Equal-Diagonal Qr Decomposition and Its Application to Precoder Design for Successive-Cancellation Detection

In multiple-input multiple-output (MIMO) multi-user detection theory, the QR decomposition of the channel matrix ${\mmb H}$ can be used to form the back-cancellation detector. In this paper, we propose an optimal QR decomposition, which we call the equal-diagonal QR decomposition, or briefly the QRS decomposition. We apply the decomposition to precoded successive-cancellation detection, where we assume that both the transmitter and the receiver have perfect channel knowledge. We show that, for any channel matrix ${\mmb H}$, there exists a unitary precoder matrix ${\mmb S}$, such that ${\mmb H} {\mmb S} = {\mmb Q} {\mmb R}$, where the nonzero diagonal entries of the upper triangular matrix ${\mmb R}$ in the QR decomposition of ${\mmb H}{\mmb S}$ are all equal to each other. The precoder and the resulting successive-cancellation detector have the following properties. a) The minimum Euclidean distance between two signal points at the channel output is equal to the minimum Euclidean distance between two constellation points at the precoder input up to a multiplicative factor that equals the diagonal entry in the R-factor. b) The superchannel ${\mmb H}{\mmb S}$ naturally exhibits an optimally ordered column permutation, i.e., the optimal detection order for the vertical Bell Labs layered space–time (V-BLAST) detector is the natural order. c) The precoder ${\mmb S}$ minimizes the block error probability of the QR successive cancellation detector. d) A lower and an upper bound for the free distance at the channel output is expressible in terms of the diagonal entries of the ${\mmb R}$-factor in the QR decomposition of a channel matrix. e) The precoder ${\mmb S}$ maximizes the lower bound of the channel's free distance subject to a power constraint. f) For the optimal precoder ${\mmb S}$, the performance of the QR detector is asymptotically (at large signal-to-noise ratios (SNRs)) equivalent to that of the maximum-likelihood detector (MLD) that uses the same precoder. Further, in this paper we consider two multiplexing schemes: time-division multiple access (TDMA) and orthogonal frequency-division multiplexing (OFDM). We design the optimal precoder for binary phase-shift keying (BPSK) with these multiplexing schemes, but outline the procedure to extend the method to nonbinary schemes such as pulse amplitude modulation (PAM), phase-shift keying (PSK), and quadrature amplitude modulation (QAM). Finally, examples are given that illustrate the performance of the precoder and the corresponding successive cancellation detector.