 i γ -σ 2 i > 0, i = 1, . . . , (5.6)
with

√ γ = i=1 λ i σ 2 i p T 2 + i=1 σ 2 i (5.7) and g i =   λ i σ 2 i γ -σ 2 i   1/2
/ λ i , i = 1, . . . , .

(5.8)

In absence of impulse error, the resulting MSE on the chunk coecients is then computed as

ε NI = n Ck i= +1 λ i + √ γ i=1 λ i σ 2 i .
(5.9)

If < n SC , n SCsubchannels are not used for the transmission of scaled chunk coecients.

They will be helpful to mitigate the eect of the impulse noise, as shown Section 5.4.

In the plain Softcast, a Hadamard transform is performed after chunk scaling. This transform may be replaced by an orthogonal random matrix [LHL + 14b], which will not modify the MSE [LP76, ZAC + 16], but may improve the robustness to losses of sub-channels. Here, this additional transform is not considered.

Impulse Noise Correction

This section introduces the considered impulse noise correction algorithm adapted to the LVC and transmission scheme presented in Section 5.3.

When,

< n SC , one observes from (5.4) that the last q = n SCrows of G are null, corresponds to sub-channels that are not used to transmit chunk coecients. Here, in the proposed impulse noise mitigation scheme, it may be necessary to discard chunks even when there is enough available transmission power, since this operation improves the robustness to impulse noise, as shown later on. Sub-channels not used to transmit chunk coecients (called provisioned subchannels in what follows), are used to estimate the characteristics of the impulse noise. Consider the matrix Ψ∈ C q×n SC formed by the q last rows of F . From (5.4), one has ΨF H G = 0. variance chunks may be also discarded in our proposed scheme even there is enough available power. A solution to this problem is detailed in Section 5.5.

Sub-channel provisioning for impulse noise mitigation

The eciency of the FBMP algorithm increases with the number q of observations of linear combinations of the impulse errors (5.11). Nevertheless, increasing q reduces the number of subchannels on which chunk coecients can be transmitted. A trade-o has thus to be found between eciency of impulse noise correction and transmission performance. This requires a model of the residual noise after impulse noise mitigation and an evaluation of the impact of subchannel provisioning on the performance of the SoftCast-based LVC and transmission scheme.

Residual noise after impulse noise mitigation

One may rewrite (5.13) as

y = u + F v r + F v g , (5.14)
where v r = v I -v I represents the impulse noise residual vector after mitigation. This residual can be seen as an additional noise component to the background Gaussian noise aecting the sub-channels. This additional noise component has to be taken into account in the design of the precoding and decoding matrices described in Section 5.3.5.

One assumes that v r and v g are uncorrelated. As for the covariance of v r , it can be observed that

Cov (v r |s) = Cov ((v I -v I ) |s) = Cov ((v I -E (v I |s)) |s) = Cov (v I |s) .
(5.15)

The covariance of the estimation error (5.15) can be closely approximated [START_REF] Schniter | Fast bayesian matching pursuit, Information Theory and Applications Workshop[END_REF] as

Cov (v I |s) ≈ δ∈∆ * p δ|s Cov v I |s, δ (5.16)
where δ is a binary vector which non-zero entries indicate the estimated locations of impulses and ∆ * is the set of the D vectors δ that achieve the largest values of p δ|s . A large value of D provides a better estimate of v I and of Cov (v I |s) but at the price of a higher complexity.

Moreover, from [START_REF] Schniter | Fast bayesian matching pursuit, Information Theory and Applications Workshop[END_REF], one has

Cov v I |s, δ = R δ -R δ Ψ H ΨR δ Ψ H + N s -1 ΨR δ , (5.17)
where R δ = Cov v I | δ is a sparse diagonal matrix, since the components of the impulse noise are iid and p I is small in general. Using the matrix inversion lemma, one gets

Cov v I |s, δ = R δ 1 2 I -R δ 1 2 Ψ H ΨR δ Ψ H + N s -1 ΨR δ 1 2 R δ 1 2 (5.18) = R δ 1 2 I + R δ 1 2 Ψ H N -1 s ΨR δ 1 2 -1 R δ 1 2 ,
(5.19)

Consider a column Ψ j , j = 1, . . . , n SC of Ψ. As shown in [START_REF] Tareq | Impulse noise estimation and removal for ofdm systems[END_REF] one has

Ψ H j Ψ j =      q n SC j = j , q n SC sin π(j-j ) q n SC qsin π(j-j ) 1 n SC j = j .
(5.20) Hence, any pair of columns Ψ j and Ψ j of Ψ, such that j -j is a multiple of n SC q , are orthogonal.

For other columns Ψ j and Ψ j with j = j , Ψ H j Ψ j is small compared to q/n SC . Hence when q

and n SC are suciently large, one may approximate Ψ H Ψ as diagonal

Ψ H Ψ ≈ q n SC I.
(5.21)

Figure 5.3 illustrates the norm of each entry of Ψ H Ψ when n SC = 400 and q = 120. It shows that the elements of Ψ H Ψ which around the diagonal are large and that the other terms are much smaller. Now, one rst assumes that σ 2 n SC -q+1 = • • • = σ 2 n SC = σ 2 g , leading to N s = 2σ 2 g I of size q × q.

Using (5.19) and (5.21), one deduces that Cov (v I |s, δ) is diagonal. In the general case, when

σ 2 n SC -q+1 • • • σ 2 n SC , Ψ H N -1
s Ψ can again be approximated by a diagonal matrix provided that n SC and q are large enough, see for example (5.23) Considering (5.1), (5.14), and (5.23), Each vector Gt 2i and Gt 2i+1 is corrupted respectively by the real and imaginary parts of F v r and F v g , with F v r ∼ CN (0, σ 2 r I) and F v g ∼ CN 0, 2diag σ 2 1 , . . . , σ 2 n SC . Assuming that F v r and F v g are uncorrelated, each component of Gt 2i and Gt 2i+1 will be corrupted by a zero-mean Gaussian noise with variance σ 2 i + σ 2 r /2.

Using this in the design of the optimal precoding matrix (5.4) and decoding matrix (5.5), the updated MSE (see (5.3) ) of the received chunk vector E t -t

2 2 is ε = n Ck i= +1 λ i + √ γ i=1 λ i σ 2 c,i , (5.24) where σ 2 c,i = σ 2 i + σ 2 r 2 and
n SC is the largest integer such that

λ i σ 2 c,i γ -σ 2 c,i > 0, i = 1, . . . , (5.25)
with

√ γ = i=1 λ i σ 2 c,i p T 2 + i=1 σ 2 c,i
.

(5.26)

The number of chunks to be transmitted which minimizes ε has to be determined account-ing for the fact that σ 2 r depends on q = n SC -, N g , σ 2 I , and p I [START_REF] Schniter | Fast bayesian matching pursuit, Information Theory and Applications Workshop[END_REF].

Estimation of σ 2 r

An explicit expression of the evolution of σ 2 r as a function of n SC , q, N g , σ 2 I , and p I is very dicult to obtain. Thus, in this section, we will resort to a phenomenological model (PM) of σ 2 r as a function of these parameters. First experiments have been conducted to characterize the structure of the model. Then the value of the model parameters are estimated via least-square estimation.

Two main channels have been considered, the rst with n SC = 256 subchannels and the second of n SC = 416 subchannels, respectively. For both channels, Gaussian background noise with N g = 2σ 2 g I and impulsive noise with σ 2 I = 100 are introduced. The variance of the background noise is adjusted in such a way that the impulsive to background noise ratio (INR) in dB, i.e., 10 log 10 σ 2 I /σ 2 g ranges from 10 dB to 30 dB with a step of 2 dB. The impulse probability p I ranges from 0.5% to 3% with a step of 0.5%. Under these channel conditions, the variance of the residual noise σ 2 r is evaluated once the FBMP algorithm has been employed for impulse estimation. It is obtained as the average of v I -v I The proposed PM of log 10 (σ 2 r ) has to be simultaneously consistent with (5.27), (5.28), and

(5.29). Starting from (5.29), one has considered the same structure for each µ i (r d , INR dB ), and can be used in (5.24) to evaluate the total distortion.

i = 0, 1, namely µ i (r d , INR dB ) = µ i,0 + µ i,1 INR dB + µ i,2 (1 -r d ) 2 +µ i,3 (1 -r d )

Optimization of sub-channel provisioning

This section describes the way the optimal proportion r d of provisioned subchannels is evaluated, as a function of the system parameters, namely the background and impulsive noise characteristics, and the characteristics of the chunks. Here, one assumes a point-to-point communication.

