Properties of optimum Mean square quantizers:

1) The quantizer output is an unbiased estimate of input i.e.

for uniform quantizer increases by 6 dB /bit

2) The quantizer error is orthogonal to the quantizer output i.e., quantization noise is uncorrelated with quantized output.

3) The variance of quantizer is reduced by the factor where denotes the mean square distortion of the B-bit quantizer for unity variance inputs.

i.e.

4)  It is sufficient to design mean square quantizers for zero mean and unity variance distributions.

Example : Suppose of image sensor takes values from 0 to 10. If samples are quantized uniformly to 256 levels then decision and reconstruction levels are:

	or	for ,...257

and note as range is (0,10).

and

Proof of property 1.

1. If is the probability of i.e.

then

As	we have

Substitute for and using the definition of , we have

Proof for property 2:

. .This implies,

An interesting model for the quantizer follows from this i.e.

is equivalent to

where is quantizer noise.

As the quantizer noise is uncorrelated with we can write

Since this implies average power of quantizer output is reduced by average power .

Also quantizer noise is dependent on input since.

Proof of property 3.

Since for any mean square B-bit quantizer,

we get from previous results,