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550 Chapter 10 ROUND-OFF EFFECTS IN DIGITAL FILTERS
Solution
The following MATLAB script illustrates calculations for r =0.9, which implements
exact as well as approximate approaches.
% Filter Parameters
N = 8; r = 0.9; b = 1; pl = r*exp(j*2*pi*[0:N-1]/N); a = real(poly(pl));
% Variance-gain (Exact)
Vg = VarGain(b,a)
Vg =
1.02896272593178
% Variance-Gain (approximate)
x = [1,zeros(1,10000)]; % Unit sample sequence
h = filter(b,a,x); % Impulse response
VgCheck = sum(h.*h)
VgCheck =
1.02896272593178
Clearly, both approaches give the same variance-gain, which for r =0.9 isabout
3% above unity. For r =0.99 the calculations are:
% Filter Parameters
N = 8; r = 0.99; b = 1; pl = r*exp(j*2*pi*[0:N-1]/N); a = real(poly(pl));
% Variance-gain (Exact)
Vg = VarGain(b,a)
Vg =
6.73209233071894
The variance-gain is more than 673%, which means that when poles are close
to the unit circle, the filter output can be very noisy.
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10.2 ROUND-OFF EFFECTS IN IIR DIGITAL FILTERS
With our insight into the quantizer operation and its simpler statistical
model, we are now ready to delve into the analysis of finite word-length
effects in both IIR and FIR digital filters. We have already studied the
effects of input signal quantization and filter coefficient quantization on
filter behavior. We will now turn our attention to the effects of arithmetic
operation quantization on filter output responses (in terms of signal-tonoise
ratios). For this study we will consider both fixed-point and floatingpoint
arithmetic. We first consider the effects on IIR filters since, due to
feedback paths, the results are more complicated—yet more interesting—
than those in FIR filters. The effects on FIR filters are studied in the next
section.
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