Analysis of decision feedback detection for MIMO Rayleigh-fading channels and the optimization of power and rate allocations

For an uncoded, K-transmit, N-receive antenna coherent narrow-band communication system employing a decorrelating decision feedback detector (D-DFD), the exact average (over channel realizations) joint error probability (JEP) as well as the average per-symbol error probabilities (SEPs) are derived without making any simplifying assumptions on error propagation. It is proved that the diversity orders of the JEP and the SEP (of every symbol) is limited by error propagation to N-K+1. Based on our exact error probability analysis, however, we suggest an optimization of JEP over nonnegative quadrature amplitude modulation (QAM) constellation sizes (rates) and average powers across transmitters which yield significant improvements over the usual equal power and equal rate assignment. In fact, the JEP of such an optimized design has the much improved diversity order of N (which is also the diversity order obtained through the optimum maximum-likelihood (ML) detector). Moreover, it is seen that these simple optimized designs can achieve a significant fraction of the ε-outage capacity even without outer codes. It is also known-but only through simulations-that when the symbols are detected in certain channel realization-dependent orders it is possible to improve substantially over fixed-order detection in the case of the equal rate and equal power assignment. We provide an analysis for a recently proposed channel-dependent ordering rule and show that it does not provide an improvement of the diversity order of the JEP beyond N-K+1. Another ordering rule that was proposed earlier to maximize the worst case post-detection signal-to-noise ratio (SNR) under the perfect feedback assumption is shown to be optimal under a more compelling criterion that does not involve that simplifying assumption. While efficiently computable, this ordering rule is seen to perform almost as well as the optimal channel-dependent ordering rule that minimizes the conditional JEP (and hence the JEP). Nevertheless, a multiple-input multiple-output (MIMO) system with an optimized rate and power allocation and a fixed order of detection is not only less complex but also has a significantly lower JEP than that of the equal-power, equal-rate system, where transmitters are detected in a chann- el-dependent order, optimal or otherwise.