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For the single-carrier M-ary differential phase-shift keying (MDPSK), the multiple-symbol differential detector, or the noncoherent maximum-likelihood sequence detector (NSD), and its three special cases, namely, the noncoherent one-shot detector, the linearly predictive decision-feedback (DF) detector, and the linearly predictive Viterbi receiver are reviewed based on a hierarchical interpretation. For the multicarrier transmission, the differential orthogonal frequency division multiplexing (OFDM) systems with diversity reception are discussed. It is well known that there are two types of differential OFDM systems, namely, the time domain differential OFDM (TD-OFDM) and the frequency domain differential OFDM (FD-OFDM). In this paper, the NSD and its special cases are incorporated to the differential OFDM systems. Furthermore, we provide a simple closed-form bit-error-rate (BER) expression for the differential OFDM systems utilizing the noncoherent one-shot detector with diversity reception in the time-varying multipath Rayleigh fading channels. Numerical results have revealed that, with multi-antenna diversity reception, the performance of the noncoherent one-shot detector is improved significantly. However, when only one or two receive antennas are available, the implementation of the linearly predictive DF detector or the linearly predictive Viterbi receiver is necessary for achieving better and satisfactory performance.