Kosai Raoof
email: kosai.raoof@lis.inpg.fr

Mohammad-Ali A Khalighi

Nuttapol Prayongpun

MIMO Systems: Principles, iterative techniques and advanced polarization

Keywords: MIMO Systems: Principles, Iterative Techniques, and advanced Polarization

come

CHAPTER 1.

Introduction

This chapter considers the principles of multiple-input multiple-output (MIMO) wireless communication systems as well as some recent accomplishments concerning their implementation. By employing multiple antennas at both transmitter and receiver, very high data rates can be achieved under the condition of deployment in a rich-scattering propagation medium. This interesting property of MIMO systems suggests their use in the future highrate and high-quality wireless communication systems. Several concepts in MIMO systems are reviewed in this chapter. We first consider "MIMO channel models" and recall the basic principles of MIMO structures and channel modelling. We next study the "MIMO channel capacity" and present the early developments in these systems, concerning the information theory aspect. "Iterative signal detection" is considered next, where we particularly consider iterative techniques for signal detection or in other words, for space-time decoding. As the capacity is inversely proportional to spatial channel correlation, MIMO antennas should be enough separated by several wavelengths. In order to minimize antennas' deploying, we then present the "Advanced polarization diversity techniques for MIMO systems" and explain how these techniques can help us to reduce the spatial correlation in order to achieve high transmission rates. We end the chapter by considering the application of MIMO systems in local area networks, as well as their potential in enhancing range, localisation, and power efficiency of sensor networks. hal-00384670, version 1 -15 May 2009

MIMO Systems and Channel Models

In this section, a briefly discussed structure based on MIMO communication system is provided. The definition of MIMO channel is described with some related characteristics of wireless communication channels. Recent MIMO channel models are also reported.

MIMO Communication Systems

Using multiple antennas [START_REF] Telatar | Capacity of multi-antenna Gaussian channels[END_REF] at both transmitter and receiver is to increase the data rate by creating multiple spatial channels. Multiple receiving antennas can also be used to combat fading without expanding the bandwidth of the transmitted signal. In particular, with M T transmitting and M R receiving antennas, it is possible to achieve an M -time capacity of single transmitting and single receiving antenna configuration where M = min{M T , M R }. filter into M T × 1 continuous-time complex baseband sequence x(t) and then the baseband signal is modulated with a transmission carrier. The transmission channel H superposes the transmitted signal due to the distortions of environment. At the reception side, under the assumption of synchronous sampling, the received signal y(t) with additive noise is downconverted to baseband and sampled to produce a discrete-time signal sequence. Finally, the estimated symbols are decoded by the space time decoding block. If the channel is time-invariant, the equivalent received signal at the receiving antenna with M T elements at the transmitter and M R elements at the receiver can be written as

y(t) = L-1 l=0 H l x(t -τ l ) + η(t) (1.1)
If the channel is time-variant, the overall MIMO relation can be formulated as

y(t) = τ H(t, τ )x(t -τ ) + η(t) (1.2)
where x(t) and y(t) represent the transmitted and received signals

x(t) = [x 1 (t), x 2 (t), . . . , x M T (t)] (1.3) y(t) = [y 1 (t), y 2 (t), . . . , y M R (t)] (1.4)
and L denotes the number of resolvable multipath, τ is the propagation delay, H is the M R × M T channel matrix at instant τ and finally η(t) is an additive noise.

MIMO Channel stationary definition

In wireless communication, stochastic time-variant linear channel usually employs wide sense stationary uncorrelated scattering (WSSUS) for stationary property [START_REF] Bello | Characterization of randomly time-variant linear channels[END_REF], [START_REF] Parsons | The Mobile Radio Propagation Channel[END_REF]. This WSSUS channel expresses uncorrelated attenuation in both time-delay and Doppler-shift domains.

The quasi-WSSUS channel [START_REF] Bello | Characterization of randomly time-variant linear channels[END_REF] is usually applied to real radio systems. It has the properties of a WSSUS channel for a limited bandwidth and for limited time or within a limited environment. This kind of assumption is exceptionally useful in communication systems.

For example, Ergodic MIMO channel performance can be given by averaging the channel performances over many independent channel realizations considering that they have the same statistics.

Since in many practical communication systems, the WSSUS property is not truly satisfied hal-00384670, version 1 -15 May 2009

Classification of MIMO channel modeling

Many different Multiple-input multiple-output channel models have been proposed in the last years. MIMO channel models can be classified as physical and analytical models as shown in Fig. 1.2. On the one hand, physical channel models focus on the characteristics of an environment and the electromagnetic wave propagation between the transmitter and the receiver, also they consider the antenna configurations at both ends. On the other hand, analytical models do not provide the site-specific descriptions so that they do not take into account the wave propagation characteristics. The model impulse response is mathematically generated and related to the statistical properties of the propagation environment. However, due to its simplicity, an analytical channel model is very useful for producing MIMO channel matrix for different kind of communication systems.

In literature, physical models can be distinguished into deterministic models [START_REF] Mcjown | Ray tracing as a design tool for radio networks[END_REF]- [START_REF] Athanasiadou | A microcellular ray-tracing propagation model and evaluation of its narrowband and wideband[END_REF] and geometry-based stochastic channel models [START_REF] Petrus | Geometrical-based statistical macrocell channel model for mobile environments[END_REF]- [START_REF] Raoof | Channel capacity performance for MIMO polarized diversity systems[END_REF]. Deterministic models (such as ray-tracing and recording impulse response models) begin by creating an artificial environment. The channel response can be consequently calculated for simulation purposes. But in this case, CHAPTER 1.

the calculation time is considerable high. Geometry-based stochastic channel models (GSCM) consider that the channel response is carried out by respecting the characteristic of wave propagation, both site-specific Tx-Rx environments, and scattering mechanism. All parameters are statistically defined to closely match the measured channel observation. The channel response can be rapidly computed for a single-bounce, double-bounce or multi-bounce scattering mechanism.

Analytical channel models can be further illustrated into correlation-based models, statistical cluster models, and propagation-based models. Correlation-based models contain the Tx and Rx correlations overall channel matrixes. For example, the i.i.d. model is proposed in the case of rich scattering environment with no spatial correlation effect [START_REF] Telatar | Capacity of multi-antenna Gaussian channels[END_REF], [START_REF] Foschini | On limits of wireless communications in a fading environment when using multiple antennas[END_REF], [START_REF] Khalighi | Capacity of wireless communication systems employing antenna arrays, a tutorial study[END_REF].

The kronecker model [START_REF] Shiu | Fading correlation and its effect on the capacity of multi-element antenna systems[END_REF]- [START_REF] Zcelik | Deficiencies of the 'kronecker' MIMO radio channel model[END_REF] assumes that the channel correlation is a product of the correlations at the transmitter and receiver sides. Statistical cluster models determine physical parameters in a random manner without referring to the geometry of a physical medium. For example, Saleh-Valenzuela model [START_REF] Saleh | A statistical model for indoor multipath propagation[END_REF] uses two exponentially decaying amplitudes varying in time and distance of the clusters, while increasing delay time with the assumptions that the direction of departure (DOD) and the angle of arrival (DOA) are independent and identically distributed. The other models are propagation-based models such as keyhole channel models [START_REF] Gesbert | Outdoor MIMO wireless channels: Models and performance prediction[END_REF], finite scattered model [START_REF] Burr | Capacity bounds and estimates for the finite scatterers MIMO wireless channel[END_REF], maximum entropy model [START_REF] Debbah | MIMO channel modeling and the principle of maximum hal-00384670[END_REF], virtual channel representation [START_REF] Sayeed | Deconstructing multiantenna fading channels[END_REF] . . . etc.

In addition, there are several organizations that proposed different MIMO channel models, systems and algorithms for example COST 207, COST 231, COST 259 [START_REF] Correia | Wireless Flexible Personalised Communication (COST 259[END_REF], COST 273 [START_REF]Mobile Broadband Multimedia Networks (COST 273 Final Report)[END_REF],

3GPP [START_REF]Spatial channel model for multiple input multiple output (MIMO) simulations[END_REF] and IEEE 802.16a,e. hal-00384670, version 1 -15 May 2009

MIMO Channel Capacity

At the end of the 1990s, pioneering works in Bell Labs showed for the first time that the use of multiple antennas at the both sides of the transmission link can result in tremendous channel capacities, provided that the propagation medium is rich scattering [START_REF] Telatar | Capacity of multi-antenna Gaussian channels[END_REF][START_REF] Foschini | Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas[END_REF][START_REF] Foschini | On limits of wireless communications in a fading environment when using multiple antennas[END_REF][START_REF] Golden | Detection algorithm and initial laboratory results using V-BLAST space-time communication architecture[END_REF][START_REF] Raleigh | Multivariate modulation and coding for wireless communication[END_REF]. This increase in capacity is obtained without any need to extra bandwidth nor to extra transmission power. Multipath propagation, previously regarded as an impediment to reliable communication, was shown to be exploitable for increasing the data throughput. In this section, we explain briefly how exploiting the spatial dimension can lead to an increase in the system spectral efficiency. Only single-user applications are considered. For more discussions and details, the reader is referred to [START_REF] Khalighi | Capacity of wireless communication systems employing antenna arrays, a tutorial study[END_REF] and the references therein.

Capacity of a fading channel

Let us first recall the definition of the capacity for a fading channel. For a time-varying channel, the capacity C becomes a random variable whose instantaneous value depends on the channel realization [START_REF] Biglieri | Fading channels, information-theoretic and communications aspects[END_REF]. In such a case, the Shannon capacity of the channel may even be zero. Indeed, if we choose a transmission rate for communication, there may be a non-zero probability that the channel realization is incapable of supporting it. The two mostly used definitions for the channel capacity are ergodic capacity and outage capacity. The ergodic capacity C erg , which is the expected value of C, is suitable for fast varying channels. The outage capacity C out is usually used when considering packet-based transmission systems where the block-fading model properly describes the channel. If the pre-assumed channel capacity is too optimistic, i.e., larger than the instantaneous capacity, a channel outage may occur. The outage capacity, or to say more correctly, the capacity-versus-outage, as seen from its name, is the channel capacity conditioned to an outage probability P out . Obviously, there is a trade-off between the expected data throughput and the outage probability.

