PEP analysis of SDP-based non-coherent signal detection

In multi-antenna communication systems, channel information is often not known at the receiver. To fully exploit the bandwidth resources of the system and ensure the practical feasibility of the receiver, the channel parameters are often estimated and then employed in the design of signal detection algorithms. However, sometimes communication can occur in the environment where learning the channel coefficients becomes infeasible. In this paper we consider the problem of maximum-likelihood (ML)-detection in single-input multiple-output (SIMO) systems when the channel information is completely unavailable at the receiver and when employed signalling at the transmitter is q-PSK. It is well known that finding the solution to this optimization requires solving an integer maximization of a quadratic form and is, in general, an NP hard problem. To solve it, we propose an approximate algorithm based on the semi-definite program (SDP) relaxation. We derive a bound on the pairwise probability of error (PEP) of the proposed algorithm and show that, the algorithm achieves the same diversity as the exact maximum-likelihood (ML) decoder. Furthermore, we prove that in the limit of large system dimension this bound differs from the corresponding one in the exact ML case by at most 3.92 dB if the transmitted symbols are from 2 or 4-PSK constellations and by at most 2.55 dB if the transmitted symbols are from 8-PSK constellation. This suggests that the proposed algorithm requires moderate increase in the signal-to-noise ratio (SNR) in order to achieve performance comparable to that of the ML decoder but with often significantly lower complexity.