C ¸agatay Dikici

Anissa Mokraoui
email: mokraoui@univ-paris13.fr

Michel Kieffer
email: kieffer@lss.supelec.fr

Pierre Duhamel
email: duhamel@lss.supelec.fr

JOINT SOURCE-PROTOCOL-CHANNEL DECODING: IMPROVING 802.11N RECEIVERS

This paper combines joint protocol-channel (JPC) and joint source-channel (JSC) decoding techniques within a receiver in the context of wireless data transmission. It assumes that demodulation and channel decoding at physical (PHY) layer can provide soft information about the transmitted bits. At each layer of the protocol stack, JPC decoding allows headers of corrupted packets to be reliably decoded and soft information on the corresponding payload to be forwarded to the correct upper layer. When reaching the application (APL) layer, packets may still contain errors and are JSC decoded, exploiting residual redundancy present in the compressed bitstream, to remove part of the residual errors. The main contribution of this paper is to show that these tools may be efficiently combined to obtain i) reliable protocol layers permeable to transmission errors and ii) improved source decoders. Performance is evaluated using an OMNET++ simulation for the transmission of compressed HTML files (HTTP 1.1) over a standard RTP/UDP-Lite/Ipv6/MAC-Lite/802.11n-PHY protocol stack, only the receiver is modified. For a given packet error rate, the proposed scheme provides gains up to 2 dB in SNR compared to a standard receiver.

INTRODUCTION

The widely used OSI layered model partitions networking tasks into distinct layers [START_REF] Kurose | Computer Networking: A Top-Down Approach Featuring the Internet[END_REF]. This facilitates network design, since each layer has not to be aware of the information introduced by other layers, allowing heterogeneous contents to be delivered via the same communication network. Moreover, each layer, assuming that the lower layers behave perfectly, attempts to provide perfect information to the upper layers. For that purpose, error-detecting codes (CRC or checksums) protecting essentially headers (and sometimes the payload) have been introduced at various places of the protocol stacks. They are combined with retransmission mechanisms (when feasible) for data packets deemed as corrupted. Moreover, since the layers work independently, but sometimes require the knowledge of identical (or correlated) information, some redundancy may be found, essentially in the packet headers that are processed at each layer. This redundancy has been recognized and used for example in ROHC [START_REF] Bormann | Robust header compression (ROHC): Framework and four profiles[END_REF] for reducing the header lengths. However, this redundancy can also be used for a better reception of the headers, therefore reducing the number of rejected (and possibly retransmitted) packets.

The role of Joint Protocol-Channel (JPC) decoding is to make an efficient (and joint) use of the redundancy present in the protocol layers as well as the redundancy introduced by the channel coding in order to obtain optimal receiver performance. Recently, JPC decoding techniques have been applied to perform Reliable Packet Header Estimation (RPHE) [START_REF] Lamy | Method of correcting an erroneous frame by a receiver[END_REF][START_REF] Nieto | Cyclic redundancy check (CRC) based error method and device[END_REF][START_REF] Marin | Robust mac-lite and soft header recovery for packetized multimedia transmission[END_REF][START_REF] Mériaux | Robust IP and UDP-lite header recovery for packetized multimedia transmission[END_REF]. Using JPC decoding, corrupted packets, which would be dropped at intermediate layers of the protocol stack by classical receivers, now may reach the APL layer [15,[START_REF] Panza | Supporting network transparency in 4G networks[END_REF]. Even if they are still corrupted, the residual redundancy present in the compressed bitstream may be exploited by JSC decoding techniques to remove residual errors, see [START_REF] Duhamel | Joint source-channel decoding: A cross-layer perspective with applications in video broadcasting[END_REF] and the ref-erences therein. These techniques do not need any change in the structure of the transmitted signal, since JPC and JSC decoding is performed within the receiver. The ability to use these tools in the context of existing standards makes it potentially very practical.

However, to the best of our knowledge, the performance gains obtained when implementing simultaneously JPC and JSC decoding techniques have not yet been evaluated in a realistic receiver. Several issues have to be addressed. First, soft-output demodulators and channel decoders [START_REF] Bahl | Optimal decoding of linear codes for minimizing symbol error rate[END_REF] have to be used to get soft information about the transmitted packets (a posteriori probabilities or loglikelihood ratios). RPHE has to be performed at each layer of the protocol stack. Soft bit information has to be transmitted between layers of the protocol stack [START_REF] Panza | Supporting network transparency in 4G networks[END_REF]. Aggregated packets have to be reliably delineated [START_REF] Chiani | Optimum synchronization of frames with unknown, variable lengths on Gaussian channels[END_REF][START_REF] Ali | Joint protocol-channel decoding for robust aggregated packet recovery at WiMAX MAC layer[END_REF]. Finally, JSC decoding has to be performed at APL layer.

This paper addresses several of previously mentioned issues and considers the reliable transmission of source-coded contents encapsulated in RTP/UDP-Lite/IPv6/MAC-Lite/PHY-802.11n headers over an AWGN channel with Rayleigh fading. Soft demodulation and channel decoding, RPHE in all protocol layers, transmission of soft information between layers, and JSC decoding of compressed HTML files [START_REF] Fielding | Hypertext transfer protocol -HTTP 1.1. RFC 2616[END_REF] have been implemented in an OMNET++ simulator [START_REF] Varga | OMNeT++, chapter Modeling and Tools for Network Simulation[END_REF].

Section 2 briefly recalls the principles involved in JPC decoding for RPHE and in JSC decoding. Section 3 introduces the considered protocol stack in which JPC and JSC decoding have been implemented, as described in Section 4. Simulation results are reported in Section 5 before drawing some conclusions in Section 6.

JOINT PROTOCOL-CHANNEL AND SOURCE-CHANNEL DECODING

This section briefly recalls estimation techniques used to perform JPC decoding for RPHE within the protocol stack and JSC decoding at APL layer. See [START_REF] Duhamel | Joint source-channel decoding: A cross-layer perspective with applications in video broadcasting[END_REF] for more details.

JPC decoding

At any layer L of the protocol stack, packets have more or less the same structure. For example, the n-th packet at layer L consists of a header h n and a payload x n . The header h n may be partitioned into up to four fields (not necessarily contiguous), depending on their properties. The constant field k n contains all bits which either do not change from one packet to the next one (they are fully determined by the standard) or which may be perfectly predicted from previously correctly decoded headers (from layer L or from other layers). The unknown field u n contains data which cannot be predicted but are important for the correct processing at Layer L. The protocol may be such that the values taken by u n are restricted to a set Ω n , where Ω n is determined from the various sources of redundancy of the protocol stack. All bits of the header which are not fully determined and are not instrumental for the processing of the packet at Layer L are gathered in the other field o n . Finally, an optional Header Error Check (HEC) field c n is often evaluated from some or all previously mentioned bits and (possibly) from the packet payload x n by some (CRC or checksum) encoding function

c n = f (k, u n , o n , x n ) . (1)
Assume that soft bit information y = [y k , y u , y o , y x , y c ] has been provided to Layer L on the header fields and on the payload by the channel decoder of the PHY layer or by Layer L -1. The RPHE considered in [START_REF] Marin | Robust mac-lite and soft header recovery for packetized multimedia transmission[END_REF][START_REF] Mériaux | Robust IP and UDP-lite header recovery for packetized multimedia transmission[END_REF] consists in evaluating the MAP estimate

u n = arg max u∈Ω n p (u|k, y) (2)
of u n taking into account k and the soft information y. Assuming that the HEC does not consider x n and that all u ∈ Ω n are equally probable a priori, one gets

u n = arg max u∈Ω n p (y u |u) ∑ o p (o) p (y o |o) p (y c |f (k, u, o)) (3)
A reduced-complexity optimal algorithm for evaluating (3) with a complexity O(

(o n ) 2 (c n ) )
, where (z) is the length of the vector z, has been proposed in [START_REF] Marin | Robust mac-lite and soft header recovery for packetized multimedia transmission[END_REF] when the HEC is a CRC and in [START_REF] Mériaux | Robust IP and UDP-lite header recovery for packetized multimedia transmission[END_REF] when it is a checksum. Suboptimal algorithms achieving a complexityefficiency trade-off have also been proposed in [START_REF] Marin | Robust mac-lite and soft header recovery for packetized multimedia transmission[END_REF][START_REF] Mériaux | Robust IP and UDP-lite header recovery for packetized multimedia transmission[END_REF]. Once u n has been estimated, the header may be processed at Layer L and y x , containing the soft information about the bits of the packet at layer L + 1 is forwarded to that layer. Floating-point numbers to represent soft information may be forwarded between layers. When Log-likelihood ratios (LLRs) are transmitted, quantized LLR values on 3 or 4 bits are accurate enough and significantly reduce memory requirements, see [START_REF] Panza | Supporting network transparency in 4G networks[END_REF].

Note that no JPC decoding is performed on packets which CRC or checksum is valid. The more complex JPC decoding process is only applied to packets which header has been deemed as erroneous. With good channel conditions, JPC is then only seldom used.

JSC decoding

Assume that the n-th packet has reached the APL layer, where it has to be processed by the source decoder. This packet may still contain errors, since CRCs and checksums at lower layers have been used to help RPHE. The decoded content d n has to be reliably estimated. It corresponds to the payload x APL n for which soft bit information y APL

x,n is provided by the lower protocol layer. For that purpose, residual redundancy left by the source coder in the compressed bitstream may be exploited. This redundancy may come from the syntax of the codewords, the semantic of the bitstream, and from the packetization process, see [START_REF] Duhamel | Joint source-channel decoding: A cross-layer perspective with applications in video broadcasting[END_REF]. This redundancy translates into the fact that x APL n can only take a limited number of values within a set denoted as Ω x n . A MAP estimate of x APL n may then be obtained from y APP

x n and Ω x n as

x APL n = arg max x∈Ω xn P(x|y APP x n ) (4)
In some cases, it is possible to structure the sequences belonging to Ω x n with a multi-dimensional trellis. Efficient channel-decoding techniques such as the Viterbi or BCJR algorithms [START_REF] Bahl | Optimal decoding of linear codes for minimizing symbol error rate[END_REF] may then be employed to evaluate x APL n . With content-adaptive data compression techniques, such as the CABAC, the CAVLC, or the deflate algorithm [START_REF] Deutsch | Deflate compressed data format specification version 1[END_REF], the number of states of the trellis may be prohibitively large. Nevertheless, in such cases, it is usually possible to determine whether a sequence x does not belong to Ω x n or does not correspond to a prefix of a sequence in Ω x n . This allows to use decoding algorithms such as the stack or the M-algorithm [START_REF] Anderson | Source and Channel Coding: An Algorithmic Approach[END_REF].

