Bounds on performance of optimum quantizers

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}
.