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Multiple-input multiple-output (MIMO) channels provide an abstract and unified representation of different physical communication systems, ranging from multi-antenna wireless channels to wireless digital subscriber line systems. They have the key property that several data streams can be simultaneously established. In general, the design of communication systems for MIMO channels is quite involved (if one can assume the use of sufficiently long and good codes, then the problem formulation simplifies drastically). The first difficulty lies on how to measure the global performance of such systems given the tradeoff on the performance among the different data streams. Once the problem formulation is defined, the resulting mathematical problem is typically too complicated to be optimally solved as it is a matrix-valued nonconvex optimization problem. This design problem has been studied for the past three decades (the first papers dating back to the 1970s) motivated initially by cable systems and more recently by wireless multi-antenna systems. The approach was to

This paper presents an in-depth analysis of the zero forcing (ZF) and minimum mean squared error (MMSE) equalizers applied to wireless multi-input multi-output (MIMO) systems with no fewer receive than transmit antennas. In spite of much prior work on this subject, we reveal several new and surprising analytical results in terms of the well-known performance metrics of output signal-to-noise ratio (SNR), uncoded error and outage probabilities, diversity-multiplexing (D-M) gain tradeoff, and coding gain. Contrary to the common perception that ZF and MMSE are asymptotically equivalent at high SNR, we show that the output SNR of the MMSE equalizer (conditioned on the channel realization) is ρ mmse = ρ zf + η snr , where ρ zf is the output SNR of the ZF equalizer, and that the gap η snr is statistically independent of ρ zf and is a non-decreasing function of input SNR. Furthermore, as snr → ∞, η snr converges with probability one to a scaled F random variable. It is also shown that at the output of the MMSE equalizer, the interference-to-noise ratio (INR) is tightly upper bounded whereas for the MMSE-V-BLAST architecture, the SNR gain due to ordered detection is even better, and significantly so. KeywordsThis work was supported in part by the National Science Foundation Grant CCF-0423842 and CCF-0434410. Zero forcing, minimum mean squared error, MIMO, error probability, V-BLAST, diversity gain, spatial multiplexing gain, tradeoff, outage capacity, outage probability.

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Abstract. Given a complex matrix H, we consider the decomposition H = QRP * , where R is upper triangular and Q and P have orthonormal columns. Special instances of this decomposition include the singular value decomposition (SVD) and the Schur decomposition where R is an upper triangular matrix with the eigenvalues of H on the diagonal. We show that any diagonal for R can be achieved that satisfies Weyl's multiplicative majorization conditions:where K is the rank of H, σ i is the i-th largest singular value of H, and r i is the i-th largest (in magnitude) diagonal element of R. Given a vector r which satisfies Weyl's conditions, we call the decomposition H = QRP * , where R is upper triangular with prescribed diagonal r, the generalized triangular decomposition (GTD). A direct (nonrecursive) algorithm is developed for computing the GTD. This algorithm starts with the SVD and applies a series of permutations and Givens rotations to obtain the GTD. The numerical stability of the GTD update step is established. The GTD can be used to optimize the power utilization of a communication channel, while taking into account quality of service requirements for subchannels. Another application of the GTD is to inverse eigenvalue problems where the goal is to construct matrices with prescribed eigenvalues and singular values.