Bertrand Gottin
email: bertrand.gottin@gipsa-lab.inpg.fr

Ioana Cornel
email: cornel.ioana@gipsa-lab.inpg.fr

Srdjan Stankovic
email: srdjan@cg.ac.yu

Ljubisa Stankovic
email: ljubisa@cg.ac.yu

Jocelyn Channussot
email: jocelyn.chanussot@gipsa-lab.inpg.fr

ON THE CONCEPT OF TIME-FREQUENCY DISTRIBUTIONS BASED ON COMPLEX-LAG MOMENTS

Keywords:

The instantaneous frequency law is an important tool for the analysis of time-frequency content of a signal. Generally, the analyzed signals are composed of several time-frequency components characterized by various non-linear contents. To deal with the nonlinearity of time-frequency content, several time-frequency distributions have been proposed, mainly based on warping techniques. Recently, we introduced the time-frequency distributions based on complex-lag arguments. Its property to strongly reduce, in the case of a single component, the inner interferences due to the time-frequency non-linearity has been shown. In this paper we prove its capabilities to deal with multi-component signals. This property is achieved by an appropriate combination of complex-lag moments computed for different complex lags sets. Numerical examples prove the benefits of the concept defined in this paper.

INTRODUCTION

The analysis of signals characterized by complex timefrequency behaviour is a very challenging topic, due to the richness of the information describing the analyzed phenomena. In a large number of applications the analysis of the time-frequency (T-F) content provides an efficient solution for the characterization of diverse physical phenomena. Wave propagation trough time-varying dispersive channels, micro-Doppler effects or mechanical signals are just three examples requiring an efficient time-frequency analysis of signals arising from these applications [START_REF]Applications in time-frequency signal processing[END_REF]. The signals associated to these applications are generally characterized by many nonlinear time-frequency structures. An efficient analysis of such signal should highlight the time-frequency energy of signal structures despite of artefacts that inherently appear when using time-frequency representations (TFR). Hence, in the case of linear TFRs, the well-known trade-off between time and frequency resolutions has to be considered. This topic has been subject to a large number of works. An alternative to linear TFR is the concept of bilinear TFRs [START_REF] Cohen | Time-frequency analysis[END_REF]. One of the major research directions concerns the interference (gener-ated by the bi-linearity) control in order to focus on timefrequency components of the signals. There are two types of interferences: inner interferences, generated in the case of non-linear time-frequency components and cross-terms, generated by the multi-component structures. The first interferences type is usually addressed by non-linear TFR designed with help of time or frequency warping concept [START_REF]Applications in time-frequency signal processing[END_REF], [START_REF] Papandreou-Suppappola | Quadratic time-frequency representations with scale covariance & generalized time-shift covariance: A unified framework for the affine, hyperbolic & power classes[END_REF]. Recently, complex time distribution concept has been introduced in [START_REF] Lj | Time-frequency distributions with complex argument[END_REF] as an efficient way to produce almost completely concentrated representations along the polynomial instantaneous frequency laws (IFL) of order 4 or less. In [START_REF] Cornu | Time-Frequency distributions with generalized complex lag argument[END_REF] we propose the generalization of the complex time distribution producing, in the mono component case, highly concentrated distributions around arbitrary polynomial IFLs.

In this paper we will show that the complex time distribution concept is able to deal with multi-component signals. This property is achieved by an appropriate choice of sets of complex lags. Combining the complex-lag moments computed for these sets produces a significant attenuation of cross terms with respect of auto-terms.

The paper is organized as follows. In Section 2 a brief presentation of the complex time distribution concept is done. The modified version of this concept having multicomponent capabilities is presented in Section 3. The theoretical benefits of the new concept are analyzed via numerical examples in Section 4. We conclude in Section 5.

TIME-FREQUENCY DISTRIBUTION BASED ON COMPLEX LAGS ARGUMENTS

The concept of complex lag distributions has been introduced in [START_REF] Lj | Time-frequency distributions with complex argument[END_REF] as a way for inner interferences reduction with respect of Wigner distribution. Recently, this concept has been generalized in order to focus on arbitrary instantaneous phase derivate of a signal [START_REF] Cornu | Time-Frequency distributions with generalized complex lag argument[END_REF]. Let consider the signal defined as ( ) ( )

j t s t Ae φ =
. The case of A depending of t can also be addressed since the effect of slowly varying amplitude is "visible" on the instantaneous phase.

By using the Cauchy's integral formula [START_REF] Rudin | Real and Complex Analysis[END_REF] it is possible to compute the K th order derivative of the instantaneous phase as

( ) ( ) ( ) ( ) 1 ! 2 K K z K t d z j z t γ φ φ π + = - ∫ (1)
This relation shows the interest of the complex time concept: the K th order derivate of function φ at instant t can be computed as the complex integral over the integration path γ defined around this point. Applying the theory of Cauchy's integral theorem [START_REF] Rudin | Real and Complex Analysis[END_REF] and after few computations the expression (1) becomes [START_REF] Cornu | Time-Frequency distributions with generalized complex lag argument[END_REF]:

( ) ( ) ( ) 2 0 ! 2 K j j K K K t t e e d π θ θ φ φ τ θ πτ - = + ∫ (2)
The discrete version of ( 2) is defined for ( ) ( )

2 2 1 0 ! p p K N j j K N N K p K t t e e N π π φ φ τ ε τ - - =   = + +     ∑ ( 3
)
where ε is the discretization error term.

( ) ( ) ( ) 1 , , 0 ! , N K N K K N p N p p K t t Q t N φ ω τ ω φ τ τ - - =   + = +       ∑ ( 4
)
where Q is the spread function defined as [START_REF] Rudin | Real and Complex Analysis[END_REF] :

( ) ( ) ( ) ( ) 1 1 1 ! , ! Np Np K K Np K p K Q t N t Np K N τ τ φ + + +∞ + =   =   +   ∑ (5)
As indicated by ( 4) and ( 5), the sum of the phase samples defined in the complex coordinates (left side of 4) is linear depending on τ if the φ's derivates of orders greater than N+K are 0. In order to exploit this property we define the generalized complex-lag moment (GCM) of s as the operation leading to (4):

[ ]( ) ( ) ( ) ( ) , 1 , 0 (4) , ! , N K N p K N K K N N p p with j t j Qt K GCM s t s t N e ω φ τ τ τ ω τ - - = +   = +       = ∏ (6)
The computation of GCMs implies the evaluation of signal samples at complex coordinates. This abstract notion is defined with help of analytical continuation of a signal defined as [START_REF] Rudin | Real and Complex Analysis[END_REF]:

( ) ( ) 2 2 mf j ft s t jm S f e e df π π ∞ - -∞ + = ∫ (7)
where S(f) is the Fourier transform of signal s.

Taking the Fourier transform of GCM with respect of τ we define the generalized complex-lag distribution (GCD):

[ ]( ) [ ]( ) ( ) ( ) ( ) ( ) , , , K K N N K j Q t GCD s t GCM s t t Ae τ τ τ ω ω τ δ ω φ   = ℑ =     = - * ℑ   (8)
As stated by this definition, the K th order distribution of the signal, obtained for N complex-lags, highly concentrates the energy around the K th -order derivate of the phase law.

This concentration is optimal if the φ's derivates of orders greater than N+K are 0, exactly like in the case of chirps represented by Wigner distribution.

The general definition (8) leads to a large number of TFRs, part of them well known in literature. For example, for K=1; N=2 the WVD is obtained whereas the case K=1; N=4 corresponds to the complex-time distribution (CTD) [START_REF] Lj | Time-frequency distributions with complex argument[END_REF]. In [START_REF] Cornu | Time-Frequency distributions with generalized complex lag argument[END_REF] we have shown that increasing the number of complex lags leads to an attenuation of inner interference due to the time-frequency non-linearity. This is illustrated by the next example for a test signal

GCD

We remark the better concentration of time-frequency energy in the case of GCD 1 6 than in the case of the other TFRs. This is analytically proved by the spread function expression (5) and illustrated by this example.

The next example points out on the derivability property of GCD. We consider a 4 th order polynomial phase signal defined by ( ) ( ) ( )

5
= + -⋅ + ⋅ = .
The figure 3 plots in the top the analytic derivatives of first, second and third orders of the IPL of this signal. We plot, in the bottom of the figure 3, the GCDs of the same orders.

We remark that the theoretical derivatives are correctly represented by the GCDs of corresponding order, justifying the derivation property of the complex-lag distribution. By derivation property it is possible to reduce a high order polynomial phase signal to a simpler one. That is, in the case of signal s 2 , the estimation of 4 th order polynomial coefficient could be done by computing the Fourier transform of GCM of order 4. This property, detailed in [START_REF] Cornu | Time-Frequency distributions with generalized complex lag argument[END_REF], has been studied so far only in the case of a single component. The next section contains an extension of GCD to multi-component signals.

COMPLEX LAGS MOMENT FOR MULTI-COMPONENTS SIGNALS

As proved in [START_REF] Cornu | Time-Frequency distributions with generalized complex lag argument[END_REF] and illustrated in the previous section, the GCD is an interesting tool for analysis of arbitrary derivates of instantaneous phase of a signal. So far, only monocomponent signals have been considered. In the case of multi-component signals cross-terms will appear due to the non-linear form of the GCMs [START_REF] Rudin | Real and Complex Analysis[END_REF]. That is, considering a signal composed by two components, , by analogy with the CTD case [START_REF] Lj | Time-frequency distributions with complex argument[END_REF], the GCM can be expressed as :

( ) ( ) ( ) ( ) ( ) 1
[ ]( ) , 1 2
1 1 , 2 , 0 , ! ! N K N p K N N K K N p N p p GCM s s t K K s t s t N N ω τ ω τ ω τ - - = + =       = + + +                   ∏ ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 2 1 12 (4) , , , , K K with j t j Q t j t j Q t e e C T NK φ τ τ φ τ τ φ φ + + = + + (9)
where CT is the terms indicating the cross terms. Because of the complicate GCM's definition, an analytical calculation of cross terms is a very difficult task. From (9) we remark that the auto-terms are sinusoids in τ with frequencies given by the K th order derivative of instantaneous phase laws φ 1 et φ 2 .

The spread functions, Q 1 and Q 2 , can be neglected if φ's derivates of orders greater than N+K are 0. Obviously, the cross terms generated by combination of signal's components according to the GCM definition affect the visibility of autoterms. This is illustrated, via the next example, in the case of a two-component signal defined as: GCD of the two-component signal contains the structures corresponding to the second order derivative and having a correct shape with respect of theoretical derivation (top side of figure 4). Nevertheless, the cross terms have the same energetic level and they have a more complicate shape than in the case of WVD. In order to reduce the level of cross terms the starting point is the remark that the auto-terms of (9), excepting Q function and lag set !/ K K N τ τ → (however, this dependence can be avoided by a simple warping procedure as we will see further), don't depend of number of lags and of their structure. It means that, computing GCMs for several lag sets, the auto-terms will have the same structure whereas the cross-terms will be differently located because of their dependence of N. Furthermore, the summation of these several GCMs will highlight the auto-terms decreasing in the same time the level of cross terms. Considering the sets of GCMs computed for several lags sets,

( ) ( ) ( ) ( ) ( ) 5
[ ] ( ) ( ) { } ( ) 1,.., , ; ! / i i i K K N i i P GCM s t K N τ τ = = ( 10
)
the multi-lags sets GCM (mlsGCM) is defined as

{ } [ ]( ) [ ]( ) 1
, ,

i i P K K N N i mlsGCM s t GCM s t τ τ = = ∑ (11)
Practically, the first step is the computation of GCMs (10). The second step is the warping of lag sets of each GCM according to

( ) { } ! K i i N K τ τ
→ . This step is very important in order to ensure that all GCMs (10) are expressed in the same lag axis. Finally, the summation (11) is computed. For the two-component signal used in (9) the mls-GCM shows analytically how the auto-terms are amplified:

[ ]( ) ( ) ( ) ( ) ( ) 1 1 12 2 , , 1 1
, ,

K i P j t j Q t K i i i mlsGCM s t P e CT N K φ τ τ φ φ τ + = = = + ∑ ∑ (12)
The Fourier transform of mlsGCM, called mlsGCD, 13) produces a cross-terms reduced distribution. As an illustrative example, for the signal used in figure 4, the results obtained by mlsGCD are illustrated in figure 5. As this figure shows, using many lag sets leads to a cross-terms level reduction. The attenuation of the cross-terms is more significant as the number of lag sets increases. For this example, eight set of lags is appropriate for an accurate representation of auto-terms.