In the expressions that we will provide in this section, C will denote the instantaneous channel capacity.

MIMO Capacity

We present the MIMO channel capacity for different cases of channel state information (CSI) at transmitter (Tx) and receiver (Rx).

General assumptions

The global scheme of the transmission link is shown in Fig. 1.3. We denote by M T and M R the number of antennas at Tx and Rx, respectively. The communication channel includes the effect of transmit/receive antennas and the propagation medium. We neglect the effect of the antenna patterns and assume the far field conditions; that is, dominant reflectors are assumed to be sufficiently far from the Tx and the Rx. We consider the simple case of frequency non-selective (flat) fading channel that is true for narrow-band communications.

This assumption is mostly valid in indoor applications [START_REF] Jakes | Microwave Mobile Communications[END_REF][START_REF] Hashemi | The indoor radio propagation channel[END_REF]. In the equivalent baseband representation, each subchannel is characterized by a complex circularly-symmetric random variable that is assumed to be normalized in power. The entire MIMO channel is described by a channel matrix H of dimension (M R ×M T ). For instance, the entry H ij of H, characterizes the subchannel between the i th receive and the j th transmit antenna. Finally, we assume that the total transmit power at each sample time (corresponding to each channel-use) is constrained to P T .

CSI known to Rx but unknown to Tx

This is the classical case that is usually considered in the literature. The channel is assumed to be known (e.g. perfectly estimated) at the Rx but unknown to the Tx. Since the Tx does hal-00384670, version 1 -15 May 2009 1.3. MIMO CHANNEL CAPACITY not know the channel, it is logical to distribute the available power uniformly on the transmit antennas. In fact, this is the optimal way for power allotment over the M T antennas in this case. We denote by ρ T the total average received signal-to-noise ratio (SNR) at the receiver array ρ T = P T /σ 2 n , where σ 2 n is the variance of the additive white Gaussian complex noise. The MIMO channel capacity in units of bps/Hz is:

C = log 2 det I M R + ρ T M T HH † . (1.5)
Here I M R is the (M R ×M R ) Identity matrix. Equation (1.5) can also be written in a different form if we consider the singular value decomposition of H:

H = U H Λ H V † H (1.6)
where . † denotes complex conjugate transpose ; U H and V H are unitary matrices of dimensions (M R × M R ) and (M T × M T ), respectively ; and Λ H is an (M R × M T ) matrix containing the singular values of H. Let us define M = min{M T , M R }. We denote these singular values by λ H,i , i = 1, • • • , M . The MIMO capacity can be written in the following form:

C = log 2 det I M R + ρ T M T Λ H Λ † H = M i=1 log 2 1 + ρ T M T λ 2
H,i (1.7)

CSI known to both Tx and Rx

This is the case when the estimated CSI at the Rx is provided for the Tx, which is practically feasible when the channel varies slowly in time. Providing the CSI for the Tx can be done using a (hopefully low bandwidth) feedback channel or via the reverse link when the communication takes place in a duplex mode. We assume that CSI is provided perfectly and without any delay to the Tx. In this case, the Tx can allot the available power on the antennas in an optimal manner in order to achieve the maximum capacity [START_REF] Khalighi | Water filling capacity of Rayleigh MIMO channels[END_REF]. This capacity is often called known-CSI capacity or water-filling (WF) capacity. For this purpose, we should weight the transmitted symbols vector x by the matrix V H , and the received signal vector by the matrix U † H [START_REF] Khalighi | Capacity of wireless communication systems employing antenna arrays, a tutorial study[END_REF][START_REF] Anderson | Array gain and capacity for known random channels with multiple element arrays at both ends[END_REF]. This can be regarded as an optimal beam forming solution. By CHAPTER 1.

this weighting, we can in fact consider an equivalent channel H eq between x and y:

H eq = U † H H V H = Λ H . (1.8)
In other words, the MIMO channel matrix H is decomposed into several parallel independent single-input single-output subchannels. The number of these subchannels is equal to rank(H), and their gain is given by the singular values of H. Let R X be the autocorrelation matrix of x, with eigenvalues λ X,i , i = 1, • • • , M . The optimal WF solution consists in distributing the available power P T , optimally over the equivalent parallel subchannels, which results in [START_REF] Khalighi | Capacity of wireless communication systems employing antenna arrays, a tutorial study[END_REF][START_REF] Raleigh | Spatio-temporal coding for wireless communication[END_REF]:

λ X,i = ψ - σ 2 n λ 2
H,i + , i = 1, • • • , M (1.9)
where, (s) + = s if s > 0, and 0 otherwise. Also, ψ is a constant which is determined so as to satisfy the constraint on the total transmit power, M i=1 λ X,i = P T . The WF solution imposes that we allocate more power to best subchannels and lower (or perhaps no) power to worse ones. Now, the WF capacity C WF is given by [START_REF] Khalighi | Capacity of wireless communication systems employing antenna arrays, a tutorial study[END_REF][START_REF] Raleigh | Spatio-temporal coding for wireless communication[END_REF]:

C WF = M i=1 log 2 1 + λ X,i λ 2 H,i σ 2 n .
(1.10)

CSI unknown to both Tx and Rx

If channel is not known to the Tx nor to the Rx, we can consider (1.5) as un upper bound on capacity. Let ∆ be the coherence interval of the channel in units of channel-uses. As ∆ tends to ∞, the channel capacity approaches this upper bound, because with greater ∆, tracking the channel variations becomes more possible for the Rx [START_REF] Marzetta | Capacity of a mobile multiple-antenna communication link in Rayleigh flat fading[END_REF]. For the same reason, there is less difference between the capacity and the upper bound for higher SNR values.

This difference becomes more considerable, however, for larger M T and/or M R [START_REF] Marzetta | Capacity of a mobile multiple-antenna communication link in Rayleigh flat fading[END_REF]. For a fast varying channel, the capacity is far less than the Rx perfect-knowledge upper bound, because practically there is no possibility to estimate the channel at the Rx. For M > ∆, no increase is achieved in capacity of MIMO channel by increase in M [START_REF] Marzetta | Capacity of a mobile multiple-antenna communication link in Rayleigh flat fading[END_REF]. It is shown in [START_REF] Zheng | Communication on the Grassmann Manifold: a geometric approach to the noncoherent multiple-antenna channel[END_REF] hal-00384670, version 1 -15 May 2009

Some numerical results

We present some numerical results, excluding the case where CSI is unknown to both the Tx and the Rx. We consider the outage capacity C out for an outage probability of P out = 0.01 and will refer to it simply by "capacity." Let us first consider the case where CSI is known only to the Rx. We will call this case no-WF. Fig. 1.4 contrasts the outage capacity of MIMO, singleinput multiple-output (SIMO), and multiple-input single-output (MISO) systems under the conditions of uncorrelated Rayleigh flat fading [START_REF] Sklar | Rayleigh fading channels in mobile digital communication systems; Part I: Characterization; Part II: Mitigation[END_REF] and P out = 0.01. SNR stands for ρ T .

For the MIMO system, we have M T = M R = M . Uncorrelated fading necessitates enough antenna spacings at the Tx and the Rx that depends, in turn, on the propagation conditions [START_REF] Salz | Effect of fading correlation on adaptive arrays in digital mobile radio[END_REF][START_REF] Shiu | Fading correlation and its effect on the capacity of multi-element antenna systems[END_REF]. The MIMO capacity increases linearly with M . It is much more considerable than that of MISO and SIMO systems. Now consider the case of known-CSI at both the Tx and the Rx. For M R ≥ M T , the improvement in capacity by performing WF, that we call the WF gain, is considerable for low SNR and a large number of transmit antennas [START_REF] Khalighi | Water filling capacity of Rayleigh MIMO channels[END_REF]. But this gain is much more considerable CHAPTER 1. system is equal to that of an (M R , M T ) system. Also, the WF-capacity of a MISO system is equal to that of the equivalent SIMO system.

The gain in capacity by WF is specially interesting for the case of correlated channels. For instance, for the case of Ricean fading [START_REF] Driessen | On the capacity formula for multiple input-multiple output wireless channels: a geometric interpretation[END_REF][START_REF] Khalighi | On capacity of Ricean MIMO channels[END_REF], curves of capacity versus the Ricean factor (RF) are presented in Fig. 1.6 a MIMO system with M T = M R = M and two cases of M = 2 and M = 4. RF represents the percentage of power received from the line-of-sight (LOS) to the total received average power [START_REF] Khalighi | On capacity of Ricean MIMO channels[END_REF]. For relatively high RF that can be regarded as a more correlated channel, the WF gain is quite considerable.

Iterative Signal Detection

No need to say, signal detection is a crucial part of the transmission system. Among the various detection techniques proposed for the case of MIMO systems, there are iterative (also called turbo) detectors. This is what we are going to focus on in this section. In effect, since the invention of turbo-codes by Berrou et al. [START_REF] Berrou | Near optimum error correcting coding and decoding: turbo-codes[END_REF] who proposed iterative decoding of parallel concatenated convolutional codes, the turbo principle has been applied to several problems in communications such as channel equalization [START_REF] Douillard | Iterative CHAPTER 1. correction of intersymbol interference: Turbo-equalization[END_REF], channel estimation [START_REF] Khalighi | Semi-blind channel estimation using EM algorithm in iterative MIMO APP detectors[END_REF], synchronization [45], multi-user detection [START_REF] Gamal | Iterative multiuser detection for coded CDMA signals in AWGN and fading channels[END_REF], and of course to MIMO signal detection [START_REF] Sellathurai | Turbo-BLAST for wireless communications: theory and experiments[END_REF]. The turbo principle consists of the exchange of soft information between two different stages of the Rx, mostly including the soft channel decoder. In MIMO systems too, iterative processing has attracted special attention as it makes a good compromise between complexity and performance. Before presenting the basics of iterative detection, we have to present a brief introduction on space-time coding that is performed at the Tx. We mostly consider frequency non-selective (flat) fading conditions and single-carrier modulation.