Iterative decoding may be considered when it is possible to get extrinsic information at APL layer, which is easy when the BCJR decoder is used. Soft output sequential decoders such as the softoutput M-algorithm (SOMA) [START_REF] Wong | The Soft-Output M-Algorithm And Its Applications[END_REF] or the soft-output stack algorithm [23] may also be employed. The role of the outer code may be played by a CRC or a checksum, e.g., at UDP layer or by the channel code at PHY layer. The price to be paid in the second case, is a large amount of additional soft information and decoded headers which have to be forwarded to the APL layer.

CONSIDERED PROTOCOL STACK

JSC decoding of various types of data has been considered: still images, video, audio, and HTML files. In this paper, we considered the transmission of HTML files over an 802.11n WiFi network [START_REF]lan medium access control (mac) and physical layer (PHY) spec. amendt 5: Enhancements for higher throughput[END_REF], with an RTP/UDP-Lite/Ipv6/MAC-Lite/802.11n-PHY encapsulation, see Fig. 1. Clearly, HTML is usually combined with TCP/IP, but RTP/UDP/IP facilitates the implementation of a permeable protocol stack (thanks to UDP-lite [START_REF] Larzon | The lightweight user datagram protocol (UDP-Lite)[END_REF]), since no retransmission of erroneous packets has to be considered at UDP layer, contrary to the connection-oriented TCP. Moreover, this may also give some insights on the behavior of the combination of JSC decoders of video contents after JPC decoding.

Transmitter side

At transmitter side, the APL layer compresses the HTML file with deflate [START_REF] Deutsch | Deflate compressed data format specification version 1[END_REF] and supplements the data with the appropriate Zlib header and tail information [START_REF] Deutsch | Zlib compressed data format specification version 3.3[END_REF]. The packet is then delivered to the RTP layer. In order to specify the compressed HTML content, the RTP layer fixes the 7 bits of the payload type field value to an unassigned value in [START_REF] Schulzrinne | RTP profile for audio and video conferences with minimal control[END_REF] (between 35 and 71).

The UDP-Lite layer [START_REF] Larzon | The lightweight user datagram protocol (UDP-Lite)[END_REF] encapsulates the RTP packet specifying the source port and destination port fields, such that the source port corresponds to the HTTP port 80, and the destination port to a value which is not known at the receiver, The checksum coverage field is fixed to 24 (bytes): the 16 bits of the checksum covers the 8 bytes of the UDP-Lite header and the 16 bytes of the considered RTP header. Then, the IPv6 layer adds the appropriate addressing and header information. Here, it is assumed that the source and destination are located in the same subnetwork hence the first 12 bytes of the IPv6 source and destination addresses are identical. According to the UDP-lite specifications [START_REF] Larzon | The lightweight user datagram protocol (UDP-Lite)[END_REF], the IPv6 pseudo-header is protected by UDP-checksum, which corresponds to the source and destination IPv6 addresses [START_REF] Hinden | Ip version 6 addressing architecture[END_REF].

The MAC-Lite layer is a MAC layer [START_REF]lan medium access control (mac) and physical layer (PHY) spec. amendt 5: Enhancements for higher throughput[END_REF] where the 32 bit MAC CRC covers only the 802.11n MAC header.

At PHY layer, IEEE 802.11n defines three PHY frame formats. The HT-mixed (HT for High Throughput) format has been chosen since the packets are transmitted with a preamble which is compatible with the legacy 802.11a/g. In HT-mixed format, the PHY In this paper, we assume that medium reservation is performed via Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) [2, S-9.2.5]. The transmitter sends a Request To Send (RTS) packet before sending information. If the receiver is not busy, it responds with a Clear To Send (CTS) packet to notify that: i) the transmitter can send the information and ii) all other stations of the same network are informed that the receiver is unavailable.

REDUNDANCY

The aim of this section is to specify the transmission conditions in order to identify the fields in the various headers which content is known, predictable, or unknown.

Assuming that the CSMA/CA medium reservation succeeds, the receiver has many information about the access point to which it is connected, the duration of the packet it will receive, etc.

802.11n Physical Layer

A 1 bit parity field covers the L-SIG header, whereas the HT-SIG header has a 7 bit CRC field. In L-SIG, the Rate field and the Reserved bit are known to the receiver. The receiver has then to estimate the 12 bits of Length in L-SIG, the 7 bits of MCS and 16 bits of HTLength in HT-SIG (see Fig. -3). Since the three unknown fields are related, RPHE is performed jointly on L-SIG and HT-SIG headers using a CRC-based correction algorithm as described in [START_REF] Marin | Robust mac-lite and soft header recovery for packetized multimedia transmission[END_REF][START_REF] Jaoua | Robust transmission of[END_REF].

MAC-Lite Layer

As explained in Section 3, the MAC-Lite CRC covers only the MAC header. Considering a non-encrypted downlink transmission of ordered MAC data packets with deactivated retransmission and power-save mode, the 2-byte Frame Control field is assumed to be known. The 6-byte Address2 field contains the MAC address of the receiver and is thus known. The 6-byte Address1 and Address3 are transmitted during the medium reservation procedure (RTS-CTS) and may be totally deduced by the receiver. Assuming that the access point is connected to a single router and that the router address has been already received in other information packets, the 6-byte Address4 field may also be predicted by the receiver. The 2-byte Sequence Control field contains two parameters: a sequence number and a fragment number. The sequence number represents the value of the current IP packet counter. The fragment number indicates the value of the current MAC data packet counter. In this study, packets are transmitted in order and these parameters can be easily determined: the sequence number is incremented by one for each RTS-CTS and the fragment number is incremented by one for each received MAC data packet, hence can be estimated by the receiver. The last field of the MAC header is reserved for local wireless networks and is composed of 6 bytes of zeros in this study.

Thus, the receiver only estimates the following unknown field (See Fig. 3): 16 bits DurationID using RPHE based on the CRC field, as in [START_REF] Marin | Robust mac-lite and soft header recovery for packetized multimedia transmission[END_REF]. Since in the encoding scheme, the concatenation of the RTP, UDP-Lite, and pseudo-IPv6 headers is protected with a 16 bit checksum introduced by the UDP-Lite header [START_REF] Larzon | The lightweight user datagram protocol (UDP-Lite)[END_REF]. RPHE of the headers introduced by the three layers is done jointly. The receiver determines the known fields using inter and intra-layer redundancy and estimates the following unknown fields (see Fig. 4): the last 32 bits of the Source Address field of IPv6, the 16 bits of Destination Port of UDP-Lite, the 7 bits of Payload type, and the 16 bits of Sequence number introduced by the RTP layer. The decoding algorithm based on the checksum field may be found in [START_REF] Mériaux | Robust IP and UDP-lite header recovery for packetized multimedia transmission[END_REF].

RTP, UDP-Lite and IPv6 layers

Application Layer

From the RTP layer, the receiver knows that the packet it receives contains a compressed HTML file. The decoded file has to be compliant with the HTML syntax as specified in [1]. The set Ω x n introduced in Section 2.2 contains thus all encoded sequences which could be produced by a compliant HTML page provided in [1].

Based on this a priori information and on the soft information provided by the lower layers of the protocol stack, a JSC decoder involving a modified soft-output M-algorithm is employed. Iterative decoding with the channel decoder at APP layer is performed, see [14] for more details.

SIMULATION RESULTS

This section evaluates the performance of the JPC and JSC decoding schemes previously described via a simulation using OMNET++ [START_REF] Varga | OMNeT++, chapter Modeling and Tools for Network Simulation[END_REF]. JPC decoding is only performed on packets which CRC or checksum is not valid.

Channel model

We assume that if the transmitted signal is p, the received signal is y = Hp + n, where H is the complex channel response matrix which is assumed to be known at the receiver, and n is a complexvalue Gaussian noise distributed as C N (0, σ 2 I).

Performance of JPC decoding

The RPHE performance at each layer of the protocol stack is evaluated for several values of E b /N 0 . The header error rate as a function of E b /N 0 is given for the PHY, MAC-Lite, IPv6, UDP-Lite, and RTP layers. For each value of E b /N 0 , enough simulations have been performed to get 100 erroneous headers.

For a target packet error rate of 10 -3 , JPC decoding for RPHE provides a gain of 1.1 dB for the PHY layer header (see Fig. 5), 2.1 dB for MAC-Lite layer (see Fig. 6), 2.3 dB for IPv6 layer (see Fig. 7), 1.8 dB for UDP-Lite layer (see Fig. 8), and 1.9 dB for RTP layer (see Fig. 9) all compared to a classical decoding scheme.

Performance of JSC at APL layer

The worldcup.html html file test is available at www.forum.nokia.com/tools. In a first step, as in [14], we assume that all headers of the lower protocol layers were without errors. Therefore, the performance of the HTML file recovery alone is determined, using a plot of the HTML symbol error rate (SER) as a function of E b /N 0 , see Fig. 10. For a target SER of 10 -4 , the JSC decoding algorithm achieves 0.8 dB gain compared to a standard decoder, see Fig. 10.

Fig. 11 provides the SER at APL layer when JPC has been employed on the noisy packet headers at all layers. 1. RPHE and JSC decoding at APL layer; 2. RPHE and classical decoding at APL layer; 3. Classical header processing and JSC decoding at APL layer; 4. Classical header and APL layer decoding.

In average, JPC and JSC decoding provides an improvement of 0.6 dB compared to classical header and APL layer decoding. Fig. 11 shows that for E b /N 0 between 5 dB and 7 dB, the performance of the robust receiver (RPHE and JSC decoding at APL layer) is better then that obtained by the other three methods. For E b /N 0 above 7 dB, RPHE and classical header processing perform similarly when combined with JSC decoding. Moreover, the results in terms of SER are similar to those in Fig. 10 where all headers were assumed error-free.

CONCLUSION

This paper combines JPC and JSC decoding techniques in a realistic communication context. Within the protocol stack, gain of up to 2 dB may be obtained with JPC decoding techniques compared to classical processing. At APL layer, JSC decoders provide a gain of about 0.8 dB in channel SNR. The gain is moderate due to the little redundancy left in compressed HTML files.

These results are directly applicable to other types of contents: speech, music, or video. JPC and JSC decoding reduces the need for retransmission of damaged packets. Moreover, they are compatible with the existing layered structure of a wireless communication network avoiding therefore modifications of the transmitters. [14] Z. Jaoua, A. Zergainoh-Mokraoui, and P. Duhamel. Robust transmission of html files: iterative joint source-channel decoding of Deflate codes. In Proc. EU-SIPCO, 2008.

