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Round-off Effects in FIR Digital Filters 583
qn = round(qn*(2^bM)/(2*qnmax)+0.5); % Normalized en (interger between -K & K)
qn = sort([qn,-K:1:(K+1)]); %
H = diff(find(diff(qn)))-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/varqn); % Computed SNR
SNR_T = 6.02 + 6.02*B + 10*log10(sum(h.*h)/Xm^2) - 10*log10(M); % Theoretical SNR
% Filter output with multi quant (1 multiplier)
yq = QFix(yn,B,’round’,’satur’);
% Output Error Analysis
qn = yn-yq;
% Outout error sequence
varyn = var(yn); varqn = var(qn); % Signal and noise power
qqmax = max(qn); qqmin = min(qn); % Maximun and minimum of the error
qnmax = max(abs([qqmax,qqmin])); % Absolute maximum range of the error
qnavg = mean(qn); qnstd = std(qn); % Mean and std dev of the error
qn = round(qn*(2^bM)/(2*qnmax)+0.5); % Normalized en (interger between -K & K)
qn = sort([qn,-K:1:(K+1)]); %
H = diff(find(diff(qn)))-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/varqn); % Computed SNR
SNR_T = 6.02 + 6.02*B + 10*log10(sum(h.*h)/Xm^2); % Theoretical SNR
The computed and theoretical SNRs as well as output error histograms for the
two models are shown in Figure 10.27. The top plot shows the histogram when
five multipliers are used. The output error has Gaussian-like distribution with
SNR equal to 65.42 dB, which agrees with the theoretical value. The bottom
plot show the histogram when one multiplier is used. As expected, the error is
uniformly distributed with SNR equal to 72.43 dB, which also agrees with the
theoretical one.
□
Cascade-form realization Let the filter be realized by a cascade of
K, 2nd-order (M =3)sections given by
H(z) =
K∑
H i (z) where H i (z) =β 0i + β 1i z −1 + β 2i z −2 (10.90)
i=1
as shown in Figure 10.28. The overall length of the filter is M =2K +1.
Figure 10.28 also shows the finite word-length model for the cascade form,
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