Space-time coding and decoding

An important aspect in the implementation of MIMO systems is to appropriately distribute redundancy in space and in time at the Tx, what is called space-time (ST) coding [START_REF] Vucetic | Space-Time Coding[END_REF]. To this date, there has been considerable work on this subject and a variety of ST schemes has been proposed for MIMO systems. The key criteria in the design of ST codes are the coding gain and the diversity gain. The first one aims at achieving high rate by capitalizing on the MIMO capacity, whereas the latter aims at profiting from the space diversity to reduce fading at the Rx. The two extreme schemes corresponding to these criteria are respectively spatial multiplexing and transmit diversity. For instance, orthogonal space-time block codes (OSTBC) [START_REF] Tarokh | Space-time block codes from orthogonal designs[END_REF][START_REF] Alamouti | A simple transmit diversity technique for wireless communications[END_REF] aim at diversity gain and some coding rate; space-time trellis codes (STTC) [START_REF] Tarokh | Space-time codes for high data rate wireless communication: performance criterion and code construction[END_REF] and linear constellation precoding (LCP) [START_REF] Xin | Space-time diversity systems based on linear constellation precoding[END_REF] aim at both coding and diversity gain; spatial multiplexing or the V-BLAST architecture maximizes the coding rate [START_REF] Golden | Detection algorithm and initial laboratory results using V-BLAST space-time communication architecture[END_REF]; and the more general family of linear dispersion (LD) codes [START_REF] Hassibi | High-rate codes that are linear in space and time[END_REF] that maximize the mutual information between the Tx and the Rx, allow flexible rate-diversity trade-off.

Apart from the problem of code construction, one important criterion in the choice of the appropriate ST scheme could be its decoding complexity. ST orthogonal designs like OS-TBCs offer full diversity while they can be decoded using an optimal decoder with linear complexity. However, these schemes suffer from low rate, especially for a large number of transmit antennas. In addition, full-rate OSTBCs exist only for restricted number of transmit antennas and modulations [START_REF] Tarokh | Space-time block codes from orthogonal designs[END_REF]. Non-orthogonal schemes, on the other hand, offer higher coding rates but their optimal decoder becomes prohibitively complex for large number of transmit antennas and large signal constellation sets. This is specially the case for STTC schemes that although offering high rates and good diversity gains, are complex to decode and, moreover, suffer from long decoding delays. For non-orthogonal schemes, instead of performing complex optimal decoding, we may use sub-optimal decoding based on simple linear-algebraic techniques such as sphere decoding [START_REF] Viterbo | A universal lattice code decoder for fading channels[END_REF] or interference-cancelling-based decoding [START_REF] Golden | Detection algorithm and initial laboratory results using V-BLAST space-time communication architecture[END_REF][START_REF] Sellathurai | Turbo-BLAST for wireless communications: theory and experiments[END_REF]. For either solution, the Rx performance can be improved considerably by performing iterative detection.

ST coding, Tx scheme

In addition to using a special ST scheme, we usually perform channel coding at the Tx.

Let us consider bit-interleaved coded modulation (BICM) [START_REF] Caire | Bit-interleaved coded modulation[END_REF] for which a typical scheme is shown in Fig. 1.7. The advantage of the BICM is its flexibility regarding the choice of the code and the bit-symbol mapping, as well as its conformity to iterative detection. In The output bits c are then mapped to symbols according to a given constellation set. We will mostly consider QAM modulation with B bits per symbol. Power-normalized symbols s are next combined according to a given ST scheme and then transmitted on M T antennas.

General formulation of LD codes

Before talking about ST decoding, let us present the general formulation of the LD codes from [START_REF] Hassibi | High-rate codes that are linear in space and time[END_REF] that can be equally used for other ST schemes as well. Let S of dimension (Q × 1)

be the vector of data symbols prior to ST coding:

S = [s 1 , s 2 , • • • , s Q ] t , (1.11)
where . t denotes transposition. By ST coding, these symbols are mapped into a (M T × T ) matrix X, where T is the number of channel-uses. We define the ST coding rate as R STC = Q/T . Corresponding to an encoded matrix X, we receive the (M R × T ) matrix Y . We separate the ℜ and ℑ parts of the entries of S and X and stack them row-wise in vectors S of dimension (2Q × 1) and X of dimension (2M T T × 1), respectively. For instance,

S = ℜ{s 1 } ℑ{s 1 }, • • • , ℜ{s Q } ℑ{s Q } t .
(1.12)

We have then, X = F S, where the (2M T T × 2Q) matrix F depends on the actual ST scheme (see [START_REF] Hassibi | High-rate codes that are linear in space and time[END_REF] for more details). Let the (M R × M T ) matrix H represent our flat channel.

Similar to X , we construct the (2M R T × 1) vector Y from Y . Vectors X and Y are then

related through a (2M R T × 2M T T ) matrix H: Y = H X + N (1.13) CHAPTER 1.
where N is the vector of real AWGN of zero mean and variance

σ 2 n . Matrix H is composed of (2T × 2T ) segments H ij , i = 1, • • • , M R , j = 1, • • • , M T , described below. H ij =        H ij 0 • • • 0 0 H ij • • • 0 . . . . . . . . . 0 • • • 0 H ij        (1.14)
The (2 × 2) elements H ij are obtained from each entry H ij of the initial matrix H :

H ij = ℜ{H ij } -ℑ{H ij } ℑ{H ij } ℜ{H ij } . (1.15)
Now, we can describe the "ST encoder and channel" input/output relationship by considering an equivalent channel matrix H eq of dimension (2M R T × 2Q):

Y = H F S + N = H eq S + N . (1.16)

Iterative detection for non-orthogonal ST schemes

We assume that H eq and σ 2 n are known at the Rx. Having received the vector Y, we should extract from it the transmitted data S. As we perform channel coding together with ST coding, the idea of iterative detection comes into mind. Indeed, by profiting in this way from the channel coding gain, we can obtain a good performance after only few iterations and approach the optimal "ST decoder + channel decoder" performance. This is, of course, the case for non-orthogonal ST schemes. In what follows, we explain the principle of iterative detection and explain in detail the ST decoding part.

The block diagram of such a Rx is shown in Fig. 1.8. Soft-input soft-output signal detection and channel decoding are performed. For MIMO signal detection or ST decoding, we can use the optimal maximum a posteriori (MAP) algorithm, or a sub-optimal solution based on sphere decoding or interference cancelling, for example. In fact, the optimal MAP de- Soft channel decoding, on the other hand, can be done using the well known forwardbackward algorithm [START_REF] Bahl | Optimal decoding of linear codes for minimizing symbol error rate[END_REF], the soft-output Viterbi algorithm (SOVA) [START_REF] Hagenauer | Iterative decoding of binary block and convolutional codes[END_REF] or a simplification of them [START_REF] Robertson | Optimal and sub-optimal maximum a posteriori algorithms suitable for turbo decoding[END_REF]. For the final decision making on the transmitted data bits, we use the a posteriori probabilities at the decoder output. Like in any other turbo processing case, extrinsic information are exchanged between the two blocks of soft detector and channel decoder. In what follows, we explain the principle of MIMO detection based on MAP, sphere decoding, and soft interference cancelling, while focusing on the third approach.

MAP signal detection

We present here the formulation of the MAP detector based on probabilities. It can also be implemented using logarithmic likelihood ratios (LLR). Remember the expression of Y from (1.16). The MIMO detector provides at its output extrinsic probabilities on coded bits c.

Let Q be the cardinality of S of size q |Q| = 2 BQ . Let also c i , i = 1, • • • , BQ be the bits corresponding to a vector of symbols S ∈ Q. The extrinsic probability on the bit c j at the MIMO detector output, P Det ext (c j ), is calculated as follows [START_REF] Boutros | Turbo coding and decoding for multiple antenna channels[END_REF]:

P Det ext (c j = 1) = K S∈Q c j =1 exp - Y -H eq S 2 σ 2 n BQ i=1 i =j P Dec ext (c i ) (1.17)
where K is the normalization factor satisfying P Det ext (c j = 1) + P Det ext (c j = 0) = 1. The probability P Dec ext (c i ) is in fact the a priori information on bit c i , fed back from the channel CHAPTER 1.

decoder. At the first iteration, where no a priori information is available on bits c i , P Dec ext (c i ) are set to 1/2. The summation in (1.17) is taken over the product of the conditional channel likelihood (the exponential term) given a vector S, and the a priori probability on this symbol, i.e., the term P Dec ext . In this latter term, we exclude the a priori probability corresponding to the bit c j itself, so as to respect the exchange of extrinsic information between the channel decoder and the MIMO detector. Also, this term assumes independent coded bits c i , which is true for random interleaving of large size.

Sphere decoding

As it was seen, in the calculation of the extrinsic probabilities, the MAP detector considers the exhaustive list of all possibly-transmitted symbol vectors. Hence, the complexity of the MAP detector grows exponentially with the number of transmit antennas M T and the number of bits per modulation symbol B. By sphere decoding, these probabilities are calculated based on a non-exhaustive list [START_REF] Viterbo | A universal lattice code decoder for fading channels[END_REF]. Corresponding to a vector Y, we only take into account those lattice points that are in a hypersphere of radius R around Y. In this way, the average complexity of the detector under flat fading conditions becomes polynomial (often sub-cubic) [START_REF] Hassibi | On the sphere-decoding algorithm I. Expected complexity[END_REF]. The main detection tasks are then setting the radius R as well as determining which lattice points are within the sphere.

Parallel-Interference-Cancelling-based detection

The block diagram of the detector based on soft-parallel interference cancellation (Soft-PIC) is shown in Fig. 1.9. We will refer to the corresponding receiver scheme as turbo-PIC. It is essentially composed of the three blocks of PIC detector, conversion to LLR, and softestimation of transmit symbols. [START_REF] Sellathurai | Turbo-BLAST for wireless communications: theory and experiments[END_REF][START_REF] Khalighi | Channel estimation in turbo-BLAST detectors using EM algorithm[END_REF][START_REF] Khalighi | Data-aided channel estimation for Turbo-PIC MIMO detectors[END_REF].

