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Round-off Effects in IIR Digital Filters 573
10.2.7 ANALYSIS USING MATLAB
We will again simulate round-off errors using the MATLAB function QFix
with quantization mode ’round’ and overflow mode ’satur’. Since a
MAC architecture is assumed, we do not have to quantize the intermediate
results and worry about overflow. Only the final sum needs to be quantized
with saturation. These operations are also simulated in sequential fashion,
which has an impact on execution speed. The simulation steps for the
2nd-order filter are detailed in the following example.
□ EXAMPLE 10.12 Consider the model given in Figure 10.22d. We will simulate this model in
MATLAB and investigate its output error characteristics. Let r = 0.9 and
θ = π/3, from which filter parameters are computed and quantized to B bits.
The input signal is uniformly distributed over the [−1, +1] interval and is also
quantized to B bits prior to filtering. The scaling factor X max is determined
using (10.58), which can be obtained in MATLAB by computing the impulse
response for a sufficiently large number of samples. Using 100, 000 signal samples
and B =6bits, the following MATLAB script computes the true output SNR,
the computed SNR, and the lower and upper bounds of the SNR.
close all; clc;
% Example Parameters
B = 12;
% # of fractional bits
N = 100000;
% # of samples
xn = (2*rand(1,N)-1); % Input sequence - Uniform
r = 0.9; theta = pi/3;% Pole locations
% Computed Parameters
p1 = r*exp(j*theta); % Poles
p2 = conj(p1); %
a = poly([p1,p2]); % Filter parameters
hn = filter(1,a,[1,zeros(1,1000)]); % Imp res
Xm = 1/sum(abs(hn)); % Scaling factor
Xm_L = (1-r)*sin(theta); % Lower bound
Xm_U = (1-r)*sqrt(1+r*r-2*r*cos(2*theta)); % Upper bound
% Local variables
bM = 7; DbM = 2^bM; % bin parameter
BB = 2^B;
% useful factor in quantization
M = round(DbM/2); % Half number of bins
bins = [-M+0.5:1:M-0.5]; % Bin values from -M to M
Q = bins/DbM; % Normalized bins
YTN = 2^(-bM);
% Ytick marks interval
YLM = 4*YTN;
% Yaxis limit
% Quantize the input and the filter coefficients
xn = QFix(Xm*xn,B,’round’,’satur’); % Scaled Input quant B bits
a = QFix(a,B,’round’,’satur’); % a quantized to B bits
a1 = a(2); a2 = a(3);
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