Space-Time Codes in Correlated Channels

Multimedia Communications Laboratory
University of Texas at Dallas



Introduction

Multiple-antenna (MIMO) signaling and coding can provide improvements either in the sense of reliability (e.g., trellis space-time codes, super-orthogonal codes) or in the sense of the capacity of the channel (e.g., linear dispersion codes, BLAST). The object of performance analysis is to derive upper bounds on error probabilities that can replace Monte Carlo simulations and thus are helpful in large-scale simulations of wireless networks. Much of the previous analysis concentrates on idealistic case where there is no correlation between fading coefficients. In reality, insufficient antenna spacing, angle spread, or the lack of rich scattering may cause spatial correlation between antennas.

We present a comprehensive analysis of multiple-antenna signaling in the presence of spatially or temporally correlated fading in [1]. We begin by calculating pairwise error probability (PEP) expressions in a variety of scenarios, including quasi-static fading, fast fading, block-fading, general Rayleigh fading, and Rician fading. We consider spatially correlated fading on the transmit or receive side (or both), as well as combined spatially and temporally correlated fading. We then use the PEP's to develop union bounds for a variety of MIMO signaling scenarios. In particular, we provide bounds for trellis space-time codes, super-orthogonal space-time codes, linear dispersion codes, and diagonal algebraic space-time codes. We also explore the uniform error property (UEP) of codes in correlated channels and show that UEP may be lost if there is transmit-side antenna correlation.

For the special case of concatenated space-time block codes and channel codes, we derive PEP expressions in [2]. Our methodology is to construct a SISO equivalent representation of the channel, resulting in a block fading channel. The analysis is usually dependent on the interleaver used. We use the concept of random (uniform) interleaver to overcome the problem of interleaver-dependent error probabilities.

System Model

We consider a MIMO system with $ nT$ transmit and nR receive antennas. The binary data is encoded by a space-time encoder. $ {\bf x}_n = [x_1^{(n)}, \ldots, x_{n_T}^{(n)} ]^T$ denotes the transmitted signal vector in the $ n$ -th time interval.  Let$ {\bf H}_n = \{h_{ji}^{(n)} \}$ denote the $ n_R \times n_T$ channel matrix at time $ n$ , where each entry $ h_{ji}$ is the channel gain between transmit antenna $ i$ and receive antenna $ j$ . The channel gains are assumed to be circularly symmetric complex Gaussian random variables, so their magnitude exhibits a Rayleigh distribution. Let $ \be_n$ denote the error vector at time $ n$ , and $ \Delta = [\be_1 ,\ldots, \be_N]$ be the codeword error, where $ N$ is the frame length. We assume that the receiver has perfect knowledge of the channel matrix and performs coherent maximum likelihood detection. The entries of channel matrix may be correlated due to several reasons. We denote the transmit side correlation matrix with $ \rtx$ , the receive side correlation matrix with $ \rrx$ , and the temporal correlation matrix with $ \bR_t$ . The spatially correlated channel can be modeled using the concept of innovations as

$\displaystyle \bH = \rrx^{1/2} \tilde \bH \rrx^{1/2} $
We now proceed to find PEP expressions. Using the methods originated by Craig, and widely applied by Simon and Alouini [3], we first calculate the conditional PEP and then integrate over channel gain coefficients. We use the alternative integral formula for Q-function
$\displaystyle Q(x) = \frac{1}{\pi} \int_{0}^{\pi/2} \exp \left ( -\frac{x^2}{2\sin^2 \theta}\right ) \; d\theta $
and subsequently integrate over the randomness of the channel coefficients to obtain the PEP. We provide PEP equations for various cases in the following (please refer to our submitted papers [1] and [2] for details). For spatially correlated quasi-static Rayleigh fading channel:
$\displaystyle P(\bx \rightarrow {\bf\hat{x}}) = \frac{1}{\pi} \!\int_0^{\frac{... ... \left( 1+ \frac{E_s}{4N_0\sin^2\theta}\lambda_j \mu_i\right)^{-1} d\theta \;,$ (1)

where $ \lambda$ and $ \mu$ are the eigenvalues of $ \rrx$ and $ \Delta \Delta^H\rtx$ respectively. For spatially correlated fast-fading channel:
$\displaystyle P(\bx \rightarrow {\bf\hat{x}}) = \frac{1}{\pi} \!\int_0^{\frac{... ... \left( 1+ \frac{E_s}{4N_0\sin^2\theta}\lambda_j \mu_n\right)^{-1} d\theta \;,$ (2)

For spatially and temporally correlated Rician fading channel:
$\displaystyle P(\bx\rightarrow{\bf\hat{x}}) = \frac{1}{\pi} \int_0^{\frac{\pi}{... ..._T}+\frac{E_s}{4(1+K)N_0\sin^2\theta} {\mathbf \Gamma} \right\vert} \; d\theta$ (3)

where $ {\mathbf \Gamma}= \bR^{1/2}(\bI_{n_R} \otimes {\mathbf \Lambda}{\mathbf \Lambda}^H) \bR^{H/2}$ , and $ \bR=\rrx \otimes \bR_t \otimes \rtx$ .