For that purpose, one assumes that the number of subchannels n SC , the noise variances aecting each subchannel 2σ 2 1 , . . . , 2σ 2 n SC , the probability p I , and the variance 2σ 2 I of the impulse noise are all known. These parameters may be estimated by the receiver and fed back to the transmitter. Moreover, the transmitter already knows the vector of chunk variances λ = (λ 1 , . . . , λ n CK ). to analyze the impact of the impulse noise and of the subchannel provisioning on the LVC coding and transmission scheme. Considering an LVC scheme with Subchannel Provisioning and Impulse Correction (LVC-WSP-IC) (Section 5.4) and a scheme with Optimal Subchannel provisioning (LVC-OSP-IC) (Section 5.5.3) allows one to study the impact of impulse correction.

The simulation parameters are detailed in Section 5.6.2. Simulation results are described in Section 5.6.3.

Compared LVC schemes

Four LVC schemes are compared in what follows.

In the LVC-NIC scheme, the number of transmitted chunks is only constrained by the bandwidth and total power constraints. Nevertheless, the eect of the impulse noise is taken into account by an increase of the variance of the background noise from σ 2 i to p I σ 2 I + σ 2 i . The precoding and decoding matrices are adapted accordingly.

In the LVC-WSP-NIC scheme, a proportion r d of subchannels is not used for chunk transmission. Consequently, the remaining chunks benet from more transmission power. At receiver, only the subchannels on which chunks have been transmitted are considered. The other subchannels are not considered for impulse noise correction. The eect of the impulse noise is again captured by an increase of the background noise from σ 2 i to p I σ 2 I + σ 2 i . The precoding and decoding matrices are adapted accordingly.

In the LVC-WSP-IC scheme, a proportion r d of subchannels is used for impulse noise correction. The value of r d is not optimized. This scheme is used to analyze the impact of the choice of r d on the reconstruction MSE.

Finally, in the LVC-OSP-IC scheme, an optimal proportion r d of subchannels is used for impulse noise correction. The optimization is performed using the PM model described in Section 5.5.3.

In all cases, metadata have to be transmitted to indicate the indexes and variances of the chunks, the subchannel noise variances of the reference channel, as well as the variance and probability of the impulse noise. With this information, each receiver (in case of multicast or broadcast) may rebuilt the precoding matrix and evaluate the decoding matrix optimized for the observed channel conditions. The amount of side information is of the same order of magnitude as that of plain SoftCast [START_REF] Jakubczak | Softcast: Clean-slate scalable wireless video[END_REF] and is neglected in what follows.

Simulation parameters

The considered channels consist of n SC = 256 or n SC = 416 have to be designed considering the n gCk chunks of same index in each groups of chunks.

Figure 5.11 illustrates the way chunks are transmitted over the available subchannels.

For impulse noise correction, the parameter D used in the FBMP (Section 5.4) is chosen equal to 5, which represents a compromise between complexity and performance as shown in [START_REF] Schniter | Fast bayesian matching pursuit, Information Theory and Applications Workshop[END_REF].

The simulation parameters are shown in p It = 0.5% is chosen, in case of mismatch, the PSNR decrease is much smoother than in absence of subchannel provisioning for impulse noise mitigation.

Conclusion

This chapter considers the transmission of SoftCast-based encoded videos over channels aected by impulse noise. Albeit LVC schemes are relatively robust to impulse noise, at high channel SNR, a signicant performance loss in terms of PSNR may be observed.

This chapter uses a Fast Bayesian Matching Poursuit algorithm for impulse noise identication and removal. This requires the provisioning of some subchannels on which no information is transmitted. At receiver side, the samples received on these subchannels are used to estimate the realizations of the impulse noise. The price to pay for the subchannel provisioning is a decrease of the nominal PSNR that may be obtained in absence of impulse noise. A trade-o has thus to be found between impulse noise correction eciency and nominal PSNR reduction.

A phenomenological model has been proposed to describe the variance of the noise residual after impulse noise estimation and removal. This model takes as input the channel and noise characteristics as well as the number of provisioned subchannels. Combined with a model of the PSNR evolution as a function of the channel and video characteristics and of the number of provisioned subchannels, it is possible to optimize the proportion of subchannel to provision.

The parameters of the PM have been adjusted for dierent channel characteristics and one has observed that it leads to accurate estimates of the optimal proportion of subchannel to provision once the characteristics of the video are known.

The performance of the optimal subchannel provisioning algorithm combined with the impulse noise mitigation algorithm has been evaluated on two reference video sequences. Provided that the amount of provisioned subchannels is not too large, the nominal PSNR decrease remains moderate, while the impulse noise can be eciently mitigated.

Future work will be dedicated to the evaluation of the optimal amount of subchannels to provision in case of LVC for video multicast over channels prone to impulse noise.

Chapter 6

Conclusions and Perspectives

Conclusions

SoftCast-based LVC schemes have attracted a lot of attention in the last decade. Such schemes provide decoded video quality at dierent receivers to be commensurate with their experienced channel quality.

Nevertheless, SoftCast-based video transmission under per-subchannel power constraints and transmission over channels prone to impulse noise have not been considered. Per-subchannel power constraints can be found in PLT channels or when considering multi-antenna transmission over dierent antennas. In the second problem, if impulse noise has high amplitude or is bursty, may lead to signicantly degraded received video.

We have addressed these two issues in this thesis. At rst, by using multi-level water lling and a solution to an inverse eigenvalue problem, we have proposed an optimal power allocation algorithm for SoftCast-based video transmission under per-subchannel power constraint.

Furthermore, inspired by multi-water lling, we also have proposed three lower complexity suboptimal power allocation algorithms. They can reduce signicantly the execution time and have negligible performance loss compared to optimal allocation algorithms. All of them have significant performance gain over a straightforward extension method of SoftCast in point-to-point communication and in multicast situation.

For the second issue, the mitigation of impulse noise, it is necessary to perform subchannels provisioning and over which there are no data transmitted. Then we use a Fast Bayesian Matching Poursuit algorithm to estimate the impulse noise samples and correct them. The problem here is that a trade-o has to be found between the subchannels provisioning for impulse noise correction and subchannels for data transmission. To address this problem, we proposed a phenomenological model of the impulse noise correction residual error. This model allows one to estimate the optimal number of subchannels to provision as a function of the chunk vector variances and channel conditions, e.g., SNR, variance and probability of impulse noise. Simulation results have shown the accuracy of this method to nd an optimal trade-o.

Moreover, the video performance with impulse noise mitigation is signicantly better than when there is no impulse noise correction. In the multicast situation, the results show that a small number of subchannels provisioned is helpful to improve the robustness of video transmission under various channel conditions.

Perspectives

Three short and medium term research directions are detailed in what follows.

Precoding matrix design for multicast

In Section 3.5, the problem of precoding matrix design in a multicast scenario has been presented. We have not provided an optimal solution to this problem. The approach we considered was to compute the precoding matrix for multicast in an analytic way (using KKT conditions to nd a closed-form expression) and several approximations, for example the SNR of subchannels is assumed to be large. Nevertheless, on some toy examples, we have seen that the solutions obtained via numerical optimization provide a better performance than those obtained using our proposed approximated solution. Numerical methods have thus to be considered to compute precoding matrix for SoftCast-based video transmission in multicast or broadcast situation.

On the other hand, the objective function to be minimized was the average of MSE of the receivers. An alternative criterion may be to minimize the worst MSE among receivers (min-max approach). In this case, under total power constraint, from (3.8), the precoding and decoding design problem can be reformulated as min G,H 1 ,...,H k max i=1,...,k

ε i = tr H i GΛG T H T i -H i GΛ -Λ T G T H T i + tr H i N i H T i + Λ s.t tr GΛG T p T , (6.1)
where k is the number of receivers.

A problem similar to (6.1) has been considered in [START_REF] Muhammad | Precoding design for mimo relay multicasting[END_REF] in the context of relay-assisted multicast. An optimal iterative method has been proposed. First, some G is chosen that satises the total power constraint. Then, optimal H i s are computed from (3.10)

Ψ i + H i GΛ + Λ T G T H T i H i GΛ 1 2 H i GΛ 1 2 T I    0 i = 1, . . . , k tr (Φ) p T    Φ G G T Λ -1    0. (6.2)
The SDP problem (6.2) may then be solved using classical tools such as the CVX toolbox.

An alternative low-complexity solution has been proposed in [START_REF] Muhammad | Precoding design for mimo relay multicasting[END_REF]. One has to determine whether this solution may also be applied in the context of SoftCast.

