Diversity–Multiplexing Tradeoff and Outage Performance for 2×2 Dual-Polarized Uncorrelated Rice MIMO Channels *

,


Introduction
Due to the space-cost of the conventional Single-polarized (SP) multiple-input multiple-output (MIMO) systems, dual-polarized (DP) MIMO has been receiving much attention as an attractive alternative for realizing MIMO architectures in compact devices [1][2][3][4][5][6][7].Compared with SP MIMO, DP MIMO exhibits many different characteristics.For instance , in [1] it has been clearly illustrated that in Rice fading, after some k-factor (defined as the ratio of the power in the fixed exponent to the power in the variable exponent), error probability of zero-forcing detection method for polarization multiplexing starts to decrease with increasing k-factor, while SP systems perform the opposite.Moreover, various literatures, such as in [1][2][3][4], an idea has been well developed that polarization diversity works well only in correlated Rayleigh fading or Rice fading channels with LOS components.It is necessary to note that measurements have been done to get real parameters of DP channels, which helps in getting more accurate polarized channel model [6].To go further, channel correlation and capac-ity are discussed in these literatures, proving that such dual polarization has de-correlation effect on correlated channels from a practical aspect.Nevertheless, this result does not extend to diversity systems, such as Almouti coded MIMO, where polarization confronts performance loss [2].In conventional MIMO systems, it is known that there exists a fundamental tradeoff between achievable diversity and multiplexing gains of any transmission over t r n n  MIMO channel, i.e., diversity-multiplexing tradeoff (DMT), as has been clearly illustrated in [8] for signal to noise ratio (SNR) approaching infinity.Moreover, it is also pointed out that DMT curve at finite SNR is quite different [9][10][11][12][13].Under realistic propagation conditions, since SNR cannot reach infinity, it would be meaningful to study DMT behavior at finite SNRs that are practical in operating regimes.Up to now there are no literatures that investigate finite-SNR DMT for dual-polarized systems.
In previous literatures [9][10][11][12][13], DMT curve is discussed based on the assumption that, elements of H HH follow Wishart distribution.However, for polarized MIMO, because of the asymmetric properties of the generalized channel matrix, random matrix theory results for Wishart matrices cannot be leveraged.Inspired by the idea proposed in [14], which used gamma, lognormal or weibull distribution to approximate outage capacity for dualpolarized MIMO in high SNR regime, by approximating mutual information exponent, we get theoretical DMT curve for DP in low and medium SNR regimes in 2×2 uncorrelated Rice channels.
The rest of this paper is organized as follow.Section2 describes the channel model developed for a 2×2 Dual-Polarized uncorrelated Rice fading channels.Section 3 discusses statistic characteristic of mutual information exponent, outage probability and DMT curve and their approximations.Section4 shows the simulation results.Finally, section5 is the conclusion.
In this paper, and represents the expectation and variation of random variable x, respectively, * stands for the element-wise conjugation, H for conjugate transpose,

 
det A is the determinant of matrix A.

System Model and Definitions
Consider a system with one dual-polarized transmit and one dual-polarized receive antenna.The channel is assumed frequency-flat over the band of interest.The channel matrix is given by 11 21 Assume that both transmitter and receiver employ the same polarization scheme, i.e. both of them employ horizontal/vertical or slanted polarization.Decomposing the channel matrix into the sum of a fixed exponent and a variable exponent as The elements of the matrix H do not vary and satisfy 1,

0; 0 E h h E h h E h h
where 0 1 In [8], conventional asymptotic definitions of multiplexing and diversity gains for a MIMO channel are given by: where I is the mutual information between received and transmitted signals over the MIMO channels, and is the mutual information exponent satisfy . The asymptotic DMT is given by the piece-wise linear function connecting the points , where r , t are numbers of receive and transmit antennas, respectively.Note that the asymptotic DMT describes situation where SNR approches infinity.However, for practical system design, it is desirable to characterize the diversity-multiplexing tradeoff at operational SNRs.The finite-SNR definitions for diversity and multiplexing gains can provide useful tool to characterize the DMT at real environment.The finite-SNR multiplexing gain r is defined as the ratio of to the capacity of an AWGN channel at SNR with array gain [11], The finite-SNR outage probability  , The finite SNR diversity gain  , d r   is defined by the negative slope of the plot

Computation of DP finite-SNR DMT
In this section, the DMT for 2×2 Dual-polarized Rice channels is examined.First, we derive an exact expression for the mean and variation of mutual information exponent, based on which some discussions on channel parameters are proposed to have a deeper insight into dual-polarization system.Second, using the expressions of statistic information derived in the first step, approximation equations of outage probability are presented.Finally, DMT for both asymptotic and finite-SNR in 2× 2 Dual-polarized Rice channels at are investigated.

