Optimum Bit-by-Bit Power Allocation for Minimum Distortion Transmission

In this paper, we consider a joint source-channel coding problem and minimize the end-to-end mean square error (MSE) distortion of a communication system through power allocation to the transmitted bits. This communication system consists of a quantizer with natural binary mapping and Binary Phase Shift Keying (BPSK) modulator at the transmitter and we consider the cases with and without coding. We show that there is an optimal transmit power allocation to the transmitted bits of the quantized word that minimizes MSE. In the first part of the paper, the case with no channel coding is considered with hard decision decoding at the receiver. The optimum power profile is determined analytically by using the Chernoff bound and through computer-based optimization methods. The optimum power allocation gives a constant MSE gain over the uniform power allocation for high SNRs. In the second part of the paper, (7,4) Hamming code with soft-decision decoding is considered. An upper bound on the MSE expression is derived and nearoptimum power allocation is obtained. It is observed that for lower SNRs, power is allocated only to the information bits showing that coding is not required while for higher SNRs, power is allocated to all the bits in a codeword and it gives a constant gain in MSE over the uniform power allocation.