{ } [ ]( ) { } [ ]( ) , , i i K K N N mlsGCD s t mlsGCM s t τ ω τ   = ℑ   (
Figure 5 -Cross-terms reduction using mlsGCM

RESULTS

The capability of mlsGCD, defined in the previous section, to perform in a multi-component case can be successfully exploited in a large number of applications dealing with complex time-frequency modulations. The first example is the estimation of a harmonically frequency modulation mixed with coherent signals. This could be the case of audio signals or micro-Doppler radar signals (corresponding to radar signal returned by helicopter blades) corrupted by coherent attacks as tones or chirps. To illustrate such application let consider a sinusoidal frequency modulation corrupted by a chirp and sinusoid (theoretical IFLs plotted in the figure 6.a). Clearly, in the spectrogram domain (figure 6.b), the SFM is masked by the chirp and the tone as well as the noise (SNR=10 dB). The mlsGCD 2 (computed for 9 lag sets, figure 6.c) removes the effect of tone and stationaries the chirp effect. Finally, only the SFM is "visible" (figure 6.d) if the third order derivate is done via mlsGCD 3 . In this domain, the filtering or parameter estimation of SFM can be easily done.

Another application of the concept of mlsGCD is the extraction of transient impulses corrupted by some coherent signals. Hence, using the mlsGCD concept, we take advantage on the infinite derivability of the transient signals.

Transient signals can be globally defined as impulsive or very short duration signals often with oscillations. These signals exist in many different applications and systems from underwater acoustic to audio signals with sound attacks or electrical systems with partial discharges and commutation switches, for example. The main processing done on this type of signal aims to detect them, characterize them and localize their source. Several signal processing tools such as higherorder statistics and wavelets coefficients have been already used and upgraded to perform the goals of detection and localization but they do not lead to real characterization of the transient signals. The concept of GCD and mlsGCD shows its convenience to perform on transient signals in so far as, for an impulse, the successive order derivates of the phase law remain impulsive in a certain way and with no end. Therefore, the distinction of the transient is marked in the instantaneous law whatever the derivative order is.

The capability of GCD and mlsGCD to perform with transient signals can be exploited in several applications. Let us consider a multi-component signal composed by a chirp and a BPSK (Binary Phase-Shift Keying) modulation. The previous example illustrates also the mlsGCD performing in the case of variable amplitudes signals. Even if the pulses have different amplitude their representation in the GCD 3 domain is correctly done.

As well as GCD, the mlsGCD concept shows its utility for transient signals analysis, especially in the case of a signal corrupted by noise. As explained in section 3, the mlsGCD for multi-components signals leads to amplify the auto-terms and reduce the level of cross terms. In the case of transients embedded in noise, the idea is the same. A high order GCD of such a signal gives a high value signature of each transient but the phase derivative of the noise strongly corrupts the distribution. By using mlsGCD, the high value signature of the transients is amplified whereas the noise effect is reduced. Figure 10 illustrates this property of mlsGCD; the same train of impulses (defined in the figure 8) is used and complex white gaussian noise is added (SNR=12 dB). On the third order GCD, the transient signatures are clearly masked by the noise phase derivative, even after thresholding to keep the highest amplitude values. The distribution resulted from the third order mlsGCD (computed for 8 lag sets) makes the transient signatures more visible. As shown in figure 10.b, the detection of transient signal can be accurately done in the mlsGCD domain.

CONCLUSION

In this paper we address the problem of complex-lag representation of multi-component signals. The derivation property of the phase of a signal has been extended to multicomponent signals. This extension is based on the property of the auto-terms of the GCM which don't depend to the complex lag sets. Thus, the summation of GCMs evaluated for several lag sets, can produce the amplification of autoterms. The new defined distribution allows focusing on arbitrary derivate of phase in a multi-component context. This property can be of great helpful to separate signals with infinitely derivate phase laws from embedded observation.

Otherwise, mlsGCD can be seen as a new methodology for polynomial phase modelling. These applications as well as theoretical subjects, related to signal-dependent choice of lag sets and 2D extension of complex lag concept, will be address in our future works.

  the figure1) where N is the number of discrete values of the angle θ.

Figure 1 -

 1 Figure 1 -Definition of complex lag coordinates Using the properties of the roots of unity, 2 / , j p N N p e π ω =

Figure 2 -

 2 Figure 2 -Inner interference for WVD, CTD and

Figure 3 -

 3 Figure 3 -Derivation of instantaneous phase by GCDs

Figure 4 -

 4 Figure 4 -Derivation IPLs for a two-component signal

Figure 6 -

 6 Figure 6 -Separation of a sinusoidal frequency modulation (SFM)

Figure 7 -

 7 Figure 7 -GCD Performance to detect transient information As shown in the figure 7, the chirp crossing the BPSK hides, in the case of spectrogram and the WVD, the phase transitions corresponding to the BPSK. Using the IPL's derivation provided by GCD (figure 7.b) we can remove the chirp effect highlighting also the transitions. We can remark that the transitions detected in GCD domain are coherent with the theoretical ones illustrated in figure 7.d.Another application is illustrated in figure9where a simulated train of impulses (plotted in figure8) is corrupted by a chirp (spectrogram plotted in figure9.a). The GCD 3 theoretically removes the chirp effect and keeps the mark of the impulses in so far as the successive phase derivatives of a transient remain impulsive. The result of the GCD 3 plotted in figure9.b consequently shows the successful separation of the four transients present in the signal.

Figure 8 -

 8 Figure 8 -Train of impulses

Figure 10 -

 10 Figure 10 -Noise effect reduction using mlsGCD This example shows the capability of the mlsGCD to extract, via derivability property, the transient signals corrupted by both noise and coherent perturbations. It performs also in the case of pulses having different amplitudes. As shown in figure 10.b, the detection of transient signal can be accurately done in the mlsGCD domain.

Acknowledgements. This work was supported by the French Ministry of Foreign Affairs/EGIDE under the grants ECO-NET 2007 and by French-Montenegrin collaboration program PAI "Pelikan".


Jun Zhang

Bertrand Gottin

Antonia Papandreou-Suppappola

Cornel Ioana

Time-frequency

Keywords: Time-frequency analysis, dispersive channels, time-frequency mode separation, underwater acoustic communications, time-dispersion diversity

INTRODUCTION

As a waveform is transmitted in the shallow water environment, it can interact with the ocean bottom and surface, resulting in dense dispersion due to the water medium and timevarying (TV) changes of the ocean environment. Specifically, the shallow water acoustic environment is a linear TV dispersive system that shifts different frequencies by different amounts in time [1].

The dispersive effect introduced by shallow water environments can severely limit the performance of underwater acoustic applications such as sonar and communications. It can be shown that the propagation characteristics of an environment can provide us with a specific nonlinear function that defines the dispersion in this environment. Thus, the corresponding signal representation based on a normal mode model can be applied to exploit the potential diversity suggested by the model when the acoustic receiver is appropriately designed to match this nonlinear function.

*This work was supported by the NSF CAREER Award CCR-0134002 and the Department of Defense Grant No. AFOSR FA9550-05-1-0443.

In [1], a shallow water characterization was considered that matched dispersive signal transformations on the transmitted waveforms and was used successfully for shallow water communications to obtain time-dispersion diversity. However, this characterization was only applicable to signals with very high bandwidth as it assumed that the transmitted waveform was an impulse.

In [START_REF] Zhang | Timefrequency based waveform and receiver design for shallow water communications[END_REF], we presented a generalized characterization that was applicable to a larger class of signals by using the normalmodel in [START_REF] Ziomek | Fundamentals of Acoustic Field Theory and Space-Time Signal Processing[END_REF] assuming perfect waveguide conditions (homogeneous fluid layer with a soft top and rigid bottom). In this paper, we extend our characterization when these conditions are not perfect by following the waveguide model with pressure-release surface and fluid boundaries in [START_REF] Frisk | Ocean and Seabed Acoustics[END_REF]. Using the new model, we propose a new time-frequency receiver design which operates on individual modes of the multi-component received signal instead of on the whole signal. This upgraded receiver design of an underwater communication system is expected to achieve better bit error rate (BER) performance for time-dispersion diversity than the one obtained in [START_REF] Zhang | Timefrequency based waveform and receiver design for shallow water communications[END_REF].

The paper is organized as follows. In Section 2, we formulate the Pekeris model for shallow water environments. In Section 3, we present the time-frequency (TF) characteristics of this model, and we investigate the impact of the environment parameters on the dispersion effect. In Section 4, we use a TF mode separation technique to separate each component of the received signal and then design a new receiver structure with a corresponding optimal detector. In Section 5, numerical results of BER performance illustrate our improved performance in terms of diversity order.

MODELING FOR SHALLOW WATER ENVIRONMENT: PEKERIS WAVEGUIDE

The normal-mode model treats the ocean as a waveguide with plane, parallel boundaries, representing the acoustic field in the ocean medium as a sum of normal modes. We first discuss the normal-mode model for shallow water environments with pressure-release surface and a fluid seabed following [START_REF] Frisk | Ocean and Seabed Acoustics[END_REF][START_REF] Deane | Internal friction and boundary conditions in lossy fluid seabeds[END_REF][START_REF] Ziomek | Fundamentals of Acoustic Field Theory and Space-Time Signal Processing[END_REF]. A simple waveguide model of the ocean is shown in This corresponds to a pressure release surface with a fluid seabed environment [START_REF] Frisk | Ocean and Seabed Acoustics[END_REF].

III correspond to air, ocean water and seabed, respectively. An omnidirectional point source with signal spectrum X(f ) is located in Medium II at r = 0 and z = z 0 . The ocean is D m deep. We consider the sound speed in Medium II as a constant c m/s and density ρ kg/m 3 , and in Medium III as a constant c B m/s and density ρ B kg/m 3 . The ocean surface (at z = 0) is modeled realistically as an ideal pressure release boundary and the ocean bottom (at z = D) is modeled as a boundary between two different fluid media. The normal mode model is given by the solution of this ocean waveguide problem, which is determined by the environment parameters and satisfies all boundary conditions, including the boundary condition at the source.

After a detailed derivation in [START_REF] Frisk | Ocean and Seabed Acoustics[END_REF] and [START_REF] Deane | Internal friction and boundary conditions in lossy fluid seabeds[END_REF], the received signal spectrum excited by X(f ) at location (r, z) is given by the Pekeris waveguide model:

Y P ekeris (f ) = X(f ) Np-1 n=0 C n (f )Θ n (f ).
(1)

Without the assumption of ideal waveguide condition as in [START_REF] Zhang | Timefrequency based waveform and receiver design for shallow water communications[END_REF], the nth mode is characterized by

Θ n (f ) = 1 k n (f ) r e -jkn(f ) r ,
where

k n (f ) = 2π c ⎛ ⎜ ⎝f 2 - n 2 c 2 4D 2 ⎛ ⎝ 1 + ρ B c ρD 1 2πf 1 -c 2 c 2 B ⎞ ⎠ -2 ⎞ ⎟ ⎠ 1 2
is the wave number of the nth mode and N p is the largest mode number. The parameter

C n (f ) = A 2 n (f )sin(k zn (f )z 0 )sin(k zn (f )z)
is a function of frequency where

A n (f ) = √ 2 1 ρ D - sin(2k zn (f )D) 2k zn (f ) - ρ ρ 2 B tan(k zn (f )D)sin 2 (k zn (f )D) k zn (f ) -1 2 (2)
and

k zn (f ) ≈ nπ D ⎛ ⎝ 1 + ρB c ρD 1 2πf Ö 1-c 2 c 2 B ⎞ ⎠ -1
.

