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Bit-interleaved coded modulation with iterative decoding (BICM-ID) is investigated, wherein a novel method of designing amplitude phase-shift keying (APSK) constellations is proposed, which is capable of outperforming both traditional quadrature amplitude modulation (QAM) and nonuniformly spaced QAM (NU-QAM). It is shown that the channel capacity can be approached by the proposed M-APSK constellation as M tends to infinity. Additionally, a new algorithm is introduced for finding the best bit-to-symbol mapping. Furthermore, when signal space diversity is also employed, our extrinsic information transfer (EXIT) chart analysis and Monte Carlo simulations demonstrate that the proposed BICM-ID schemes exhibit a near-Shannon-capacity performance for transmission over both additive white Gaussian noise (AWGN) and Rayleigh fading channels. For a block length of 64 800 bits, the bit-error-rate (BER) curve of the proposed BICM-ID 16/64-APSK scheme is only about 0.8 and 1.0 dB away from the Gaussian-input Shannon limit over AWGN and Rayleigh fading channels, respectively, at the BER of 10-5 and for a code rate of 1/2.