Optimization of the amount of Metadata

Another issue is that we do not know yet the impact of accuracy of metadata on the whole performance. It is necessary to optimize the allocation between the metadata and chunk coecients. For example if they are transmitted together under bandwidth limitation, we may increase the chunk size to reduce the metadata, or in another way to keep the chunk size and decrease the bandwidth for metadata.

Application of Deep Learning to SoftCast schemes

On the whole, joint source-channel coding scheme is a prominent coding technology. The key point is to exploit the channel conditions in source compression and use source information in channel error protection. This research domain has been considered since a while and has a lot of successful results.

Recently, deep learning and convolutional neural network (CNN) have been applied to joint source-channel coding problems. For example to text transmission [START_REF] Farsad | Deep learning for joint source-channel coding of text[END_REF] and to image transmission [START_REF] Bourtsoulatze | Deep joint sourcechannel coding for wireless image transmission[END_REF], where the latter has used the linearity property of SoftCast. In [START_REF] Yin | Convolutional neural networks based soft video broadcast[END_REF],

CNN are used at decoder side of SoftCast-based video transmission schemes. At encoder side, since SoftCast does not use the digital scheme for compression, there is still redundant information and consequently the number of symbols need to be transmitted is large. To address this problem, the Shannon-Kotel'nikov Mapping is applied as shown in Section 2.2.6.

We also can try to use Linear factor model [GBCB16, Chapter 13] to nd a sparse vector that can reconstruct the original vector in a linear way. In this case, the number of symbols need to be transmitted can be reduced. This thesis considers first the channel precoding and decoding matrix design problem for LVC schemes under a per-subchannel power constraint. Such constraint is found, e.g., on Power Line Telecommunication (PLT) channels and is similar to per-antenna power constraints in multi-antenna transmission system. An optimal design approach is proposed, involving a multi-level water filling algorithm and the solution of a structured Hermitian Inverse Eigenvalue problem. Three lower-complexity alternative suboptimal algorithms are also proposed. Extensive experiments show that the suboptimal algorithms perform closely to the optimal one and can reduce significantly the complexity. The precoding matrix design in multicast situations also has been considered. A second main contribution consists in an impulse noise mitigation approach for LVC schemes. Impulse noise identification and correction can be formulated as a sparse vector recovery problem. A Fast Bayesian Matching Pursuit (FBMP) algorithm is adapted to LVC schemes. Subchannels provisioning for impulse noise mitigation is necessary, leading to a nominal video quality decrease in absence of impulse noise. A phenomenological model (PM) is proposed to describe the impulse noise correction residual. Using the PM model, an algorithm to evaluate the optimal number of subchannels to provision is proposed. Simulation results show that the proposed algorithms significantly improve the video quality when transmitted over channels prone to impulse noise.

Universit é Paris-Saclay

Espace Technologique / Immeuble Discovery Route de l'Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France

Figure 1 . 1 :

 11 Figure 1.1: Multimedia broadcast to dierent users with dierent channel characteristics.

(Figure 1 . 2 :

 12 Figure 1.2: An illustration of HomePlug AV2

Figure 1

 1 Figure 1.4: Power Spectrum Density (PSD) no power back-o in PLT. The gure is from [YAA + 13].

  4. The alternative low-complexity suboptimal algorithms named Simple Chunk Scaling (SCS), Power Allocation with Inferred Split Position (PAISP), PAISP with Dichotomy, and Power Allocation with Local Power Adjustment (PALPA) are presented in Chapter 4. The dierent solutions are compared in Section 4.6. The transmission of several videos over realistic PLT channel models

Figure 2 . 1 :

 21 Figure 2.1: Communication channel. message W is drawn from index set {1, 2, . . . , M } . The gure comes from [CT06].

  the average distortion achieved by this combined source and channel coding scheme. Then the distortion D is achievable if and only if the rate R (D) is less than the channel capacity C R (D) < C.

( 2 . 2 )

 22 This theorem enable us to design source encoder and channel encoder separately. The source encoder achieves the rate distortion by encoding the source sequence of length n into one of the 2 nR(D) messages. Then a channel encoder protects each one of these 2 nR(D) message from channel noises by encoding it into a sequence of n input symbols of channel with capacity C.

Figure 2 . 2 :

 22 Figure 2.2: The achievable distortion. The horizontal axis ∆ 1 and vertical axis ∆ 2 represent respectively the distortion of Receiver 1 and Receiver 2. The circle represents the achieved distortion of analog coding, while the shallow region represents the achievable region of separation coding scheme. To have these distortions, the parameters are set with P = 1, σ 2 1 = 0.1, σ 2 2 = 0.2. The gure comes from [GRV03].

( 2 .

 2 10) where K is the number of transmitted source elements with analog communication or with digital communication, which depends on the channel SNR and bandwidth. (2.9) represents the distortion (MSE) of analog coding based communication under total power constraint. Since these are AWGN channels and the variance of noise is unity, the total power constraint can be represented as M SNR. The minimum MSE computation involves water-lling as shown in Section 3.3. Moreover, Eq. (2.10) represents the distortion (MSE) of N Gaussian random variables which are transmitted over M AWGN channels with a specic SNR by under digital communication scheme. To obtain (2.10), at rst by using [CT06, Theorem 10.3.3], one gets

Figure 2 . 3 :

 23 Figure 2.3: SoftCast transmitter (a) and receiver (b)

Figure 2

 2 Figure 2.4: PSNR (dB) as a function of the channel SNR for SoftCast (in black) and for single-layer MPEG-4. Figure taken from [JK10b].

Figure 2 .

 2 5b), after inverse of coset and inverse of DCT, the reconstructed pixels and predicted pixels are combined through LMMSE to reconstruct video. In this way, they improve the performance of SoftCast by 1.5dB at low channel SNR (see Figure 2.6). Similar work of Dcast can be found at [FWZ + 12, ZFXZ13, FXZW15].

Figure 2 . 5 :

 25 Figure 2.5: Dcast encoder and decoder. The gure comes from [FWZA13].

Figure 2 . 6 :

 26 Figure 2.6: Comparison between Dcast, SoftCast, and H.264. DCast encoder is optimized for targer channel SNR of 5 dB. The gure comes from [FWZA13].

  (a) the encoder part (b) the decoder part

Figure 2 . 8 :

 28 Figure 2.8: The framework of WSVC. The gure comes from [YLL14] .

Figure 2 . 9 :

 29 Figure 2.9: The chunk size adaptation. (a) The energy of DCT coecients (F (u, v), where (u, v) is the coordinate) in log domain; (b) equal size chunk in SoftCast; (c) adaptive chunk division; (d) curve-tting based modeling scheme, F (u, v) is the estimated version of F (u, v).

  comparing to the other components and subchannels matching schemes. The framework of video coding and transmission in ParCast+ is shown in Figure 2.10. There are other papers which also work on fading channel [CSY + 13, CLCW14b, CLCW14a, CXL + 15, ZLCW16, HLL + 17, ZLMW17], at which the video transmission under more complicated transmission conditions are considered, for example the channel state prediction and multicast for dierent users.

Figure 2 .

 2 Figure 2.10: Framework of video coding and transmission in ParCast+

Figure 2

 2 Figure 2.11: The components of distortion by using 2 : 1 Shannon-Kotel'nikov (bandwidth reduction) mapping on Archimedes' spiral. The gure comes from [HFR09].

Figure 2 .

 2 Figure 2.12: Bandwidth allocation of n SC original chunks and 2n SK SK-mapped chunks. The gure comes from [CK15].

  scheme to increase the performance (Dcast, WSVC, WaveCast). The improvement of the

Figure 3 . 1 :

 31 Figure 3.1: Vectorization of the chunks

  to nd G, H = arg min G,H tr Λ -2HGΛ + HGΛG T H T + HN H T s.t. tr(GΛG T ) p T .

From ( 3 .

 3 10), one obtains HGΛG T + HN = ΛG T . (3.11) Right multiplying both sides of (3.11) by H T , one gets

Figure 3 . 2 :

 32 Figure 3.2: Optimal subchannel power allocation under total power constraint

2

 2 When the conditions of Theorem 2 are satised In this case, Z can be computed using the approach described in [ZZ95, VA99]. Combining (3.44) and (3.36), the optimal precoding matrix for the initial problem becomes G = N 1 2 Z G.

( 3 .