TIME-FREQUENCY CHARACTERISTICS OF SHALLOW WATER ENVIRONMENT MODEL

The TF characteristic of the acoustic signal is determined by the modal group velocity (MGV) g n (f ), due to the propagating delay of the frequency f 0 in the nth mode is determined by τ n (f 0 ) = r gn(f0) . The modal group velocity is shown in Fig. 2. From Fig. 2, we notice that, for the waveguide model with fluid seabed, the MGV approaches c when the frequency approaches infinity, and approaches c B when the frequency approaches the cutoff frequency of this mode. We analyzed further the TF characteristics of the received signal using the spectrogram TF representation (TFR) and investigated the effect of changing the parameters of the environment. Fig. 3(a) represents the spectrogram for three modes with environment parameters: D = 100 m and r = 15 km. Fig. 3(b) uses the same set of parameters except that r = 30 km. Fig. 3 shows each mode as a dispersive curve in the TF plane. We note that the curves stop at the cutoff frequency with no asymptotical behavior to the cutoff frequency. Fig. 3 shows that longer range between the receiver and transmitter causes more dispersion in the received signal. Fig. 4 shows that with larger ocean depth, the modal cutoff frequency is lower, and with a smaller depth, the dispersive effect is more visible at higher frequencies.

SYSTEM DESIGN FOR COMMUNICATION DIVERSITY

According to the Pekeris waveguide in (1), the received signal spectrum is the superposition of multiple normal Considering each mode as a communication subchannel, the time-dispersive channel can be interpreted as a superposition of subchannels, and using their TF characteristics, we can separate each subchannel and design the corresponding receiver to exploit time-dispersion diversity.

Separation of normal modes

Fig. 3 illustrates that each mode of the received signal appears as a dispersive curve in the TF plane, which makes the separation of the modes possible. In this paper, we use a two-step TF mode separation technique based on warpings to separate the modes. The two steps are described as follows.

During the first step, we design N p -1 instantaneous frequency (IF) curve separators in the TF plane, represented as e n (t), n = 0, • • • , N p -2. An IF curve separator is a curve situated between two successive modes in the TF plane as shown in Fig. 5(a) for the example in Fig. 3. In this problem, all environmental parameters are assumed to be known, so that the TF structures of the received signal are also known using the model in (1). Knowing the TF structures, we appropriately set M TF points (t i , f i ), i = 1, 2, . . . , M in the middle of the space between two successive modes. The M points between the nth mode and the (n + 1)th mode will constitute the IF curve separator e n (t). In the next step, the IF curves will be processed with the received signal. Thus, the separation lines must have the same duration as the received signal. To ensure that, all the IF curves begin at the starting point of the TF structure and are extended with a straight horizontal line until the end of the TF window.

During the second step, we process the different modes in the TF plane using a warping technique [START_REF] Jarrot | A class of linear time-varying filters based on non-unitary timewarping operators. Part I: principles and examples[END_REF][START_REF] Jarrot | A timefrequency characterization framework for signals issued from underwater dispersive environments[END_REF] based on the use of the local harmonic convolution operator [START_REF] Jarrot | A class of linear time-varying filters based on non-unitary timewarping operators. Part I: principles and examples[END_REF]. Assume that the received noisy signal in (1) is r(t) with Fourier transform (FT) R(f ) = Y P ekeris (f ) + W (f ), where W (f ) is additive white Gaussian noise. We then obtain the warping functions from the IF curve separators as ζ n (t) = 1 fc t -∞ e n (u)du, where f c > 0, and compute the corresponding generalized FT of r(t) as [START_REF] Papandreou-Suppappola | Time-Varying Processing: Tutorial on Principles and Practice[END_REF] M

r (λ) = ℘ r(t) dζ n (t) dt e -j2πλ ζn(t) dt, (3)
where ℘ contains the values of t in the domain of the warping function ζ n (t), and λ is a real and unitless parameter. To obtain the first mode, we compute the inverse generalized FT over the range (-∞, f c /f r ] as follows:

r 0 (t) = fc/fr -∞ M r (λ)e j2πλζ0(t) dλ, (4)
where f r is a normalization frequency. Then we subtract the first mode from the received signal, i.e, r(t) = r(t)r 0 (t), and the remaining modes are contained in r(t).

To obtain the seconde mode, we apply the above procedure to r(t) using the IF curve separator e 1 (t). Repeating the above procedure for each IF curve separator, we can then separate each mode from the received signal.

For the example in Fig. 5, we use the mode separation technique on the received signal excited by the waveform X(f ) = √ f for N P = 3 modes. The IF curve separators are shown in Fig. 5(a) and the separated components are shown in Fig. 5(b).

Receiver structure

Under the Pekeris waveguide assumption, the model in (1) can be seen as an inherent frequency domain transfer function. However, due to the distortion and signal propagation uncertainty in realistic shallow water environments, a more suitable model of random signal embedded in random noise consists of

R(f )=Y (f ) +W (f )= X(f ) Np-1 n=0 D n C n (f )Θ n (f )+W (f ), (5)
where R(f ) is the spectrum of the received signal, D n is a random process incorporating amplitude and phase variability of the source, and W (f ) is additive white Gaussian noise with variance σ 2 W . We propose a filter bank receiver scheme based on this signal model which can exploit the frequency domain dispersion diversity existing in the normal modes. The received signal spectrum R(f ) in ( 5) is first processed as described in Section 4.1 to separate each mode, and then the outputs are combined in the minimum-probability-of-error sense to obtain the transmitted information symbol b. This receiver structure is illustrated in Fig. 6.

We design the transmission waveform as X(f ) = √ f for f ≥ f 0 where f 0 is the cutoff frequency of the first mode, and then we employ the corresponding frequency domain matched filter to each separated mode component of the received signal. The nth mode component obtained from the separation processing is expressed as

R n (f ) ≈ bX(f )D n C n (f )Θ n (f ) + W n (f ), (6)
where W n (f ) is non-white Gaussian noise, with variance σ 2 Wn , that resulted from W (f ) when the nth mode is separated.

We first define the receiver matched waveform U n (f ) = X(f )C n (f )Θ n (f ) for the nth mode. Then, as each mode is separated, we can perform individual matched filtering on the n 0 th mode by D n0 U n0 (f ); this can avoid interference with the other modes. The output of the n 0 th matched filter can be TF mode separation

( ) 0 R f ( ) 1 R f ( ) 1 p N R f - ( ) 0 0 D U f ( ) 1 1 D U f ( ) 1 1 Np Np D U f - - Combination Σ ... ( ) R f Estimated Symbol

Matched fitering Matched fitering

Matched fitering Fig. 6. Receiver structure scheme.

expressed as

Z n0 = R n0 (f ), D n0 U n0 (f ) = f R n0 (f )D * n0 U * n0 (f )df. (7)
Using the expression of R n (f ) in ( 6), we can rewrite [START_REF] Jarrot | A timefrequency characterization framework for signals issued from underwater dispersive environments[END_REF] as

Z n0 ≈ b|D n0 | 2 f U n0 (f )U * n0 (f )df + f W n0 (f )D * n0 U * n0 (f )df.
As the output of this filter bank is given by the concatenation Z 0 . . . Z Np-1 , the combination in the minimumprobability-of-error sense is defined as the arithmetical mean of the filtering outputs. Thus, the estimation of the transmitted information symbol can be expressed as

b = ⎛ ⎝ Np-1 n=0 Z n ⎞ ⎠ . (8)
where (•) is the operator that takes the real part. Assuming an antipodal symbol transmission, i.e, b = +1 or b = -1, and that the probabilities of transmitting +1 and -1 are equal, the minimum error probability detector can be expressed as:

decide b = +1 if b > 0, decide b = -1 if b < 0.

PERFORMANCE ANALYSIS

As the modes are separated, each mode can be considered as a subchannel, and the aforementioned matched filter receiver can avoid cross-correlation interferences between different modes of the received signal. Hence, this matched filter receiver can exploit the potential diversity existing among the different modes. In this section, we first derive the theoretical probability of error for the aforementioned receiver, and then we obtain the BER results from numerical simulations. We also compare the performance of this receiver with the performance of a receiver that processes the complete received signal.

Let us consider the matrix of frequency domain mode autocorrelations defined as A = diag(A 0 , A 1 , . . . , A NP -1 ) where

A i = f U i (f )U * i (f )df, i = 0, . . . , N P -1
and the matrix of modal noise variance σ = diag(σ 2 Wi ) for i = 0, . . . , N P -1. Defining

= σ -1 √ AQ √ A † and m = √ σ -1 A E[D]
, where E[•] is the expectation operator and Q is the covariance matrix of the coefficients vector D = [D 0 , D 1 , . . . , D NP -1 ] T , we can derive the theoretical BER as the probability of error given by [START_REF] Zhang | Timefrequency based waveform and receiver design for shallow water communications[END_REF]:

P err = 1 π π 2 0 det sin 2 θ + I -1 e -m † ( È +sin 2 θ I) -1 m dθ, (9)
where we denote T and †, respectively, as the transpose and conjugate transpose operations. The BER obtained from the simulation and the theoretical expression are shown in Fig. 7, using the environment parameters D = 100 m, r = 15 km, c = 1500 m/s, c B = 1800 m/s, ρ = 1000 kg/m 3 and ρ B = 1800 kg/m 3 (as in Figures 3 and5). The results show the BER as a function of the signal-to-noise ratio (SNR) for the transmission frequency band 5 -250 Hz.

We notice from Fig. 7 that the simulation results are worse than the theoretical results. This is because the mode separation technique caused loss of information, especially at higher frequencies, which leads to less accurate matched filtering. However, if we assume no information is lost in the procedure of separation, i.e, the separation can generate the exact modes, then the numerical simulation results are close to the theoretical results, as expected.

We also compared the BER performance of the new receiver with the performance obtained in [START_REF] Zhang | Timefrequency based waveform and receiver design for shallow water communications[END_REF] where the matched filtering was applied to the whole multi-component received signal. Fig. 7 shows that the new receiver performed better according to the numerical simulation results.

CONCLUSION

We developed a frequency domain characterization of realistic shallow water environments and analyzed its dispersive characteristics. Following this model, we developed the appropriate receiver design based on using a warping technique for modal separation. This receiver structure aims to exploit the diversity existing in the system. Simulation results show that BER performance and diversity order were improved by the aforementioned system design.

[1] Y. Jiang and A. Papandreou-Suppappola,"Discrete timefrequency characterizations of dispersive time-varying

Fig. 1 .

 1 Fig. 1. Waveguide model with point source in Medium II at r = 0, z = z 0 , D m deep. This corresponds to a pressure release surface with a fluid seabed environment [4].

Fig. 2 .

 2 Fig. 2. Modal group velocity (MGV) function.

Fig. 3 .

 3 Fig. 3. Spectrograms with environment parameters (a) r = 15 km, D = 100 m, and (b) r = 30 km, D = 100 m.

Fig. 4 .

 4 Fig. 4. Spectrograms with environment parameters (a) r = 15 km, D = 30 m, and (b) r = 15 km, D = 100 m.

Fig. 5 .

 5 Fig. 5. (a) Spectrogram of received signal with the IF curve separators, (b) separated TF modes.

Fig. 7 .

 7 Fig.7. Theoretical BER performance of the new receiver (diamonds), BER performance of the new receiver obtained by numerical simulation (squares), BER performance of the new receiver obtained by numerical simulation using exact modes (line) and BER performance of the former receiver (circles).


