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562 Chapter 10 ROUND-OFF EFFECTS IN DIGITAL FILTERS
Distribution of e 1
Distribution of e 2
4128
3128
2128
1128
4128
3128
2128
1128
SAMPLE SIZE N = 500000
ROUNDED TO B = 6 BITS
MEAN = 1.7026e–006
MIN PROB BAR HEIGHT = 0
MAX PROB BAR HEIGHT = 0.017902
SIGMA = 0.0045105
0
−0.5 −0.375 −0.25 −0.125 0 0.125 0.25 0.375 0.5
Normalized Error e 1
SAMPLE SIZE N = 500000
ROUNDED TO B = 6 BITS
MEAN = 1.6953e–006
MIN PROB BAR HEIGHT = 0
MAX PROB BAR HEIGHT = 0.017606
SIGMA = 0.0031059
0
−0.5 −0.375 −0.25 −0.125 0 0.125 0.25 0.375 0.5
Normalized Error e 2
FIGURE 10.14 Multiplication quantization error distribution for the sinusoidal
signal in Example 10.9, B =6bits
The plots of the resulting histogram are shown in Figure 10.14. For the sinusoidal
signal, when B =6bits, the error samples are not uniformly distributed
and the samples are not independent. The means of e(n) and [e(n)+e(n−1)]/2
are small. Their standard deviations are 0.0045105 and 0.0031059, which do
not agree with (10.10). The corresponding plots for B =12bits are shown in
Figure 10.15 from which we observe that the quantization error sequence appears
to satisfy the model assumptions for B ≥ 12 bits. The means of e(n) and
[e(n)+e(n − 1)]/2 are very small, and their standard deviations agree closely
with (10.10).
□
□ EXAMPLE 10.10 Let x(n) beanindependent and identically distributed random sequence whose
samples are uniformly distributed over the [−1, 1] interval. Using 500,000 samples
to minimize any statistical variations, analyze normalized errors.
Solution
The following MATLAB script computes the distributions for B =6bits.
clear; close all;
% Example parameters
B = 6; N = 500000; xn = (2*rand(1,N)-1); bM = 7; c = 1/sqrt(2);
% Signal and Coefficient Quantization
xq = (round(xn*(2^B)))/(2^B); c = (round(c*(2^B)))/(2^B);
cxq = c*xq;
% Multiplication of constant and signal
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