Multiple Symbol Differential Detection of MPSK which is Invariant to Frequency Offset using Maximum-likelihood detection

Abstract

It is well known that classical differential detection of MPSK signals, wherein the information is encoded as the first order phase difference, is a simple and robust form of communication in environments not subject to frequency variation. For channels that introduce into the carrier a random frequency shift, eg., those associated with moving vehicles, classical differential detection as above may yield poor performance, particularly if the frequency shift is an appreciable fraction of the data rate. In such situations, one must resort to a form of differential detection that encodes the information as higher order (second order for constant frequency shift) phase difference process. It is shown that the proposed receiver is robust to the distortions caused by the random frequency variation. A lower bound on the error probability of the proposed MSDD receiver is also derived and compared to that of an autocorrelation demodulator for the case where the observation interval approaches infinity. I. Introduction Differential detection of phase-shift keying (PSK) signals is a well-known strategy for mitigating the performance degradation due to unknown phase offset. The constellation rotation caused by the phase offset can be removed using a differential PSK (DPSK) modulation scheme along with a differential detector. However, this detector suffers from a signal-to-noise power ratio (SNR) loss compared to a coherent detector. An effective means to mitigate this SNR loss is known as multiple-symbol differential detection (MSDD). The MSDD scheme is, indeed, a more general case of the conventional differential detection in which more than two consecutive samples are utilized to detect the information symbols. It is shown in that by increasing the number of r e c e i ve d sa m pl es in MSDD , th e r ec ei ver per for m an ce approaches that of coherent demodulation of DPSK signals. However, the MSDD receiver analysis assumes that the fr equen cy offset equals zero. In the ca se of nonzero frequency offset, conventional MSDD must take the frequency offset into account. Otherwise, increasing the number of the received samples in MSDD degrades the performance very quickly. A double DPSK (DDPSK) modulation scheme (also referred to as second-order phase difference modulation) has been proposed for the case when the frequency offset is unknown. In the two types of demodulators for this modulation scheme have been introduced, the autocorrelation demodulator (ACD) and the optimum I-Q demodulator. This reference also proposed an MSDD technique based on ACD for DDPSK signals and showed that this structure is frequency offset invariant. Nevertheless, even under the most optimistic conditions, i.e., an infinite number of received samples and vanishingly small noise power, the proposed receiver still requires 3 dB more SNR than coherent detection of differentially encoded PSK signals. In this paper, we study the effect of frequency variations on a PSK signal transmitted over an additive white Gaussian noise (AWGN) channel. We show that frequency offset attenuates the amplitude of the transmitted signal and rotates its constellation points about the origin by a time-varying phase. Then, we derive a MSDD scheme to demodulate a DDPSK signal and show that this scheme is not sensitive to constellation rotation caused by the frequency offset. The proposed demodulator suffers from a SNR loss compared to a conventional MSDD with DPSK modulation when frequency offset is not present. However, as will be seen in the sequel, the SNR loss can be less than 3 dB for some modulation schemes resulting in a net performance gain relative to the ACD-based MSDD scheme proposed. This paper is organized as follows. In Section II, we present the signal model for the cases where a rectangular pulse- shaping filter and a bandlimited pulse-shaping filter are used at the transmitter. In Section III, we propose a new MSDD scheme for DDPSK signals and derive a lower bound on its error probability. Numerical results are presented in Section IV. In Section V, some conclusions are drawn