Concatenated Coding for the AWGN Channel With Noisy Feedback

The use of open-loop coding can be easily extended to a closed-loop concatenated code if the transmitter has access to feedback. This can be done by introducing a feedback transmission scheme as an inner code. In this paper, this process is investigated for the case when a linear feedback scheme is implemented as an inner code and, in particular, over an additive white Gaussian noise (AWGN) channel with noisy feedback. To begin, we look to derive an optimal linear feedback scheme by optimizing over the received signal-to-noise ratio (SNR). From this optimization, a linear feedback scheme is produced that is asymptotically optimal in the sense of blocklength-normalized SNR; it is then compared to other well-known schemes. Then, the linear feedback scheme is implemented as an inner code to a concatenated code over the AWGN channel with noisy feedback. This code shows improvements not only in error exponent bounds, but also in bit error rate (BER) and frame error rate (FER). It is also shown that if the concatenated code has total blocklength L and the inner code has blocklength, N, the inner code blocklength should scale as N = O(^C/_R), where C is the capacity of the channel and R is the rate of the concatenated code. Simulations with low-density parity-check (LDPC) and turbo codes are provided to display practical applications and their error rate benefits.