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A new suboptimal demodulator based on iterative decision feedback demodulation (DFD), and a singular value decomposition (SVD) for estimation of unitary matrices, is introduced. Noncoherent communication over the Rayleigh flat-fading channel with multiple transmit and receive antennas, where no channel state information (CSI) is available at the receiver is investigated. With four transmit antennas, codes achieving bit-error rate (BER) lower than 10-4 at bit energy over the noise spectral density ratio (Eb/No) of -0.25 dB up to 3.5 dB, with coding rates of 1.6875 to 5.06 bits per channel use were found. The performance is compared to the mutual information upper bound of the capacity attaining isotropically random (IR) unitary transmit matrices. The codes achieve BER lower than 10-4 at Eb/No of 3.2 dB to 5.8 dB from this bound. System performance including the iterative DFD algorithm is compared to the one using Euclidean distance, as a reliability measure for demodulation . The DFD system presents a performance gain of up to 1.5 dB. Uncoded systems doing iterative DFD demodulation and idealized pilot sequence assisted modulation (PSAM) detection are compared. Iterative DFD introduces a gain of more than 1.2 dB. The coded system comprises a serial concatenation of turbo code and a unitary matrix differential modulation code. The receiver employs the high-performance coupled iterative decoding of the turbo code and the modulation code. Information-theoretic arguments are harnessed to form guidelines for code design and to evaluate performance of the iterative decoder.