Robust Walsh–Hadamard transform-based spatial modulation

Abstract

This paper presents a combination of spatial modulation (SM) and Walsh–Hadamard transform (WHT) for achieving robustness against wireless fading channels. The WHT operation merge together SM transmitting symbols, where the number of symbols depends on the order of WHT matrix. A WHT matrix of order two merges two transmitting symbols by their sum and difference, therefore both the symbols mutually carry their information which results in a diversity gain. A bad noise sample might corrupt completely a symbol, while transmitting the combined symbols allows to average out the noise and have a lower noise variance in average. This kind of approach has been used for generalized frequency division multiplexing (GFDM) and space–time block coding (STBC) with WHT matrix of order two, and thus achieves a diversity gain of two. Unlike, the conventional communication schemes like GFDM and STBC, the use of WHT in SM is not straight forward due to the two information carrying domains, i.e. antenna index, and transmitted symbols. A simple WHT of the SM symbols causes a degraded bit error rate (BER) performance due to the fact that the WHT of these symbols results in some transmit symbols having zero energy. To overcome this problem, we use the phase shifting trick. The required phase shift depends on the order of the WHT matrix and M-ary phase shift keying (M-PSK) modulation scheme. Before the WHT of the transmit symbols, the phase of each constellation point is shifted by a certain angle that is different for each symbol and thus overcomes the problem of transmit symbols having zero energy after the WHT operation. The proposed scheme is evaluated under maximum received ratio combining (MRRC) and optimal Maximum likelihood (ML) detector. The complexities of both MRRC and ML are calculated for WHT–SM. Extensive simulations are presented for the proposed WHT–SM system with and with-out the phase shifting approach. It is shown, how the WHT matrix order and modulation level M effects the performance of WHT–SM system with-out the phase shift approach. The impact of high diversity gains is discussed and shown in the simulation results. The proposed scheme achieves a superior BER performance compared to the conventional SM scheme.