 3 47) 3.4.3 When the conditions of Theorem 2 are not satisedIn that case, the multi-level water-lling approach proposed in [PLC04, Section VI] is used to split the vector of variances and the vector of SNR constraints into subvectors on which the conditions of Theorem 2 are tested again. If they are not satised the subvectors are split again in a recursive way. A solution necessarily exists since these conditions are satised when the size of the subvectors is 1.

  m j = p T , one gets

  .70) Minimizing (3.70) under total power constraint n Ck i=1 m i p T and p T = n SC i=1 p i σ -2 L,i is a Linear Programming problem. There is no simply analytical solution, but this problem can be solved numerically [BV04, Page 6]. Once we have found m i , we can compute g i .Then use the solution of Structured Hermitian Inverse Eigenvalue problem to nd an orthogonal matrix Z L to adapt the per-subchannel power constraints in the equivalent channel. At the end

Figure 3 . 3 :

 33 Figure 3.3: SNR as a function of the subchannel index for the considered PLT channel

Figure 3 . 4 :

 34 Figure 3.4: Organizations of the transmission of chunks of the i-th GoP: (top) each subchannel is duplicated into v Ck virtual subchannels, each being able to transmit a single chunk per GoP; (bottom) chunks of similar variance are gather into groups of v Ck chunks, each group of chunk being transmitted over a dedicated subchannel

Figure 3 . 5 :

 35 Figure 3.5: Evolution of the PSNR with the channel SNR for the Kimono1 sequence considering a total power constraint

Figure 3 . 7 :

 37 Figure 3.7: The gure from [YAA + 13]. Power Spectrum Density (PSD) no power back-o in PLT

Figure 3

 3 Figure 3.8: SNR as a function of the subchannel index

  (a) no mismatch, PSNR =44.42dB (b) p i /σ 2 o,i , PSNR =44.05dB (c) Ave. SNR, PSNR = 40.61dB (d) Chan. 1, PSNR = 36.40dB

Figure 3 . 9 :

 39 Figure 3.9: Reconstructed rst frames of BQMall for receiver 2 in multicast by considering dierent per-subchannel power constraints as a reference in equivalent channel. The probability of ipping is 10%. (a) no mismatch, (b) p i /σ 2 o,i , (c) Average SNR, (d) Channel 1.

3 (

 3 Figure 4.2: (a) Standard deviations of a chunk vector of the rst GoP of Kimonol (after reordering) and (b) Modied SNR constraints, i.e., s i +1, associated to the channel in Figure 3.3 (after reordering).

Figure 4 .

 4 3a represents the evolution of a k and b k as functions of k. One observes that the conditions (4.1) are not satised for k = 293 up to k = n SC -1 = 1382.

  Figure 4.3 represent the values of the vectors a and b obtained for several τ (i) s. The optimal τ (i) is 535, larger than 293, but much less than τ (1) = n SC -1 which would have been chosen in the next iteration of the loop 10-16 of Algorithm 3.1.

Figure 4 . 3 : 2 G 3 [G = Z G 7 else % Uses dichotomy nd split position c 8 a 2 ,

 4323782 Figure 4.3: a k and b k for dierent values of τ

  takes initially λ = (λ 1 . . . λ n SC ) and s = (s 1 . . . s n SC ) of length µ = n SC as inputs. At rst the largest index τ that violates Conditions (3.41) is evaluated. Then PAISP searches the optimal split position at the interval [a, b] = [1, τ ] by dichotomy. First, the midpoint c = a+b 2 is considered. If Conditions (3.41) are satised for (λ 1 . . . λ c ) and (s 1 . . . s c ), then PAISP updates a = c; Else the largest index τ that violates Conditions (3.41) is evaluated and PAISP updates b = τ. These iterations are repeated until the dierence between two successive midpoints is not larger than 1, see Algorithm 4.2.

Figurei

  Figure 4.4: Initial ( m (0) i ) and updated ( m (1)i ) allocated powers when ∆ (0) is small (left) and when ∆ (0) is large (right)

p i + 1 Figure 4

 14 Figure 4.5: A toy example where the PAISP fails to estimate the splitting position.

Fig

  Fig.4.5.

p i + 1 Figure 4

 14 Figure 4.6: A toy example where the PALPA works less better than optimal method. (4.6a) a vector of standard deviations; (4.6b) modied power constraint vector.

  method is : [1012.5 826.5 754.4 543.8 212.7 150.1 10 10 4 0]; under PALPA it is [1012.5 826.5 754.4 543.8 212.7 150.1 10 10 2 2]; under PAISP (α = 0.75 β = 0.5) is [967.2 789.6 720.7 519.4 203.1 300 10 10 4 0]

Figure

  Figure 4.8: PSNR of Class C

Figure

  Figure 4.9: PSNR of Class D

Figure 4 .

 4 Figure 4.10: PSNR of Class E and Class F

Figure 4 .

 4 Figure 4.11: Reconstructed rst frames of Kimonol with PAISP and SCS. (a) PAISP, (b) SCS

Figure 4

 4 Figure 4.12: Evolution of the value of the largest index τ at which Condition (3.41) is violated and the test positions for Dichotomy as a function of the iteration index for a chunk vector of the rst GoP of Kimonol (Class B): (a) for the optimal allocation algorithm, (b) for PAISP, (c) for PAISP Dichotomy, and (d) for PALPA.

Figure 4 . 13 :

 413 Figure 4.13: Evolution of the value of the largest index τ at which Condition (3.41) is violated and the test positions for Dichotomy as a function of the iteration index for a chunk vector in the 7th GoP of BasketballDrill (class C): (a) for the optimal allocation algorithm, (b) for PAISP, (c) for PAISP dichotomy, and (d) for PALPA.

Figure 5

 5 Figure 5.1: SoftCast-based linear video coding, transmission, and decoding architecture

Figure 5 . 2 :

 52 Figure 5.2: Modied SoftCast-based LVC (with discarded chunks), transmission, and decoding architecture (with impulse noise correction)

Figure 5 . 3 :

 53 Figure 5.3: Ψ H Ψ

Figure 5

 5 Figure 5.4: Ψ H N -1s Ψ. n SC = 400, q = 120, and N s = diag (0.1, 0.2, . . . , 11.9, 12).

2 2 ,

 2 Figure 5.5: log 10 (σ 2 r ) as a function of the INR in dB for dierent values of p I and r d

Figure 5

 5 Figure 5.7: log 10 (σ 2 r ) as a function of log 10 (p I ) for dierent values of the INR in dB and of r d

  r

Figure 5 .

 5 Figure 5.10: MSE as predicted by the model (5.24) and measured on simulations for one chunk vector of BQSquare and RaceHorses. SNR= 9 dB, p I = 0.01, and σ 2 I = 100. (a) BQSquare when τ = 1; (b) BQSquare when τ = 7; (c) RaceHorses when τ = 1; (d) RaceHorses when τ = 7.

Figure 5

 5 Figure 5.11: Organizations of the transmission of chunks of the i-th GoP: (top) A vector of chunks; (bottom) chunks of similar variance are gathered into groups of ν Ck = 2 chunks, each group of chunk being transmitted over a dedicated subchannel.

SNRFigure 5 SNRFigure 5 . 13 : 2 IFigure 5 2 IFigure 5 . 16 :

 5513252516 Figure 5.12: PSNR evolution as a function of SNR (dB) and r d for BQSquare when σ 2 I = 100

Figure 5

 5 Figure 5.18: Eect of the mismatch between the target impulse noise probability p It and the actual impulse noise probability p I when σ 2 I = 100 and SNR = 20 dB (for the RaceHorses video sequence)

Titre:

