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Round-off Effects in IIR Digital Filters 563
Distribution of e 1
Distribution of e 2
4128
3128
2128
1128
4128
3128
2128
1128
SAMPLE SIZE N = 500000
ROUNDED TO B = 12 BITS
MEAN = –9.0218e–008
MIN PROB BAR HEIGHT = 0.005792
MAX PROB BAR HEIGHT = 0.008184
SIGMA = 7.0526e–005
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 = 12 BITS
MEAN = –9.034e–008
MIN PROB BAR HEIGHT = 9.2e 005
MAX PROB BAR HEIGHT = 0.01578
SIGMA = 4.9902e–005
0
−0.5 −0.375 −0.25 −0.125 0 0.125 0.25 0.375 0.5
Normalized Error e 2
FIGURE 10.15 Multiplication quantization error distribution for the sinusoidal
signal in Example 10.9, B =12bits
% Quantization error analysis
[H1,H2,Q, estat] = StatModelR(cxq,B,N);
H1max = max(H1); H1min = min(H1);
H2max = max(H2); H2min = min(H2);
% Max and Min of H1
% Max and Min of H2
The plots of the resulting histogram are shown in Figure 10.16. Even for B =6
bits, the error samples appear to be uniformly distributed (albeit in discrete
fashion) and are independent of each other. The corresponding plots for B =12
bits are shown in Figure 10.17. It is clear for B =12bits that the quantization
error samples are independent and uniformly distributed. Readers should verify
the statistics of these errors given in (10.7), (10.9), and (10.10).
□
From these two examples, we conclude that the statistical model for
the multiplication quantization error, with its stated assumptions, is a
very good model for random signals when the number of bits in the quantizer
is large enough.
10.2.5 STATISTICAL ROUND-OFF NOISE—FIXED-POINT ARITHMETIC
In this and the next section, we will consider the round-off effects on IIR
filters using the multiplication quantization error model developed in the
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