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Round-off Effects in IIR Digital Filters 561
where e(n) isamultiplication quantization error. For analysis purposes
we assume that the conditions on e(n) are similar to those for the A/D
quantization error:
1. The random signal e(n) isuncorrelated with the sequence x(n) for
rounding operation (or two’s-complement truncation operation) in the
quantizer.
2. The signal e(n) isanindependent process (i.e., the samples are independent
of each other).
3. The probability density function (pdf) f E (e) ofe(n) for each n is uniformly
distributed over the interval of width ∆ = 2 −B , which is the
quantizer resolution.
We will emphasize the rounding operation for the rest of this section.
Based on the above model assumptions, the results given in (10.7), (10.9),
and (10.10) are also applicable for the multiplication quantization error
e(n).
We offer the following two MATLAB examples to illustrate this
model. A more thorough investigation of this error can be found in
Rabiner and Tukey [25].
□ EXAMPLE 10.9 Consider the sequence given in Example 10.1, which is repeated here.
x(n) = 1 [sin(n/11) + sin(n/31) + cos(n/67)]
3
This signal is multiplied by c =1/ √ 2, quantized to B bits and the resulting
multiplication is quantized to B bits with rounding. Using the StatModelR
function and 500,000 samples, compute and analyze normalized errors e 1(n)
and e 2(n), defined in (10.2) and (10.3), respectively.
Solution
The following MATLAB script computes error distribution, for B =6bits.
clear; close all;
% Example parameters
B = 6; N = 500000; n = [1:N]; bM = 7;
xn = (1/3)*(sin(n/11)+sin(n/31)+cos(n/67)); clear n;
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
% Quantization error analysis
[H1,H2,Q, estat] = StatModelR(cxq,B,N);
H1max = max(H1); H1min = min(H1);
% Max and Min of H1
H2max = max(H2); H2min = min(H2);
% Max and Min of H2
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