Iterative Demodulation and Decoding of Non-Square QAM

The labels of the remaining 2²ⁿ⁺¹ points are the same as in the square 2²ⁿ⁺²-QAM constellation. This means that bits are used to label each of the 2²ⁿ⁺¹ points in the (NS)2²ⁿ⁺¹-QAM constellation: ⁿ⁺¹ bits for the I-dimension labeling, and ⁿ⁺¹ bits for the Q-dimension labeling. However, each point in an (NS)2²ⁿ⁺¹-QAM constellation represents only bits of information. Hence, the bits used for labeling an (NS)2-QAM signal point are not independent.

Close examination has shown that for any signal point in the (NS)2²ⁿ⁺¹-QAM constellation, the last bit of its ²ⁿ⁺² labeling bits can be viewed as a parity-check bit of the other 2²ⁿ⁺¹ bits. Therefore, each (NS)2²ⁿ⁺¹-QAM symbol can be generated by first encoding the corresponding bits with a (²ⁿ⁺² , 2 n +1)single-parity- check (SPC)block encoder and then using the ²ⁿ⁺² encoded bits to select one of the ²ⁿ⁺¹
points on the (NS)2²ⁿ⁺¹-QAM. The I-dimension position and the Q-dimension position of a signal point on an (NS)2²ⁿ⁺¹-QAM constellation can be independently determined by the first ⁿ⁺¹ encoded bits and the remaining ⁿ⁺¹ encoded bits, respectively. The ( ²ⁿ⁺² 2²ⁿ⁺¹) SPC block code can be generated as a recursive, systematic, terminated convolutional code with two states. In other words, the decomposition of (NS)2²ⁿ⁺¹-QAM into a block encoder and a memoryless modulator leads to a showing that (NS)2²ⁿ⁺¹-QAM is, by itself a form of coded modulation.

When concatenated with a forward-error-correcting (FEC) code (see figure), this decomposition can be applied to obtain joint iterative demodulation and decoding algorithms that exploit the inherent memory of (NS)2²ⁿ⁺¹-QAM so as to achieve better coding gains. In addition, because of the independent I and Q mapping of (NS)2²ⁿ⁺¹-QAM, the decoding complexity can be reduced to that of one-dimensional PAM. Moreover, because the signal constellation of (NS)2²ⁿ⁺¹ -QAM is a subset of the square 2²ⁿ⁺²-QAM constellation, it should be possible, in practice, to implement (NS)2²ⁿ⁺¹-QAM by use of 2²ⁿ⁺²-QAM equipment already in existence.

Results of some computational simulations have shown that with iterative demodulation and decoding according to this scheme, coded (NS)8-QAM performs 0.5 dB better than does coded standard 8-QAM and 0.7 dB better than does coded 8-ary phase-shift keying (8 PSK) when the FEC code is the (15,11) Hamming code concatenated with a rate-1 accumulator code. Other simulation results show that coded (NS)32-QAM performs 0.25 dB better than does coded standard 32-QAM.