Claude Oestges
email: claude.oestges@uclouvain.be

Bruno Clerkx

Maxime Guillaud
email: guillaud@ftw.at

Merouane Debbah
email: mer-ouane.debbah@supelec.fr

Bruno Clerckx
email: bruno.clerckx@samsung.com

Mérouane Debbah

FROM PROPAGATION MODELS TO SYSTEM PERFORMANCE EVALUATION

Keywords: MIMO, polarization, propagation, mutual information

published or not. The documents may come

I. INTRODUCTION

I N RECENT years, increasing attention has been paid to Multiple-Input Multiple-Output (MIMO) broadband wireless communication systems. However, antenna spacings of at least half a wavelength at the subscriber unit and ten wavelengths at the base station are usually required for achieving significant multiplexing and/or diversity gains. Hence, the use of possibly co-located orthogonally-polarized antennas appears as a space-and cost-effective alternative [START_REF] Neubauer | Simultaneous characterization of polarization matrix components in pico cells[END_REF]. Indeed, orthogonal polarizations ideally offer a much better separation between channels, through a large decorrelation at both transmit (Tx) and receive (Rx) sides. In this paper, we consider both Two-Input Two-Output (TITO) and MIMO dualpolarized systems, and we use as a reference the corresponding TITO or MIMO uni-polarized system. In the TITO case, both the transmit and receive arrays are made of two spatially separated antennas with orthogonal polarizations, in order to combine both spatial and polarization diversity/multiplexing. The reference scenario is then a TITO configuration with similarly separated uni-polarized antennas. MIMO n r ×n t schemes (for even values of n t and n r ) are modeled by considering that the transmit (resp. receive) array is made of n t /2 (resp. n r /2) dual-polarized sub-arrays, extending over a given length L t (resp. L r ). The reference scenario is then a n r × n t unipolarized scheme based on antenna arrays extending over the same lengths. As can be noticed, we carry out comparisons by keeping constant the dimensions of the MIMO matrix. Indeed, these dimensions equal the number of required RF chains, which are the most expensive components in a system. Note that using two co-located antennas with orthogonal polarizations at both ends of the link already constitute a 2 × 2 system. Another definition, motivated by the spatial extension of the antenna arrays, consists in considering the above scenario as a Single-Input Single-Output (SISO) system with dual-polarized antennas. This last definition, which has often been used in the literature, however leads to unfairly comparing a uni-polarized 2×2 system with a so-called 2×2× two-polarization system, although the latter is actually a 4 × 4 scheme, which needs to be compared with a uni-polarized 4×4 system if we want to keep the number of RF chains equal. Despite a number of recent studies focusing on spatial channel models, only a limited number of papers have addressed the polarization issue [START_REF] Neubauer | Simultaneous characterization of polarization matrix components in pico cells[END_REF]- [START_REF] Asplund | Propagation characteristics of polarized radio waves in cellular communications[END_REF], theoretically or experimentally, mainly because the (de-)coupling effect between orthogonal polarizations is a complex mechanism. On the one hand, a number of geometry-based models are available for simulating dual-polarized transmissions, the most recent model being used by the 3GPP group [START_REF] Calcev | A wideband spatial channel model for system-wide simulations[END_REF]. Although these models are very useful in simulating signal processing techniques, they do not allow for analytical manipulations (e.g. in expresions of the mutual information or the error probability). One the other hand, analytical formalisms are needed to design space-time coding and precoding schemes. One can make a comparison with single-polarized schemes: whereas geometrybased models (e.g. the one-ring model of [START_REF] Shiu | Fading correlation and its effect on the capacity of multielement antenna systems[END_REF]) have been heavily used in simulations, analytical approaches such as the Kronecker [START_REF] Chuah | Capacity scaling in MIMO wireless systems under correlated fading[END_REF] or eigenbeam [START_REF] Weichselberger | A stochastic MIMO channel model with joint correlation of both link ends[END_REF] models have proved very useful in code design. Hence, there is an obvious appeal to develop an analytical formalism to model dual-polarized 1536-1276/08$25.00 c 2008 IEEE multi-antenna channels. The first goal of this paper is precisely to provide a sufficiently general analytical framework using a reduced number of physically sound parameters. Based on this model, it is then possible to clearly identify the benefits of multiple polarizations from a performance point of view. This constitutes the second objective of this paper.

In Section II, we carry out a thorough literature overview of experimental or electromagnetic results regarding dualpolarized channels. These results are then used in Section III to build an analytical model combining the effects of space and polarization separations. In Section IV, we investigate whether dual-polarized systems may possibly increase the system throughput. Finally, in Section V, we analyze the performance of Orthogonal Space-Time Block Coding (O-STBC) and Spatial Multiplexing from an error probability viewpoint.

In this paper, E{x} = x represents the expectation of random variable x, * stands for element-wise conjugation, H , for conjugate transpose, x → a indicates that x tends toward a, vec(A) is the operator that forms a vector from successive columns of matrix A, |x| is the absolute value of x, A F is the Frobenius norm of A, det {A} is the determinant of matrix A, Tr {A} is the trace of matrix A, is the Hadamard product, ⊗ is the Kronecker product, I n stands for the n × n identity matrix, and 1 n×m denotes a n × m matrix with unit entries.

II. CHANNEL CHARACTERIZATION OF DUAL-POLARIZED SYSTEMS

A. Mechanisms and Parameters

Ideally, the cross-polar transmissions (e.g. from a verticallypolarized Tx antenna to a horizontally-polarized Rx antenna) should be equal to zero. This is actually not the case owing to two depolarization mechanisms: the use of imperfect antenna cross-polar isolation (XPI) and the existence of a cross-polar ratio (XPR) in the propagation channel. Both effects combine to yield a global cross-polar discrimination (XPD).

The first mechanism is well-known in antenna theory and is easily accounted for by means of the cross-polar antenna pattern. Analytically, this can be approximated by a coupling matrix at transmit or receive sides

M t = 1 √ χ a,t √ χ a,t 1 and M r = 1 √ χ a,r √ χ a,r 1 ,
where the scalar antenna XPI is defined as χ -1 a,t and χ -1 a,r at Tx and Rx respectively. Note that antenna XPI is the only mechanism affecting line-of-sight (LOS) components. Scattered components (either coherent or non coherent) are affected by both XPI and XPR, although these are well separated effects if the antenna cross-coupling is represented by M t and M r . Throughout this paper, channel matrices are denoted differently, depending on the scenario and the antenna quality, as detailed below:

1) H × designates the dual-polarized channel matrix combining spatial separation, dual-polarized arrays using antennas with infinite XPI;

2) H ×,a = M r H × M t designates the dual-polarized channel matrix combining spatial separation, dual-polarized arrays and finite antenna cross-polar isolation (the subscript a means that antenna XPIs are accounted for); 3) H designates the uni-polarized channel matrix (all antennas are identically polarized) accounting only for spatial correlations at both ends, 4) G is the 2 × 2 matrix representing the dual-polarized TITO channel for orthogonally-polarized co-located antennas (there are no spatial correlation effects) with infinite XPI, whereas G a = M r GM t is the equivalent matrix, but for antennas with finite XPI.

Furthermore, for Ricean fading channels, the above matrices can be decomposed into a coherent part (we then write H× , H, Ḡ, etc. in agreement with our notation) and a non-coherent Rayleigh-fading part, proportional to H× , H, G, etc.

Quite arbitrarily, we decided to model the downlink channel (from the base station to the user terminal). Naturally, the uplink channel matrix (from the user terminal to the base station) is simply obtained by transposition of the downlink channel matrix. For dual-polarized arrays with co-located vertically and horizontally-polarized (abbreviated as VH) antennas, we write the downlink channel matrix as

G = g vv g vh g hv g hh . ( 1
)
Denoting by p ij = |g ij | 2 the instantaneous gain on channel ij, we may define various cross-polar ratios [START_REF] Neubauer | Simultaneous characterization of polarization matrix components in pico cells[END_REF] (XPDs could be similarly defined based on G a rather than G):

1) uplink cross-polar ratios (uplink XPR)

XPR U v = p vv /p vh (2) XPR U h = p hh /p hv (3)
2) downlink cross-polar ratios (downlink XPR),

XPR D v = p vv /p hv (4) XPR D h = p hh /p vh . ( 5
)
We may also define a unique co-polar ratio as

CPR = p vv /p hh . ( 6
)
Clearly, all these parameters are not independent. In particular, the following relationships hold true:

CPR = XPR U v XPR D h = XPR D v XPR U h . ( 7
)
The above definitions only concern channel gains. To define the phase relationships, G can be characterized by its correlation matrix, given by E vec G H vec G H H . The diagonal elements of this 4 × 4 matrix are the various average gains E{p ij }, while the off-diagonal elements express the different co-, cross-and anti-polar correlations. The crosspolar correlations (XPC) correspond to the classical transmit and receive correlations, the correlation between g vv and g hh is defined here as the co-polar correlation (CPC), and the correlation between g vh and g hv , as the anti-polar correlation (APC).

B. Overview of Experimental Results

Most experimental results provide a partial characterization of G a , or even of H ×,a , providing values of the global XPD on the up-or downlink, and rarely of the CPR. 1 Hence, it is sometimes difficult to isolate the various contributions (antenna XPI, channel XPR and spatial correlations). However, in most cases, one may expect that experimental arrays are made of co-located antennas with a large XPI, and that only significant results (i.e. with XPD lower than the XPI) are reported in these papers. Hence, this implies that for these values, XPD ≈ XPR, and it should be possible to characterize G, at least partially, from the results summarized below. This is the reason why we use XPR denominations in the following, but it should be reminded that these are formally XPD values.

1) Outdoor Scenarios: As far as uplink single-input multiple-output (SIMO) results are concerned, these usually consider a vertically polarized transmitter and a base station using an orthogonally-polarized array with vertical and horizontal polarizations (VH) or 45-degree tilted polarization (slant scheme). For a VH reception, the measured parameters are therefore XPR U v and the receive cross-polar correlation:

• the experimental uplink XPR in [START_REF] Kozono | Base station polarization diversity reception for mobile radio[END_REF] reaches values of 6 dB and 7.4 dB respectively in macro-and microcells (at 900 MHz), with mean envelope cross-polar (receive) correlation coefficients around 0.1; • in [START_REF] Vaughan | Polarization diversity in mobile communications[END_REF], the uplink XPR is about 12 dB in suburban areas and 7 dB in urban areas at 463 MHz, while the power receive correlation is quasi zero; • in [START_REF] Lempainen | The performance of polarization diversity schemes at a base station in small/micro cells at 1800 MHz[END_REF], at 1800 MHz, the power receive correlations are below 0.3 and the uplink XPR values are around 7 to 9 dB in various urban and suburban non-LOS scenarios. In [START_REF] Lotse | Base station polarization diversity reception in macrocellular systems at 1800 MHz[END_REF], experimental data recorded at 1800 MHz in urban and suburban microcells with handheld terminals highlight that

• the uplink XPR levels approximately equal 7 dB in urban areas and vary from 8 to 16 dB in suburban areas for vertical Tx polarization, whereas XPR U h varies from -4 to -1 dB in urban areas and from 2 to -6 dB in suburban areas, • the estimated corresponding CPR levels vary between 5.5 and 4 dB in urban areas and from 3 to 11 dB in suburban areas, • the estimated downlink XPR values are independent from the Tx polarization, and equal approximately 1.5 dB and 5 dB respectively in urban and suburban areas. For ±45 degree reception scheme, it is found [START_REF] Vaughan | Polarization diversity in mobile communications[END_REF] that the XPR is reduced to 0 dB, but at the expense of a larger receive power correlation (≈ 0.44).

Other experiments have dealt with handheld terminals [START_REF] Sorensen | Performance of two-branch polarisation antenna diversity in an operational GSM network[END_REF] and have yielded XPR values as low as 2 dB, with mean correlation levels below 0.1 for a VH reception scheme, and around 0.15 for a slant reception scheme. The low measured XPR is explained in the paper by interaction between the antenna and the human head.