  Prise en compte des contraintes de canal dans les sch émas de codage vid éo conjoint du source-canal Mots cl és : codage conjoint du source-canal, transmission vid éo, codage vid éo lin éaire, allocation de la puissance, r éduction de l'impact du bruit impulsif R ésum é : Les sch émas de Codage Vid éo Lin éaire (CVL) inspir és de SoftCast ont émerg é dans la derni ère d écennie comme une alternative aux sch émas de codage vid éo classiques. Ces sch émas de codage source-canal conjoint exploitent des r ésultats th éoriques montrant qu'une transmission (quasi-)analogique est plus performante dans des situations de multicast que des sch émas num ériques lorsque les rapports signal-à-bruit des canaux (C-SNR) diff èrent d'un r écepteur à l'autre. Dans ce contexte, les sch émas de CVL permettent d'obtenir une qualit é de vid éo d écod ée proportionnelle au C-SNR du r écepteur. Une premi ère contribution de cette th èse concerne l'optimisation de la matrice de pr écodage de canal pour une transmission de type OFDM de flux g én ér és par un CVL lorsque les contraintes de puissance diff èrent d'un sous-canal à l'autre. Ce type de contrainte apparait en sur des canaux DSL, ou dans des dispositifs de transmission sur courant porteur en ligne (CPL). Cette th èse propose une solution optimale à ce probl ème de type multi-level water filling et n écessitant la solution d'un probl ème de type Structured Hermitian Inverse Eigenvalue. Trois algorithmes sous-optimaux de complexit é r éduite sont également propos és. Des nombreux r ésultats de simulation montrent que les algorithmes sous-optimaux ont des performances tr ès proches de l'optimum et r éduisent significativement le temps de codage. Le calcul de la matrice de pr écodage dans une situation de multicast est également abord é. Une seconde contribution principale consiste en la r éduction de l'impact du bruit impulsif dans les CVL. Le probl ème de correction du bruit impulsif est formul é comme un probl ème d'estimation d'un vecteur creux. Un algorithme de type Fast Bayesian Matching Pursuit (FBMP) est adapt é au contexte CVL. Cette approche n écessite de r éserver des sous-canaux pour la correction du bruit impulsif, entrainant une diminution de la qualit é vid éo en l'absence de bruit impulsif. Un mod èle ph énom énologique (MP) est propos é pour d écrire l'erreur r ésiduelle apr ès correction du bruit impulsif. Ce mod èle permet de d'optimiser le nombre de sous-canaux à r éserver en fonction des caract éristiques du bruit impulsif. Les r ésultats de simulation montrent que le sch éma propos é am éliore consid érablement les performances lorsque le flux CVL est transmis sur un canal sujet à du bruit impulsif. Title : Accounting for Channel Constraints in Joint Source-Channel Video Coding Schemes Keywords : joint source-channel coding, video transmission, linear video coding, power allocation, impulse noise mitigation Abstract : SoftCast based Linear Video Coding (LVC) schemes have been emerged in the last decade as a quasi analog joint-source-channel alternative to classical video coding schemes. Theoretical analyses have shown that analog coding is better than digital coding in a multicast scenario when the channel signal-to-noise ratios (C-SNR) differ among receivers. LVC schemes provide in such context a decoded video quality at different receivers proportional to their C-SNR.

  CONTENTS 3.3.3 A toy example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Per Subchannel Power Constraints . . . . . . . . . . . . . . . . . . . . . . . . . Evaluation of m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 When the conditions of Theorem 2 are satised . . . . . . . . . . . . . . Total Power Constraint . . . . . . . . . . . . . . . . . . . . . . 3.5.1.2 Per-subchannel power constraint . . . . . . . . . . . . . . . . . 3.5.2 General multicast channels . . . . . . . . . . . . . . . . . . . . . . . . . .

		3.4.1
	3.4.3 Chapter 1 When the conditions of Theorem 2 are not satised . . . . . . . . . . . .
	3.5	Precoding Matrix Design in multicast scenarii . . . . . . . . . . . . . . . . . . .
		3.5.1	Multicast scenario with linearly degraded multicast channels . . . . . . .
	3.5.1.1 Introduction
	3.5.2.1 3.5.2.2 1.1 Context and motivation Eigenvalues of GΛ 1 2 Eigenvalues of GΛ 1 2	T T	N -1 i N -1 i	GΛ GΛ	1 2 2 1	larger than one . . . . . .

are small . . . . . . . . . . 3.6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Simulation conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Metadata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Total Power Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . .

  Theorem 1. (Source channel separation theorem with distortion) Let V 1 , V 2 . . . V n be a nite alphabet iid source which is encoded as a sequence of n input symbols X n of a discrete memoryless channel with capacity C. The output of channel Y n is mapped onto the reconstruction alphabet

The rate distortion function R (D) is the inmum of rates R such that (R, D) is achievable for a given distortion D. The distortion rate function D (R) is the inmum of all distortion D such that (R, D) is achievable for a given rate R.

The source-channel separation theorem with distortion [CT06, Theorem 10.4.1] reported below shows in which condition this distortion D can be achieved with a given channel capacity C.

  Section 2.2.1) and SoftCast for video sequence Foreman. The target channel SNR for encoder design is 5 ∼ 25 dB. The bandwidth is 1.33MHz. The gure comes from[START_REF] Yu | Wireless scalable video coding using a hybrid digitalanalog scheme[END_REF].the reconstructed LR and the decoded enhancement layer source give the high resolution (HR) video. The procedure to transmit these two streams is shown in below.

	motion compensated temporal lter (MCTF) [ST03, CCA + 07, ACAB07] to reduce the in-
	ter frame redundancy. At low channel SNR, WaveCast increases the video PSNR by 2dB
	Figure 2.7: The performance (PSNR) of received HR (high resolution) video by using WSVC,
	compared to SoftCast. MCTF is a lter that uses motion trajectories in lifting-based transform performed over on a sequence of frames of video. For M -level MCTF, there are M output high-pass subbands and 1 low-pass subband. In WaveCast, after MCTF, 2D discrete wavelet transform is used to exploit the spatial redundancy of these M + 1output frames. Then the output wavelet coecients are Dcast (The digital stream at rst is protected by forward error correction (FEC) code and then
	scaled under total power constraint (See (2.14) ). modulated by BPSK. Next each modulated component is allocated with average power P d .
	For the enhancement layer, WSVC introduces a power allocation unit (PAU) whose role is 2.2.3 WSVC similar to that of chunk in SoftCast. Let us assume that there are N p PAUs in enhancement
	Wireless scalable video coding (WSVC) framework [YLL14] is a SoftCast-based hybrid digital-layer within a GoP, among which there are N LR p PAUs coming from the residual of the LL
	analog (HDA) coding scheme [PWS94, Sch95, MP02, SPA02, SPA06]. WSVC uses 2D-DWT subband. Then, the N p PAUs are sorted in decreasing order of standard deviations σ k , where
	(discrete wavelet transform) instead of using 2D-DCT, thus it has more spatial scalability. k = 1, . . . , N P . Moreover the average allocated power for PAU's component is P a /2, where the
	Moreover it achieves a PSNR gain up to 3.3dB over DCast (see Figure 2.7). In the following, factor of 2 comes from the I/Q modulation as shown later. The components of each PAU are
	WSVC is briey presented. scaled with a scaling factor g k under total power constraint N p	pa 2	. The forms of g k are the
	The architecture of WSVC is shown in Figure 2.8. At the encoder (Figure 2.8a), at rst same as in SoftCast (2.14), that is
	each frame within a GOP is transformed by 2D-DWT to get four dierent subbands: LL, LH,
	HL, HH. The LL subband are then compressed by conventional video codec (e.g. H.264). Then
	the residual of LL subband which is the dierence between the original and the reconstructed
	in H.264 and the other three high pass subbands LH, HL, HH are compressed by the SoftCast
	codec (temporal DCT and power allocation). The output stream of H.264 is considered as a
	base layer of video source, while the output of SoftCast codec is analog and is considered as an
	enhancement layer of video source. In the receiver (Figure 2.8b), the decoded base layer and
	the reconstructed LL subband residual allow to obtain a low resolution (LR) video. Finally,

  There are n T chunks and n C subchannels and n T > n C . The n SC chunks among n T are mapped on n SC subchannels, the other 2n SK chunks are mapped on the remaining n SK suchannels by

	2 : 1 SK mapping. The problem is how to compute n SK and the power allocation between
	n SC original chunks and 2n SK SK mapped chunks by given a channel condition.[CK15] resolved
	the power allocation problem under total power constraint by given a bandwidth allocation.
	Then [LLX + 17] proposes a scheme to resolve the bandwidth allocation problem given a power
	allocation. By using this scheme, [LLX + 17] proposes an algorithm called SK-Cast which is

Table 3

 3

	Chunk 1	Chunk 2	Chunk Ck

.1 gathers the main notations used throughout this and next chapter. Random quantities are in bold, matrices in capital letters, vectors and scalars in small letters. R + refers

  3. Since the resulting precoding matrix does not necessarily satisfy the per-subchannel power constraints (3.39), one searches an orthogonal matrix Z such that Z G satises the per-subchannel power constraints (3.39). Sucient conditions on the vector of eigenvalues m = ( m 1 , . . . , m n SC ) T of GΛ G T are provided in[START_REF] Marshall | Inequalities: Theory of majorization and its applications[END_REF] 9.B.2] to guarantee the existence of such matrix Z. Introducing the vector s =

	p 1 /σ 2 1 , . . . , p n SC /σ 2 n SC

T

, the conditions are expressed in the following theorem. Theorem 2. [MOA11, 9.B.2] If the entries of s and m, arranged in non-increasing order