Figure 1 :

 1 Figure 1: Encapsulations on top of 802.11n

Figure 3 :

 3 Figure 3: MAC-Lite packet format

Figure 4 :

 4 Figure 4: IPv6, UDP-Lite, and RTP header formats

Figure 6 :

 6 Figure 5: PHY header decoding performance

  Figure 7: IPv6 header decoding performance

Figure 9 :Figure 10 :Figure 11 :

 91011 Figure 9: RTP header decoding performance


S M Popoff

A Aubry

G Lerosey

M Fink

A C Boccara

S Gigan

Exploiting the Time-Reversal Operator for Adaptative Optics, Selective Focusing and Scattering Pattern Analysis

Keywords: numbers: 42.25.-p, 07.60.-j, 78.67.-n, 42.15.Fr

We report on the experimental measurement of the backscattering matrix of a weakly scattering medium in optics, composed of a few dispersed gold nanobeads. The DORT method (Decomposition of the Time Reversal Operator) is applied to this matrix and we demonstrate selective and efficient focusing on individual scatterers, even through an aberrating layer. Moreover, we show that this approach provides the decomposition of the scattering pattern of a single nanoparticle. These results open important perspectives for optical imaging, characterization and selective excitation of nanoparticles.

The propagation of waves in inhomogeneous media is a fundamental problem which implies applications in many domains of wave physics, ranging from optics or acoustics to solid-state physics, seismology, medical imaging, or telecommunications. Conventional focusing and imaging techniques based on the Born approximation generally fail in aberrating or strongly scattering media due to the phase distorsions and the multiple scattering events undergone by the incident wavefront. Recent advances in light manipulation techniques have allowed great progresses in optical focusing and imaging through complex media. Following pioneering works in ultrasound [1,2], Vellekoop et al. [3] showed how light can be focused spatially through a strongly scattering medium by actively shaping the incident wavefront with a spatial light modulator (SLM). Several groups [4][5][6] have recently extended this approach to the time domain and demontrated the spatio-temporal focusing of femtosecond pulses through highly scattering media.

A general treatment of such linear complex media is provided by the scattering matrix formalism, as defined in [7] for instance. For ultrasound or microwaves, such a formalism is particularly appropriate as the wave field can be generated and detected by arrays of independent elements. Popoff et al. [8,9] recently extended this method to optics and measured the transmission matrix for light between a SLM (emission) and a CCD camera (reception) through a scattering medium. The latter is a subpart of the scattering matrix linking the forward scattered field to the incident field.

The experimental access to this transmission matrix is particularly important since it contains many informations on the propagation medium. In particular, it should give access to the open channels that allow to transmit light through the scattering sample [7,10,11].

In this Letter, we study another subpart of the scattering matrix : the backscattering matrix K (BM). The knowledge of the BM brings more fundamental insight into the medium. Initially developed for ultrasound [12][13][14] and then extended to microwaves [15,16], the DORT method (French acronym for Decomposition of the Time Reversal Operator) takes advantage of the BM to focus iteratively by time reversal processing on each scatterer of a multi-target medium. Derived from the theoretical study of the iterative time reversal process, this approach relies on the diagonalization of the time reversal operator KK † . A one-to-one association between each eigenstate and each scatterer allows focusing on each scatterer by transmitting the computed eigenvectors.

Technically, a singular value decomposition (SVD) of the BM is used to compute the eigenstate of the time reversal operator. In this letter, we will demonstrate the DORT method in the optical regime and show its efficiency in weakly scattering media. An experimental setup is proposed to record the BM in optics. Using the DORT method, we perform selective focusing on 100 nm gold nanoparticles through an aberrating medium, opening an original alternative to conventional adaptive optics [START_REF] Tyson | Principles of adaptive optics[END_REF]. In a second part, we show how the DORT method can be used to discriminate and selectively excite the different scattering patterns of a single nanoparticle, hence paving the way towards a better characterization, imaging and control of nano-objects for instance in plasmonics [START_REF] Gjonaj | [END_REF] or for super-resolution [19].

The experimental setup used for the measurement of the BM is displayed in Fig. 1. To generate the incident wavefront, the beam from a single mode laser at 532 nm is expanded and spatially modulated by a liquid crystal SLM. Using a polarizer and an analyzer, we choose an appropriate combination of polarizations before and after the SLM to achieve a 2π-phase modulation with a residual amplitude modulation below 15%. Light is focused onto the surface of a glass slide on which is deposited a dilute solution of 100 nm isotropic gold beads. We image the output backscattered field pattern onto a first CCD camera (CCD1) thanks to a beamsplitter. To minimize specular reflections, the two polarizers are in a cross-polarized configuration. A reference arm can be used to make the backscattered field interefere with a plane wave onto CCD1. The system is set such that a pixel of the SLM corresponds approximately to an incident k-vector. An aberrating slab is placed 0.5 mm from the image plane between the scattering sample and the objective. It consists in an irregular PDMS polymer layer with a mean thickness of 200 µm deposited on a cover glass.

The CCD1 records the image of the intensity in the plane containing the gold particles, with and without the aberrator. For control purposes, another CCD camera (CCD2) images the same plane in transmission, through a high NA objective.

In the experimental setup shown in Fig. 1, the input and output modes correspond to SLM and CCD1 macropixels, respectively. The BM corresponds to a matrix K whose complex coefficients k mn connect the optical response (in amplitude and phase) between the n th SLM pixel to the m th CCD1 pixel. In other words, the complex optical field on the m th outgoing pixel reads E out m = n k mn E in n with E in n the field on the n th input pixel. K is of dimension M × N , with M and N the number of pixels contained by CCD1 and the SLM, respectively. The inter-element responses k mn are measured following the procedure described in [8], i.e using phase-shifting interferometry and a Hadamard input basis.

A singular value decomposition (SVD) of the BM consists in writing K = UΣV † where the subscript † denotes the transpose conjugate. Λ is a diagonal matrix containing the real positive singular values λ i in a decreasing order images the same plane in transmission with a 60x oil immersion objective (NA = 1.4). Analysers P 2 and P 3 are oriented in a cross-polarized configuration. L, lens. BS, beamsplitter. method consists in using the SVD of the BM to efficiently detect and separate the responses of several scatterers in homogeneous or weakly heterogeneous media [13]. Indeed, under the single scattering approximation and for pointlike scatterers, each scatterer is associated mainly to one significant eigenstate. The latter is linked to a non-zero singular value λ i that relates to the scatterer reflectivity. In first approximation, each singular vector V i of K corresponds to the complex amplitude mask which, when applied to the SLM, focuses onto the associated scatterer.

λ 1 > λ 2 > • • • > λ N .
Figs. 2 and3 show the experimental results of the DORT method applied to the BM matrix when several 100 nm gold beads are dispersed on the slide placed behind an aberrating layer. For this experiment, we have used the field generated by the unmodulated part of the illumination as the reference for the interferometric measurement rather than the plane wave from the reference arm. This method detailed in [8] allows for a better stability without being detrimental to focusing applications. The numbers of pixels have been fixed to N = M = 1024. Fig. 2(a) shows the image measured by CCD1 in reflection when a plane wave is sent from the SLM. The image of the medium is drastically damaged by the aberrating layer : the number of gold beads as well as their position and reflectivity cannot be deduced from this image. In contrast, the DORT method

gives an unambiguous answer to these questions. The singular value distribution (Fig. 2(b)) of the BM exhibits two parts. The continuum of low singular values is associated to the noise in So far, we made the assumption that each scatterer is only associated to one single singular value. However, this is not strictly true : there can be several singular values associated to a single scatterer, each one corresponding to a different scattering pattern of the particle [20][21][22]. To investigate the different eigenstates associated to one nanoparticle, we now study analytically and experimentally the DORT analysis of the BM without any aberrating layer and with only one gold particle in the observation area.

Let us assume that the incident field is linearly polarized along x, z is the propagation axis and y the polarization observed in reflection (cross-polarized configuration). The BM can be decomposed into the product of three matrices : K = K .ᾱ.K . Here K is a 3 × N matrix describing the propagation of light from the N pixels of the SLM to the scatterer. The three lines of K correspond to the different orientations of the electric field at the particle position. ᾱ is the polarisability tensor of the particle linking the local electric field to the dipole moment p of the particle. K is a M × 3 matrix which connects the dipole moment p to the y-component of the electric field measured on the M pixels of CCD1. We will assume that the particle is isotropic : ᾱ = αI with α the particle polarizability and I the identity matrix.

The electric field illuminating the particle in the focal plane can be deduced from [23,24] and leads to the following expression for the coefficients of K :

k 1j = I 0 (θ 1 j , ρ s ) + I 2 (θ 1 j , ρ s ) cos(2φ s ) negligible ( 1
)
k 2j = I 2 (θ 1 j , ρ s ) sin(2φ s ) (2)
k 3j = 2jI 1 (θ 1 j , ρ s ) cos(φ s ) (3)
φ s and ρ s are the polar coordinates of the particle in the focal plane (see Fig. 4(a)). θ 1 j < θ max 1 is the angle made by the k-vector associated to the j th element of the SLM with the optical axis. The functions I 0 , I 1 and I 2 are given by :

I 0 (θ, ρ) = √ cos θ sin θ(1 + cos θ)J 0 (2πρ sin θ) (4)
I 1 (θ, ρ) = √ cos θ sin 2 θJ 1 (2πρ sin θ) (5)
I 2 (θ, ρ) = √ cos θ sin θ(1 -cos θ)J 2 (2πρ sin θ) (6)
with J 0 , J 1 and J 2 the zero, first and second order spherical Bessel functions of the first kind, respectively.

The y-component of the backscattered field can be also deduced from the calculations of [23] and leads to the following expression for K :

k k1 = θ max 2 0 dθ 2 I 2 (θ 2 , ρ k ) sin(2φ k ) (7)
k k2 = θ max 2 0 dθ 2 [I 0 (θ 2 , ρ k ) + I 2 (θ 2 , ρ k ) cos(2φ k ) negligible ] (8)
k k3 = 2j θ max 2 0 dθ 2 I 1 (θ 2 , ρ k ) sin(φ k ) (9)
ρ k and φ k are the polar coordinates of the k th pixel of CCD1 and θ 2 is the angle made by a beam diffracted by the particle with the optical axis before going through the microscope objective (see ). θ max 2 is directly given by the numerical aperture of the optical system, but θ max 1 can be smaller if the area of illumination is smaller than the area of the pupil of the microscope objective. Theory predicts that the x, y and z components of the particle dipole moment p give rise to different scattering patterns : monopolar for y (Eq.8), bipolar for z (Eq.9)

and quadrupolar for x (Eq.7) (see Fig. 4(d)). Those scattering patterns are strictly orthogonal if they are observed over a solid angle Ω = 2π. Hence, in theory, the output singular vectors U i of K should correspond to the different scattering patterns of the nanoparticle (i.e the columns of K ).

In practice, this is not strictly true since only a limited part of the half space is accessible in our experiment.

Our theoretical predictions are now confronted to the experimental results. As an exact measurement of the output singular vectors is needed, the complex wave field has to be measured directly.