γp = W † p Y = h † p H eq H † eq + σ 2 n I -1 Y (1.18)
where W p denotes the filter, h p is the p th column of H eq , and (.) † stands for transpose complex conjugate. Note that the entries of H eq are real values, and hence, (.) † is equivalent to transposition. From the second iteration, we can calculate soft estimates S of the transmitted data using the soft decoder outputs. Using these estimates, we perform interference cancelling followed by a simple inverse (zero-forcing) or MMSE filtering:

Ŷp = Y -H p Sp , γp = W † p Ŷp (1.19) ZF: W p = 1 h † p h p h p , MMSE: W p = 1 (h † p h p + σ 2 n ) h p (1.20)
where Sp of dimension ((2Q -1) × 1) is S with its p th entry removed, and H p of dimension

(2M R T × (2Q - 1
)) is the matrix H with its p th column removed. Notice that, compared to the exact MMSE filtering proposed in [START_REF] Sellathurai | Turbo-BLAST for wireless communications: theory and experiments[END_REF], (1.20) are simplified solutions that assume almost perfect estimation of data symbols and permit a considerable reduction of the computational complexity. Thanks to iterative processing, the performance loss due to this simplification would be negligible. In the results that we present later, we will consider the simplified ZF solution.

• Conversion to LLR For QAM modulation with B (an even number) bits per symbol, we can attribute m = B/2 bits to the real and imaginary parts of each symbol. Let for instance, the bit c i corresponds CHAPTER 1.

to the real (imaginary) part of the symbol s q . Let also a 1,j and a 0,j , j = 1, • • • , B 2 denote the real (imaginary) part of the signal constellation points, corresponding to c i = 1 and c i = 0, respectively. Remember that the signal constellation points have normalized average power.

The LLR corresponding to c i is calculated as follows [START_REF] Khalighi | Channel estimation in turbo-BLAST detectors using EM algorithm[END_REF]:

LLR i = log 10 2 m-1 j=1 exp -1 2σ 2 p (γ p -a 1,j ) 2 2 m-1 j=1 exp -1 2σ 2 p (γ p -a 0,j ) 2 , i = 1, • • • , m (1.21)
where σ 2 p is the variance of noise plus the residual interference (RI) that intervenes in the detection of γp , and assumed to be Gaussian. Note that, as the detection is performed on blocks of Q complex symbols, or in other words on blocks of 2Q real symbols in our model, the RI comes in fact from (2Q -1) other real symbols in the corresponding channel-use [START_REF] Khalighi | Should MIMO orthogonal space-time coding be preferred to non-orthogonal coding with iterative detection?[END_REF].

In LLR calculation, we need the variances σ 2 p . These variances can be calculated analytically as shown in [START_REF] Wang | Iterative (turbo) soft interference cancellation and deoding for coded CDMA[END_REF], or estimated at each iteration and for each one of 2Q real symbols, as done in [START_REF] Khalighi | Should MIMO orthogonal space-time coding be preferred to non-orthogonal coding with iterative detection?[END_REF]. To simplify the detector further, we may neglect the RI and take only into account the noise variance. For not too large signal constellation sizes, this simplification causes a negligible performance loss [START_REF] Khalighi | Choice of appropriate space-time coding scheme for MIMO systems employing channel coding under BICM[END_REF][START_REF]Contrasting orthogonal and non-orthogonal space-time schemes for perfectlyknown and estimated MIMO channels[END_REF].

• Soft estimation of transmit symbols

Each element γp of the vector S is obtained by taking a summation over the all possible values of the real part (or imaginary part) of the signal constellation, multiplied by the corresponding probability calculated using the soft decoder output [START_REF] Khalighi | Channel estimation in turbo-BLAST detectors using EM algorithm[END_REF][START_REF] Khalighi | Data-aided channel estimation for Turbo-PIC MIMO detectors[END_REF]. It is preferable to use the a posteriori information from the decoder output rather than extrinsic information in the calculation of γp . This has the advantage of permitting a better and faster convergence of the Rx.

Case study

As we focus here on the turbo-PIC detector as the sub-optimal solution, we just compare the performance of this detector with that of turbo-MAP. We consider the simplified implementation of Soft-PIC based on ZF filtering given in (1.20). For this comparison, we consider

Orthogonal versus non-orthogonal ST schemes

In practice, to attain a desired spectral efficiency, we should adopt the most appropriate scheme by fixing the degrees of freedom of the system, that is, the signal constellation, the channel coding rate, and the ST coding scheme. The answer to the question "what is the most suitable combination" is not obvious for moderate to high spectral efficiencies. In effect, if a low spectral efficiency is required, an OSTBC scheme together with a powerful turbo-code would be a suitable solution, as the reduction in the overall coding rate is best invested in turbo channel codes [START_REF] Le Nir | Combination of space-time block coding with MC-CDMA technique for MIMO systems with two, three and four transmit antennas[END_REF]. To attain high spectral efficiencies with OSTBC schemes, however, we have to use large signal constellations and to increase the channel coding rate by puncturing the encoder output bits. Use of larger signal constellations complicates the tasks of synchronization and detection at the Rx and also results in a higher SNR required to attain a desired BER. On the other hand, puncturing results in a reduced channel code robustness against noise. Higher ST coding rates are offered by non-orthogonal schemes, hence, relaxing the conditions on signal constellation and channel coding. Here, a simple (sub-optimal) iterative detector can be used for ST decoding, as explained in the previous subsection, and we may approach the optimal detection performance after few iterations. Nevertheless, the detector remains more complex, as compared to the OSTBC case. However, this increased Rx complexity is quite justified; using such an appropriate non-orthogonal ST scheme and iterative detection, we obtain a considerable gain in performance with respect to OSTBC choice [START_REF] Khalighi | Should MIMO orthogonal space-time coding be preferred to non-orthogonal coding with iterative detection?[END_REF][START_REF] Khalighi | Choice of appropriate space-time coding scheme for MIMO systems employing channel coding under BICM[END_REF][START_REF]Contrasting orthogonal and non-orthogonal space-time schemes for perfectlyknown and estimated MIMO channels[END_REF]. Results in [START_REF] Khalighi | Choice of appropriate space-time coding scheme for MIMO systems employing channel coding under BICM[END_REF][START_REF]Contrasting orthogonal and non-orthogonal space-time schemes for perfectlyknown and estimated MIMO channels[END_REF] have also confirmed that the gain obtained by using non-orthogonal with respect to orthogonal schemes is still considerable and even more important when channel estimation errors are taken into account.

• Case study

We consider again the case of a (2×2) MIMO system, Gray bit/symbol mapping and random interleaving, as well as the Rayleigh flat block-fading channel model with N c = 32 independent fades per frame. The number of channel-uses corresponding to a frame is 768. The NRNSC channel code (133, 171) 8 is considered with rate R c = 1/2. ST schemes we consider are resumed in Table I, where η is the spectral efficiency in units of bps/Hz. As the OSTBC scheme, we consider the Alamouti code [START_REF] Alamouti | A simple transmit diversity technique for wireless communications[END_REF]. Using the formulation of LD codes that we hal-00384670, version 1 -15 May 2009

Q = T = 2, R STC = 1
, and:

X = s 1 s 2 -s * 2 s * 1 . (1.22)
For the OSTBC case, the decoding is performed once using (1.18). We also consider two nonorthogonal schemes. The first one is the simple V-BLAST scheme described by X = [ s 1 s 2 ] t for which Q = 2, T = 1 and R STC = 2. The second one is the optimized scheme proposed in [START_REF] Belfiore | The golden code: a 2 × 2 full-rate spacetime code with nonvanishing determinants[END_REF] and called Golden code that we denote by GLD. For this code that offers full-rate and full-diversity with the property of non-vanishing determinant, we have Q = 4, T = 2, and R STC = 2:

X = 1 √ 5 α (s 1 + θ s 2 ) α (s 3 + θ s 4 ) γ α (s 3 + θ s 4 ) α (s 1 + θ s 2 ) , (1.23)
where

θ = 1+ √ 5 2 , α = 1 + j(1 -θ), θ = 1 -θ, α = 1 + j(1 -θ), γ = j = √ -1.
The factor 1/ √ 5 ensures normalized transmit power per channel-use. For η = 2 bps/Hz, performance curves are shown in Fig. 1.11, where again perfect channel knowledge is assumed at the Rx.

For V-BLAST and GLD schemes, BER curves are shown for the second and the fourth iteration where almost full Rx convergence is attained. We see that, by using V-BLAST scheme, we gain about 3.3 dB and 3.75 dB in SNR at BER= 10 -4 after two and four iterations, respectively, compared to Alamouti coding. The corresponding gains by using GLD code are about 3.5 dB and 4.3 dB, respectively. We note that even when for the reasons of complexity and/or latency, only two iterations are to be performed, the gain in SNR compared to Alamouti scheme is still considerable.

Advanced Polarization Diversity Techniques for MIMO Systems

For next-generation wireless communication systems, multiple antennas at both transmitter and receiver could be used to achieve a higher capacity and reliability of wireless communication channels under rich scattering environments. With multiple antenna communication systems, the performance could moreover be improved on a limited bandwidth and transmission power.

Due to the potential of MIMO systems, the initial research demonstrates that the channel capacity based on uncorrelated channel model can be proportionally increased with respect to the number of antennas. However, in practice, the performance of MIMO communication channel are affected by the spatial correlation, which is dependent on antenna array configurations (such as radiation pattern, antenna spacing and array geometry), and propagation channel characteristics, which are dependent on the environment, (such as number of channel paths, distribution and properties of scatterers, angle spread and cross-polarization discrimination). Thus, the antenna arrays at transmitter and receiver should be properly designed to reduce the spatial correlation effects and to improve the communication performance. However, it is possible to reduce these effects by increasing antenna array spacing but this solution is not always suitable in some wireless applications where the array size is limited. Therefore, in order to eliminate the spatial correlation effects and remain with high transmission performance, there are essentially two diversity techniques such as angular [START_REF] Vaughan | Beam spacing for angle diversity[END_REF] and polarization diversity techniques [START_REF] Kyritsi | Effect of antenna polarization on the capacity of a multiple element system in an indoor environment[END_REF]. For pattern diversity technique, the radiation of antennas should be generated in a manner to isolate the radiation pattern. For polarization diversity techniques [START_REF] Eiceg | Dual-polarization versus single-polarization MIMO channel measurement results and modeling[END_REF], the antennas are designed to radiate with orthogonal radiation po- Finally, there are also other diversity techniques; such as multimode diversity [START_REF] Svantesson | On the potential of multimode antenna diversity[END_REF] that exploits the difference of high order modes to obtain low correlated channels across the modes and a combination of pattern and polarization diversity techniques [START_REF] Svantesson | Analysis of electromagnetic field polarizations in multiantenna systems[END_REF], [START_REF] Andersen | The mimo cube -a compact MIMO antenna[END_REF] that take together the advantages of orthogonal radiation patterns and polarizations.