  CheckSuCond veries whether the sucient conditions (3.41) in Theorem 2 are satised. If this is not the case, it returns the largest index k such that k i=1 s i > k i=1 m i . SHIE (Structured Hermitian Inverse Eigenvalue) computes the orthogonal transform matrix

in Algorithm 3.1. Its inputs are the vectors λ = (λ 1 . . . λ n SC ) of subband variances and s = (s 1 . . . s n SC ) of SNR constraints 1 , both with components sorted in decreasing order. Its output is the opti- mal precoding matrix G . We assume that the following four algorithms are available. OptTo-talPower computes the optimal precoding matrix (3.44) and power allocation (3.46) under total power constraint.

  large chunk variance and large channel SNR) is not less than that of a lower level subblock (corresponding to smaller chunk variance and channel SNR). Contrary to the total power constrained case, the optimal precoding matrix is in general block-diagonal, see(3.48). Consequently, chunks may be mixed together and transmitted over several subchannels.The OptimalPrecoding algorithm may be relatively complex, since the search for the transform subblock Z (i) G (i) always starts with the subvectors λ k (i) , . . . , λ nsc and s k (i) , . . . , s nsc with k (i) = τ (i-1) + 1. In what follows, k (i) is called the i-th split position. The size of these vectors is progressively reduced until (3.41) and (3.42) are satised. In the worst case, this may require n sc -τ (i-1) iterations, and as many evaluations of the corresponding optimal precoding matrix under total power constraint. As a consequence, the complexity to nd all the split positions in the worst case (when λ and s are split into n SC components) is O (n 3

Lagrange multiplier γ in

(3.45

) represents the water level. Consider a transform subblock Z (i) G (i) associated to the subvectors λ (i) and s (i) for which the conditions of Theorem 2 are satised. In [PLC04, Appendix D], it is shown that the water level of an upper level subblock (corresponding to SC ), see

[START_REF] Palomar | Optimum linear joint transmitreceive processing for MIMO channels with qos constraints[END_REF] AppendixD]

.

Table 3 .

 3 2 summarizes the additional notations used in this section.In the following simulations, one assumes that video has to be transmitted over an inhome power line channel to one or several receivers with dierent channel characteristics. The frequency range is from 1.8 MHz to 86.13 MHz, which is the same range considered by the HomePlug Alliance in the HomePlug AV2 specication [YAA+ 13]. The spacing between sub- channels is f SC = 24.414 kHz and the maximum number of subchannels that may be used for data transmission is η SC = 3217. Not all subchannels are allowed for data transmission. In

	Variable	Value				Signication
	n F	8					nb of frames per GoP
	n C × n R							Frame size
	n c × n r							Chunk size
	n Ck						nb of chunks in a GoP
	f SC	24.414 kHz			Spacing between subcarriers
	η SC	3217			Nb of available subchannels
	β r	30%				Nyquist lter roll-o
	n VSC					number of virtual subchannels
	r SC	37560			per-subchannel rate in symb/s
	r Ck					source chunk rate in chunk/s
	ρ Ck			per-subchannel chunk rate in chunk/s/subchannel
	v Ck			nb of chunks a subchannel can transmit per GoP
	n gCk						nb of group of chunks
			Table 3.2: Additional notations
			30
			20
		Sub-channel SNR (dB)	-10 0 10
			-20
			-30	500	1000	1500	2000	2500	3000	3500
						Sub-channel index

OFDM-based PLT systems like AV2, typically SNRs per subchannel are available. A realization of the individual subchannel SNRs is represented in Figure

3

.3, which relates to a bad SISO link from ETSI STF 477 database. Assuming that each subchannel is corrupted by independent

  .72) and the per-subchannel chunk rate is obtained from (3.71) as Ck > η SC ρ Ck , a certain amount of chunks in each GoP can not be transmitted due to channel bandwidth constraints. Considering GoPs of constant size n F , the number of chunks a subchannel can transmit for the duration of a GoP is

	ρ Ck =	r SC n r n c	.	(3.73)
	Clearly, if
	r

Table 3 .

 3 r divides n R and n c Ck , the chunks Ck Class B: Frames of 1920 × 1080 px, 13824 chunks of 40 × 30 px Class E: Frames of 1280 × 720 px, 6144 chunks of 40 × 30 px Class F: Frames of 1280 × 720 px, 6144 chunks of 40 × 30 px 3: Characteristics of the considered video sequences and corresponding chunk organization. of least variance are dropped. The values of the parameters n gCk and v Ck are also provided in

	Name n gCk × v Kimonol Frame rate [Hz] Nb Frames 24 240 1393 × 10
	BasketballDrive	50	500	2765 × 5
	BQ Terrace	60	600	3217 × 4
	Cactus	50	500	2765 × 5
	ParkScene	24	240	1393 × 10
	Class C: Frames of 832 × 480 px, 3328 chunks of 32 × 30 px
	PartyScene	50	500	555 × 6
	BQMall	60	600	666 × 5
	BasketballDrill	50	500	555 × 6
	RaceHorses	30	300	333 × 10
	Class D: Frames of 416 × 240 px, 832 chunks of 32 × 30 px
	BQSquare	60	600	167 × 5
	RaceHorses	30	300	84 × 10
	BlowingBubbles	50	500	139 × 6
	BasketballPass	50	500	139 × 6
	FourPeople	60	600	1536 × 4
	Jonny	60	600	1536 × 4
	KristenAndSara	60	600	1536 × 4
	SlideShow	20	500	512 × 12

divides n C . When r Ck η SC ρ Ck , only the best subchannels are selected. Moreover, for each GoP, n F r F r SC nrnc chunks are transmitted on each subchannel. When r Ck > η SC ρ

Table 3 .

 3 3. In the simulations, always the best n gCk subchannels are used.

Table 3

 3 .3, one obtains a metadata rate of 13824 × (1 + 8 + 8) × 3 × 2 = 1.41 Mb/s. If the channel characteristics are refreshed at the GoP rate, one gets an additional metadata rate of 3217×8×3×2 = 0.15 Mb/s.

Considering video transmission over PLT channels, the transmission of channel-coded metadata would require dedicated subchannels and a rate of about 1 % of the total rate available in the context of HomePlug AV2 [YAA

+ 13]

.

Table 3 .

 3 7: Mismatch For video sequence BQMall in general multicast channel

	receiver 1	receiver 2	Precoder	receiver1 PSNR(dB)	receiver2 PSNR(dB)
	Chan.1	5% ip from Chan.1	p i /σ 2 o,i Ave. SNR Chan.1	44.66 41.76 44.91	44.66 41.78 38.00
	-	10% ip from Chan.1	p i /σ 2 o,i Ave.SNR Chan.1	44.38 39.80 44.91	44.38 39.71 35.32
	-	15% ip from Chan.1	p i /σ 2 o,i Ave.SNR Chan.1	44.09 38.29 44.91	44.09 38.29 33.65
	-	20% ip from Chan.1	p i /σ 2 o,i Ave.SNR Chan.1	43.77 37.16 44.91	43.77 37.17 32.34

  at each recursion of PAISP and when λ is split into n SC components at the end. In such case, there are n SC recursions and the total complexity is proportional to Power Allocation with Local Power Adjustment Power Allocation with Local Power Adjustment (PALPA) is an algorithm that takes initially λ = (λ 1 . . . λ n SC ) and s = (s 1 . . . s n SC ) as inputs. PALPA evaluates rst the power allocated with a total power constraint. The resulting allocated power vector has entries m Otherwise, let τ be the largest index for which Condition (3.41) is violated.As shown in what follows, the powers allocated to the chunk subvector (λ τ +1 , . . . , λ n SC ) are then easily updated to match the total power constraint of the n SC -τ last subchannels, while satisfying Conditions(3.41). The corresponding part of the precoding matrix is then build, see Section 3.4.2. PALPA is then called iterative on (λ 1 . . . λ τ ) and (s 1 . . . s τ ) to build the remaining parts of the precoding matrix.

	4.4 (0) i , i =
	1, . . . , n SC . If Conditions (3.41) and (3.42) are satised, the m (0) i s are used to build the solution
	to the power allocation problem with per-subchannel power constraints using [ZZ95, VA99],
	see Section 3.4.2.
	n SC µ=1 µ and hence is
	O (n 2 SC ).

