Block Quantization of Correlated Gaussian Random Variables

The paper analyzes a procedure for quantizing blocks of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> correlated Gaussian random variables. A linear transformation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(P)</tex> first converts the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> dependent random variables into <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> independent 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 of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> symbols is fixed. Finally, a second <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N × N</tex> linear transformation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(R)</tex> constructs from the quantized values the best estimate (in a mean-square sense) of the original variables. It is shown that the best choice of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R = p⁻¹</tex> , regardless of other considerations. If <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R = P⁻¹</tex> , the best choice for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</tex> is 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 the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> quantized 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.