Antennas

In pratice, not only the propagation environment for the multiple antenna systems has an important role in determining the transmission performance, but also the proper implementation of the antennas plays another dominant role. For example uniform linear arrays, uniform circular arrays and cube antenna arrays give different performances in terms of channel capacity. Moreover, different array configurations produce different correlation effects. In this part, we will analyze five types of antennas [START_REF] Balanis | Antenna Theory[END_REF] such as x -, y-and z -oriented dipole antennas, azimuth, and elevation isotropic antennas are applied to a uniform linear array.

Table 1.2: Patterns of different electric dipoles

E x E y E z E φ E θ E θ (θ, φ) -cosθcosφ -cosθsinφ -sinθ 0 1 E φ (θ, φ) sinφ -cosφ 0 1 0
The radiating patterns of the antennas are considered in the far field case, and also simplified by neglecting path loss and distance phase. Hence, these radiating patterns are simply dependent on the azimuth and elevation angle direction as shown in table 1.2. A general expression of radiation patterns is given by [75]

E = E θ (θ, φ) θ + E φ (θ, φ) φ (1.24)
where E θ (θ, φ) and E φ (θ, φ) are the amplitudes of polarization vector at θ-and φ-direction,

|E θ (θ, φ)| 2 + |E φ (θ, φ)| 2 sin θdθdφ E θ (θ, φ) θ + E φ (θ, φ) φ (1.25)
where G is the antenna gain that is used for the computation of the channel matrix.

Cross polarization discrimination

In wireless communications, due to the interactions of environment such as diffractions, reflection and refraction, the transmitted signals are generally not only attenuated, but also depolarized. Depolarization is the change of the original state of the polarization of the electromagnetic wave propagated from the transmitter.

Cross-polarization discrimination (XPD) is defined as the power ratio of the co-polarization and the cross-polarization components of the mean incident wave. The higher the XPD, the less energy is coupled in the cross-polarized channel. Therefore, there are two transmission cases namely azimuth transmission (χ θ ) and elevation transmission (χ φ ) as follows

χ θ = E |E θθ | 2 E |E θφ | 2 , χ φ = E |E φφ | 2 E |E φθ | 2 (1.26)
where E θφ denotes the θ-polarized electric field which is propagated from a transmitter and received in the φ polarization. It has the same explanation for E θθ , E φφ and E φθ . In the literature, some measurement campaigns have been carried out and they concluded that the XPD depended on the physical obstacles, the distance between transmitter and receiver and the delay of multipath components of each environment [START_REF] Soma | Analysis and modeling of multipleinput multiple-output (MIMO) radio channel based on outdoor measurements conducted at 2.5GHz for fixed BWA applications[END_REF]. Therefore, for simplicity, XPD can be approximated by a Gaussian statistical distribution with average µ and variance σ 2

[77] , [START_REF] Lempiainen | Experimental results of cross polarization discrimination andsignal correlation values for a polarization diversity scheme[END_REF]. For urban environment, the mean value of XPD can vary from 0 to 16 dB and the standard deviation can change from 3 to 9 dB. In case of far field transmission without the line-of-sight channel, the flat fading transmission channel between the antenna p at the transmitter and the antenna m at the receiver can be expressed as

h mp (t, f ) = 1 √ N S N S i=1 a (i) m a (i) p exp -j k (i) • v Rx t -j k ′(i) • v T x t + ϕ mp G m θ (θ i , φ i ) G m φ (θ i , φ i ) S (i) mp G p θ (θ i , φ i ) G p φ (θ i , φ i ) (1.27)
where t is time;

f is frequency;

-N S is the number of scatterers at the receiver and the transmitter;

-v Rx and v Rx are the velocity vector of the transmitter and the receiver;

-k ′(i) and k (i) are the vectors of wave number in the direction of the i th transmit scatterer and the i th receive scatterer where

⇀ k (i) = ⇀ k ′(i) = 2π/λ;
-G p θ (θ i , φ i ) and G p φ (θ i , φ i ) are the gain in the θ and φ directions of the pth transmit antenna in the direction of the i th transmit scatterer; -G m θ (θ i , φ i ) and G m φ (θ i , φ i ) are the gain in the θ and φ directions of the mth receive antenna in the direction of the i th receive scatterer; -a (i) m is the mth element of the local vector of the receive antenna, so that the local receive vector can be expressed as a

(i) Rx = 1 exp -j k (i) • r 1 • • • exp -j k (i) • r M R -1 ; -a (i)
p is the pth element of the local vector of the transmit antenna, so that the local transmit vector can be expressed as a

(i) Tx = 1 exp -j k ′(i) • r ′ 1 • • • exp -j k ′(i) • r ′ M T -1 ; -S (i)
mp are a 2 × 2 scattering matrix for the i th transmit scatterer and the i th receive scatterer for i = 1 . . . N S wave components. Scattering matrix contains the polarization mechanism as defined by

S (i) mp =   χ θ 1+χ θ exp jβ (θθ) mp 1 1+χ φ exp jβ (φθ) mp 1 1+χ θ exp jβ (θφ) mp χ φ 1+χ φ exp jβ (φφ) mp   (1.28) CHAPTER 1.
where β (φθ) mp denotes phase offset of i th incident wave which changes from φ directions to θ directions superposing on m-p channel and χ θ and χ φ denote the ratio of the co-polarized average received power to the cross-polarized average received power. In [START_REF] Vaughan | Polarization diversity in mobile communications[END_REF], after sufficiently reflecting the propagation signal between transmitters and receivers, the polarization state of the signal will be independent of the transmitted signals.

Spatial correlation and angle spread effects

The consecutive MIMO systems based on spatial diversity technique are directly influenced by spatial correlation effect (or antenna correlation effect) [START_REF] Shiu | Fading correlation and its effect on the capacity of multi-element antenna systems[END_REF]. This effect is drastically dependent on array configurations and environment characteristics. Therefore, the antenna arrays at both transmitter and receiver should be properly designed or adapted to decrease spatial correlation effects.

As illustrated in Fig. 1.13, the spatial correlation of 2 × 2 uniform linear antenna array depends on the antenna spacing and the angle spread (AS). The general expression of spatial correlation [START_REF] Yong | A three-dimensional spatial fading correlation model for uniform rectangular arrays[END_REF] between two antenna elements can be written as

ρ ij = ∆θ,∆φ a i •a * j sin (θ) p (θ, φ) dθdφ ∆θ,∆φ |a i | 2 sin (θ) p (θ, φ) dθdφ • ∆θ,∆φ |a j | 2 sin (θ) p (θ, φ) dθdφ (1.29)
where a i is local value of the i th transmit or receive antenna and a j is local value of the Although there are only two diversity branches, it does allow the antenna elements to be colocated without the correlation effect. However, there is considerable interest in many di-CHAPTER 1.

j
versity branches by applying a combination of the pattern and polarization diversity [START_REF] Andrews | Tripling the capacity of wireless communication using electromagnetic polarization[END_REF], [START_REF] Andersen | The mimo cube -a compact MIMO antenna[END_REF].

Capacity of polarized channels

Antenna polarization diversity is very useful in MIMO systems for enhancing channel capacity. Indeed, employing polarization diversity can reduce the antenna array size and also the spatial correlation, then we can obtain a better capacity. That is why the multi-polarized antennas become more and more interesting in MIMO transmission systems. In this section, MIMO systems are investigated to show the potential of using multi-polarized antennas for differently oriented dipole antennas.

We consider that channel state information (CSI) is perfectly known to the receiver but unknown to the transmitter. This is in theory what happens to signals propagating through an urban and an indoor environment. In the case of a random channel model, the channel matrix (1.27) is stochastic and then the capacity given by (1.5) is also random. In this situation, the ergodic capacity can be obtained by taking the expectation of capacity over all possible channel realizations. can represent a combination of angular and polarization techniques. We found that for the triple-polarization case, the average power of subchannel can be unfortunately loosed when the angular spread is not large enough until covering all antennas. As seen in Fig. 1.16, the single-polarized channel capacity can be superior to the triple-polarized channel capacity because single polarization has a low spatial correlation and triple polarization loses the subchannel power due to insufficient angular spread. However, generally speaking for most scenarios, triple polarizations can maintain a higher capacity with respect to the single polarization case.

Impact of depolarization effect on MIMO configurations

In this section, we investigate the impact of the depolarization effect on 4 × 4 MIMO systems with single-and dual-polarization configurations. While pattern and polarization diversity techniques are employed, the special correlation effect can be reduced or eliminated when there is no-pattern interference. Nevertheless, the cross-polarization discrimination (XPD) becomes the most important parameter because XPD represents the ratio of the co-polarized CHAPTER 1. In the previous sections, we defined the scattering mechanisms which are used to represent not only the attenuation of traveling waves but also the polarization of the electromagnetic wave. The achieved performance in capacity is calculated under the assumption that the average received power is normalized and the channel attenuation is neglected. Subsequently, CHAPTER 1.

the variation of polarization is characterized only by the XPD effects.

As shown in Fig. 1.17, this phenomenon affects directly the performance of non-polarized MIMO systems. In single-polarization communications case, low XPD causes higher losses in channel power, in other words there is some sort of mismatch in polarization. That is why we will apply adaptive techniques to reduce this mismatch in the polarization of MIMO systems. This technique is called adaptive MIMO polarized-antenna selection technique (AMPAS). The principle of this method is to choose properly the antenna polarizations that optimize the receiving signal power while minimizing fading correlation antenna effects. In nearly the same position, high correlation is produced. Subsequently, the channel capacity is reduced even if the polarizations are correctly matched. However, it has better performance in the situation of polarization diversity when the antenna rotation has the difference of 90 • .

Moreover, we observe the MIMO capacity with spatial diversity as illustrated in Fig. 1.20d

where the antenna correlation is reduced and the channel capacity is improved.

In order to achieve a better transmission performance, the polarized antenna selection can exploit together all diversity techniques such as pattern, spatial and polarization diversity.

