Low Resolution Scalar Quantization for Gaussian and Laplacian Sources with Absolute and Squared Error Distortion Measures

This report considers low resolution scalar quantization. Specifically, it considers entropyconstrained scalar quantization for memoryless Gaussian and Laplacian sources with both squared and absolute error distortion measures. The slope of the operational rate-distortion functions of scalar quantization for these sources and distortion measures is found. It is shown that in three of the four cases this slope equals the slope of the corresponding Shannon rate-distortion function, which implies that asymptotic low resolution scalar quantization with entropy coding is an optimal coding technique for these three cases. For the case of a Gaussian source and absolute error distortion measure, however, the slope at rate equal zero of the operational-rate distortion function of scalar quantization is infinite, and hence does not match the slope of the corresponding Shannon rate-distortion function. Consequently, scalar quantization is not an optimal coding technique for Gaussian sources and absolute error distortion measure. The results are obtained via analysis of uniform and binary scalar quantizers, which shows that in low resolution their operational rate-distortion functions, in all four cases, are the same as the corresponding operational rate-distortion functions of scalar quantization in general. Lastly, the slope of the Shannon rate-distortion function (the function itself is not known) at rate equal zero is found for a Laplacian source and squared error distortion measure.