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574 Chapter 10 ROUND-OFF EFFECTS IN DIGITAL FILTERS
% Filter output without multiplication quantization
yn = filter(1,a,xn); % output using filter routine
% Filter output with multiplication quantization
yq = zeros(1,N); % Initialize quantized output array
yq(1) = xn(1); % sample yq(1)
yq(2) = QFix((xn(2)-a1*yq(1)),B,’round’,’satur’); % sample yq(2)
for I = 3:N;
yq(I) = xn(I)-a1*yq(I-1)-a2*yq(I-2); % Unquantized sample
yq(I) = QFix(yq(I),B,’round’,’satur’); % Quantized sample
end
% Output Error Analysis
en = yn-yq;
% Output error sequence
varyn = var(yn); varen = var(en); % Signal and noise power
eemax = max(en); eemin = min(en); % Maximum and minimum of the error
enmax = max(abs([eemax,eemin])); % Absolute maximum range of the error
enavg = mean(en); enstd = std(en); % Mean and std dev of the error
en = round(en*(2^bM)/(2*enmax)+0.5); % Normalized en (integer between -M & M)
en = sort([en,-M:1:(M+1)]); %
H = diff(find(diff(en)))-1; % Error histogram
H = H/N;
% Normalized histogram
Hmax = max(H); Hmin = min(H);
% Max and Min of the normalized histogram
% Output SNRs
SNR_C = 10*log10(varyn/varen);
% Computed SNR
SNR_T = 6.02 + 6.02*B + 20*log10(Xm); % Theoretical SNR
SNR_L = 6.02 + 6.02*B + 20*log10(Xm_L); % Lower SNR bound
SNR_U = 6.02 + 6.02*B + 20*log10(Xm_U); % Upper SNR bound
The part of the script not shown above also computes and plots the normalized
histogram of the output error and prints the statistical values in the plot, as
shown in Figure 10.23. The error again has a Gaussian distribution. The exact
value of the output SNR is 25.22 dB, which agrees with the computed value of
25.11 dB and lies between the lower bound of 20.89 dB and the upper bound
of 26.47 dB. Similar results done for B =12bits are shown in Figure 10.24.
Again, the simulation results agree with the model results.
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10.2.8 HIGHER-ORDER FILTERS
The analysis of the quantization effects in a second-order filter can be
applied directly to higher-order filters based on a parallel realization.
In this case each 2nd-order filter section is independent of all the other
sections, and therefore the total quantization noise power at the output
of the parallel structure is simply the linear sum of the quantization noise
powers of each of the individual sections. On the other hand, the cascade
realization is more difficult to analyze because the noise generated in any
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