Statistic Information of Mutual Information Exponent
Consider that channel state information (CSI) is perfectly known at the receiver.The MIMO mutual information I conditioned on the channel realization is given by where w h e r e , a n d for the sake of space saving.
The distribution of the mutual information exponent provides information about the available diversity in the system.
describes the ergodic mutual information exponent, which can be used to get upper bound of mutual information I.And presents some information about outage probability, i.e., the smaller the variance, the lower the probability of the outage error is when transmitting at a fixed rate [8].
From the analytical expression of and 14)-( 15 Then k-factor for where In suburban area, where XPD is measured to be 8-15dB range, let =0.4,0.
. We find that in SP, mean of mutual information exponent decrease fast with the increase in k-factor, while for DP, declension is less.At 0dB

 
, required k-factor to fill the gap between DP and SP is k= -0.5754 or -2.4376; 10dB

Approximating of Outage Probability
Motivated by the work [14], in this section, we derive the approximation curve for outage probability at finite SNR for 2×2 dual-polarized uncorrelated Rice channels.
The steps begin with the approximation of statistical information of mutual information exponent W.
If we assume gamma distribution for W, i.e.

   
Then outage probablity at given multiplexing gain and SNR is where is the incomplete gamma function.

 
Then where both and are given in section2.Note that format parameters of W are directly related to polarization parameters as well as k-factor, SNR.Corollary: When , for outage probability of DP, In contrast, outage probability in SP is given by [12] Proof: As , for conventional SP, the Rice fading channel approaches a rank-one AWGN channel, such that the outage probability is 1 for , and 0 for ; However, for DP Rice fading channels, as , thanks to polarization orthogonality, channel matrix remains full rank.Thus, as k increases, channel approach two rank-one AWGN channels.Therefore as , outage probability for both

Asymptotic DMT for Rice Dual Polarized Channels
Theorem: The asymptotic DMT curve for dual-polarized channels is independent of  , f  , which is identical to conventional asymptotic DMT in SP channels as described in (9) [8].
Proof: The proof is given in appendix 2.

Diversity and Multiplexing Trade-off at Finite SNR
Simulated by the method in [11], we get finite-SNR DMT using (11).
where for gamma approximation, For lognormal approximation, calculation step is similiar, which is omitted for the sake of space.

Impact of k-factor and SNR on Mutual Information Exponent
As it has been known that DP and SP systems perform rather diffident in Rice channels.In order to study how such a difference occurs, be explained by the effect of eigenvalue of H HH .At medium SNR, minimum eigenvalue begins to affect channel information exponent.For conventional SP 2×2 Rice systems, when k-factor increases, channel matrix tends to be a rank-deficient matrix, leading the minimum eigenvalue to be smaller even approaching zero.In contrast, eigenvalues of DP systems nearly stays constant, without being hugely affected by varying k-factor.Hence, channel matrix does not become ill conditioned, i.e., not badly affected by LOS component.Thus, in seminars with strong LOS component, we suggested DP be used.To illustrate the different eigenvalues we can see Figure 2.

Impact of XPD on Mutual Information Exponent
As a final parameter dependency study, we examine on mutual information exponent as a function of the XPD in LOS component.Using the analytical formation in sec-tion2, Figure 3 plots plots and as a function of the   From Figure 3, it is clear that at low SNRs increases with EW f  .However, at moderate SNR, starts to drop with improving EW f  .Conclusions can be made that XPD and SNR have impacts on the mutual information at the same time.It is then meaningful to find the optimal SNRs for different DP systems for optimal code design.