A reference arm is now used to make the backscattered wave interfere with a reference plane wave on CCD1 (see Fig. 1). The output singular vectors U i correspond then to the complex scattering . The position of the particle in the focal plane has been arbitrarily fixed (φ s = 7/8π, ρ s = 665 nm) such that the theoretical and experimental ratios between the first and the second singular values are equal.Our theoretical calculations predict that only the three first singular values are non-zero, each one corresponding to the independent scattering patterns of the three orthogonal dipolar radiations (along x, z and y resp.). The experimental results display a continuum of low singular values corresponding to noise and three higher singular values (Fig. 4(d)). The two first output singular vectors match the scattering patterns of the xand z-components of the nanoparticle dipole moment, as predicted theoretically. The third output singular vector has a geometry comparable to the second one, but is more affected by the experimental noise since a significant part of its energy is observed outside the particle. This singular vector might correspond to a coupling between the second singular state (z-component) and eigenmodes of the noise background.

The scattering pattern associated to the y-component of the dipole moment is drowned in noise and could not be observed.

The DORT method allows to excite independently each component of the nanoparticle dipole moment : such an experiment would require to increase the number of SLM segments for quantitative results and a near field scanning optical microscope (NSOM) to probe the near-field of the nanoparticle. For larger scatterers, the singular spectrum may be even richer : more radiation modes would be excited and a better characterization of the nanoparticle could be obtained (size, position, permittivity etc.). In particular, one could discriminate the quadrupolar radiation of the particle, which may open important perspectives in plasmonics.

In summary, we have performed an experimental measurement of the BM of a weakly scattering medium. We have used this BM to demonstrate the DORT method in optics and proved that it allows selective focusing on nanoparticles and their caracterization. This approach could lead to many applications in nano-optics, sensing or imaging and entails the interest of the BM in optics.

We are confident that measuring the BM in more complex media will raise interesting problems in optics. For instance, the measurement of the BM in highly scattering media may allow the separation of the single and multiple scattering contributions [25] as well as the study of the coherent backscattering cone and of the diffusive halo [26].

We wish to thank Claire Prada and Sébastien Bidault for inspiring discussions as well as Laurent Boitard and Gilles Tessier for their help in the preparation of the samples. This work was made possible by financial support from "Direction Générale de l'Armement" (DGA), Agence Nationale de la Recherche via grant ANR 2010 JCJC ROCOCO, Programme Emergence 2010 from the City of Paris and BQR from Université Pierre et Marie Curie and ESPCI.

FIGURE 1 :

 1 FIGURE 1: Schematic of the apparatus. The 532 nm laser is expanded and reflected off a Holoeye LC-R 2500 liquid cristal SLM. Polarization optics (polarizer P 1 and analyser P 2 ) allow phase modulation. The modulated beam is focused on the plane containing the gold particules by a 20x objective (NA = 0.5) and the backscattered intensity is imaged by a first CCD-camera (CCD1). The second CCD-camera (CCD2)

FIGURE 2 : 1 (

 21 FIGURE 2: Experimental results for selective focusing on nanobeads through an aberrating layer using the DORT method. a. aberrated image of the object plane on CCD1 in reflection for a plane wave illumination b., singular value distribution of the measured BM. c.-f. ( resp. g.-j.) ; images in reflection by camera CCD1 ( resp. in transmission by the control camera CCD2), when sending the four first singular vectors. All images from CCD1 (resp. CCD2) share the same intensity scale.

FIGURE 3 :

 3 FIGURE 3: Phase of the first input singular vector V 1 (b, c) and the corresponding focal spots recorded by CCD1 (a, d) in free space and in presence of an aberrating layer, respectively.

Fig. 4 ( 1 and θ max 2

 412 Fig. 4(b)). Hence, the BM depends on the following parameters : the position of the particle in the focal plane (φ s ,ρ s ), the numbers of SLM and CCD elements (N and M ) and the input and output maximum angles (θ max 1

FIGURE 4 :θ max 1 and θ max 2 have been used for our theoretical calculations (θ max 1 =

 4121 FIGURE 4: Singular value decomposition of the experimental and theoretical BM with a single scatterer. (a) and (b) represent the emission and the recording geometry of the experiment.(c) and (d) show the singular value distributions of the measured and calculated BM. The three first output singular vectors are shown in the insets.


Usman Ali

Michel Kieffer

Pierre Duhamel

Sliding-Trellis based Frame Synchronization

Keywords: CRC, Cross-layer decoding, Frame synchronization, Joint decoding, Segmentation

Frame Synchronization (FS) is required in several communication standards in order to recover the individual frames that have been aggregated in a burst. This paper proposes a low-delay and reducedcomplexity Sliding Trellis (ST)-based FS technique, compared to our previously proposed trellis-based FS method. Each burst is divided into overlapping windows in which FS is performed. Useful information is propagated from one window to the next. The proposed method makes use of soft information provided by the channel, but also of all sources of redundancy present in the protocol stack. An illustration of our STbased approach for the WiMAX Media Access Control (MAC) layer is provided. When FS is performed on bursts transmitted over Rayleigh fading channel, the ST-based approach reduces the FS latency and complexity at the cost of a very small performance degradation compared to our full complexity trellis-based FS and outperforms state-of-the-art FS techniques.

I. INTRODUCTION

FS is an important problem arising at various layers of the protocol stack of several communication systems. The most obvious being, at Physical (PHY) layer, to recover the payload and the side information (headers, etc...) of PHY packets or frames. First results [START_REF] Barker | Group synchronization of binary digital systems in Communication Theory[END_REF], [START_REF] Massey | Optimum frame synchronization[END_REF] considered data streams in which regularly spaced fixed patterns or Synchronization Words (SW) are inserted to delimit fixed-length frames, as is the case, e.g., at the PHY layer of Digital Video Broadcasting -Handheld (DVB-H) for MPEG2 transport stream frames. Synchronization tools based on the maximization of the correlation between the SW and the received data have been proposed in [START_REF] Barker | Group synchronization of binary digital systems in Communication Theory[END_REF]. This has been improved in [START_REF] Massey | Optimum frame synchronization[END_REF], where the optimal statistic for FS has been proposed for the AWGN case, taking into account the presence of data around the SW. This was further extended in [START_REF] Lui | Frame synchronization for Gaussian channels[END_REF] for more sophisticated transmission schemes and in [START_REF] Choi | Frame synchronization in the presence of frequency offset[END_REF], [START_REF] Lee | Robust frame synchronization for low signal-to-noise ratio channels using energy-corrected differential correlation[END_REF] to provide robustness against frequency and phase errors.

In many communication systems, frames are of variable-length, see, e.g., the 802.11/802.16 standards [START_REF]Wireless LAN Medium Access Control (MAC) and Physical layer (PHY) specifications: Enhancements for Higher Throughput[END_REF], [START_REF]IEEE standard for local and metropolitan area networks, air interface for fixed broadband wireless access systems[END_REF] for Wireless Local Area Networks (WLANs). In absence of SW, the Header Error Control (HEC) field of the header has been employed in [START_REF] Ueda | Reducing misframe frequency for HEC-based variable length frame suitable for IP services[END_REF] to perform FS with an automaton adapted from [START_REF]B -ISDN user-network interface -Physical layer specification: General characteristics[END_REF]. A length field, assumed present in the frame header, facilitates the FS when the noise is moderate. In [START_REF] Chiani | On sequential frame synchronization in AWGN channels[END_REF], [START_REF] Martini | Optimum metric for frame synchronization with Gaussian noise and unequally distributed data symbols[END_REF], several hypothesis testing techniques have been proposed to perform FS in presence of SW, which are further extended in [START_REF] Martini | Cross-layer frame synchronization for H.264 video over WiMAX[END_REF] to exploit a priori information on the prevalence of ones and zeros in the payload at the price of a small additional signaling and computational complexity. However, insertion of SW requires some modification to the transmitter, which might not be standard compliant. Most of these FS techniques work on-the-fly, i.e., at each time, only few data samples are processed to perform FS and almost no latency is introduced.

Initially, FS has mainly been considered at PHY layer, albeit this problem may also occur at upper layers of the protocol stack. In fact, some frame aggregation techniques at intermediate protocol layers have been proposed recently in order to reduce the signalization overhead, see, e.g., [START_REF] Sidelnikov | Fragmentation/aggregation scheme for throughput enhancement of IEEE 802.11n WLAN[END_REF] in the context of 802.11 standard. Efficient FS in this context is thus very important, since if some frames are not correctly delineated, a large amount of bits has to be retransmitted. In this context, processing a whole burst at each step may significantly improve the FS performance. This was evidenced in [START_REF] Ali | Joint protocol-channel decoding for robust aggregated packet recovery at WiMAX MAC layer[END_REF] for the segmentation of MAC frames aggregated in WiMAX PHY bursts, where Joint Protocol-Channel Decoding (JPCD) is performed using a modified BCJR algorithm [START_REF] Bahl | Optimal decoding of linear codes for minimizing symbol error rate[END_REF] to obtain the frame boundaries. It exploits all available information: soft information at the output of the channel (or channel decoder) as well as the structure of the protocol layers (SW, known fields in headers, presence of Cyclic Redundancy Check (CRC) or checksums, etc).

This paper proposes an adaptation of the reduced-complexity Sliding Window (SW) [START_REF] Benedetto | Soft-output decoding algorithms for continuous decoding of parallel concatenated convolutional codes[END_REF] variant of the BCJR algorithm, presented for the decoding of convolutional codes, to develop a low-delay and reduced-complexity version of the FS technique presented in [START_REF] Ali | Joint protocol-channel decoding for robust aggregated packet recovery at WiMAX MAC layer[END_REF].

The FS problem is first stated as a maximum a posteriori (MAP) estimation problem in Section II. Then, Section III reformulates the trellis-based technique for FS introduced in [START_REF] Ali | Joint protocol-channel decoding for robust aggregated packet recovery at WiMAX MAC layer[END_REF]. The proposed ST-based algorithm is presented in Section IV and is illustrated in Section V with the FS of WiMAX MAC frames aggregated in bursts.

II. MAP ESTIMATION FOR FRAME SYNCHRONIZATION

A. Frame structure

Consider the n-th variable-length frame at a given protocol layer. This frame is assumed to contain λn = ℓ h + ℓp,n bits, where the leading ℓ h bits represent the frame header, of fixed length, and the remaining ℓp,n bits constitute the variable-length payload. In the header, ℓc bits are some HEC bits: CRC or checksum. The length λn is assumed to be a realization of a stationary memoryless process Λ characterized by

π λ = Pr (Λ = λ) = 0 for ℓmin λ ℓmax, (1)
where ℓmin and ℓmax are the minimum and maximum length in bits of a frame. The header hn of the n-th frame can be partitioned into four fields. The constant field k, contains all bits which do not change from one frame to the next. It includes the SW indicating the beginning of the frame, and other bits which remain constant [START_REF] Marin | Robust header recovery based enhanced permeable protocol layer mechanism[END_REF] once the communication is established. The header is assumed to contain a length field un, indicating the size of the frame in bits λn, including the header. Our task is to estimate the successive values taken by this quantity in all frames of the burst. The other field on, gathers all bits of the header which are not used to perform FS. Finally, the HEC field cn is assumed to cover the ℓ hℓc "working" bits of the header, i.e., cn = f (k, un, o), where f is some (CRC or checksum) encoding function. The payload (assumed not protected by the HEC field) of the n-th frame is denoted by pn. It is modeled here as a binary symmetric sequence.

