In Example 10.6.1 we designed a 60-level two-dimensional quantizer by taking the two-dimensional representation of an 8-level scalar quantizer, removing 12 output points from the 64 output points, and adding 8 points in other locations. Assume the input is Laplacian with zero mean and unit variance, and A = 0.7309.

(a) Calculate the increase in the probability of overload by the removal of the 12 points from the original 64.

(b) Calculate the decrease in overload probability when we added the 8 new points to the remaining 52 points.

Example 10.6.1

Let us design a two-dimensional uniform quantizer by keeping only the output points in the quantizer of Example 10.3.2 that lie on or within the contour of constant probability given by

If we count all the points that are retained, we get 60 points. This is close enough to 64 that we can compare it with the eight-level uniform scalar quantizer. If we simulate this quantization scheme with a Laplacian input, and the same step size as the scalar quantizer, that is, A = 0.7309, we get an SNR of 12.22 dB. Comparing this to the 11.44 dB obtained with the scalar quantizer, we see that there is a definite improvement. We can get slightly more improvement in performance if we modify the step size.