For the power allocation update, since τ is the largest index for which Condition (3.41) is violated, one has τ i=1

  n SC to get m

	(1) i ,

i = τ +1, . . . , n SC in such a way that the conditions (3.41) and (3.42) are valid for the subvectors SC SC

  Table4.1. For PAISP and PALPA, the PSNR gap to optimality is never larger than 0.03 dB. In Figures 4.7, 4.8, 4.9 and 4.10, the evolution of the PSNR of some videos as a function of the frame index is shown. We observe that PAISP (with α = 0.75 and β = 0.5), PAISP

	Class	Name	SCS	PSNR (dB) Opt.Alloc. or		PAISP or	Gain
				PAISP		PALPA
				(Dichotomy)
		Kimonol	42.79	47.57		47.56	4.78
		BasketballDrive	38.83	39.54		39.53	0.71
	B	BQ Terrace	34.83	35.86		35.85	1.03
		Cactus	36.53	38.47	38.46/38.44	1.94
		ParkScene	41.83	44.06	44.06/44.03	2.23
	Av. PSNR Class B	38.96	41.10	41.09/41.08	2.13
		PartyScene	40.89	42.94		42.94	2.05
	C	BQMall BasketballDrill	41.24 44.96	44.91 47.32	44.90/44.91 47.31	3.67 2.36
		RaceHorses	42.81	46.21		46.21	3.4
	Av. PSNR Class C	42.48	45.35		45.34	2.87
		BQSquare	39.38	44.55		44.55	5.17
	D	RaceHorses BlowingBubbles	43.89 42.26	49.03 47.90		49.03 47.90	5.14 5.64
		BasketballPass	45.03	49.55		49.55	4.52
	Av. PSNR Class D	42.64	47.76		47.76	5.12
		FourPeople	40.74	47.13	47.11/47.13	6.39
	E	Jonny	40.56	48.43	48.40/48.43	7.87
		KristenAndSara	39.77	46.95	46.94/46.95	7.18
	Av. PSNR Class E	40.36	47.50	47.48/47.50	7.14
	F	SlideShow	35.28	46.83	46.82/46.80	11.55
	with Dichotomy and PALPA have very close PSNR performance. Only the results of PAISP
	are thus represented. All approaches clearly outperform the SCS allocation of Section (4.1),
	which can be considered as a natural extention of Parcast [LHL	+ 14b].

For sequences of class E (video conference content), where many chunks have very small variance (because of low spatial or temporal activity), the performance gain of the proposed approaches is really signicant compared to SCS. The many small variance chunks prevent SCS to achieve good allocation performance and justify the large gains in these cases. High gains Table 4.1: Simulation results with per-subchannel power constraints

Table 4 .

 4

3: PAISP: Inuence of α and β matrix.

Table 4

 4

	Video	receiver 1 PSNR(dB) Point-to-Point MultiCast Point-to-Point MultiCast receiver 2 PSNR(dB)
	Kimonol	47.56	47.40	42.14	42.14
	BQMall	44.90	44.84	38.29	38.29
	Table 4.4: linearly degraded multicast channel per-subchannel power constraint
	receiver 1	receiver 2	Precoder	receiver1 PSNR(dB)	receiver2 PSNR(dB)
	Chan.1	5% ip from Chan.1	p i /σ 2 o,i Ave. SNR Chan.1	44.63 41.78 44.90	44.63 41.75 38.08
	-	10% ip from Chan.1	p i /σ 2 o,i Ave.SNR Chan.1	44.37 39.74 44.90	44.37 39.79 35.27
	-	15% ip from Chan.1	p i /σ 2 o,i Ave.SNR Chan.1	44.07 38.25 44.90	44.07 38.26 33.48
	-	20% ip from Chan.1	p i /σ 2 o,i Ave.SNR Chan.1	43.75 37.20 44.90	43.75 37.19 32.26

.5: Mismatch For video sequence BQMall in general multicast channel First, one considers a channel model where the covariance of channel noise of Receiver 2

Table 5

 5

	.1: Main notations

  ) as a function of (1 -r d ) 2 for dierent values of the INR in dB and p I

					INR=10dB							INR=14dB
		1									1
	log 10 σ 2 r	0 0.2 0.4 0.6 0.8	r d =0.15 r d =0.3 r d =0.45 r d =0.6 r d =0.75						log r 10 σ 2	-0.5 0.5 0	r d =0.15 r d =0.3 r d =0.45 r d =0.6 r d =0.75
		-0.2
		-0.4									-1
		-2.4		-2.2	-2	-1.8	-1.6		-1.4	-2.4		-2.2	-2	-1.8	-1.6		-1.4
					log 10 (p I )								log 10 (p I )
					INR=18dB
		1
			r d =0.15
		0.5	r d =0.3
	log 10 σ 2 r	-0.5 0	r d =0.45 r d =0.6 r d =0.75						log 10 σ 2 r
		-1
		-1.5
		-2.4		-2.2	-2	-1.8	-1.6		-1.4	-2.4		-2.2	-2	-1.8	-1.6		-1.4
					log 10 (p I )								log 10 (p I )
					INR=10dB							INR=14dB
		1									1
	log 10 σ 2 r	0 0.2 0.4 0.6 0.8	p I =0.5% p I =1% p I =1.5% p I =2% p I =2.5% p I =3%						log r 10 σ 2	-0.5 0 0.5	p I =0.5% p I =1% p I =1.5% p I =2% p I =2.5% p I =3%
		-0.2
		-0.4									-1
		0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8
						2								(1 -r d ) 2
					INR=18dB							INR=22dB
		1									1
			p I =0.5%								p I =0.5%
		0.5	p I =1%							0.5	p I =1%
	log 10 σ 2 r	-0.5 0	p I =1.5% p I =2% p I =2.5% p I =3%						log r 10 σ 2	-0.5 0 -1	p I =1.5% p I =2% p I =2.5% p I =3%
		-1									-1.5
		-1.5									-2
		0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8
					(1 -r d ) 2								(1 -r d ) 2
	Figure 5.6: log 10 (σ 2

r

  Figure5.9: log 10 (σ 2 r ) obtained from the PM (5.29) (mod) and from the simulations (sim) for the channel with 416 subchannels (5.19), the covariance of impulse noise residual depends on the average INR, where the average is evaluated considering the noise variance of the last n SCsubchannels which are used for Step 3, the actual number of subchannels used for chunk transmission may be less than the target number t .The minimization of ε (r d ) may then be performed, e.g., by exhaustive search, or by gradient To illustrate the accuracy of the PM in the subchannel provisioning approach, one considers rst a simple example where a group of chunks of the rst GoP of the BQSquare and RaceHorses videos (their characteristics are detailed in Section 5.6.2) is transmitted considering dierent SNR and p I with σ 2 I = 100. The background channel noise is assumed zero-mean complex circular Gaussian with covariance matrix τ diag (0.08, 0.084, 0.088, . . . , 0.08 + (n SC -1) × 0.004), Figure 5.10 illustrates the evolution of the predicted and measured MSE as a function of actual values of r d . Two values of τ are considered, which leads to two dierent values of the total power constraint p T and of the INR. The match is good, especially when the background noise is large (corresponding to large values of τ ). One sees that using the PM, one is able to get a very good estimate of r d , without the need for a time-consuming exhaustive search for the value of r d that minimizes the MSE.

