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Bounds on performance of optimum quantizers

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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 } .

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IEEE Transactions on Information Theory  (Volume:16 ,  Issue: 2 )