In what follows, the length of a vector z (in bits) is denoted as ℓ (z) and its observation (soft information) provided either by a channel, a channel decoder, or a lower protocol layer is denoted as yz. z b a represents the sub-vector of z between indexes a and b (in bits).

B. Aggregated frames within a burst

Consider a burst of L bits consisting of N aggregated frames. This burst contains either N -1 data frames and an additional padding frame containing only padding bits, or N data frames. Assume that each of these frames, except the padding frame, contains a header and a payload and follows the same syntax, as described in Section II-A.

Assuming that L is fixed before frame aggregation and that N is not determined a priori, the accumulated length in bits ℓ of the n first aggregated frames can be described by a Markov process, which state is denoted by Sn. With this representation, the successive values taken by Sn, for n = 0, 1, . . . can be described by a trellis [START_REF] Ali | Joint protocol-channel decoding for robust aggregated packet recovery at WiMAX MAC layer[END_REF] such as that of Figure 1, with a priori state transition probabilities

p (Sn = ℓ | Sn-1 = ℓ ′ ) deduced from (1). If ℓ < L, then P Sn = ℓ | Sn-1 = ℓ ′ = π ℓ-ℓ ′ if ℓmin ℓ -ℓ ′ ℓmax 0 else, (2)
and if ℓ = L, then

P Sn = L | Sn-1 = ℓ ′ =        0, if L -ℓ ′ > ℓmax 1, if 0 < L -ℓ ′ < ℓmin ℓmax k=L-ℓ ′ π k , else. (3)
In the trellis, dashed transitions correspond to padding frames and plain transitions correspond to data frames.

C. Estimators for the number of frames and their boundaries

Consider a burst x L 1 of N aggregated frames and some vector y L 1 containing soft information about the bits of x L 1 . Here, one assumes that the first entry of y L 1 corresponds to the first bit of x L 1 . This assumption is further discussed in Section III-C.

The suboptimal MAP estimator presented in [START_REF] Ali | Joint protocol-channel decoding for robust aggregated packet recovery at WiMAX MAC layer[END_REF] consists in first estimating N and then in estimating the locations of the beginning and of the end of the frames. The MAP estimate NMAP of N is given by NMAP = arg max

N min n Nmax P Sn = L|y L 1 , (4)
with Nmin = ⌈L/ℓmax⌉, Nmax = ⌈L/ℓmin⌉, and ⌈•⌉ denoting the upward rounding. Once NMAP is obtained, the MAP estimate for the index ℓn of the last bit of the n-th (n = 1, ..., NMAP) frame is

ℓn = arg max ℓ P Sn = ℓ|y L 1 , (5)
and the length λn of the n-th frame is estimated as

λn = arg max ℓ P Sn = ℓ|y L 1 -arg max ℓ P Sn-1 = ℓ|y L 1 . (6)

III. TRELLIS-BASED FS ALGORITHM

In ( 4), ( 5), and ( 6), one has to evaluate P (Sn = ℓ|y L 1 ) for all possible values of n and ℓ. This can be performed efficiently using the BCJR algorithm [START_REF] Bahl | Optimal decoding of linear codes for minimizing symbol error rate[END_REF], by evaluating first

P (Sn = ℓ, y L 1 ) = αn (ℓ) βn (ℓ) (7)
where

αn(ℓ) = P (Sn = ℓ, y ℓ 1 ) = ℓ ′ αn-1(ℓ ′ )γn(ℓ ′ , ℓ), (8)
βn(ℓ) = P (y L ℓ+1 |Sn = ℓ) = ℓ ′ βn+1(ℓ ′ )γn+1(ℓ, ℓ ′ ), (9)
and

γn ℓ ′ , ℓ = P (Sn = ℓ, y ℓ ℓ ′ +1 |Sn-1 = ℓ ′ ). (10)
Classical BCJR forward and backward recursions allow to evaluate α and β. The initial value S0 of the forward recursion is known, leading to α0(ℓ = 0) = 1 and α0(ℓ = 0) = 0. For the backward recursion, assuming that all allowed final states are equally likely (which is a quite coarse approximation), one gets

βn(L) = 1 Nmax -Nmin + 1 , Nmin n Nmax. (11)
All other values of βn(ℓ), for ℓ < L are initialized to 0.

A. Evaluation of γn

Two cases have to be considered for evaluating γn (ℓ ′ , ℓ). When ℓ < L, the transition corresponding to the n-th frame cannot be the last one, thus corresponds to a data frame. Assuming that ℓmin ℓ-ℓ ′ ℓmax, the bits between ℓ ′ +1 and ℓ may be interpreted as

x ℓ ℓ ′ +1 = [k, un, o, c, p],
where un = u (ℓℓ ′ ) is the binary representation of ℓ-ℓ ′ . The corresponding observation can be written as y ℓ ℓ ′ +1 = [y k , yu, yo, yc, yp]. With these notations, for ℓ = L, γn (ℓ ′ , ℓ) = γ d n (ℓ ′ , ℓ), with

γ d n ℓ ′ , ℓ = p Sn = ℓ|Sn-1 = ℓ ′ ϕ d y ℓ ℓ ′ +1 , x ℓ ℓ ′ +1 , (12)
where

ϕ d y ℓ ℓ ′ +1 , x ℓ ℓ ′ +1 = P (y k |k) P yu|u ℓ -ℓ ′ o P (yo|o) P yc|c = f k, u ℓ -ℓ ′ , o P (o) p P (yp|p) P (p) . (13)
Under the assumptions described above and for a memoryless AWGN channel with variance σ 2 , we get

P yu|u ℓ -ℓ ′ = P (yu|un) = ℓ(un) i=1 1 √ 2πσ e -(yu(i)-un (i)) 2 /2σ 2 .
Assuming that the values taken by p are all equally likely, one gets P (p) = 2 -ℓ(p) . In (13), the sum over all possible o can be evaluated with a complexity O(ℓ (o) 2 ℓ(c) ), as proposed in [START_REF] Marin | Robust header recovery based enhanced permeable protocol layer mechanism[END_REF], see also [START_REF] Ali | Joint protocol-channel decoding for robust aggregated packet recovery at WiMAX MAC layer[END_REF] for more details.

When ℓ = L and Lℓ ′ < ℓmax, the n-th frame is the last one, and x L ℓ ′ +1 = 1 has also to be considered in γn (ℓ ′ , L), leading to

γn ℓ ′ , L = p=0,1 P (Sn = L, y L ℓ ′ +1 , Pn = p|Sn-1 = ℓ ′ ) = γ d n ℓ ′ , L P Pn = 0|Sn-1 = ℓ ′ + γ p n ℓ ′ , L P Pn = 1|Sn-1 = ℓ ′ , ( 14
)
where Pn is a random variable indicating whether the n-th frame is a padding frame and its a priori probability is given by

P Pn = 1|Sn-1 = ℓ ′ =        0, if L -ℓ ′ ≥ ℓmax 1, if 0 < L -ℓ ′ < ℓmin ℓmax λ=L-ℓ ′ +1 π λ , else. (15)
In [START_REF] Ali | Joint protocol-channel decoding for robust aggregated packet recovery at WiMAX MAC layer[END_REF],

γ p n ℓ ′ , ℓ = P Sn = L, y L ℓ ′ +1 |Sn-1 = ℓ ′ , Pn = 1 = P y L ℓ ′ +1 |Pn = 1, Sn-1 = ℓ ′ , Sn = ℓ (16)
accounts for the padding frame, while

γ d n ℓ ′ , L = P Sn = L, y ℓ ℓ ′ +1 |Sn-1 = ℓ ′ , Pn = 0 = P Sn = L|Sn-1 = ℓ ′ , Pn = 0 ϕ d y ℓ ℓ ′ +1 , x ℓ ℓ ′ +1
accounts for the data frame, with

P Sn = L|Sn-1 = ℓ ′ , Pn = 0 = π L-ℓ ′ L-ℓ ′ λ=ℓ min π λ .

B. Complexity evaluation

The complexity of the FS algorithm described in Section II is proportional to the number of nodes or to the number of transitions within the trellis on which FS is performed. From Figure 1, one sees that the trellis is lower-bounded by the line ℓ = nℓmin and upperbounded by the lines ℓ = nℓmax and ℓ = L. This region may be divided into two triangular sub-regions, one with 0 ≤ n ≤ Nmin -1, bounded between ℓ = nℓmin and ℓ = nℓmax, and the other with Nmin ≤ n ≤ Nmax -1, bounded between ℓ = nℓmin and ℓ = L. Thus, summing up the number of nodes in each sub-region, one gets the number of nodes in the trellis

Nn = N min -1 n=0 n (ℓmax -ℓmin) + Nmax n=N min (L -nℓmin) . ( 17
)
Taking Nmin ≈ L/ℓmax and Nmax ≈ L/ℓmin, (17) simplifies to

Nn = L 2 2 ℓmax -ℓmin ℓmaxℓmin = O L 2 . ( 18
)
From each node, at most ℓmaxℓmin transitions may emerge. Thus, from [START_REF] Panza | Supporting network transparency in 4G networks[END_REF], the number of transitions Nt may also approximated as

Nt = O L 2 .

C. Limitations

The hold-and-sync technique presented in [START_REF] Ali | Joint protocol-channel decoding for robust aggregated packet recovery at WiMAX MAC layer[END_REF] for performing FS is based on the knowledge of the beginning and length of the burst. This requires an error-free decoding of the headers of lower protocol layers, which contain this information. This may be done using methods presented in [START_REF] Marin | Robust header recovery based enhanced permeable protocol layer mechanism[END_REF], which enable the lower layer to forward the burst to the layer where it is processed. The main drawback of this FS technique in terms of implementation is the increase in memory requirements for storing soft information. This is estimated in [START_REF] Panza | Supporting network transparency in 4G networks[END_REF], [START_REF] Woo | Beyond the bits: cooperative packet recovery using physical layer information[END_REF] to be three to four times more. However, the plain trellis-based FS algorithm, as described above, requires buffering the whole burst which induces some buffering and processing delays proportional to L 2 , see [START_REF] Panza | Supporting network transparency in 4G networks[END_REF]. To alleviate these problems, a new low-delay and lesscomplex variant of the previously presented FS technique is now proposed.

