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
\lim_{N \rightarrow \infty}mse/(\Delta^{2}/12) \geq 1
, where
N
is the number of output levels and
\Delta
(a function of
M
) 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
\lim_{N \rightarrow \infty}N^{2} \cdot
mse
=(b-a)^{2}/12
, where
(b-a)
is the measure of the smallest interval that contains the support of the input random variable.