Uniform Error Property under Transmit Correlation

In general, space-time signaling is nonlinear, but holds a property called uniform error property (UEP). UEP codes have symmetries in the code (and its corresponding trellis) that lead to a condition where the error event probabilities as well as their multiplicities are independent of the transmitted codeword. The past analyses have mostly relied on this property. Interestingly, we found that transmit side correlation may destroy the UEP property of a code. For example, consider the eight-state QPSK STT code of, which is ordinarily a UEP code. In Figure 1, we show the error performance over a spatially correlated channel, for the all-zero codeword with numeral (I), the codeword 02, 20, 02, 20... with numeral (II), and a random codeword with numeral (III). Clearly the error probability depends on the transmitted codeword, thus the UEP property has been destroyed.


Figure 1: STT code, 2-Tx, 2-Rx, fast Rayleigh fading, $ \rho_t=\rho_r=0.7$
\begin{figure} \centerline{\epsfxsize=3.1in\epsfbox{sttcFast2RxFER.eps}} \end{figure}


This phenomenon can be explained as follows. When the channel is spatially uncorrelated, the phase of the received signals are random, so the signals from the multiple transmit antennas may add constructively or destructively with equal probabilities. But a positive correlation coefficient means that signals add constructively more often than destructively. For example consider the case of two transmit antennas, one receive antenna, and QPSK modulation, where the modulation symbols are denoted 0, 1 2, 3. In this case transmission vectors that send the same signal from two antennas (e.g. 00) have an advantage over vectors that send opposite values (e.g. 02 or 13). Worst-case codewords are those that are made entirely of these worst-case segments. Best case codewords are those made entirely of the same symbol, e.g., the all-zero codeword. A random codeword will contain a mixture of both, and therefore will fall somewhere in between. To further demonstrate this effect, we study the case where the two antennas are perfectly correlated. Assuming channel knowledge at the receiver, the equivalent receive constellation is as shown in Figure 2. Although the minimum distance for all codewords is the same, some codewords (e.g., 00) have a smaller number of nearest neighbors compared to some others (e.g., 02).This effect is further demonstrated in Figure 3, which depicts the scaled Chernoff bound of PEP's, as well as the corresponding multiplicities. On the left hand side we see that when there is no correlation, the PEP multiplicities are the same for the two codewords 00 and 02. When there is full correlation, the multiplicities of PEP's for the two codewords are different, hence the UEP does not hold.
 
Figure 2: Received constellation with transmit correlation $ \rho_t=1$ .
\begin{figure} \centerline{\epsfxsize=2.9in\epsfbox{constellation.eps}} \end{figure}
Figure 3: PEP versus multiplicities for transmitted codewords $ 00,00$ and $ 02,20$ , and $ \rho_t=0$ , i.e. $ \rtx=\bI$ , and $ \rho_t=1$ , codewords with lengths upto $ 4$ , SNR $ =10$ dB
\begin{figure} \centerline{\epsfxsize=3.5in\epsfbox{histo2.eps}} \end{figure}

Applications

The PEP expression dervied so far can be used to evaluate performance of space-time codes using union bounds. We provide many results on space-time trellis codes, linear dispersion codes, and diagonal algebraic space-time codes in [1], and coded space-time block codes in [2] under variety of channel conditions. As an example we show below the results for 2-state super-orthogonal space-time codes.

Figure 4 shows results for a 2-state super orthogonal space-time code with 2 transmit and 4 receive antennas and BPSK modulation in quasi-static fading. The solid lines represent union bounds, while the dotted lines represent Monte Carlo simulation. We note that there is a loss of 1.2 dB when there is transmit correlation of 0.7, and a further 1.5 dB loss when both the transmit and receive side has correlation of 0.7.  Note that the bounds are tight in high-SNR regime.

Figure 4: SOSTT code, 2-Tx, 4-Rx, quasi-static Rayleigh fading, $ \rho_t=\rho_r=0.7$
\begin{figure} \centerline{\epsfxsize=3.5in\epsfbox{soSlow4RxFER.eps}} \end{figure}

Acknowledgments

This research was made possible in part by the National Science Foundation through grant CCR-9985171.

References

  1. A. Hedayat, H. Shah and A. Nosratinia, "Analysis of Space-Time Coding in Correlated Fading els", submitted to IEEE transactions on wireless comm. Oct. 2003.
  2. H. Shah, A. Hedayat and A. Nosratinia, "Performance of Concatenated Channel Codes and Orthogonal Space-Time Block Codes", submitted to IEEE transactions on wireless comm. Jan. 2004.
  3. M. K. Simon and M. S. Alouini, Digital Communication over Fading Channels: A Unified Approach to Performance Analysis. New York: John Wiley and Sons, 2000.
  4. H. Shah, A. Hedayat, and A. Nosratinia, "Performance of Concatenated Channel Codes and Orthogonal Space-Time Block Codes," Allerton Conference on Communications, Control, and Computing, October 2003.
  5. H. Shah, A. Hedayat, and A. Nosratinia, "Performance of Concatenated Channel Codes and Orthogonal Space-Time Block Codes Under Independent and Correlated Rician Fading," WNCG Wireless Networking Symposium, Austin, TX, October 2003.
  6. H. Shah, Performance Analysis of Space-Time Codes. Masters' Thesis, Nov. 2003

Last modified June 2004

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