Let us now consider dual-polarized (MIMO) results. In [START_REF] Neubauer | Simultaneous characterization of polarization matrix components in pico cells[END_REF], non-LOS picocell scenarios have been investigated at 1900 MHz. However, the use of directional antenna patterns has resulted in different results depending on the array orientation (the orthogonal arrays were placed along a vertical or a horizontal baseline). For the former case, we have that

XPR U v ≈ XPR D h >> XPR D v ≈ XPR U h , (8)
implying that the CPR is close to 0 dB via [START_REF] Kotterman | Cross-correlation values for dual-polarised indoor MIMO links and realistic antenna elements[END_REF], and that, on the average, p vh << p hv . For the horizontal baseline, the situation is as follows:

XPR U v ≈ XPR D v >> XPR U h ≈ XPR D h , (9)
implying a large CPR and p vh ≈ p hv on the average. In [START_REF] Kainulainen | Polarization behavior in different urban radio environments at 5.3 GHz[END_REF], downlink measurement results are presented for a VH-to-VH array scheme at 5.3 GHz in various non-line-ofsight urban environments (corresponding to Rayleigh fading situations). Values of XPR U v range from 7.6 (macrocells) to 8 dB (microcells) while the values of XPR U h range from 2.3 (macrocells) to 6.9 dB (microcells). Meanwhile, corresponding CPR values range from 1.6 (macrocells) to 0.4 dB (microcells). Surprisingly, that translates approximately into downlink XPR values quasi-independent of the Tx polarization, especially in microcells (the agreement in microcells is almost perfect). Estimated values of XPR D v = XPR D h range from 4 to 6 dB in macrocells and equal 7.5 dB in microcells. Transmit and receive cross-polar correlations are almost all equal to 0.3, but no indication is given about the co-and antipolar correlations.

Results in [START_REF] Chizhik | Spatial and polarization characterization of MIMO channels in rural environment[END_REF] concern rural areas at 2.5 GHz. The XPR averaged over XPR D v and XPR D h (in dB scale) varies from 2 to 19 dB, with a median of 8.5 dB.

In [START_REF] Asplund | Propagation characteristics of polarized radio waves in cellular communications[END_REF], macrocell propagation data at 1.9 GHz suggest that the ratio between the mean values of p vv and p hh (averaged over the small scale fading) is close to 0 dB, with a standard deviation of 3 dB. Note that this ratio is different from the average ratio between the instantaneous values of p vv and p hh . As an example, it is observed that the instantaneous CPR can be as significant as ±15 dB, due to spatial fading decorrelation. The downlink XPR varies from 5 to 15 dB, with an average value of 9 dB, and decreases with the pathloss, reflecting higher XPR in LOS than in NLOS scenarios. The co-, anti-, and cross-polar correlations are very low, but one must mention that co-and anti-polar correlations include a spatial decorrelation effect, as antennas are not all co-located.

2) Indoor Scenarios: In [START_REF] Neubauer | Simultaneous characterization of polarization matrix components in pico cells[END_REF], in non-LOS picocells at 1800 MHz, downlink XPR values are about 6.5 dB, irrespective of the Rx polarization, and CPR levels range from -4 to 4 dB. That implies that uplink XPR levels clearly depend on the Tx polarization. In [START_REF] Kyritsi | Correlation analysis based on MIMO channel measurements in an indoor environment[END_REF], XPR values as low as 3 dB are measured in an office environment at 2.4 GHz. In [START_REF] Wallace | Modeling the indoor MIMO wireless channel[END_REF], crosspolar correlations are found to be very small in a typical indoor environment.

In [START_REF] Hämäläinen | Analysis and measurements for indoor polarization MIMO in 5.25 GHz band[END_REF], uplink XPR values at 5.2 GHz vary between 7 and 15.7 dB (for XPR U v ) and 8.6 to 14.4 (for XPR U h ). Furthermore, the difference XPR U v -XPR U h is not necessarily positive, but may vary from -3 to +3 dB.

In [START_REF] Kotterman | Cross-correlation values for dual-polarised indoor MIMO links and realistic antenna elements[END_REF], [START_REF] Kotterman | Power ratios and distributions in indoor NLOS channels for dual-polarized 2 x 2 MIMO systems[END_REF], measurements have been carried out at 5.1 GHz using a VH scheme at both link ends. The CPR varies from 2 to 6 dB, and the vertical XPR levels are higher than the horizontal XPR values, although the difference

[XPR U v -XPR U h ],
which can be as high as 8 dB, is sensibly higher than [XPR D v -XPR D h ], which varies between 1 and 4 dB. All correlation coefficients are on average below 0.6, but show a large standard deviation. Finally, indoor results of [START_REF] Asplund | Propagation characteristics of polarized radio waves in cellular communications[END_REF] are very similar to their outdoor results reported above, but with average downlink XPR around 7 dB.

3) Summary: Table I, which summarizes the most complete results, reveals that there is a clear lack of data for dualpolarized channels. Hence, it is somehow not straightforward to infer general conclusions from the above analysis. Yet, some results ( [START_REF] Kotterman | Cross-correlation values for dual-polarised indoor MIMO links and realistic antenna elements[END_REF], [START_REF] Kotterman | Power ratios and distributions in indoor NLOS channels for dual-polarized 2 x 2 MIMO systems[END_REF], [START_REF] Lempainen | The performance of polarization diversity schemes at a base station in small/micro cells at 1800 MHz[END_REF], [START_REF] Lotse | Base station polarization diversity reception in macrocellular systems at 1800 MHz[END_REF], [START_REF] Kainulainen | Polarization behavior in different urban radio environments at 5.3 GHz[END_REF], [START_REF] Kyritsi | Correlation analysis based on MIMO channel measurements in an indoor environment[END_REF], [START_REF] Wallace | Modeling the indoor MIMO wireless channel[END_REF] and [START_REF] Neubauer | Simultaneous characterization of polarization matrix components in pico cells[END_REF]-indoor) share, at least approximately, the following properties:

• the average CPR usually varies between 0 and 6 dB in outdoor environments (with maximum values of 11 dB in suburban areas), and between -4 and 6 dB indoors; this non-zero value can be easily explained by the polarization selectivity of the reflection and diffraction processes [START_REF] Oestges | Propagation modeling of multipolarized MIMO fixed wireless channels[END_REF], [START_REF] Kyritsi | Correlation analysis based on MIMO channel measurements in an indoor environment[END_REF],

• there is a clear difference between the vertical and horizontal uplink XPR levels, which are given according to [START_REF] Kotterman | Cross-correlation values for dual-polarised indoor MIMO links and realistic antenna elements[END_REF] once the downlink XPR and CPR are known, • from the data, the downlink XPR levels XPR D v and XPR D h are often equal (or at least close), with measured values of 4 to 8 dB in NLOS outdoor cases, up to 15 to 19 dB in LOS urban and rural areas; 3 to 8 dB in NLOS indoor cases, and up to 15 dB in LOS indoor scenarios (note that when the downlink XPR values have not been measured, we have estimated them based on ( 7)), • the cross-polar correlations (XPC) are usually small, • the co-and anti-polar correlations (CPC/APC) seem to be small (but the results are really scarce).

C. Output from Theoretical Models

Electromagnetic models yield some precious insight into dual-polarized propagation. As an example, a ray-tracing tool [START_REF] Oestges | Deterministic channel modeling and performance simulation of microcellular wideband communication systems[END_REF] has been applied in a macrocellular urban environment at 2.4 GHz, with the receiver moving along a street perpendicular to the link axis. The ray-tracing tool accounts for reflections up to the third order, as well as wedge and corner diffraction. It is found that the narrowband channel for a 2 × 2 dual-polarized VH-to-VH transmission has the following properties:

• the CPR is lognormally distributed with an average value of 4 to 7 dB, • the downlink XPRs are identical for both v and h polarizations, being lognormally distributed with an average value of 12 dB, • the transmit and receive cross-polar correlations are small, • the co-polar and anti-polar correlations vary between 0.8 and 0.9.

Naturally, ray-tracing cannot account for smaller and nonspecular obstacles, which are expected to provide lower CPR and XPR levels. Other papers have reached similar conclusions, using rough scattering models [START_REF] Lee | Modeling the indoor channel for the MIMO system using dual polarization antennas[END_REF]. Note that the first three properties are in line with the experimental campaigns, while the last one is not (but there are only a few available data to compare with).

III. CHANNEL MODELS FOR DUAL-POLARIZED SYSTEMS

A. Dual-Polarized TITO Rayleigh Fading Channels

In Rayleigh fading, it has been observed [25, p. 109] that the spatial correlation properties (modeled by H and R ) are independent of the polarization, especially in macro-and microcells, provided that the direction spectra are similar for all antennas. This namely implies that, if antennas are directional, they should be similarly oriented. In other words, the CPR (resp. XPR) measured on H× should be equal to the CPR (resp. XPR) measured on G. If this assumption holds true, the dual-polarized Rayleigh channel matrix with spatially-separated antennas may be rewritten as

H× = H X. (10)
In [START_REF] Shiu | Fading correlation and its effect on the capacity of multielement antenna systems[END_REF], H is modeled as a uni-polarized correlated Rayleigh channel, while X models both the correlation and power imbalance impacts of the channel depolarization. It is important to note that it would be incorrect to write H× = H G. Indeed, this expression contains twice a fading term, in both H and G. Hence, we have introduced a new matrix X, whose goal is to model only the power imbalance and the phase-shifts between the four channels. The relationship between X and G is however very simple, as G = g X, where g is a scalar complex Gaussian term representing fading.

In [START_REF] Shiu | Fading correlation and its effect on the capacity of multielement antenna systems[END_REF], multipath fading is entirely modeled by H, using the classical expression [START_REF] Paulraj | Introduction to Space-Time Wireless Communications[END_REF]

vec( HH ) = R 1/2 vec( HH w ), (11)
where Hw is the classical i.i.d. complex Gaussian matrix and

R = E{vec( HH )vec( HH ) H } is a 4 × 4 spatial correlation matrix, R = ⎡ ⎢ ⎢ ⎣ 1 t * r * s * 1 t 1 s * 2 r * r s 2 1 t * s 1 r t 1 ⎤ ⎥ ⎥ ⎦ (12)
with t and r being the transmit and receive antenna correlations, and

s 1 = E H(1, 1) H * (2, 2) and s 2 = E H(1, 2) H * (2, 1
) being the cross-channel correlations [START_REF] Oestges | Beneficial impact of channel correlations on MIMO capacity[END_REF].

