Some properties of uniform step size quantizers (Corresp.)

Some properties of the optimal mean-square error uniform quantizer are treated. It is shown that the mean-square error (mse) is given by the input variance minus the output variance. Furthermore <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">limN → ∞mse/(Δ²/12) ≥ 1</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> is the number of output levels and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</tex> (a function of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> ) is the step size of the uniform quantizer, with equality when the support of the random variable is contained in a finite interval. A class of probability densities is given for which the above limit is greater than one. It is shown that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">limN → ∞N² ·</tex> mse <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">=(b-a)²/12</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(b-a)</tex> is the measure of the smallest interval that contains the support of the input random variable.