Block Quantization of Correlated Gaussian Random Variables

Abstract

The paper analyzes a procedure for quantizing blocks ofNcorrelated Gaussian random variables. A linear transformation(P)first converts theNdependent random variables intoNindependent random variables. These are then quantized, one at a time, in optimal fashion. The output of each quantizer is transmitted by a binary code. The total number of binary digits available for the block ofNsymbols is fixed. Finally, a secondN \times Nlinear transformation(R)constructs from the quantized values the best estimate (in a mean-square sense) of the original variables. It is shown that the best choice ofRisR = p^{-1}, regardless of other considerations. IfR = P^{-1}, the best choice forPis the transpose of the orthogonal matrix wich diagonalizes the moment matrix of the original (correlated) random variables. An approximate expression is obtained for the manner in which the available binary digits should be assigned to theNquantized variables, i.e., the manner in which the number of levels for each quantizer should be chosen. The final selection of the optimal set of quantizers then becomes a matter of a few simple trials. A number of examples are worked out and substantial improvements over single sample quantizing are attained with blocks of relatively short length.