Outage Probability in Finite SNR
In this part, we study some plots of outage probability versus SNR in uncorrelated Rice fading with In Figure 4, given a fixed multiplexing gain 1 r  , outage probability versus SNR curves are plotted for SP and DP at 5,12 k  . It is seen that contrary to SP, outage probability of DP always drops as k-factor improves.At some SNR, negative gap of outage probability between SP and DP turns into positive, coinciding with previous analysis.
In Figure 5, gamma or lognormal approximation are  plotted as well as results of Monte Carlo simulation for outage probability in various multiplexing gain.The dash curves represent the approximation value, while the circle, square symbol represent the DP systems for respectively.When ( in the plot), the gamma approximation matches the simulation well.At ( ), we can use the lognormal distribution instead, which works well in medium SNR 0-15dB.Note that the higher multiplexing gain, the more accuracy of the gamma approximation.By using this approximation method, it becomes simple to estimate the outage probability of DP Rice channels at or medium SNRs without time-cost simulation.Obviously, the approximation curve agrees with the Monte Carlo simulation.For , as it has been indicated in [11], the diversity gain in SP Rice fading channels approaches zero rapidly since the rank-one LOS matrix limits the effective degrees of freedom in the channels.However, for DP Rice fading, a relatively high diversity gain can still be observed in .At , diversity gain can be as high as one.Explanations can be found from minimum eigenvalue of DP systems, that thanks to the de-correlation effect, minimum eigenvalue of DP do not approach zero despite of the exists of strong LOS component.

k 
For very high k-factor, the channel matrix only depends on the Rice exponent.As , the channels tend to be AWGN, and the capacity increases only with SNR.For DP, asymptotic diversity gain becomes infinite.

Conclusions
In this paper, outage probablity and DMT for asymotic and finite-SNR are studied in 2×2 dual-polarized uncorrelated Rice fading channels.Exact expression mean and varition of mutual information in DP Rice channels are derived, based on which how channel paremeters as k-factors, SNR or XPD influence channel infromation exponent are discussed.Results show that in subran environments where 0.4 is required to fill the gap between erogotic mean of mutual information exponent of SP and DP.Outage probablity as well as asympotic and finite-SNR DMT are compared between of SP and DP.Using the gamma or lognormal distribution, their appromaxition curves for 2×2 dual-polarized uncorrelated Rice channels at 10 k  are given.The result in this paper, helps in finding the inner difference between DP and SP channels.And the appromaxiton approach for DMT in this paper, alough not so accurate in low multiplexing gain, can provide references in pratical code design in dual-polarized Rice systems, expecially in systems with large amouts of antennas.

 
According to the distribution of channel elements (5), an  11 22 12 21

r js r js H h h h h r js
Finally, using (15), exact expression of   D W is given.According to (14), is derived.For , as

Appendix 2
Using the method prosed in [9] we derive the proof for asymptotic DMT curve in dual-polarized uncorrelated Rice channels.The proof begins with Rayleigh fading cases.
According to [9], let . Firstly ,we decompose H as: As is a variable with a higher order than , and that small is mainly due to small , i.e., the main event that causes  Eventually, as LOS component do not affect the high-SNR diversity gain [12], the asymptotic DMT analysis here hold on for Rice channels.Therefore, asymptotic DMT curve for DP Rice channels at infinite SNR is the same as the conventional conclusion(9) in [8].
related to the XPD for the fixed and variable exponent of the channel, respectively.Good discrimination of orthogonal polarizations amounts to small values of  and f  , and vice versa.becomes the conventional SP (single polarization) channel.For 2 × 2 Dual-Polarized uncorrelated Rice MIMO channels ij are complex Gaussian random variables whose parameters are: ), we find that both of them are influenced by k-factor and SNR.With the existences of po-larization indicators f  and  , the influence are dif- information exponent of SP and DP, we get If we assume lognormal distribution for W, i.e.

Figure 1 .
Figure 1.Comparison of mean and variation of W for SP and DP.

Figure 5 .
Figure 5. Outage probability approximation for different multiplexing gains and k-factor DP.

Figure 6
Figure 6 is a plot of diversity and multiplexing gain tradeoff in finite SNR for DP Rice channels, 10 k  and .[0.75, 2] r Obviously, the approximation curve agrees with the Monte Carlo simulation.For , as it has been indicated in[11], the diversity gain in SP Rice fading channels approaches zero rapidly since the rank-one LOS matrix limits the effective degrees of freedom in the channels.However, for DP Rice fading, a relatively high diversity gain can still be observed in .At
-central chi-square distribution.Mean we get and variance of i   can be derived as :   3 r   4， without any relationship with  .