`max L m 1 - Window -1 m Window m L m L o m-1 `_ " m-1 n _ n
Fig. 2. ST for the m-th decoding window, the original trellis is in gray

IV. SLIDING TRELLIS-BASED FS

In classical SW-BCJR methods, decoding is done within a window, which at each step is shifted bit-by-bit [START_REF] Benedetto | Soft-output decoding algorithms for continuous decoding of parallel concatenated convolutional codes[END_REF] or by several bits [START_REF] Gwak | Reduced complexity sliding window BCJR decoding algorithms for turbo codes[END_REF]. From one window to the next, the results obtained during the forward iteration are reused, contrary to those of the backward iteration. The number of bits the window is shifted at each iteration determines the trade-off between complexity and efficiency.

Contrary to the trellis for a convolutional code, the trellis considered in Figure 1 has a variable number of states for each value of the frame index n. One may apply directly the SW ideas, but due to the increase of the size of the trellis (at least for small values of n), this would still need very large trellises to be manipulated, with an increased computation time. Here, a ST-based approach is introduced: a reduced-size trellis is considered in each decoding window. As in [START_REF] Gwak | Reduced complexity sliding window BCJR decoding algorithms for turbo codes[END_REF], some overlapping between windows is considered, in order to allow better reuse of already computed quantities and to allow complexity-efficiency trade-offs.

A. Sliding Trellis

In the proposed ST-based approach, a burst of L bits is divided into M overlapping windows with sizes Lm, m = 1, ..., M . For each of these windows, the bits from εm-1 +1 to εm-1 +Lm are considered, where εm-1 is the bit index of the last bit of the last frame deemed reliably synchronized in the m -1-th window.

A ST moves from window to window to perform decoding. One such ST is illustrated in Figure 2. Let n and l be the local trellis coordinates. Once P (S m n = l|y

ε m-1 +Lm ε m-1 +1
) is evaluated, one can apply the estimators (4), [START_REF] Lee | Robust frame synchronization for low signal-to-noise ratio channels using energy-corrected differential correlation[END_REF], and (6) to determine the number of frames Nm in the m-th window (including the last truncated frame), the beginning, and the length of each frame. For the m-th window (m < M ), among the Nm decoded frames, only the first N c m frames are considered as reliably synchronized, since enough data and redundancy properties have been taken into account. Truncated frames, especially when the HEC has been truncated, and the frame immediately preceding such frames, are not considered reliable. Thus, only the N c m complete frames ending in the first Lmℓmaxℓ h bits of the window are considered as reliable. The unreliable region towards the boundary of the window is dashed in Figure 2.

The initialization of β is performed as in Section III, since no knowledge from the previous window can be exploited. The initialization of α and the evaluation of γ towards the window boundary may depend on the location of the window inside a burst. Three types of window locations are considered: the first window at the start of a burst, the intermediate windows in the middle of the burst, and the last window at the end of the burst.

The first window (m = 1) contains Lm < L bits and starts at ε0 = 0. The decoding approach, including the initialization of α for this first window is similar to that presented for the trellisbased approach. An exception is the computation of γn l′ , l , where two cases have again to be considered. The first corresponds to normal data frames, leading to γ d n l′ , Lm . The second accounts for truncated data frames towards the boundary of the window, leading to γ t n l′ , Lm , which is detailed in Section IV-B.

The intermediate windows (1 < m < M ) contain the bits from εm-1 + 1 to εm-1 + Lm < L, see Figure 2. The bit index εm-1, of the last bit of the last frame deemed reliably synchronized (i.e., the N c m-1 -th frame) in the m -1-th window, corresponds in the local coordinates of the m -1-th ST to l = εm-1εm-2. The m-th ST starts at the local coordinates

(n = N c m-1 , l = εm-1 -εm-2) of m -1-th ST.
The computation of γn l′ , l for an intermediate window is identical to that of the first window. For the initialization of α m n l , following the idea of the SW-BCJR decoder [START_REF] Benedetto | Soft-output decoding algorithms for continuous decoding of parallel concatenated convolutional codes[END_REF], [START_REF] Benedetto | Algorithm for continuous decoding of turbo codes[END_REF], up to ℓmax initial values for α m 0 l are propagated from the m -1-th window to the m-th window, see Section IV-C. This allows a better FS in case of erroneous FS in the m -1-th window.

The last window (m = M ) has no incomplete frame at its end. Only the presence of a padding frame has to be taken into consideration. The decoding is performed as in the trellis-based approach (Section III), except for the initialization of α M 0 l , which is similar to that of the intermediate window case.

Note that the m-th and m + 1-th windows overlap over L o m bits, with

ℓ h + ℓmax L o m < ℓ h + 2ℓmax.

B. Evaluation of γn

When m < M, transitions corresponding to truncated frames have to be considered at the end of the window. When the size of the truncated frame is larger than ℓ h , the header is entirely contained in the truncated frame. In this case γn l′ , Lm = γ t n l′ , Lm , with

γ t n l′ , Lm = p S m n = Lm|S m n-1 = l′ ϕ t y Lm l′ +1 , x Lm l′ +1 . (19)
In [START_REF] Woo | Beyond the bits: cooperative packet recovery using physical layer information[END_REF], since truncated frames have to be considered, p S m n = Lm|S m n-1 = l′ is given by (3). Moreover, the length of the frame, i.e., the content of the length field un, is now only known to be between max(Lm -l′ , ℓmin) and ℓmax bits. Thus

ϕ t y Lm l′ +1 , x Lm l′ +1 = P (y k |k) p P (yp|p) P (p) ℓ=ℓmax ℓ=max(Lm-l′ ,ℓ min ) P (u (ℓ))
o (P (yu|u (ℓ)) P (yo|o)

P (yc|c = f (k, u (ℓ) , o)) P (o)) . (20)
When the size of the truncated frame is strictly less than ℓ h , for the sake of simplicity, we assume that all bits of the truncated header are equally likely. In such case γn l′ , Lm = γ e n l′ , Lm , with

γ e n l′ , Lm = p S m n = Lm|S m n-1 = l′ x Lm l′ +1 P y Lm l′ +1 |x Lm l′ +1 P x Lm l′ +1 , (21)
where p S m n = Lm|S m n-1 = l′ is still given by (3) and

P (x Lm l′ +1 ) = 2 -ℓ(x Lm l′ +1
) , since all beginning of headers are assumed equally likely.

For m = M, the evaluation of γ M n is as in Section III-A.

C. Initialization of α in the sliding trellises

In the SW-BCJR algorithm proposed in [START_REF] Benedetto | Soft-output decoding algorithms for continuous decoding of parallel concatenated convolutional codes[END_REF], the α m s evaluated in the m-th window are deduced from those evaluated in the m -1-th window. Here, since the number of states Sn evolves with n, α m n cannot be obtained that easily from α m-1 n .

In the m -1-th window, one has evaluated α m-1 n l , with 0 l Lm-1 and 0 n ⌈Lm-1/ℓmin⌉. We choose to propagate at most ℓmax values of α from n = N c m-1 in the m -1-th window to n = 0 in the m-th window (for l = 0, ..., ℓmax -1) as follows

α m 0 ( l) = κ α m-1 N c m-1 εm-1 -εm-2 + l , ( 22
)
where κ is some normalization factor chosen such that the α m 0 ( l)s sum to one. This allows the first frame of the m-th window to start at any bit index between εm-1 + 1 and εm-1 + ℓmax.

D. Complexity

Consider M windows of approximately the same size L w , which are overlapping on average on L o = ℓ h + 1.5ℓmax bits. For sufficiently large L, one has to process M ≈ L L w -L o overlapping windows, each one with

N w n ≈ (L w ) 2 2 ℓmax -ℓmin ℓmaxℓmin
nodes. The total number of nodes to process is then

M N w n ≈ L L w -L o (L w ) 2 2 ℓmax -ℓmin ℓmaxℓmin . ( 23
)
This decoding complexity is smaller than that of Section III-B. A comparison for different values of burst size L is provided in Table I for window size L w = 480+L o . One can observe that the complexity gain increases with the size of the burst.

Choosing small values for L w reduces the latency as well as the complexity at the cost of some sub-optimality in the decoding performance. Note that L w cannot be chosen too small (smaller than ℓ h + 2ℓmax) to ensure at least one reliable FS in each window.

V. SIMULATION RESULTS

In the WiMAX standard [START_REF]IEEE standard for local and metropolitan area networks, air interface for fixed broadband wireless access systems[END_REF], the downlink (DL) sub-frames are divided into bursts. Each burst can contain multiple concatenated fixed-size or variable-size MAC frames: it is filled with several MAC frames, until there is not enough space left. Padding bytes (0xFF) are then added [START_REF]IEEE standard for local and metropolitan area networks, air interface for fixed broadband wireless access systems[END_REF] at the end of the burst. Each MAC frame begins with a fixed-length header, followed by a variable-length payload and ends with an optional CRC. As we are considering only the DL case, where the connection is already established, MAC frames belonging to a burst contain only Convergence Sublayer (CS) data, so only the Generic MAC header [START_REF]IEEE standard for local and metropolitan area networks, air interface for fixed broadband wireless access systems[END_REF] is possible inside a burst. Some assumptions are made in what follows for the sake of simplicity. CRC, ARQ, packing, fragmentation, and encryption are not used for the MAC frames inside the burst. Some fields are already fixed in a MAC header, and with the considered situation, fields such as Header Type (HT), Encryption Control (EC), sub-headers The considered simulator consists of a burst generator, a BPSK modulator, a channel, and a receiver. Simulations are carried over Rayleigh fading channels, where the modulated signal is subject to zero mean and unit variance fast (bit) Rayleigh fading plus zero-mean AWGN noise. For performance analysis, Erroneous Frame Location Rate (EFLR) is evaluated as a function of the channel Signal-to-Noise Ratio (SNR). It should be noted that in order to recover a frame correctly both ends of the frame have to be correctly determined.

In our simulations, L = 1800 bytes. Since WiMAX MAC frames are byte-aligned, i.e., the LEN field of the frame is in bytes and all MAC frames contain an integer number of bytes, α, β, and γ are evaluated for ℓs corresponding to the beginning of bytes. Data frames are randomly generated with a length uniformly distributed between ℓmin = 50 bytes and ℓmax = 200 bytes. If there is not enough space remaining in the burst, a padding frame is inserted to fill the burst. The burst is then BPSK modulated and sent over the channel.

Simulation results for the trellis-based FS technique (Section II), the ST-based approach (Section IV), a FS based on hard decision on the received bits, and a state-of-the-art on-the-fly technique (denoted by MU, for modified Ueda's method) described in [START_REF] Ueda | Reducing misframe frequency for HEC-based variable length frame suitable for IP services[END_REF] are shown in Figure 3. MU technique involves a three-state automaton for FS and uses the HCS as an error-correcting code. Here, the method in [START_REF] Ueda | Reducing misframe frequency for HEC-based variable length frame suitable for IP services[END_REF] has been modified to cope with short (8-bit) HCS, where several candidates for two-bit error syndromes are possible.