Pattern diversity should be employed when a large angle spread is detected. Spatial diversity could be exploited when a high antenna correlation is observed. Polarization diversity should be used when a low XPD occurs. Therefore, the employed diversity should be properly selected according to the propagation environment. There are also some motivations in order to improve the performance and the transmission rate in MIMO-OFDM systems when the CSI is available at the transmitter. As mentioned in previous sections, water-filling technique is used to optimize the distribution of the total transmit power over transmit antennas. Therefore, information symbols and power could be optimally allocated over space and frequency in MIMO-OFDM communication.

The number of antennas utilized in a MIMO (WLAN) router for example can vary; typical MIMO router contains three or four antennas. These systems are driving the need for the next broadband revolution focused on home networking. Such systems will cover next-generation game consoles, Video-on-Demand, HDTV and other new products. These services are creating a more sophisticated home entertainment environment, together with a high level of Quality of Service (QoS) to facilitate multimedia connectivity.

The intelligence behind the antenna polarization is described as "adaptive polarization".

CHAPTER 1.

The router receives feedback from the client adapter and has the ability to focus the polarization of the signals. As signal travels between an access point and wireless card, it will bounce off of walls, ceilings and any other obstacle, resulting in multiple reflections of the original signal arriving by different paths and different polarizations. By applying adaptive antenna polarization algorithms (AMPAS), these reflections can be used to improve the signal to noise ratio, as instead of having just one copy of the original signal.

This adaptive transmitting feature provides a more reliable signal at extreme ranges.

Moreover, MIMO can eliminate dead spots, delivering reliable whole home coverage with all the speed you need for application in the future. Today, one can say, wireless is faster than leased wire systems.

Finally, MIMO networking has the potential to increase communications data rate by 10-20 times above current systems. Such systems will use multipath reflections to create parallel channels in the same frequency bandwidth, thereby increasing spectral efficiency.

In the next section sensor networks will be discussed as an application of MIMO based system.

MIMO for Cooperative Sensor Networks

A sensor network can be considered as a self contained circuit with its sensor and RF interface, as shown in Fig. 1.21. Recent hardware advances allow more signal processing functionality to be integrated into a single chip. For example it is possible to integrate an RF transceiver, sensor interface, and base band processors into one device that is as small as a piece of coin and can be used as a fully-functional wireless sensor node. Such wireless nodes typically operate with small batteries, that is why these sensor nodes have limited power capabilities. In many scenarios, the wireless nodes must operate without battery replacement for many years. Consequently, minimizing the energy consumption is a very important design consideration and energy-efficient transmission schemes must be used for the data transfer in sensor networks hal-00384670, version 1 -15 May 2009 In addition, because sensor nodes will be deployed in remote and oftentimes dangerous locations, their maintenance (in particular, battery replacement) will be unlikely [START_REF]A survey on sensor networks[END_REF].

Sensor Networks are a new attraction for many potential applications, such as industrial, military, geolocalization, surveillance, intrusion detection, and environmental monitoring [START_REF]A survey on sensor networks[END_REF], [START_REF] Chong | Sensor networks: Evolution, opportunities, and challenges[END_REF].

Robust communications between sensor nodes are highly demanded at low power. As it was shown, MIMO communication promises performance enhancements over conventional single-input single-output (SISO) technology without increasing the bandwidth consumed by the system or the total power radiated from a transmitter. MIMO technology has promising characteristics that make it a serious candidate for sensor network communication technology.

Signal processing techniques that use multiple transmit and receive antennas, such as spacetime coding (ST Coding), have been shown to increase transmission reliability.

In a surveillance application, the ability of sensor nodes to relay data is critical to the utility and effectiveness of the sensor network.

For a given node density, nodes are more likely to be out of range, thus inhibiting communication. In a situation such as this, the extended range of MIMO is of greater importance because it enables cohesion (the ability of the sensor nodes to form a completely connected network), which guarantees the success of the final application [START_REF] Burdin | Cohesion of wireless sensor networks with MIMO communications[END_REF].

CHAPTER 1.

New protocols for target reporting and a procedure for target localization which conserve energy is recently developed [START_REF] Burdin | Cohesion of wireless sensor networks with MIMO communications[END_REF], [START_REF] Godard | Localisation des objets communicants[END_REF]. In [START_REF] Jiang | Routing protocols for sensor networks[END_REF], the authors summarize and compare several routing MIMO technology.

Mean path length provides a measurement of the impact of MIMO communications on a wireless sensor network. Mean path length provides a rough estimate of the amount of time and energy expended in a data transmission from one node to another in the network.

Most significant mean path length reduction is provided by MIMO in the low or midrange of node densities because the inter-node spacing is such that MIMO can reliably form some links that SISO cannot. In the elongated region scenario this trend holds, though it is less apparent [START_REF] Burdin | Enhancing the performance of wireless sensor networks with MIMO communications[END_REF].

There is also an increasing need for mobile networks with distributed transmitters and receivers, typically referred to as mobile ad hoc networks (MANET). There, transmitters and receivers do not pool their information together, either due to geographical dispersiveness, the bandwidth and resource limitation, or due to security/privacy concern.

Recognizing that multiple antennas at the transceivers provide inherent multiplexing capability due to their spatial selectivity, it is attractive to study MIMO communication in ad hoc networks with "interference" transmission.

Energy-efficient communication techniques typically focus on minimizing the transmission energy only, which is reasonable in long-range applications where the transmission energy is dominant in the total energy consumption.

In cooperative sensor networks, we allow the cooperation among sensors for information transmission and/or reception so as energy consumption as well as transmission delays over some distance ranges can be reduced.

In conclusion, for the same throughput requirement, MIMO systems require less transmission energy than SISO systems. However, direct application of multi-antenna techniques to sensor networks is impractical due to the limited physical size of a sensor node which typically can only support a single antenna. If individual single-antenna nodes allowed co-hal-00384670, version 1 -15 May 2009 operating on information transmission and/or reception, a cooperative MIMO system can be constructed such that energy-efficient MIMO schemes can be deployed [START_REF] Cui | Energy-efficiency of MIMO and cooperative MIMO techniques in sensor networks[END_REF].

Finally, MIMO can provide significant network performance improvements in power consumption, latency, and network robustness.

Fig. 1 .

 1 Fig.1.1 demonstrates a general system employing multiple transmitting and multiple receiving antennas to increase the data rate. A sequence of input symbols is encoded by a space-time encoding function into M T × 1 discrete-time complex baseband sequence x[n](n is a discrete time index). The x[n] sequence is subsequently transformed by pulse shaping

Figure 1 . 1 :

 11 Figure 1.1: General configuration of a MIMO communication system

Figure 1 . 2 :

 12 Figure 1.2: MIMO channel and propagation models

Figure 1 . 3 :

 13 Figure 1.3: Global scheme of a MIMO communication structure

Figure 1 . 4 :

 14 Figure 1.4: Capacity for MIMO, SIMO, MISO structures; uncorrelated Rayleigh flat fading; SNR = 10 dB, P out = 0.01.

Figure 1 . 5 :

 15 Figure 1.5: WF and no-WF capacities of a MIMO system with M R > M T = 4 (left) and M T > M R = 4 (right), Rayleigh flat fading, SNR = 3 dB, P out = 0.01.

Figure 1 . 6 :

 16 Figure 1.6: WF and no-WF capacities (left) and WF gain in capacity (right) for a MIMO system with M T = M R = M ; Ricean flat fading, SNR = 10 dB, P out = 0.01.

Figure 1 . 7 :

 17 Figure 1.7: Block diagram of the BICM transmission scheme.

Fig. 1 . 7 ,

 17 hal-00384670, version 1 -15 May 2009 the binary data b are encoded by a channel code C, before being interleaved (the block Π).

Figure 1 . 8 :

 18 Figure 1.8: Block diagram of the receiver.

•Figure 1 . 9 :

 19 Figure 1.9: Block diagram of the Soft-PIC detector

1 M R = 2 M R = 3 M R = 4 Figure 1 . 10 :

 1234110 Figure 1.10: Comparison of turbo-PIC and turbo-MAP detectors, M T = 4, (5, 7) 8 channel code, QPSK modulation, 64 channel-uses per frame.

Figure 1 . 11 :

 111 Figure 1.11: (2 × 2) MIMO system, Turbo-PIC detection, (133, 171) 8 channel code, η = 2 bps/Hz hal-00384670, version 1 -15 May 2009

  larizations to create uncorrelated channels across different array elements. The polarization diversity techniques could be applied in point-to-point communication systems such as Inter Mobile Base Station Communications, Mobile Satellite Communications, High Resolution Localization Systems, Military Communications, etc.

  and x, y and z are the antenna orientations. As different types of antenna are employed in this paper, it is necessary to normalize the radiation pattern when comparing all channel performances. Thus all radiation patterns of an antennas are normalized by that of isotropic hal-00384670, version 1 -15 May 2009 1.5. ADVANCED POLARIZATION DIVERSITY TECHNIQUES FOR MIMO SYSTEMS27 antenna, and they can be written as

Figure 1 . 12 :

 112 Figure 1.12: Geometries of MIMO channel

Figure 1 . 13 :Figure 1 . 14 :

 113114 Figure 1.13: Spatial correlation of uniform linear antenna array for single-polarized configuration

Fig. 1 .

 1 Fig.1.15 demonstrates the 2× 2 MIMO channel capacity of isotropic antennas with singlepolarization in (a) and with dual-polarization in (b). In this case, the single polarization configuration exploits only azimuth isotropic antenna and the dual polarization configuration applies azimuth and elevation isotropic antennas. The channel capacity is examined in function of angular spread (AS) with 20 scatterers distributed around the transmitter and receiver. As mentioned in the previous section, the XPD is defined for the urban case by χ θ and χ φ ∼ N (0, 9). As shown in Fig.1.15a, the MIMO channel capacity increases as the angle spread increases at transmitter and receiver for the same polarization antennas. In contrast, dual polarization improves the channel capacity due to the lower antenna correlation as shown in Fig.1.15b. However, when the spatial correlation of the single-polarized antennas is lower, the channel capacity is proportional until 6.7 dB for AS> 80 • . It should be noted that the MIMO channel capacity is significantly dependent on the antenna correlation.

Fig. 1 .