To model X for VH-to-VH downlink transmissions, we resort to the results of Section II. Hence, a fairly general model accounting for the generally observed trends summarized above and satisfying ( 7) is given by

vec XH = ⎡ ⎢ ⎢ ⎣ 1 √ μχϑ * √ χσ * √ μδ * 1 √ μχϑ μχ √ μχδ * 2 μ √ χσ * √ χσ √ μχδ 2 χ √ μχϑ * √ μδ 1 μ √ χσ √ μχϑ μ ⎤ ⎥ ⎥ ⎦ 1/2 vec XH w , (13)
where

• μ and χ represent the inverse of, respectively, the co-polar and the downlink cross-polar ratios (CPR and XPR D v = XPR D h ) and are assumed to be constant in this paper, • σ and ϑ are the receive and transmit correlation coefficients for co-located antennas (i.e. the correlation

Scenario

Freq. [START_REF] Vaughan | Polarization diversity in mobile communications[END_REF] 0.46 12 ---≈ 0 -Urban [START_REF] Vaughan | Polarization diversity in mobile communications[END_REF] 0.46 7 ---≈ 0 -Macrocell [START_REF] Kozono | Base station polarization diversity reception for mobile radio[END_REF] 0.9 6 ---0.1 -Microcell [START_REF] Kozono | Base station polarization diversity reception for mobile radio[END_REF] 0.9 7.4 ---0.1 -Suburban/urban NLOS [START_REF] Lempainen | The performance of polarization diversity schemes at a base station in small/micro cells at 1800 MHz[END_REF] 1.8 7 to 9 ---≤ 0.3 -Urban microcell [START_REF] Lotse | Base station polarization diversity reception in macrocellular systems at 1800 MHz[END_REF] 1.8 7 -4 to -1 4 to 5.5 1.5 --Suburban microcell [START_REF] Lotse | Base station polarization diversity reception in macrocellular systems at 1800 MHz[END_REF] 1.8 8 to 16 -6 to -2 3 to 11 5 --Urban NLOS microcell [START_REF] Kainulainen | Polarization behavior in different urban radio environments at 5.3 GHz[END_REF] 5.3 8 2.3 1.6 7.5 ≈ 0.3 -Urban NLOS macrocell [START_REF] Kainulainen | Polarization behavior in different urban radio environments at 5.3 GHz[END_REF] 5.3 7.6 6.9 0.4 4 to 6 ≈ 0.3 -Urban macrocell [START_REF] Asplund | Propagation characteristics of polarized radio waves in cellular communications[END_REF] 1.9 9 9 0 9 ≈ 0 ≈ 0 Indoor [START_REF] Asplund | Propagation characteristics of polarized radio waves in cellular communications[END_REF] 1.9 7 7 0 7 ≈ 0 ≈ 0 Indoor NLOS picocell [START_REF] Neubauer | Simultaneous characterization of polarization matrix components in pico cells[END_REF] 1.8 2.5 to 10.5 --4 to 4 6.5 --Indoor picocell [START_REF] Hämäläinen | Analysis and measurements for indoor polarization MIMO in 5.25 GHz band[END_REF] 5.2 7 to 15.7 8.6 to 14.4 ----Indoor corridor/hall [START_REF] Kotterman | Cross-correlation values for dual-polarised indoor MIMO links and realistic antenna elements[END_REF], [START_REF] Kotterman | Power ratios and distributions in indoor NLOS channels for dual-polarized 2 x 2 MIMO systems[END_REF] 5.1 7 to 9.5 1.5 to 6.7 2.5 to 6 8 to 11 (v) ≤ 0.6 ≤ 0.6 2 to 6 (h) coefficients between vv and hv, hh and hv, vv and vh or hh and vh),

[GHz] XPR U v XPR U h CPR XPR D XPC CPC/APC Suburban
• δ 1 and δ 2 are the co-and anti-polar correlation coefficients, • Xw is a 2×2 matrix whose four elements are independent circularly symmetric complex exponentials of unit amplitude, e jφ k , k = 1, . . . , 4, the angles φ k being uniformly distributed over [0, 2π).

Note that this model does not make any simplification on the polarization-induced correlations, but only assumes that the downlink XPD levels are independent from the transmit polarization. Values of μ and χ naturally depend on the considered environment, as outlined in Section II.

In the performance analysis, we consider (only for simplicity) that the cross-polar correlation coefficients σ and ϑ are equal to zero, as often observed experimentally. As far as δ 1 and δ 2 are concerned, our overview does not allow us to assign definite values. Hence, we will consider only two cases in this paper, which seem both reasonable assumptions from the literature overview:

|δ 1 | = 1, |δ 2 | = 1,
as found by the ray-tracing simulations [START_REF] Oestges | Deterministic channel modeling and performance simulation of microcellular wideband communication systems[END_REF], or δ 1 = δ 2 = 0, as suggested by a few results as well as some recent studies [START_REF] Coldrey | Modeling and capacity of polarized MIMO channels[END_REF] (note that the system performance is usually not very sensitive to correlation for correlations lower than 0.6).

Since X is a random matrix, the two eigenvalues of X XH , denoted as η 1 and η 2 are also random, and can be explicitly written as a function of the deterministic parameters μ, χ, δ 1 and δ 2 on the one hand, and of the random phase shifts φ 1 to φ 4 on the other hand. As an example, if μ = 1, these eigenvalues read as

η 1,2 = A ± A 2 + B, ( 14
)
where

A = 1 + χ + χ|δ 2 | 1 -|δ 2 | 2 cos φ 2 -φ 3 + arg{δ 2 } + |δ 1 | 1 -|δ 1 | 2 cos φ 1 -φ 4 + arg{δ 1 } , ( 15
)
and

B = 2χ|δ 2 ||δ 1 | cos 2φ 1 -2φ 2 + arg{δ 1 } -arg{δ 2 } + χ 2 |δ 1 | 1 -|δ 2 | 2 cos 2φ 1 -φ 2 -φ 3 + arg{δ 1 } + 2χ|δ 2 | 1 -|δ 1 | 2 cos φ 1 -2φ 2 + φ 4 -arg{δ 2 } + 2χ 2 1 -|δ 2 | 2 1 -|δ 1 | 2 cos φ 1 -φ 2 -φ 3 + φ 4 -2|δ 1 | 1 -|δ 1 | 2 cos φ 1 -φ 4 + arg{δ 1 } -1 -χ 2 -2χ 2 |δ 2 | 1 -|δ 2 | cos φ 2 -φ 3 + arg{δ 2 } . ( 16
)
Note that if δ 1 = δ 2 = 0, the eigenvalues further simplify into

η 1,2 = 1 + χ ± 2χ(1 + cos ψ), ( 17
)
where ψ = φ 1φ 2φ 3 + φ 4 is defined as a random angle uniformly distributed over [0, 2π). Interestingly, the model of ( 10) differs from the model used in [START_REF] Lozano | Multiple-antenna capacity in the low-power regime[END_REF], where H× is decomposed by extracting the impact of depolarization on the channel gains, yielding

H× = Ĥ X , ( 18
)
where X depends on the polarization scheme. What is important to notice is that Ĥ still includes two correlation mechanisms (space and polarization). Hence, it is generally not equal to an equivalent uni-polarized transmission matrix H (i.e. with the same antenna spacings, all polarizations being then identical). As a result, Ĥ is some hybrid matrix, modeling the correlation aspects of both spacing and polarization. In that sense, it is impossible to use this model to compare uniand dual-polarized schemes. By contrast, our model operates the decomposition into two matrices based on the physical mechanisms (space versus polarization) rather than on their impact (gain versus correlation).

B. Dual-Polarized TITO Ricean Fading Channels

Defining the Ricean K-factor on the vv reference transmission as K, and assuming the LOS is the only coherent contribution (typically, in mobile scenarios), the global Ricean fading channel matrix, also accounting for antenna XPIs, is given by

H ×,a = M r H× M t H×,a + 1 1 + K M r H× M t H×,a (19)
where H× reads, for a system having the same VH polarization scheme at both Tx and Rx, as

H× X = K 1 + K e jςvv 0 0 e jς hh , (20)
with ς vv and ς hh related to the baseline array orientation (e.g. ς vv = ς hh = 0 for broadside arrays at both ends, so X = K 1+K I 2 in such cases). Note that [START_REF] Chizhik | Spatial and polarization characterization of MIMO channels in rural environment[END_REF] does not result in equal K-factors for all channels, as the powers of the individual Rayleigh channels are not equal via [START_REF] Kozono | Base station polarization diversity reception for mobile radio[END_REF].

C. Dual-Polarized MIMO Fading Channels

As already mentioned, we analyze particular n r × n t schemes (for even values of n t and n r ) for which the Tx and Rx array are made of n t /2 and n r /2 dual-polarized sub-arrays (each sub-array is identical and is made of two co-located antennas with orthogonal polarizations). If we further assume that all sub-arrays are identically oriented, the Rayleigh channel matrix is easily written as

H×,nr×nt = Hnr/2×nt/2 ⊗ X, (21)
where the covariance of Hnr/2×nt/2 is the spatial covariance related to the spacing between the sub-arrays, and X is the 2 × 2 dual-polarized matrix modeled by [START_REF] Kozono | Base station polarization diversity reception for mobile radio[END_REF]. Again, X only models the differential attenuation and the correlated phaseshifts between the four dual-polarized channels. Note that ( 21) is valid if and only if the joint transmit-receive direction spectrum is identical for all sub-arrays (they "see" the same scattering environment). This implies e.g. that the channel as seen by PIFA-like designs [START_REF] Hunukumbure | MIMO channel measurements and analysis with prototype user devices in a 2 GHz urban cell[END_REF] cannot be modeled by [START_REF] Wallace | Modeling the indoor MIMO wireless channel[END_REF]. The Ricean contribution is represented (for a LOS component and broadside linear arrays) by

H×,nr×nt = 1 nr/2×nt/2 ⊗ X. (22)
Eventually, antenna XPIs are included in the above model as follows

H ×,a,nr×nt = 1 nr/2×nt/2 ⊗ M r XM t + 1 1 + K Hnr/2×nt/2 ⊗ M r XM t .(23)
Finally, a few words about power normalization [START_REF] Coldrey | Modeling and capacity of polarized MIMO channels[END_REF]. The above channel models consider that the channel energy on the vv link is one, so that the receive SNR does not depend on the amount of depolarization. If we use a fixed transmit power constraint, we define the so-called transmit SNR on the vv reference link as ρ 0 , so that the receive SNR is equal to

ρ = ρ 0 1 1 + χ a,r K + 1 K + χ + 1 1 1 + χ a,t , (24)
which expresses that if the transmit power is fixed, the receive SNR will decrease for smaller XPI and/or XPR. Naturally, this normalization issue only appears when comparing dualpolarized transmissions with different XPI and/or XPR.

IV. MUTUAL INFORMATION OF DUAL-POLARIZED CHANNELS

A. TITO Channels

For very high K-factors, the channel matrix only depends on the Ricean component. Assuming that the Tx antennas have a perfect cross-polar isolation (χ a,t = 0), we have that

H ×,a ≈ M r H× = 1 √ χ a,r √ χ a,r 1 . ( 25
)
Let us consider two particular cases, i.e. χ a,r = 0 (infinite XPI) and χ a,r = 1 (no isolation). The mutual information with equal power allocation and fixed transmit power is then given by [START_REF] Coldrey | Modeling and capacity of polarized MIMO channels[END_REF] where I × refers to dual-polarized schemes, and I, to unipolarized schemes. It is clear that, for both uni-and dualpolarized schemes, the mutual information increases with antenna XPI, as a small XPI means a constant channel energy but a lower rank for dual-polarized schemes, a and a constant rank (of one) but a lower channel energy for uni-polarized schemes. This also implies that for large XPI (i.e. χ a,r → 0), the mutual information is larger for dual-polarized schemes than for uni-polarized schemes when the SNR is larger than a given threshold [START_REF] Paulraj | Introduction to Space-Time Wireless Communications[END_REF].

I × (χ a,r = 0) = 2 log 2 1 + ρ 0 2 (26)
I ( χ a,r = 0) = log 2 (1 + 2ρ 0 ) (27) I × (χ a,r = 1) = I(χ a,r = 1) = log 2 (1 + ρ 0 ),
We assume now a finite K-factor, and use the model of (13) to highlight the impact of co-and cross-polar ratios on a mutual information upper bound. For convenience, we take χ a,t = χ a,r = 0 and consider two scenarios:

• spatially correlated antennas and

δ 1 = 1, |δ 2 | = 1, H× = K K+1 I 2 H× = 1 √ μχ e j(φ+arg{δ2}) √ χ e jφ √ μ H (29)
where H is given by [START_REF] Chuah | Capacity scaling in MIMO wireless systems under correlated fading[END_REF] and φ is a random angle uniformly distributed over [0, 2π), • well separated (uncorrelated) antennas,

H× = K K+1 I 2 H× = 1 √ μχ √ χ √ μ Hw (30)
which is valid irrespective of δ 1 and δ 2 .