For the ST-based approach, the burst is divided into three windows, with L1 = 600 bytes, L2 = 600 + L o 1 bytes, and L3 = 600 + L o 2 bytes. Compared to the trellis-based FS, the ST-based approach shows a slight performance degradation of 0.5 dB, but reduces the delay and the computational complexity. On average, the overlap is about 277 bytes and a decrease in complexity by a factor of 1.7 is observed. The computational complexity of the on-the-fly MU technique is much smaller than that of the ST-based approach, at the cost of a noticeable loss in efficiency.

VI. CONCLUSION

A reduced-complexity, low-delay, and efficient ST-based technique for the robust FS of aggregated frames is presented. Compared to the trellis-based FS algorithm proposed in [START_REF] Ali | Joint protocol-channel decoding for robust aggregated packet recovery at WiMAX MAC layer[END_REF], the FS delay and computational complexity are significantly reduced. The price to be paid is a slight performance degradation.

This technique has been applied to the FS of WiMAX MAC bursts. A significant gain in performance is obtained for Rayleigh fading channels compared to classical FS. Extensions to perform on-thefly FS, by utilizing the features of Ueda's method and at the same time exploiting the structural properties of the headers of frames, are currently under investigation.

Fig. 1 .

 1 Fig. 1. L = 16 bits with ℓ min = 4 bits and ℓmax = 7 bits

Fig. 3 .

 3 Fig. 3. FS methods for transmission over Rayleigh channel

ACKNOWLEDGMENTS

This work was partly supported by Microsoft Research through its PhD European Scholarship program and by the NoE NEWCOM++.


Nesrine Changuel

Nicolas Mastronarde

Mihaela Van Der Shaar

Bessem Sayadi

⋆ Michel Kieffer

ADAPTIVE SCALABLE LAYER FILTERING PROCESS FOR VIDEO SCHEDULING OVER WIRELESS NETWORKS BASED ON MAC BUFFER MANAGEMENT

In this paper, the problem of scalable video delivery over a timevarying wireless channel is considered. Packet scheduling and management in both Application and Medium Access Control (MAC) layers are jointly considered. Various levels of knowledge of the state of the channel are considered. The control is performed via scalable layer filtering (some scalability layers may be dropped). In all cases, the problem is cast in the context of Markov Decision Processes which allows the design of foresighted policies maximizing some long-term reward. Without channel state observation, the control has to rely on the observation of the level of the MAC buffer only. Experimental results show that even with a lack of knowledge of the channel state, the foresighted control policy provides only a moderate loss in received video quality.

INTRODUCTION

The increase of available bandwidth in wireless networks allows a larger diversity of services provided to users. Mobile television, video on demand (payed or via YouTube), telephony via data channel are applications that are currently emerging. Providing goodquality compressed data to mobile receivers with widespread largedisplay devices, such as smart-phones or tablets is becoming more and more difficult. Apart from the bandwidth scarcity, due to an increased number of users, multimedia delivery to wireless receivers is particularly challenging to the time-varying characteristics of wireless channels (error-rate and bandwidth) and to delivery delay constraints of some applications such as mobile television or telephony.

Scalable coders, such as H.264/SVC [1] for video, have been developed to partially address these issues. Packets with various priority levels are generated. Receiving only the highest-priority packets allows to get an acceptable quality, which increases with the number of lower-quality packets correctly received. One of the main drawback of this approach is that any low-priority packet is useless unless all associated higher-priority packets have been received. Therefore, an efficient packet scheduling scheme has to be designed either in unicast or in multicast scenarios.

This paper focuses on Quality of Service (QoS) optimization in the context of scalable video streaming applications to mobile users. When considering unicast applications, various types of feedback from the receiver may be obtained, for example at the application layer via RTCP feedback [START_REF] Schulzrinne | RTP profile for audo and video conferences with minimal control[END_REF] to get information about the level of buffers at application layer or at MAC layer via HARQ ACK/NACK [START_REF] Sesia | The UMTS Long Term Evolution: From Theory to Practice[END_REF] to get information about the channel conditions.

Using information fed back to the controller, [START_REF] Kumwilaisak | A cross-layer quality-of-service mapping architecture for video delivery in wireless networks[END_REF] have designed scheduling schemes for time-varying channel conditions. In [START_REF] Kumwilaisak | A cross-layer quality-of-service mapping architecture for video delivery in wireless networks[END_REF], a cross-layer scheduling architecture for video transmission is presented; the time-varying characteristics of the wireless channel are modeled by a discrete-time Markov model, which state is fed back to the controller. Scalable video coders have been considered in [START_REF] De Cuetos | Optimal streaming of layered video: joint scheduling and error concealment[END_REF][START_REF] Changuel | End-to-end stochastic scheduling of scalable video over time-varying channels[END_REF], focusing on the application layer. A Markov Decision Process (MDP) framework [START_REF] Sutton | Reinforcement Learning: An Introduction[END_REF] is considered to design a long-term control policy accounting for the level of the playback buffer. In [START_REF] Baldo | RTCP feedback based transmission rate control for 3g wireless multimedia streaming[END_REF][START_REF] Liebl | Radio link buffer management and scheduling for wireless video streaming[END_REF][START_REF] Orlov | Enhancement of video streaming QoS with active buffer management in wireless environments[END_REF] and [START_REF] Yerima | Dynamic buffer management for multimedia qos in beyond 3g wireless networks[END_REF], focus is more on the MAC layer, since these papers address buffer management problems within the Radio Access Network (RAN). Application packets may be dropped depending on their priority and on the level of the buffers at the MAC layer.

The main difficulty when designing such control schemes is that feedback comes with delay. This delay may be of the order of tens to hundreds of milliseconds for HARQ ACK/NACK messages to one or several seconds for RTCP packets, which may cause stability problems. Our aim in this paper is to design a cross-layer control mechanisms in an MDP framework [START_REF] Fu | A new systematic framework for autonomous cross-layer optimization[END_REF][START_REF] Mastronarde | Designing autonomous layered video coders[END_REF] to control the level of buffers at MAC and at Application layers at the transmitter side only of a communication chain. Feedback at MAC layer is implicitly used, but no feedback at Application layer is considered, avoiding the use of delayed measurements. The considered scalable video transmission scheme is presented in Section 2. The control is performed by a filtering of the scalability layers of the encoded video. A Markov model of the channel is considered and three hypotheses concerning the knowledge of the state of the channel are considered, namely, known state, known state with delay, and unknown state. Even in the last two cases, the design of a control law may be cast in a classical MDP framework, see Section 3. Experimental results show that when a foresighted policy is considered, the absence of knowledge of the channel state results in a moderate degradation of the behavior of the system, thanks to the availability of the MAC buffer state, which provides a reasonable estimate of the state of the channel, see Section 4.

SYSTEM

The considered single user unicast streaming system is sketched in Figure 1. The core network consists of a streaming server, hosting a scalable video coder, a proxy, and a base station. Packets are transmitted through a wireless channel and received by a mobile client. Among the components of the base station (Node B), this paper considers mainly the MAC buffer. The MAC scheduler of Node B, as well as its physical layer, its radio front-end, the wireless channel, the physical layer of the receiver, and the part of the MAC layer at receiver side managing ACK/NACK procedures are denoted in the remainder of the paper as the channel.

No

System description

In the streaming server, the video sequence is segmented into frames encoded into a base layer and a set of L -1 enhancement layers. Access Units (AU) represent basic processing units, macroblock(s), slices, or frame(s), consisting of the base layer and its corresponding enhancement layers. Here an AU corresponds to a single frame identified by its time index t. AU are generated with a constant period of time ∆t. An H.264/SVC Medium-Grain quality Scalable (MGS) coding scheme [START_REF] Mansour | Rate and distortion modeling of medium grain scalable video coding[END_REF] is considered in this paper. Its encoding parameters (quantization steps, frame rate, etc.) are controlled by the streaming server (not by the controller designed in this paper), independently of the remainder of the transmission chain. Moreover, a base layer only control scheme [1] for the encoder is adopted to minimize the drift when refinement layers are lost. Each SNR layer of each encoded frame is packetized into RTP/UDP/IP packets, which are delivered via an over-provisioned core network (assumed lossless) to L post-encoder buffers (one per layer) situated in the proxy.

The controller performs layer filtering within the proxy: for each layer, packets may be sent, kept, or dropped. Sent packets are fed to the MAC buffer of the base station after being segmented into Packet Data Units (PDUs) of constant size. PDUs are then transmitted to the mobile receiver, which stores correctly received PDUs in its own MAC buffer. Packet de-encapsulation and buffering in one of the L at application layer is done as soon as all corresponding PDUs have been received. Complete or incomplete AU are then processed by the video decoder. Outdated packets are dropped, without being decoded. When one of the post-encoder, MAC, or receiver buffers is full, packets in the queue are dropped in a head-of-line order.

System requirements and constraints

Ideally, the layer filtering process should be designed to maximize the QoS at receiver side. Some necessary conditions have to be satisfied to perform this task. First, and most important, is avoiding overflow of the MAC buffer to prevent PDUs from being dropped. Underflow should also be prevented to use the channel in a optimal way. Second, overflow of the post-encoder buffers should be avoided to limit the delay introduced by the system. Underflow is not satisfying too, especially at the base layer, since this indicates that too much importance has been given to the base layer compared to the other layers.

MODEL OF THE SYSTEM

design of an efficient layer filtering process is translated in the framework of discrete-time MDP [START_REF] Sutton | Reinforcement Learning: An Introduction[END_REF]. An MDP is a 4-tuple (S, A, P, R), where S is the set of states of the system, A is the set of actions, P (s, s ′ , a) is the transition probability from s ∈ S at time t to s ′ ∈ S at time t + 1, when the action a ∈ A is applied to the system. Finally R (s, s ′ , a) the immediate reward received after a transition from to s ′ obtained by using action a. Some policy π (s) ∈ A, s ∈ S the immediate reward (myopic policy) or a discounted sum of future rewards (foresighted policy) has then to be found.

States

The state of the controlled system consists of s e gathering the levels s e ℓ , ℓ=1, . . . , L of the post-encoder buffers, s m corresponding to the level of the MAC buffer in the base station, and h the channel state. The state of the system is thus s = (s e , s m , h). Each components of the state vector is more clearly defined in what follows.

