Transfer function bounds on performance of binary turbo-coding followed by M-ary orthogonal signal mapping through interleaver

We obtain upper bounds on the bit-error rate (BER) for maximum-likelihood (ML) decoding of binary turbo-codes followed by M-ary orthogonal signal mapping through a uniform interleaver. We use the transfer function bounding techniques to obtain the above bounds. Thus, the upper bounds correspond to the average bound over all interleavers of a given length. We apply the techniques to parallel concatenated coding (PCC) schemes where recursive convolutional codes are used as constituent codes. We present the average bounds on the BER of turbo codes with constraint length 3 for M-ary orthogonal signals on additive white Gaussian noise (AWGN) channels. We show that for a given increase in interleaver length, the improvement in BER is larger for turbo-coded 4-ary orthogonal signals than for turbo-coded binary orthogonal signals. We also show that the coding gains of turbo code for M-ary orthogonal signals becomes smaller as M is increased when the block length N is short