Two alternative scenarios (i.e. δ 1 = δ 2 = 0 with close or well separated antennas) do not need being considered, as they are indeed both covered by the second scenario (all three scenarios cause elements of H× to become fully uncorrelated). For both analyzed scenarios, we resort to an upper bound [START_REF] Oestges | Beneficial impact of channel correlations on MIMO capacity[END_REF] of the ergodic mutual information with equal power allocation Ī, outlined by

Ī ≤ log 2 (κ) = log 2 E det I 2 + ρ 2 H × H H × , ( 31
)
where the receive SNR is given by ρ = ρ 0 (K +1)/(K +χ+1). In the considered cases, the upper bounds become

log 2 (κ) = log 2 1 + ρ 2 2K + (1 + μ)(1 + χ) K + 1 + ρ 2 2 1 (K + 1) 2 K K + 2 [s 1 δ 1 ] √ μ + μ + 1 + μ 1 + |s 1 δ 1 | 2 + χ 2 (1 + |s 2 δ 2 | 2 ) . ( 32
)
We first observe that if K → ∞, [START_REF] Martin | Asymptotic eigenvalue distributions and capacity for MIMO channels under correlated fading[END_REF] approaches the result of (26) in both considered cases. Then we also note that the mutual information is higher in the first scenario (i.e. |δ 1 | = 1, |δ 2 | = 1). Indeed, in the latter, the channel matrix behaves as a diagonal channel (for μ = χ = 1 and K = 0, H × = H× is exactly a 2 × 2 diagonal channel [START_REF] Özcelik | Some remarkable properties of diagonally correlated MIMO channels[END_REF]). Note however that the difference between achieved mutual informations in both scenarios remains limited. Finally, for a fixed transmit SNR ρ 0 , the ergodic mutual information of Rayleigh channels decreases in [START_REF] Martin | Asymptotic eigenvalue distributions and capacity for MIMO channels under correlated fading[END_REF] as μ and χ increase, as illustrated in Figure 1 as far as χ is concerned (taking μ = 0.5 and close antennas with

|δ 1 | = |δ 2 | = 1)
. This is caused by the fixed transmit power constraint (the opposite result is found when keeping the receive SNR constant). We also observe that dual-polarized transmissions only offer a larger mutual information for Ricean or highly correlated Rayleigh fading channels.

B. MIMO Channels

When considering an arbitrary array size, the number of parameters of the spatial correlation matrix rapidly increases. To obtain meaningful results, we therefore make the following simplifications:

• we use the same number of antennas on both sides (n t = n r = n), • we restrict our analysis to uniform linear arrays of lengths L t and L r (respectively for the transmit and receive arrays), with antenna correlations being exponential functions of the spacings (d t and d r ) [START_REF] Martin | Asymptotic eigenvalue distributions and capacity for MIMO channels under correlated fading[END_REF], i.e. they are

proportional to e -dt/Δt at the Tx side, and e -dr/Δr at the Rx side (Δ t and Δ r are characteristic distances proportional to the spatial coherence distance at each side), • we take K = 0, • we finally assume that the spatial correlation matrix R is separable (i.e. the well-known Kronecker model may be used to represent R).

As a consequence of the above assumptions, the Tx and Rx correlation matrices are respectively expressed as

Θ t = ⎡ ⎢ ⎢ ⎢ ⎣ 1 e -dt/Δt . . . e -(n-1)dt/Δt e -dt/Δt 1 . . . e -(n-2)dt/Δt . . . . . . e -(n-1)dt/Δt e -(n-2)dt/Δt . . . 1 ⎤ ⎥ ⎥ ⎥ ⎦ (33)
Θ r = ⎡ ⎢ ⎢ ⎢ ⎣ 1
e -dr/Δr . . . e -(n-1)dr/Δr e -dr/Δr 1 . . . e -(n-2)dr/Δr . . . . . .

e -(n-1)dr/Δr e -(n-2)dr/Δr . . .

1 ⎤ ⎥ ⎥ ⎥ ⎦ , (34)
while the full correlation matrix simply reads as R = Θ r ⊗Θ t . It can also be shown that the determinant of, say, Θ t reads as

detΘ t = 1 -e -2dt/Δt n-1 = 1 -e -2L t (n-1)Δ t n-1 , ( 35
)
where d t = L t /(n -1) is the element spacing for n antennas over a length L t . Since our goal is to compare uni-polarized with dualpolarized systems, we consider two systems: the reference system is made of uni-polarized arrays with n equi-spaced antennas whereas the second system is made of n/2 dualpolarized equi-spaced sub-arrays. The channel matrices therefore read as

• for the first system,

H = Θ 1/2 r Hw Θ 1/2 t , ( 36
)
• for the second system,

H× = Θ 1/2 r H w Θ 1/2 t H ⊗ X, (37)
Note that Θ r , Hw and Θ t are n × n matrices, while Θ r , H w and Θ t are n/2 × n/2. It is also interesting to note that

H× HH × = H H H ⊗ X XH (38)
Finally, we may arbitrarily use a fixed receive or transmit power constraint, as our goal is to compare uni-and dualpolarized schemes with given χ and μ.

1) High SNR Analysis: The mutual information of channel H using an identity transmit covariance is well approximated at high SNR [START_REF] Paulraj | Introduction to Space-Time Wireless Communications[END_REF] by

I ≈ log 2 det ρ n HH H . ( 39
)
For uni-polarized systems, (39) can be developed as follows:

I ≈ log 2 ρ n n det H HH , (40) = n log 2 ρ n + log 2 detΘ r + log 2 detΘ t + log 2 det Hw HH w (41) = n log 2 ρ n + (n -1) log 2 1 -e -2L t (n-1)Δ t + (n -1) log 2 1 -e -2Lr (n-1)Δr + log 2 det Hw HH w . ( 42
)
The ergodic mutual information is then given by

Ī = E{I} = n log 2 ρ n + (n -1) log 2 1 -e -2L t (n-1)Δ t 1 -e -2Lr (n-1)Δr + 1 log 2 n k=1 n-k l=1 1 l -nγ , (43)
where γ ≈ 0.57721566 is Euler's constant. For dual-polarized systems in the high SNR regime, we have:

I × ≈ n log 2 ρ n + n 2 log 2 η 1 η 2 + 2log 2 det H w H w H + 2 n 2 -1 log 2 1 -e -4L t (n-2)Δ t + 2 n 2 -1 log 2 1 -e -4Lr (n-2)Δr , (44)
so that the ergodic mutual information becomes

Ī× = E{I × } = n log 2 ρ n + n 2 E log 2 η 1 η 2 + 2 log 2 n/2 k=1 n/2-k l=1 1 l - n 2 γ + 2 n 2 -1 log 2 1 -e -4L t (n-2)Δ t + 2 n 2 -1 log 2 1 -e -4Lr (n-2)Δr . ( 45
)
We are now able to calculate the normalized difference Δ Ī/n = ( Ī× -Ī)/n assuming that n is large. To simplify the notations, let us define ξ t = nΔ t /L t and ξ r = nΔ r /L r , which can be thought of as normalized antenna densities at Tx and Rx. This yields

Δ Ī/n ≈ 1 + log 2 1 -e -4/ξt 1 -e -2/ξt + log 2 1 -e -4/ξr 1 -e -2/ξr + 1 2 E log 2 η 1 η 2 . ( 46
)
Furthermore, if δ 1 and δ 2 are sufficiently small (typically, lower than 0.25), numerical evaluations show that E log 2 η 1 η 2 ≈ 0, irrespective of χ and μ. As an example, if δ 1 = δ 2 = 0 and μ = 1, the term 0.5 E log 2 η 1 η 2 is always lower2 than 7 • 10 -3 for 0 ≤ χ < 1: it can therefore be neglected with respect to the first term in (46), which is always equal to 1. For low δ 1 and δ 2 , (46) is therefore well approximated by

Δ Ī/n ≈ 1 + log 2 1 -e -4/ξt 1 -e -2/ξt + log 2
1e -4/ξr 1e -2/ξr (47)

2) Arbitrary SNR Analysis: At arbitrary SNR, the asymptotic mutual information of uni-polarized spatially correlated channels is well-known, and can be calculated using the Stieltjes transform [START_REF] Dumont | On the asymptotic performance of MIMO correlated Ricean channels[END_REF]. Alternative methods can also be used (see [START_REF] Martin | Asymptotic eigenvalue distributions and capacity for MIMO channels under correlated fading[END_REF] as an example). We eventually obtain that the asymptotic average mutual information per receive antenna Ī/n is given by

Ī n = 1 n log 2 det(I n +β t Θ r )+ 1 n log 2 det(I n +β r Θ t )- 1 ρ β t β r , (48)
where β t and β r are the solutions of ⎧ ⎨ ⎩

β t = ρ n Tr Λ Θt I n + β r Λ Θt -1 β r = ρ n Tr Λ Θr I n + β t Λ Θr -1 (49)
and Λ Θt and Λ Θr are diagonal matrices containing the eigenvalues of Θ t and Θ r . Both correlation matrices have the form

Θ = ⎛ ⎜ ⎜ ⎜ ⎝ 1 . . . n-1 1 . . . n-2 . . . . . . . . . . . . n-1 n-2 . . . 1 ⎞ ⎟ ⎟ ⎟ ⎠ , ( 50
)
and it is known (see [34, p. 38]) that the eigenvalues of Θ converge uniformly (as n → ∞) to

λ Θ (x) = ∞ k=0 k e jk2πx + ∞ k=1 k e -jk2πx (51) = 1 1 -e j2πx + e -j2πx 1 -e -j2πx
(52)

for x ∈ [0, 1] and = e -d t Δ t
or e -dr Δr . Therefore, the asymptotic ergodic mutual information per antenna is given by lim n→∞ Ī/n

= 1 0 log 2 1 + α t 1 -e -dr Δr e j2πx + e -dr Δr α t e -j2πx
1e -dr Δr e -j2πx dx

+ 1 0 log 2 1 + α r 1 -e -d t Δ t e j2πx + e -d t Δ t α r e -j2πx 1 -e -d t Δ t e -j2πx dx - 1 ρ α t α r , (53)
where α t and α r are solutions of

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ α t = ρ 1 0 1 1-e - d t Δ t e j2πx + e -d t Δ t e -j2πx
1-e

- d t Δ t e -j2πx 1 + αr 1-e - d t Δ t e j2πx
+ αr e -d t

Δ t e -j2πx
1-e

- d t
Δ t e -j2πx dx α r = ρ dx.

(54) For dual-polarized schemes, assume first that the eigenvalues of X XH are fixed. In this case, the n eigenvalues of H× HH × can be expressed as the product of the n/2 eigenvalues of H H H by η 1 and η 2 respectively. Hence, we may decompose the conditional ergodic mutual information per antenna as Ī×

n η1,η2 = 1 2 log 2 1 + ρη 1 λ p λ (λ) dλ + 1 2 log 2 1 + ρη 2 λ p λ (λ) dλ, ( 55
)
where λ designates the eigenvalues of H H H /n and p λ (λ) is the limit probability density of λ when n → ∞. The latter can be quite easily evaluated, e.g. as described in [START_REF] Martin | Asymptotic eigenvalue distributions and capacity for MIMO channels under correlated fading[END_REF]. When η 1 and η 2 are random, the quantities ρη 1 and ρη 2 can be thought of as randomly varying effective SNRs. The randomness is represented by the four phase-shifts φ k , k = 1, . . . , 4, which are uniformly distributed over [0, 2π). The ergodic mutual information per antenna is finally given by

Ī× n = 1 32π 4 2π 0 . . . 2π 0 2 k=1 log 2 1 + ρη k λ p λ (λ) dλ dφ 1 . . . dφ 4 . (56)
Simulation results are illustrated in Figure 2, which assumes that ξ t = ξ r = ξ and that δ 1 = δ 2 = 0. The minimum normalized antenna density ξ min for which Δ Ī/n ≥ 0 is plotted for various values of χ (ranging from 0 to 0.9). The value of ξ min decreases as the SNR increases, and reaches its asymptotic value of 2.27 at high SNR. Note that the approximation of (47) is indeed well verified, in the sense that ξ min is quasi-independent of χ at high SNR. The impact of χ is also pretty intuitive: for small XPR values, uni-polarized schemes remain attractive for larger densities, as the dualpolarized transmissions are heavily penalized by the energy loss, especially at low SNR levels. At low SNR, it is indeed well known that the mutual information is essentially linked to the channel energy [START_REF] Paulraj | Introduction to Space-Time Wireless Communications[END_REF]. Finally, note that the above discussion would yield the same values if one had considered the transmit SNR.

