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
. 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 . 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.
We consider a MIMO system with transmit and nR receive antennas. The binary data is encoded by a space-time encoder. denotes the transmitted signal vector in the -th time interval. Let denote the channel matrix at time , where each entry is the channel gain between transmit antenna and receive antenna . The channel gains are assumed to be circularly symmetric complex Gaussian random variables, so their magnitude exhibits a Rayleigh distribution. Let denote the error vector at time , and be the codeword error, where 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 , the receive side correlation matrix with , and the temporal correlation matrix with . The spatially correlated channel can be modeled using the concept of innovations as
We now proceed to find PEP expressions. Using the methods originated by Craig, and widely applied by Simon and Alouini , we first calculate the conditional PEP and then integrate over channel gain coefficients. We use the alternative integral formula for Q-function
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  and  for details). For spatially correlated quasi-static Rayleigh fading channel:
where and are the eigenvalues of and respectively. For spatially correlated fast-fading channel:
For spatially and temporally correlated Rician fading channel:
where , and .
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.
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.
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 , and coded space-time block codes in  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.
This research was made possible in part by the National Science Foundation through grant CCR-9985171.
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