	log 10 σ 2 r At descent to nd -2.5 -2 -1.5 -1 -0.5 0 0.5	0.75(mod) r d =0.15(sim) p I =0.005 0.15(mod) r d =0.45(sim) 0.45(mod) r d =0.75(sim)						log 10 σ 2 r	-2 -1 0 1	0.75(mod) r d =0.15(sim) p I =0.010 0.15(mod) r d =0.45(sim) 0.45(mod) r d =0.75(sim)
		10 -3	15	INR(dB) 20	25	30 r d = arg min r d	10 -3 ε (r d ) .	15	INR(dB) 20	25	30	(5.35)
					p I =0.015									p I =0.020
		1												1
	0.5 The version of the LVC scheme implementing the Optimal Subchannel Provisioning (OSP) with 0.5 10000 10000
		0												0
	INR(dB) 20 the Impulse noise Correction (IC) is denoted LVC-OSP-IC in what follows. 10 15 25 30 log 10 σ 2 r -2.5 -2 -1.5 -1 -0.5 r d =0.15(sim) 0.15(mod) r d =0.45(sim) 0.45(mod) r d =0.75(sim) 0.75(mod) INR(dB) 10 15 20 -2.5 MSE 7000 7000 MSE 0.75(mod) -2 -1.5 0.45(mod) r d =0.75(sim) 8000 Measured 8000 log 10 σ 2 r r d =0.45(sim) Model -1 -0.5 r d =0.15(sim) 0.15(mod) 9000 9000	25	30
		6000										6000
		5000										5000	Model
															Measured
		4000										4000
		0.2	0.3	0.4	0.5	0.6	0.7	0.8					0.2	0.3	0.4	0.5	0.6	0.7	0.8
				r d actual										r d actual
			(a) BQSquare when τ = 1									(b) BQSquare when τ = 7
	the impulse estimation. One thus introduces ×10 4 1.8		1.8	×10 4
		1.6										1.6
		1.4				INR = Measured Model	(n SC -) σ 2 I n SC i i= +1 σ 2 1.4	,	Model Measured	(5.33)
	MSE	1.2									MSE	1.2
	which will be used in the PM (5.29) in place of INR in what follows. A consequence of this 1 1
	substitution is that INR depends on , the number of subchannels allocated for the chunk 0.8 0.8
	transmission. One also introduces the average SNR dened as 0.6 0.6
		0.2	0.3	0.4	0.5	0.6	0.7	0.8					0.2	0.3	0.4	0.5	0.6	0.7	0.8
				r d actual										r d actual
			(c) RaceHorses when τ = 1			SNR =	p T n SC i=1 σ 2 i	,	(d) RaceHorses when τ = 7	(5.34)
	to evaluate the channel quality.
		Then for a given value of r d ,
	5.6 Simulation
	1. one evaluates the target number t of subchannels available for chunk transmission and
		INR using (5.33),
	2. σ 2

Contrary to Section 5.5.2, where all σ 2 i , i = 1, . . . , n SC are assumed equal to σ 2 g and the INR equal to σ 2 I /σ 2 g , here, one considers that the INR depends on the subchannel index. As seen in r is then deduced from the PM (5.29), 3. the chunk reconstruction MSE ε (r d ) is nally obtained from (5.24). where τ is a scaling coecient. At transmitter, various target values of r d are chosen. The precoding and decoding matrices are updated accordingly to account for σ 2 r , as evaluated by the PM, see Section 5.5.1. At receiver side, the MSE (5.3) as estimated by (5.24) and the measured MSE for dierent values of r d and dierent channel conditions are compared. Note that the actual value of r d may be dierent from the target value, due to the total power constraint. This is especially true at low SNR. Several variants of LVC schemes described in Section 5.6.1 are considered in what follows. The rst are the baseline LVC with No Impulse noise Correction (LVC-NIC) as well as an LVC With Subchannel Provisioning but No Impulse noise Correction (LVC-WSP-NIC). The aim is

  subchannels. Each subchannel has a bandwidth f SC = 24.414 kHz. Using analog QAM and root-raised-cosine Nyquist lters with β r = 30 % roll-o, one obtains a per-subchannel transmission rate SC = 37.56 × 10 3 real-valued symbols per second. The two video sequences are taken from the MPEG test set used for the standardization of HEVC [OSS + 12], namely BQSquare (Class D) and RaceHorse (Class C). One considers only the luminance component of each video. The frame rate is r F . Considering GoPs of constant size n F , and chunks of n c × n r pixels, each GoP contains For the typical values of the parameters considered in the simulations, v Ck > 1, i.e., several chunks have to be transmitted on the same subchannel for the duration of a GoP. Therefore, given the number of subchannels n SC , at most n TrCk = ν Ck n SC chunks can be transmitted. For that purpose, the n TCk chunks are ordered by decreasing

	r SC =	2f SC 1 + β r	,	(5.36)
	here equal to r n Ck =	n C n R n c n r	n F
	chunks. The number of chunks a subchannel can transmit for the duration of a GoP is
	ν Ck =	n F r F	r SC n r n c	.	(5.37)
	variance and are partitioned into
	n gCk =	n Ck ν Ck

groups of ν Ck chunks of similar variance. Consequently, ν Ck precoding (and decoding) matrices

Table 5

 5 Impact of r d on the eciency of impulse noise correctionThe average PSNR of the rst 5 GoPs of BQSquarre and RaceHorses is evaluated for SNRs ranging from 0 dB to 20 dB (the SNR is evaluated without taking into account the impulse noise). The power p T for one OFDM symbol is set equal to 2560. The variance and the probability of impulse noise are σ 2 I = 100 and p I = 0.01 or p I = 0.02. Then INR is computed from SNR, p T and n SC . R × n C 240 × 416 480 × 832

	.2.

are denoted as LVC Without Impulse noise (LVC-WoI) and LVC With Subchannel Provisoning and Without Impluse noise (LVC-WSP-WoI), respectively. Simulation results are shown in Figures 5.12 and 5.13. Compared to the reference situation n

Table 5 .

 5 2: Simulation parameters LVC-WoI, when p I = 0.01, without correction, the impulse noise leads to a PSNR decrease ranging from 0.6 dB at low channel SNR to 13 dB at high SNR. When p I = 0.02, the PSNR decrease is between 1 dB at low channel SNR and 15 dB at high SNR. Depending on the number of provisioned subchannels, the impulse noise correction may only be partial (when p I is large and r d is small) or complete (when r d is large enough). In the second case, LVC-WSP-IC and LVC-WSP-WoI perform similarly. Nevertheless, r d should not be chosen too large, since the PSNR in absence of impulse noise at high SNR decreases when r d increases, due to the additionally dropped chunks.Figures 5.14 and 5.15 represent the gains obtained by LVC-WSP-IC compared to LVC-NIC at dierent SNRs and for dierent target values of r d taken in R = {0.25, 0.33, 0.41, 0.5, 0.66, 0.75}. observes that the optimal value of r d depends on the value of the channel SNR. At low SNRs, r d should be large, whereas at large SNRs, r d may be reduced. This is mainly due to the fact that at low SNR, the INR is low and impulse noise identication is dicult with few

	One

syndrome samples. At high SNR, the INR increases, and it becomes easier to perform impulse noise identication .

Table 5 .

 5 2.PSNR results for actual impulse noise probabilities ranging from 0% to 4% are shown in Figure5.18. As expected, the PSNR decreases as p I increases. The performance is best when p I matches p It . Choosing a large p It improves the robustness to a larger p I , but the price to be paid is a lower PSNR when p I is smaller than p It . Figure5.18 shows that even if a small

  . Next, given the H i s, (6.1) is converted to a convex Semi-Denite Programming (SDP) problem. Let us Here, A B indicates that B -A is positive semi-denite.

	introduce the constraints
		GΛG T Φ
	Then using Shur's complement, (6.1) can rewritten as
	min G,Ψ i ,Φ	t
	s.t.

and

H i GΛG T H T i -H i GΛ -Λ T G T H T i Ψ i

and a real valued slack variable t.

tr (Ψ i ) + tr H i N i H T i + Λ t   

13 Else, if τ αµ, then τ is chosen as the split position. PAISP is then recursively called with inputs subvectors λ (i) and s (i) of length µ (i) , i = 1, 2 of λ and s. This avoids the repeated tests performed at Lines 13-15 of Algorithm 3.1.

To evaluate the complexity of PAISP to nd all the split positions, the worst case is when τ = 1 at Line 3 of Algorithm 4.1 at each recursion. In such case, there are n SC recursions and in each recursion the complexity (mainly due to OptTotalPower ) is proportional to the length µ of each subvector being considered. The total complexity is proportional to Combining (5.2), (5.10), and (5.1), one may evaluate the syndrome vector

(5.11)

Since the rows of Ψ are the q last rows of F , Ψv g ∼ CN (0, N s ), with N s = 2diag σ 2 n SC -q+1 , . . . , σ 2 n SC .

Therefore to mitigate the eect of the impulse noise, one has to estimate the sparse vector v I from noisy measurements of Ψv I . This is a typical compressive sensing estimation problem for which many solutions have been proposed. Here, one considers the Fast Bayesian Matching

Pursuit algorithm (FBMP) to get an estimate

(5.12) of v I , see [START_REF] Schniter | Fast bayesian matching pursuit, Information Theory and Applications Workshop[END_REF] for more details. Finally, the vector y' after impulse noise mitigation is

(5.13)

The components of y are then used to get LMMSE estimates of t 2i-1 and t 2i , i = 1, . . . , n r ×n c /2 using (5.5), where N is replaced by the covariance matrix of F (v I -v I ) + F v g divided by 2 to account for (5.1). The main diculty lies in the optimization of the number q of sub-channels provisioned for impulse noise mitigation. The number q is not only due to the power constraint, the q lowest Considering this structure, using (5.27), one deduces