A geometric treatment of the source encoding of a Gaussian random variable

Abstract

This paper gives a geometric treatment of the source encoding of a Gaussian random variable for minimum mean-square error. The first section is expository, giving a geometric derivation of Shannon's classic result [1] which explicitly shows the steps in source encoding and the properties that a near optimum code must possess. The second section makes use of the geometric insight gained in the first section to bound the performance that can be obtained with a finite block length of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> random variables. It is shown that a code can be found whose performance approaches that of the rate distortion function as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/L</tex> in mean-square error and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(\ln L)/L</tex> in rate.