Buffers. The state of the ℓ-th post-encoder buffer is e ℓ ∈ S e ℓ and the state of the MAC buffer is s m ∈ S m . Various granularity levels may be considered to represent the content of a buffer: number of bits, number of packets, or even coarser description such as underflow/normal/overflow, see [START_REF] Mastronarde | Designing autonomous layered video coders[END_REF][START_REF] Fu | A new systematic framework for autonomous cross-layer optimization[END_REF] for more details. In this paper, the state of the post-encoder buffer will describe the amount of packets corresponding to parts of stored AU. This help to track the delay introduced within the system. The state of the MAC buffer correspond to the number of PDUs or of bits (since PDUs have all the same length). Channel model. The channel state ht the channel conditions (rate, probability of error, capacity, etc.) assumed constant between t -1 and t. The channel is modeled by a first-order Markov process with n states, with known transition probability p(ht+1|ht) stationary probability p (ht). In this paper, ht ∈ Rc represents the rate available during the considered time slot, with Rc = {R1, . . . , Rn} bits/s. Three hypotheses concerning the knowledge of the state of the channel are considered. Hyp. 1: instantaneous channel state, where ht is assumed available when choosing the action to apply between time t and t + 1; this is realistic only when feedback with very short delay is possible. Hyp. 2: delayed channel state, where only h t-δ (δ is a strictly positive integer) is assumed available when choosing the action to apply between t and t + 1; represents a more realistic situation, where the channel state is only known with some delay δ. Hyp. 3: unknown channel state which is scenario where no channel state feedback is considered.

Actions

For each layer, the proxy may send, hold, or drop packets. The action a ℓ,t ∈ A taken for the ℓ-th layer between time t and t + 1 represents the number of transmitted packets from the post-encoder buffer to the MAC buffer (when a ℓ,t > 0), or the number of dropped packets (when a ℓ,t < 0). If a l,t = 0, packets are kept in the post-encoder buffer. The vector gathering all actions is a = (a1, . . . , aL) ∈ A L .

Transition matrices and reward function

Once all states and actions have been identified, one has to determine the transition probability matrix P (s, s ′ , a) has to be evaluated. For that purpose, models of the lengths of packets delivered for each layer by the streaming server have to be considered first. Packet model. The length of each encoded packet at each layer depends on the type of frame that has been encoded (I, P, or B), on the activity of the frame, and of the chosen encoding parameters. A Markov model has been proposed in [START_REF] Wan | A simple, two-level markovian traffic model for IPTV video sources[END_REF] to capture the dependency between the length of consecutive packets. This model could be used in an MDP framework by supplementing the state with a state vector representing the length of packets stored in the buffers. Nevertheless, this would significantly increase the complexity of the problem. To limit the dimension of the problem, such state is not introduced and the lengths of packets stored in the ℓ-th post-encoder buffer is described by the probability distribution q ℓ (λ), ℓ = 1, . . . , L assumed to be known (it may be provided by the streaming server). Transition matrices. Now, the transition probability matrix may be evaluated in the three hypotheses concerning the knowledge of the state of the channel described in Section 3.1.

Hyp. 1: In this case, ht is available at time t when applying at to the system. The state transition matrix then P1 (st, st+1, at) = Pr (s e t+1 , s m t+1 , ht+1| s e t , s m t , ht, ,

which may be easily evaluated using the fact that p(ht+1|ht) q ℓ (λ), ℓ = 1, . . . , L are known. Hyp. 2: This case is a special class of partially delayed MDP. The current state of the channel is available without noise with a unit delay. In [START_REF] Altman | Closed-loop control with delayed information[END_REF], the case δ = 1 was considered and the delayed MDP model is transformed into a standard fully observable MDP. Here, the situation is simpler, and one may obtain a new state transition matrix as follows

The last line is obtained thanks to the fact that ht is known, s e t , s m t , or at do not provide additional information on ht. Thus P2 may be easily derived from P1.

Hyp. 3: In this case, no channel state is available to the controller. The state transition matrix may then be written as

= h t+1 h t P1 (st, st+1, at) p(ht) (3)
since s e t , s m t , or at do not provide additional information on ht. In order to limit the size of these transition matrices, it is possible to quantize the values which may be taken by the states in a more or less coarse way to get a compromise between complexity and description accuracy.

Reward function.

Concerning the reward function, a time t, the layer filtering process chooses an action that maximizes the QoS at receiver side. One of the QoS to satisfy is the video quality. To maximize the received video quality, one may maximize the average Peak Signal to Noise Ratio (PSNR) of the decoded frames. However building a reward function using this quality metric is not easy due to the variability of the delay between the time at which an AU is filtered and the time at which it is displayed. This time varying delay is linked to the time-varying channel conditions. alternative reward function is then built that penalize dropped packets, as well as buffer overflow and underflow, according to the constraints described in Section 2.2. The expected value of the reward is

Rt(st+1, st, at) = E[ L ℓ=1 γ ℓ a ℓ,t transmission + L ℓ=1 λ ℓ ρ ℓ (s e ℓ,t+1 , a ℓ,t ) post encoder buffer + βρ2(s m t+1 , a ℓ,t ) MAC buffer ]. (4)
positive parameters γ ℓ , λ ℓ , β, with ℓ = 1 . . . L, trade off the importance of the various constraints. The reward function (4) involves several parts, the first linked to the number of transmitted SNR layers, the others to the post encoder and the MAC buffers constraints.

Assuming that increasing the amount of transmitted packets increases the received quality, the transmission reward should help to maximize the amount of transmitted packets. The parameters γ l allow to give a higher priority to packets belonging to the base layer compared to those of the enhancement layers. For post encoder and MAC buffer constraints, ρ1(.) and ρ2(.) provide positive rewards for satisfying buffer states and negative rewards for states that should be avoided.

The proposed reward function depends on the actions and on the states of the post encoder and the MAC buffers. Assuming that theses states do not depend on the state of channel, the three MDP problems would have the same reward function and different transition matrix.

Myopic and foresighted policies

The policy π as a mapping from joint states to joint actions. In the considered system, the policy π simply indicates the number of SNR layers to transmit knowing the state of the post encoder buffer, of the MAC buffer, and possibly of (delayed) state of the channel. The optimal foresighted policy consists in finding the optimal stationary Markov policy π * corresponding to the optimal state-value function defined as

V * (st) = max a∈A ∞ k=0 α k R t+k+1 |st, (5)
where 0 < α < 1 is the discount factor, which defines the relative importance of present and future rewards. The optimal foresighted policy may be obtained by value or policy iteration algorithms, as detailed, e.g., in [START_REF] Sutton | Reinforcement Learning: An Introduction[END_REF]. When α = 0,one gets a myopic policy, maximizing only the immediate reward.

EXPERIMENTAL RESULTS

The performance of the proposed layer filtering process has been evaluated on several video sequences. Here, due to lack of space, only the results for Foreman.qcif at = 30 fps are reported. The H.264/SVC encoder using L = 3 MGS scalability layers per frame has been used. The cumulated average rates (and PSNR for luminance) are kbits/s (32.3 dB) for Layer 1, 165.0 kbits/s (34.7 dB) for Layer 1 and 2, and kbits/s (36.82 dB) for all layers. Known average source and encoder characteristics have been considered, leading to known average packet lengths in each layers.

The wireless channel is modeled as a 2-state Markov model (n = 2): Good state ht = 1 and bad state ht = 0. The channel rates are Rc = {0, 240} kbit/s. The channel state transition probabilities are p11 = 0.9 and p00 = 0.8, resulting in an average channel rate of 190 kbits/s. Four possible actions per layer are considered at each time instant A = {-1, 0, 1, 2}. To minimize complexity and facilitate the satisfaction of the constraints described in Section 2.2, the levels of all buffers are quantized into three possible values: 1 representing underflow, 2 for a satisfying level, and 3 for overflow. The post-encoder buffers are assumed to have a maximum size (in term of number of packets) S e = 55; the over and underflow levels are S e max = 50 and S e min = 10. For the MAC buffer, a maximum size of S m = 220 equal size PDU of 200 bits each corresponding to a maximum size of 44 kbits, and the levels at which it is considered in underflow and overflow are S m min = 10 kbits and S m max = 25 kbits. The values of the parameters in the reward function [START_REF] Kumwilaisak | A cross-layer quality-of-service mapping architecture for video delivery in wireless networks[END_REF], have been set to reflect the importance of the various constraints: γ1,2,3 = {150, 60, 15}, λ1,2,3 = {100, 40, 10} and β = 300. They may be much better adjusted using Lagrange optimization.

In what follows, the channel state is assumed either known, or totally unknown. In the two considered cases, the evolution of the PSNR of the decoded sequence obtained with a myopic policy (α = 0) and that with a foresighted policy (α = 0.9) are represented in Figure 2.

When the channel state is available, an average gain of about 1.2 dB is obtained with the foresighted policy compared to the myopic one. This gain is mainly due to more packets of the first enhancement layer reaching the receiver. When the channel state is not available, with the myopic policy, about 46 % of the time, the MAC buffer in the state exceeding some time the maximum buffer size. This situation results in the loss of some PDUs which induces a notable decrease of the received video quality. With the foresighted policy, the MAC buffer is in overflow state for about 25 % of the time but never looses PDU packets.

Without channel information and using the foresighted policy results in a loss of 0.5 dB in PSNR compared to the case of an available channel state. The availability of the state of the MAC buffer provides thus a reasonable estimate of the state of the channel, allowing a satisfying regulation of the received video quality.

CONCLUSIONS AND PERSPECTIVES

This paper presents a scalable video streaming system over a timevarying wireless channel. This problem is addressed in the framework of MDP. Foresighted policies are developed for two models of the system characterized by the availability or unavailability of the state of the channel to the controller. The level of video packet buffers and that of the MAC buffer of the base station are controlled.

Experimental results show that without channel state information, the performance of the control system is only slightly degraded compared to a case where the channel state information is available, provided that a foresighted policy is used. The observation of the level of the MAC buffer provides a satisfying estimate of the channel state. With a myopic policy, the degradation is much larger. This result may be very useful, since using channel state in the control system requires processing ACK/NACK or RTCP feedback messages, which usually only provide a delayed estimate of the channel state.

[1] H. Schwarz, D. Marpe, and T. Wiegand, "Overview of the scalable video coding extension of the H.264/AVC standard," IEEE Trans. on Circuits and Systems for Video Technology, vol. 17, no. 9, pp. 1103-1120, Sept. 2007.

Fig. 1 .

 1 Fig. 1. End-to-end video streaming scheme

Fig. 2 .

 2 Fig. 2. PSNR of the decoded sequences, myopic and forsighted policies, for known and unknown channel state

  RTCP feedback is considered in what follows. Only feedback provided by the ACK/NACK

			Wired network	Base	Wireless network
		Streaming server	Proxy	station		Mobile User
	Video content	Scalable Encoder	Layers Filtering Process	MAC Scheduler	Wireless channel	Video Decoder
				MAC buffer	Channel	MAC buffer
			Post encoder buffers		Channel states feedback
					ACK/NACK
			Mac buffer states feedback