مرکز منطقه ای اطلاع رساني علوم و فناوري - Bounds on performance of optimum quantizers

Abstract :

A quantizer $Q$ divides the range [0, 1] of a random variable $x$ into $K$ quantizing intervals the $i$ th such interval having length $\\Delta x_i$ . We define the quantization error for a particular value of $x$ (unusually) as the length of the quantizing interval in which $x$ finds itself, and measure quantizer performance (unusually) by the $r$ th mean value of the quantizing interval lengths $M_r (Q) = \\overline {\\Delta x^{r^{1/r}}}$ , averaging with respect to the distribution function $F$ of the random variable $x$ . $Q_1$ is defined to be an optimum quantizer if $M_r (Q_1) \\leq M_r (Q)$ for all $Q$ . The unusual definitions restrict the results to bounded random variables, but lead to general and precise results. We define a class $Q^{\\ast }$ of quasi-optimum quantizers; $Q_2$ is in $Q^{\\ast }$ if the different intervals $\\Delta x_i$ make equal contributions to the mean $r$ th power of the interval size so that $\\Pr { \\Delta x_i } \\Delta x_{i^{r}}$ is constant for all $i$ . Theorems 1, 2, 3, and 4 prove that $Q_2 \\in Q^{\\ast }$ exists and is unique for given $F, K$ , and $r$ : that $1 \\geq KM_r (Q_2) \\geq KM_r (Q_1) \\geq I_r$ , where $I_r = {\\int_0^{1} f (x)^p dx}^ {1/q}, f$ is the density of the absolutely continuous part of the distribution function $F$ of $x, p = 1/(1+ r)$ , and $q = r /(1 + r)$ : that $\\lim KM_r (Q_2) = I_r$ as $K \\rightarrow \\infty$ ; and that if $KM_r (Q) = I_r$ for finite $K$ , then $Q=Q^{\\ast }$ .