Tradeoff between diversity and decodable-rate: Diversity multiplexing tradeoff for fixed encoding schemes
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In [1], Zheng and Tse formulated the problem of determining the diversity multiplexing tradeoff for MIMO systems. In their formulation they considered a family of coding schemes {C(p)} for an M N MIMO channel where C(p) denotes the coding scheme corresponding to an SNR of p. For any coding scheme family with a multiplexing rate of r, i.e., whose rate grows as r log(1 + p), they characterized the diversity order, d*DMT(r), the largest possible rate of decay for the probability of outage [1]. Here we consider a problem where the encoding scheme C at the transmitter is fixed (independent of SNR) and the receiver attempts to recover information bits at a rate of rlog(1 + p) from the received signal, where p is the receive SNR. r is termed the decodable multiplexing rate (DMR). We define diversity of a scheme as the rate of decay of the probability that a receiver with receive SNR p decodes a rate less than r log(1 + p). We ask the question, what is the best possible diversity for a given DMR? We present a superposition scheme and show that for the MISO / SIMO channel, i.e. when min(M, N) = 1, the superposition scheme has a diversity order equal to max(M, N)(1 - r) which is also the best possible decay rate in this case. For the MIMO channel we show that the superposition scheme achieves a diversity of MN(1 - r) for r < 1. For the block fading SISO channel with coding over L blocks, the scheme achieves a diversity of L(1 - Lr) for r < 1/L. 2007 IEEE.