Robust Quantization of Discrete-Time Signals with Independent Samples

The
N
-level uniform quantizer on
[-c,c]
plus the assignment of
y_{0}\deg = -(a _{s}+ c)/2
and
y_{N+1}\deg = (a_{s}+ c)/2
to signal values falling in the saturation regions
[-a_{s},- c) and (c,a_{s}]
, respectively, is shown to be the minimax
(N + 2)
-level quantizer with a nonsaturating input range
[-c,c]
. The performance criterion considered is the mean weighted quantization error and the input signals are only required to be amplitude bounded by
\pm a_{s}
where
a_{s} > c > 0
. The worst case input signal marginal probability distributions are shown to be discrete. From the derivation of this result, the minimax error can be computed. An example is given which illustrates the performance of the minimax quantizer for several input ranges against different input signal probability distributions.