This letter presents a technique of combining multilevel coded modulation and product coding to form product modulation codes which achieve low bit error rates with reduced decoding complexity. Three multistage decoding algorithms are presented, and good codes for both the additive white Gaussian noise (AWGN) and Rayleigh fading channels have been constructed.

The first part of this paper presents a simple and systematic technique for constructing multidimensional M-ary phase shift keying (MPSK) trellis coded modulation (TCM) codes. The construction is based on a multilevel concatenation approach. In which binary convolutional codes with good free branch distances are used as the outer codes and block MPSK modulation codes are used as the inner codes (or the signal spaces). Conditions on phase invariance of these codes are derived and a multistage decoding scheme for these codes is proposed. The proposed technique can be used to construct good codes for both the additive white Gaussian noise (AWGN) and fading channels as is shown in the second part of this paper.

For pt.I see ibid., vol.45, no.1, p.64-72, 1997. We use the construction technique proposed in part I to construct multidimensional trellis coded modulation (TCM) codes for both the additive white Gaussian noise (AWGN) and the fading channels. Analytical performance bounds and simulation results show that these codes perform very well and achieve significant coding gains over uncoded reference modulation systems. In addition, the proposed technique can be used to construct codes which have a performance/decoding complexity advantage over the codes listed in literature.

The error performance of a modulation code over a channel depends on several distance parameters and the path multiplicity of the code. For the AWGN channel, the error performance of a modulation code depends mainly on its minimum squared Euclidean distance and path multiplicity. For the Rayleigh fading channel, however, the error performance of a modulation code depends strongly on its minimum symbol distance, minimum product distance, and path multiplicity. It depends on the minimum squared Euclidean distance in a lesser degree. This paper is concerned with the construction of block and trellis MPSK modulation codes for the Rayleigh fading channel. In each construction, the distance parameters are chosen to achieve good error performance with reduced decoding complexity.

Nested concatenated coded 8-PSK modulation schemes are proposed to achieve good error performance, large coding gains, and high spectral efficiency with reduced decoding complexity. In the proposed schemes, Reed-Solomon (RS) codes, including the NASA standard (255, 223) RS code, are used as the outer codes and multidimensional trellis 8-PSK codes are used as the inner codes. The inner codes are constructed from convolutional codes with good free branch distances and multidimensional 8-PSK coset codes through concatenation. These schemes are analyzed and upper bounds on their error performances are derived. Analytical and simulation results show that these schemes achieve large coding gains over uncoded reference systems for bit-error-rates below 10/sup -3/, with reduced decoding complexity.>