Max Lloyed Quantizer

In this case quantizers are designed minimizing MSQE and tables are developed for input governed by standard distribution functions such as gamma, Laplacian, Gaussian, Rayleigh, Uniform. Quantizers can also be designed tailored to histograms. For a well designed predictor, the histrogram of predicted errors tend to follow Laplacian distribution.

Given the range of input f as from to , and the number of output levels as J, we design the quantizer s.t MSQE is minimum.

Figure (3.3.3)

Let = decision level ( level)

and = reconst level ( level)

This is shown in the above figure (3.3.3).

If then and

where = Probability density function of random variable f

The error can also be written as

and are variables. To minimize we set

Since and solution that is valid is

or

i.e. input level is average of two adjacent output levels. Also

or ,

Hence

where

Output level is the centroid of adjacent input levels. This solution is not closed form. To find input level , one has to find and vice versa. However by iterative techniques both and can be found using Newton 's method.

When number of output levels s large, the quantizer design can be approximated as follows: Assuming is constant over each quantization level, or a piecewise constant function

As shown in Figure (3.7), we have, a ;

We have,

i.e. each reconstruction level lies mid-way between two adjacent decision levels.