Delay limited transmission of a uniform source over an AWGN channel

Delay limited transmission of a uniform source over an additive white Gaussian noise (AWGN) channel under an average power constraint is considered. Assuming that the channel can be used only once, mean squared error (MSE) distortion is studied for both the bandwidth matched, and the 2:1 bandwidth compression cases. In the bandwidth matched scenario, simply scaling the source sample, i.e., analog transmission, performs better than transmitting the scalar quantized source samples. For the bandwidth compression scenario, a hybrid digital analog transmission scheme that quantizes the first source sample and superimposes the quantized sample with the scaled version of the second sample is studied. It is shown that, in this scheme, as opposed to the bandwidth matched case, a finite number of quantization indices minimizes the achievable distortion. The performance of this hybrid scheme is then compared with a numerically optimized encoder using the steepest decent algorithm iteratively. It is observed that the performance of the hybrid scheme is reasonably close to the numerically optimized scheme, while having a significantly lower computational complexity. The theoretical Ziv-Zakai (ZZ) bound on the average distortion is also considered to better understand the gap between the optimal performance and the proposed scheme.