 1 Fig.1.[START_REF] Shiu | Fading correlation and its effect on the capacity of multi-element antenna systems[END_REF] shows the difference between the triple-polarized and the single-polarized channel

Figure 1 . 16 :

 116 Figure 1.16: Difference between the triple-polarized and the single-polarized channel capacity of 3 × 3 MIMO system employing the dipole antennas average received power to the cross-polarized average received power. Then, for a high XPD value, less energy is coupled between the cross-polarized wireless channels. At lower XPD and higher K-factor values[START_REF] Prayongpun | MIMO channel capacities in presence of polarization diversity with and without line-of-sight path[END_REF], multi-polarized antenna arrays can give high capacity. However, at higher XPD and lower spatial correlation, a single-polarized antenna array can provide even better results.

Figure 1 . 17 :Fig. 1 .

 1171 Figure 1.17: Difference between the dual-polarized and the single-polarized channel capacity of 4 × 4 MIMO systems in the functions of XPD and AS Fig.1.17 explains the difference between the dual-polarized and single-polarized channel capacity (∆C = C dual-polar -C single-polar ) of 4 × 4 MIMO systems versus XPD and AS. Same angle spreads at transmitter and receiver is considered in the simulation. For high XPD and sufficiently large angle spread, we can note that the MIMO channel capacity of the single-polarized antenna is superior to that of the dual-polarized antenna because of the

Fig. 1 .

 1 Fig.1.[START_REF] Zcelik | Deficiencies of the 'kronecker' MIMO radio channel model[END_REF] shows the XPD impact on 4×4 MIMO systems with single-and dual-polarization configurations. We employ four azimuth isotropic antennas for the single-polarization configuration and two elevation and two azimuth isotropic antennas for the dual-polarization configurations, which are applied to a uniform linear antenna array with a λ/2 antenna separation. It shows that the channel power of single-polarization configuration augments significantly with respect to XPD until H F =4 × 4=16, while for that of dual-polarization configuration keeps around H F =2 × 2+2 × 2=8.

Figure 1 . 18 :

 118 Figure 1.18: Frobenius norm of 4 × 4 MIMO systems in terms of XPD and AS with: (a) a single-polarization configuration and (b) a dual-polarization configuration

Fig. 1 .

 1 Fig.1.19, an example of an adaptive polarization system employing 4 z -oriented dipoles at transmitter and 4 pairs of y-and z-oriented dipoles at receiver is illustrated. Simulation results based on 3D ray-tracing techniques show that the channel capacity obtained by an adaptive polarization increases 7%-13% in comparison to the single polarization channel capacity.

Figure 1 . 19 :

 119 Figure 1.19: Adaptive polarization technique (AMPAS) at Rx

Fig. 1 .

 1 Fig.1.20 demonstrates the obtained performances of 1 × 1 SISO, 1 × 2 SIMO and 2 × 2MIMO communication systems while the receiving antenna is rotating on y-z plane with χ θ and χ φ ∼ N (0, 5). Performance can be better enhanced if polarization at the receiver is

Figure 1 . 20 :

 120 Figure 1.20: Performances of (a) 1 × 1 SISO, (b) 1 × 2 SIMO, (c) 2 × 2 MIMO as the same antenna position, (d) 2 × 2 MIMO as the separated antenna position while the receiving antenna is rotating on plane y-z

Figure 1 . 21 :

 121 Figure 1.21: Typical wireless sensor node

Table 1 . 1 :

 11 Different ST schemes for a (2 × 2) MIMO system with η = 2 bps/Hz ST scheme R STC Modulation R c

	Alamouti	1	16-QAM	1/2
	V-BLAST	2	QPSK	1/2
	GLD	2	QPSK	1/2
	presented in Subsection 1.4.2, we have

List of Figures


Ciro Scognamillo

Sebastien Fregonese

Senior Member, IEEE Thomas Zimmer

Vincenzo Daalessandro

Antonio Pio Catalano

Vincenzo D'

A Technique for the in-situ Experimental Extraction of the Thermal Impedance of Power Devices

Keywords: Electrothermal simulation, finite-element method simulation, in-situ thermal characterization, power devices, thermal impedance measurement2

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Nowadays, the working conditions of power circuits are limited by their heat disposal capability. Utmost importance must be given to the self-heating thermal impedance (ZTH) of devices [START_REF]Transient dual interface test method for the measurement of the thermal resistance junction-to-case of semiconductor devices with heat flow through a single path[END_REF], which fully describes their individual dynamic thermal behavior and allows defining power ratings and safe operating areas [START_REF] Swan | A fast loss and temperature simulation method for power converters, Part II: 3-D thermal model of power module[END_REF]. The ZTH can be experimentally extracted through (i) an indirect temperature assessment with the measurement of a temperature-sensitive electrical parameter (TSEP), the temperature dependence of which (thermometer) has to be well established [START_REF] Russo | Influence of layout design and on-wafer heatspreaders of the thermal behavior of fully-isolated bipolar transistors: Part II -Dynamic analysis[END_REF]- [START_REF] Gonzalez | An investigation of temperature-sensitive electrical parameters for SiC power MOSFETs[END_REF], or (ii) a direct temperature measurement by means of sensors (e.g., thermocouples) [START_REF] Berthou | Monolithically integrated temperature sensor in silicon carbide power MOSFETs[END_REF], [START_REF] Kim | Miniature piezoelectric sensor for in-situ temperature monitoring of silicon and silicon carbide power modules operating at high temperature[END_REF] or IR cameras [START_REF] Riccio | An equivalent time temperature mapping system with a 320×256 pixels full-frame 100 kHz sampling rate[END_REF].

Unfortunately, both indirect and direct approaches are difficult or even impossible to apply for the evaluation of the overall junction-to-ambient ZTH (ZTH,j-a) of a power device operating in its application environment (in situ). The calibration of the thermometer is unviable in most cases, since the temperature of the backside of the system embedding the device under test is typically not controlled. Likewise, the adoption of sensors or IR imaging is inhibited, as the top surface of the device under test is often not exposed; moreover, direct methods are more suited for detecting slow or even dc thermal responses. Lastly, the evaluation of the in-situ ZTH,j-a with the above approaches also requires a nontrivial estimation of the ambient temperature. France (e-mails: sebastien.fregonese@ims-bordeaux.fr; thomas.zimmer@ims-bordeaux.fr).

)

In this letter, a technique is proposed for the in-situ experimental extraction of the ZTH,j-a, which relies on simple measurements of electrical signals and does not require a thermometer definition, costly equipment for the detection of temperature maps, or the assessment of the ambient temperature. The theory behind the technique is inspired by Müller [START_REF] Müller | Internal thermal feedback effects in four-poles especially in transistors[END_REF] and Rinaldi [START_REF] Rinaldi | Small-signal operation of semiconductor devices including self-heating, with application to thermal characterization and instability analysis[END_REF], who laid the theoretical foundations for the frequency-domain thermal characterization of RF transistors from low-frequency y-parameters.

The technique is first presented along with the associated measurement setup, and then is unambiguously validated not by comparing it with traditional methodsunviable for in-situ extractionsbut through the 'simulated experiments' strategy applied to a SiC-based multi-chip power module (PM) [START_REF] Hussein | Dynamic performance analysis of a 3.3 kV SiC MOSFET half-bridge module with parallel chips and body-diode freewheeling[END_REF]. The validation is performed by emulating the experimental procedure with circuit-based electrothermal (ET) simulations accounting for a known ZTH,j-a used as a reference, and comparing the extracted ZTH,j-a with the reference counterpart.

II. IN-SITU EXTRACTION TECHNIQUE

Regardless of the nature of the electronic component, the critical temperature impacting the I-V characteristics is conventionally referred to as junction temperature (Tj). As accurately described in [START_REF] Diebold | Transient thermal impedance of semiconductor devices[END_REF], [START_REF] Blackburn | Transient thermal response measurements of power transistors[END_REF], the junction-to-ambient self-heating thermal impedance ZTH,j-a(t) [K/W] of a device embedded in a system is defined as , () ()

j amb TH j a D T t T Zt P - - = (1)
where PD is the amplitude of a dissipated power step applied at t=0 and Tamb is the ambient temperature. ZTH,j-a depends on (i) the geometry of the domain and of the heat source, (ii) thermal properties of the materials, and (iii) boundary conditions.

The thermal behavior in the frequency domainanalyzed in the power electronics scenario by Ma et al. in [START_REF] Ma | Frequency-domain thermal modeling and characterization of power semiconductor devices[END_REF], [START_REF] Ma | Modeling and characterization of frequency-domain thermal impedance for IGBT module through heat flow information[END_REF] is characterized by the junction-to-ambient power pulse thermal response Zj-a(f), which is the F-transformation of the time derivative of the thermal impedance dZTH,j-a(t)/dt [START_REF] Gerstenmaier | Rigorous model and network for transient thermal problems[END_REF]. A simple fitting with a Foster thermal network allows converting any Z(f) to the corresponding ZTH(t) and vice versa.

A. Extraction Procedure

The extraction procedure is described for the specific case of a MOS transistor, yet, mutatis mutandis, it can be simply generalized to other power devices. The drain current (ID) can be expressed as a function of the input voltages and junction temperature:

( ) ,, D D I DS GS j I F V V T =
(2) which in the frequency domain becomes (3) Let us consider an operating condition in which VGS=VGSdc, and VDS is given by the superposition of a dc value and a small ac signal (VDS=VDSdc+VDSac); in this case, the VDS derivative of ( 3) is given by

DD jdc DSdc II j D DS DS j DS T V FFdT dI dV V T dV  = +   (4)
where Tjdc is the Tj unaffected by the small signal (i.e., only determined by the dc bias). Let us denote as go the frequency-dependent output conductance of the transistor and as go|T jdc the output conductance at Tj=Tjdc.