V. AVERAGE PAIRWISE ERROR PROBABILITY IN DUAL-POLARIZED TITO CHANNELS

We are now interested in the average pairwise error probability (PEP) as a function of the SNR ρ, i.e. P (C, E) represents the probability that the receiver decodes the codeword E = [e 0 . . . e T-1 ] instead of codeword C = [c 0 . . . c T-1 ] (T is the duration of the space-time block code). In 2 × 2 dualpolarized Rayleigh and LOS Ricean slow fading channels, the expressions of the PEP [START_REF] Oestges | MIMO Wireless Communications: From Real-World Propagation to Space-Time Code Design[END_REF] are given by

P (C, E) = 1 π π/2 0 r(CR ×,a ) i=1 1 + Ξλ i C R×,a -1 exp -ΞKvec HH ×,a H I nr ⊗ Ë I nrnt + ΞC R×,a -1 vec HH ×,a dβ (57)
where Ξ = ρ/(4(1 + K) sin 2 β), C R×,a = R ×,a I nr ⊗ Ë , R ×,a is the correlation matrix of vec H×,a and Ë = (C -E) (C -E) H . In this paper, we restrict our analysis to two simple schemes:

• the Alamouti O-STBC (T = 2), which extends the principle of transmit/receive diversity, • the Spatial Multiplexing scheme, which consists in sending different data streams over each antenna, thereby increasing the system throughput. In Rayleigh fading channels with vertical and horizontal antennas at both ends, neglecting antenna XPI (χ a,t = χ a,r ≈ 0) and using the same models as those used when estimating the mutual information (see Section IV), we may write the dual-polarized correlation matrix as

R × = ⎡ ⎢ ⎢ ⎣ 1 0 0 s * 1 δ * 1 √ μ 0 μχ s * 2 δ * 2 √ μχ 0 0 s 2 δ 2 √ μχ χ 0 s 1 δ 1 √ μ 0 0 μ ⎤ ⎥ ⎥ ⎦ .
(58)

A. Performance of Orthogonal Space-Time Block Coding

Restricting our analysis to the Alamouti scheme, the codeword matrix is given by

C = 1 √ 2 c 0 -c * 1 c 1 c * 0 ( 59
)
where c 0 and c 1 are two symbols of a given constellation, and we denote the codeword errors as d 0 = (c 0e 0 )/ √ 2 and d 1 = (c 1e 1 )/ √ 2. 1) Rayleigh Fading Channels: In Rayleigh fading channels, the matrix C R (×) is rewritten for O-STBC as

C R (×) = R (×) |d 0 | 2 + |d 1 | 2 (60)
where the notation R (×) means that the relationship is valid for both R in [START_REF] Weichselberger | A stochastic MIMO channel model with joint correlation of both link ends[END_REF] and R × in (58). Therefore, the impact of the propagation channel on the error probability is directly given by the four eigenvalues of the correlation matrix. To minimize the average PEP, it is sufficient to maximize the following Chernoff bound [START_REF] Oestges | MIMO Wireless Communications: From Real-World Propagation to Space-Time Code Design[END_REF],

det I 4 + ρ 4 C R (×) = det I 4 + ρ e R (×)
where we define for better legibility an effective SNR as ρ e = ρ |d 0 | 2 + |d 1 | 2 /4. Dual-polarization is then preferred when

det (I nrnt + ρ e R × ) > det (I nr nt + ρ e R) . (61)
In the high SNR regime, the condition simply becomes

det (R × ) > det (R) . ( 62
)
Note however that the SNR level for which one reaches the so-called high SNR regime depends on the eigenvalues of R × . Hence, the high SNR regime assumption might not be realistic when χ and/or μ are small, i.e. when some eigenvalues of R × are small. The above conditions are easily expressed in terms of the correlation coefficients, as well as μ and χ. As an example, let us consider a Kronecker-structured channel uncorrelated at the receiver (hence, r = s 1 = s 2 = 0). When μ = 1 (i.e. when co-polarized waves are equally attenuated), the above conditions are respectively rewritten as

(1 + ρ e ) 2 (1 + χρ e ) 2 > (1 + ρ e ) 4 + ρ 2 e |t| 2 ρ 2 e |t| 2 -2(1 + ρ e ) 2 (63)
and

|t| > 1 -χ 2 . ( 64
)
A similar condition on |r| would be found for Kroneckerstructured channels uncorrelated at the transmitter. This indicates that dual-polarization can increase the performance at high spatial transmit/receive correlation levels, despite the reduction of the average energy of the channel caused by χ. Naturally, the higher χ, the lower the transmit/receive correlation for which dual-polarized schemes outperform unipolarized schemes. Note also that transmit and receive correlations equally affect the performance of diversity schemes. It might also seem sensible to use the effective diversity order N div as a natural metric to decide whether dual-polarized systems should be preferred to uni-polarized systems [START_REF] Nabar | Performance analysis and transmit optimization for general MIMO channels[END_REF]. Defining

N div = Tr R R F 2 , ( 65
)
the condition under which dual-polarized transmissions (with same polarization scheme at both Tx and Rx) perform better than uni-polarized transmissions is given

• if δ 1 = 1 and |δ 2 | = 1 by (1 + μ + μχ + χ) 2 1 + μ 2 + μ 2 χ 2 + χ 2 + 2μ(|s 1 | 2 + χ 2 |s 2 | 2 ) > 4 1 + |t| 2 + |r| 2 + |s1| 2 +|s2| 2 2
, (66)

• if δ 1 = δ 2 = 0 by (1 + μ + μχ + χ) 2 1 + μ 2 + μ 2 χ 2 + χ 2 > 4 1 + |t| 2 + |r| 2 + |s1| 2 +|s2| 2 2 .
(67) If we consider the same example as above (r = s 1 = s 2 = 0 and μ = 1), (66) and (67) are both rewritten as

|t| > 1 -χ 1 + χ . ( 68
)
This condition is actually looser than (64) with respect to favoring dual-polarized transmissions, as it only implies that the number of degrees of freedom offered by dual-polarized channels be higher than that offered by uni-polarized channels.

It does not imply that the error probabililty is lower. When

1-χ 1+χ < |t| ≤ 1 -χ 2
, dual-polarized schemes exhibit a larger diversity order, but are still penalized by the energy loss since Tr{R × } < Tr{R}.

2) Ricean Fading Channels: We analyze this case assuming that the Ricean component is the LOS, that the K-factor is high (so that the Rayleigh component can be neglected) and that the transmit antenna XPI is large enough to be neglected. The performance solely depends on the Frobenius norm of the coherent component. Naturally, this norm is always smaller for dual-polarized schemes (since χ a,r ≤ 1). The same conclusion is reached if we consider the effective diversity order (neglecting the Rayleigh component): N div = 2K for uni-polarized transmissions, while N div = (2 + χ a,r )K/2 for dual-polarized systems. Therefore, the use of dual-polarized transmissions is not recommended for diversity schemes in Ricean channels.

B. Performance of Spatial Multiplexing

For a simple uncoded spatial multiplexing scheme, we have (76)

(C -E) (C -E) H = |d 2 0 | d 0 d * 1 d * 0 d 1 |d 1 | 2 (69
An estimate of the average symbol error rate may be easily obtained by weighting the average PEP over the different possible symbols, using a union bound approximation. Various simulation results are illustrated in Figure 3 for a QPSK modulation, as a function of t (assuming that r = s 1 = s 2 = 0) or as a function of r (assuming that t = s 1 = s 2 = 0), for a constant receive SNR ρ = 15 dB. Note that the comparison between various dual-polarized channels should be made with care, as a fixed transmit power constraint would vary the receive SNR. Yet, the comparison between each uni-polarized channel and each dual-polarized channel does not depend on the power constraint, as the channel depolarization is constant for each comparison.

Considering first uni-polarized systems (solid lines), we observe that transmit correlation is more harmful than receive correlation, as already noticed in [START_REF] Nabar | Performance of multiantenna signaling techniques in the presence of polarization diversity[END_REF]. As for the use of dual-polarized arrays (dashed lines), it is clear that these are only beneficial with respect to uni-polarized schemes when the transmit and/or receive correlations are higher than a certain level. This level is directly related to the CPR and XPR. The smaller μ and/or χ, the larger this correlation level. Naturally, when both μ and χ are equal to one, dual-polarized systems always perform better than uni-polarized systems, as the decorrelation effect is not hampered by a decrease of power. For μ and/or χ smaller than one, the required correlation increases rapidly. As a consequence, dual-polarized spatial multiplexing schemes perform well in Rayleigh fading channels when the uni-polarized spatial correlations are high enough.

2) Ricean Fading Channels: In LOS Ricean fading channels, the error probability is obtained through (57). Simulation results are displayed in Figure 4 for χ = 0.1, μ = 1, δ 1 = δ 2 = 0 and ρ 0 = 10 dB. The cases of uni-polarized transmissions approximately correspond here to χ a,r = 1 (the higher the K-factor, the better the approximation). Clearly, the use of dual-polarized antennas with large XPI significantly increases the performance, even when considering a constant transmit power.

VI. CONCLUSIONS

In this paper, we have presented a simple analytical model of dual-polarized TITO and MIMO transmissions. The channel model efficiently separates the spatial and polarization effects, which makes it particularly easy to parametrize, thanks to a thorough overview of the literature, whose conclusions are summarized at the end of Section II.B.

As far as the benefits from dual-polarized arrays are concerned, the derivations point out that large multiplexing (i.e. capacity) gains are achievable in Ricean or highly correlated Rayleigh fading channels, but that diversity gains are only possible in highly correlated Rayleigh fading channels (and not in Ricean channels). Furthermore, our derivations have highlighted that there is a fundamental trade-off between the array density and the capacity gain, i.e. for a given array geometrical extension, there is a critical antenna density above which dual-polarized arrays should be preferred. Finally, our analysis identifies quantitatively for which channels (correlations, K-factor), schemes (O-SBTC or Spatial Multiplexing) and SNR levels the use of multiple polarizations might become beneficial.

Fig. 1 .

 1 Fig. 1. Mutual information of uni-and dual-polarized 2 × 2 channels for different K-factors, correlations and cross-polar ratios

Fig. 2 .

 2 Fig.2. Normalized antenna density ξ min above which dual-polarization should be favored as a function of the SNR (χ -1 is the channel XPR, χ ∈ {0, 0.1, 0.3, 0.5, 0.7, 0.9} and δ 1 = δ 2 = 0).

) 1 )Fig. 3 .

 13 Fig. 3. Performance of QPSK Spatial Multiplexing for uni-(solid lines) and dual-polarized (dashed lines) transmissions as a function of t or r.

Fig. 4 .

 4 Fig.[START_REF] Oestges | Propagation modeling of multipolarized MIMO fixed wireless channels[END_REF].Performance of QPSK Spatial Multiplexing in various Ricean channels as a function of Rx antenna XPI for a transmit SNR ρ 0 = 10 dB.

TABLE I OVERVIEW

 I OF EXPERIMENTAL RESULTS (XPR AND CPR VALUES ARE EXPRESSED IN DECIBELS)

Many denominations also exist: some papers define XPD/XPR values as horizontal-to-vertical ratios, etc. In all cases, we translated the denominations used in these papers into our own denominations.

Using Jensen's inequality to obtain an analytical upper-bound is not adequate here, as the upper-bound is very loose.

The associate editor coordinating the review of this paper and approving it for publication was Y. Gong. This work was supported in part by the Belgian Fonds de la Recherche Scientifique, France Telecom R&D, the EU Network of Excellence NEWCOM and the I0 strategic project of ftw. ftw. is supported by the Austrian Government and by the City of Vienna within the competence center program COMET.

C. Oestges is with the Microwave Laboratory,