Research On Semi-Definite Relaxation Detection For Multiple-Symbol Differential Phase Shift Keying (PSK) Modulation

M-ary phase shift keying (MPSK) is a popular digital modulation technique in digital communication. Generally speaking, with MPSK modulation, if the coherent detection is adopted, the channel estimation is required. However, when the channel varies too fast or there are a larger number of antennas in the communication system, the channel estimation becomes very difficult. Although differential detection eliminates the need for explicit channel phase and amplitude estimation at the receiver side, it suffers from a performance penalty when compared to ideal coherent detection. In addition, a severe error floor occurs if differential detection is applied to rapid-fading channels. In order to improve the performance of differential detection, the so-called multi-symbol differential detection (MSDD) algorithm has been proposed.It is known that MSDD with respect to the maximum-likelihood (ML) criterion (ML-MSDD) is optimum in the performance. But the detection complexity grows exponentially with the modulation order and dimension size, if brute-force search is applied. Therefore, it's very significant to reduce the complexity of ML-MSDD. The semi-definite relaxation (SDR) technique has shown great significance to reduce the complexity, and now it has been widely studied and applied in signal processing and communications, such as transmit beam-forming, sensor network localization and MIMO signal detection etc. Recently, a novel MSDD detection algorithm based on SDR (SDR-MSDD) has been proposed, which can reduce the complexity significantly and achieve excellent performance. However, this SDR-MSDD approach is used in system only based on differential QPSK modulation. In this paper, we first study the SDR technology and its performance in the MIMO system, and then extend SDR-MSDD from only based on differential QPSK modulation to be suitable for higher order differential MPSK (MDPSK) modulation. At last, the performance of the SDR-MSDD is studied in the time-varying Rayleigh fading channel. Analysis and simulation results verify that, for higher modulation alphabet size, the performance of the SDR-MSDD algorithm is very close to that of ML detection and the computational complexity is lower than that of sphere decoder.Considering the soft decision decoding (SDD) can provide better performance than that of the hard decoding (HDD), simultaneously taking into account the excellent performance of MSDD, we combine low-rank SDR-MSDD with the soft decision decoding and present a new algorithm for MSDD based on maximum a posteriori (MAP) and SDR, which is called MAP-SDR-MSDD. In this paper, a mathematical model of the MAP-SDR-MSDD detection algorithm is derived. Also, the principle and process is described in detail. At last, combining with convolutional codes and turbo codes in a time-varying Rayleigh fading channel, the performance of the proposed detection algorithm is simulated and analyzed. It is validated through simulations that the proposed MAP-SDR-MSDD algorithm can greatly enhance the performance of the system.