( )

, ;

D j dc jdc DI o o T DS DS T dI F g f g dV V  ==  (5)
By making use of ( 5) in ( 4), it follows:

( ) ( ) jdc D DSdc oo T j I DS j V g f g dT f F dV T - =   (6)
According to the perturbative approach shown in [START_REF] Müller | Internal thermal feedback effects in four-poles especially in transistors[END_REF],

( ) ( ) ( )

D D Ddc DSdc Ddc DSdc o DS DS dP dI f I V f I V g f dV dV = +  = +  (7)
Since Tj(f)=Zj-a(f)•PD(f)+Tamb, the term dTj/dVDS is given by

( ) ( ) ( ) j j a Ddc DSdc o DS dT f Z f I V g f dV -  =  +   (8)
from which

( ) ( ) ( ) jdc D DSdc oo T ja I Ddc DSdc o j V g f g Zf F I V g f T - - =    +    (9)
In ( 9), the term ∂FI D /∂Tj|V DSdc is a thermometer that should be in principle determined in a temperature-controlled environment. To make the Zj-a(f) evaluation only relying on in-situ measurements of electrical signals, such a term is simplified by normalizing [START_REF] Riccio | An equivalent time temperature mapping system with a 320×256 pixels full-frame 100 kHz sampling rate[END_REF] to Zj-a(0), which is the static value of Zj-a(f) and represents the junction-to-ambient thermal resistance (R TH,j-a ). The normalized junction-to-ambient power pulse thermal response zj-a(f) is given by

( ) ( ) ( ) ( ) ( ) ( ) 0 (0) 0 jdc jdc oo T Ddc DSdc o ja oo j Ddc D a ja Sdc o T g f g Zf I V g zf Z g g I V g f - - - - + = =  - +  (10)
In [START_REF] Müller | Internal thermal feedback effects in four-poles especially in transistors[END_REF], go|T jdc is easily obtainable through a fitting of go, while go(0) is the measured value at f=0 Hz. Further details are provided in [START_REF] Sahoo | Thermal impedance modeling of Si-Ge HBTs from low-frequency small-signal measurements[END_REF].

Once zj-a(f) is extracted, the corresponding Zj-a(f) can be evaluated through a de-normalization process, which makes use of the junction-to-case power pulse thermal response Zj-c(f) (i.e., the Z(f) of the device not embedded in the assembly) taken from the device datasheet. First, a high-frequency matching range is identified, where the Zj-c(f) (known) and Zj-a(f) (to be determined) coincide, as the heat is still confined in the device region [START_REF] Rencz | Measuring partial thermal resistances in a heat-flow path[END_REF], [START_REF] Esposito | Thermal penetration depth investigations and BEOL metal impact on the thermal impedance in SiGe HBTs[END_REF]. Subsequently, RTH,j-a=Zj-a(0) is tuned to make Zj-a(f)=zj-a(f)•RTH,j-a coincide with Zj-c(f) in the matching range. This de-normalization procedure is schematically depicted in Fig. 1. Finally, ZTH,j-a(t) is obtained by (i) fitting Zj-a(f) with a Foster thermal network and (ii) computing its power step response.

B. Measurement setup

The measurement circuit sketched in Fig. 2 is devoted to the extraction of go vs. frequency by operating in the time domain. The circuit requires: (i) two voltage supplies (i.e., VGGdc and VDDdc) making the transistor work at the desired bias point, that is, VDSdc and IDdc; (ii) a voltage generator (VDDac) applying a small-signal piecewise sinusoidal waveform; (iii) two voltage probes and a shunt resistor (Rshunt) allowing the sensing of VDS(t) and ID(t). For each frequency, the evaluation of go is performed according to the following equation:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) - == - D Ddc Dac o DS DSdc DSac RMS I t I RMS I t g RMS V t V RMS V t ( 11
)
where RMS stands for root mean square.

III. 'SIMULATED EXPERIMENTS' STRATEGY

As other experimental methods are not suited for an in-situ extraction, the proposed technique was validated by resorting to the 'simulated experiments' strategy (e.g., [START_REF] Russo | Evaluating the self-heating thermal resistance of bipolar transistors by DC measurements: A critical review and update[END_REF], [START_REF] Catalano | Using EMPHASIS for the thermography-based fault detection in photovoltaic plants[END_REF]), which also allows for an unambiguous accuracy verification since the ZTH,j-a to extract is known.

A PM embedding four SiC-based VDMOS transistors was considered as a case-study; the devices replicate the bare-die CREE product CPMF-1200-S080B (rated 1200 V, 50A [START_REF]Datasheet of CREE[END_REF]).

To emulate the in-situ ZTH,j-a extraction, the 'simulated experiments' strategy was applied to one of the devices mounted on the PM. The procedure is articulated as follows:

• First, the reference ZTH,j-a of the device was obtained through detailed purely-thermal 3-D FEM simulations in the COMSOL Multiphysics environment [START_REF][END_REF], in which the exact replica of the PM was reproduced (Fig. 3). The boundary conditions were accounted for through a heat transfer coefficient h=2×10 3 W/m 2 K applied to the bottom surface of the thick copper baseplate, which describes the contact with an efficient heat-sink [START_REF] Catalano | Numerical simulation and analytical modeling of the thermal behavior of single-and double-sided cooled power modules[END_REF]. • The obtained reference ZTH,j-a was used to build a SPICE-compatible thermal feedback network (TFN) with a Foster topology [START_REF] Jakopović | Identification of thermal equivalentcircuit parameters for semiconductors[END_REF]; the TFN was then coupled to the electrical model of the VDMOS transistor, the temperature-sensitive parameters of which were allowed to vary during the simulation run. The electrical model was calibrated on experimental data [START_REF] Riccio | A temperature-dependent SPICE model of SiC power MOSFETs for within and out-of-SOA simulations[END_REF] and is thoroughly described in [START_REF]Circuit-based electrothermal simulation of multicellular SiC power MOSFETs using FANTASTIC[END_REF]. • Transient simulations were conducted in the OrCAD Capture software package [START_REF]PSPICE User's Manual[END_REF] on the ET model to emulate the experimental procedure presented in Section II.B to extract the zj-a. • The Z j-c was determined by simulating the bare-die device with isothermal backside at 300 K in COMSOL. • The de-normalization process and the time domain conversion were then carried out to obtain the thermal impedance ZTH,j-a. • The reference and extracted ZTH,j-as were finally compared.

IV. RESULTS AND DISCUSSION

The device was biased with VDDdc=15 V, VGGdc=10 V, and Rshunt=200 mΩ. For the given bias point, a preliminary investigation allowed observing that nonlinear thermal effects do not play a relevant role in defining the dynamic response of the assembly. Concerning the piecewise waveform, 3-period sinusoids were adopted for each frequency. The logarithmically-spaced frequency range was 3 mHz-20 kHz with six points per decade. The V DDac waveform was approximately 1 hour long, while its amplitude was set at 0.5 V. The large thermal time constants introduced by the thick copper baseplate led to a relatively long duration of the experiment, this being the main drawback of the in-situ characterization of the assembly under analysis.

The ID and VDS vs. time waveforms were computed according to [START_REF] Rinaldi | Small-signal operation of semiconductor devices including self-heating, with application to thermal characterization and instability analysis[END_REF], and zj-a(f) was extracted through [START_REF] Müller | Internal thermal feedback effects in four-poles especially in transistors[END_REF]. Results are shown in Fig. 4, where the reference and extracted zj-a(f) are compared. An excellent agreement was achieved: considering the percentage mean-square error as a figure of merit, the mismatch between the two curves in the high-frequency matching range turned out to be 1.72%, while 0.34% was the discrepancy in the overall frequency range.

The de-normalization process discussed in Section II.A was then applied. Results are shown in Fig. 5; RTH,j-a was calibrated to 0.35 K/W (to be compared with the COMSOL reference RTH,j-a=0.354 K/W) to overlap the curves in the 3-20 kHz matching range. Fig. 5 witnesses the agreement between the extracted and reference thermal responses Z j-a (f) and impedances Z TH,j-a (t). V. CONCLUSIONS This letter presents an innovative technique allowing the measurement of the overall junction-to-ambient self-heating thermal impedance of power devices mounted in a generic assembly and working under any environmental condition (in-situ extraction). The technique is applicable to any power semiconductor device without the need for temperature sensing; it only relies on the measurement of electrical signals and requires the knowledge of the junction-to-case thermal impedance of individual devices (i.e., not embedded in the assembly). The thermal impedance extraction can take from tens of minutes to few hours to be performed, depending on how slow the dynamic thermal response of the assembly is.

Since traditional methods do not allow in-situ measurements, their accuracy cannot be compared with that ensured by the proposed technique; therefore, a successful and unambiguous validation of the technique has been performed through the 'simulated experiments' strategy. The thermal behavior of a SiC-based VDMOS transistor embedded in a state-of-the-art multi-chip power module has been considered as a case-study. Electrothermal simulations in a SPICE-like software package have been used to emulate the experimental application of the proposed technique. An excellent assessment of the junction-to-ambient thermal impedance was obtained. Therefore, this technique proposes itself as a valid solution for the in-situ thermal characterization of power devices.

Fig. 1 .

 1 Fig. 1. Schematic representation of the de-normalization process: the RTH,j-a value is tuned to overlap Zj-a(f) (solid blue line) and Zj-c(f) (yellow triangles) in the high-frequency matching range; z j-a(f) is also shown (solid red line).

Fig. 2 .

 2 Fig. 2. Circuit conceived to perform the measurements of the electrical signals VDS(t) and ID(t). The gray box shows the small-signal piecewise sinusoidal waveform (solid blue line).

Fig. 3 .

 3 Fig. 3. 3-D representation of the PM under investigation reproduced in COMSOL Multiphysics. Evidenced are: the direct-bonded copper (DBC) substrate in the aluminum nitride technology, the copper baseplate, and the die corresponding to the VDMOS transistor on which the extraction technique was applied.

Fig. 4 .

 4 Fig. 4. Comparison between extracted (solid blue line) and reference (dashed magenta) zj-a(f). Also reported are the VDS (solid green line) and ID (dashed orange) vs. time waveforms at f=3 mHz and f=6 Hz (top-left and bottom insets, respectively).

Fig. 5 .

 5 Fig.5. (bottom-left axes) comparison between extracted (solid blue line) and reference (dashed magenta) Zj-a(f)s. The inset highlights the matching range and the overlap between Zj-a(f) and Zj-c(f) (yellow triangles). (top-right axes) comparison between extracted and reference ZTH,j-a(t)s.

ACKNOWLEDGMENT

The funding for the Ph.D. activity of Ciro Scognamillo was generously donated by the Rinaldi family in the memory of Niccolò Rinaldi, a bright Professor and Researcher of University of Naples Federico II, prematurely passed away in 2018.