Chapter 13 Analysis of finite wordlength effects in fixed-point DSP
systems
function ADCNoiseGain=ADCNoise(b,a,n,FM)
[B,A] = sos2tf([b a]); %form A(z) and B(z)
[h,t] = impz(B,A,n);
ADCNoiseGain = sum(h.^2)/12.0;
%Output the results
fprintf('ADC noise gain is %f\n\n',ADCNoiseGain);
if FM~=1
fprintf('ADC noise is %g^2*%g*q^2\n',[FM ADCNoiseGain]);
else
fprintf('ADC noise is %g*q^2\n',ADCNoiseGain);
end
==========================================
function CoeffQuantizeErr(b,a,maxbits,ftype,f,Fs)
%COEFFICIENT QUANTIZATION ERROR ANALYSIS
n=256;
for nbits=2:maxbits
[B,A]=QuantizeCoeff(b,a,nbits);
% quantize all coeffs
[B,A] = sos2tf([B A]); %form A(z) and B(z)
[h,w] = freqz(B,A,n);
amag = abs(h);
% obtain the amplitude response
amag = amag/max(amag); % normalized amplitude response
dev(nbits-1,:) = RippleAtten(ftype,f,amag,n,Fs); % obtain the
pass and stopband deviations
end
fprintf('nbits\tband1\t\tband2\t\tband3\n');
fprintf('%d\t%f\t%f\t%f\n',reshape([(2:maxbits)' dev]',maxbits-1,4));
fprintf('\nfrequency response with quantization noise for desired
wordlength:\n');
nbits=input('
wordlength (32 for unquantized coefficients): ');
[B,A] = sos2tf([b a]); %form A(z) and B(z)
freqz(B,A,n);
%compute and plot the frequency response with
unquantized coefficients
hold on;
[B,A] = QuantizeCoeff(b,a,nbits);
% compute response of nbit filter
[B,A] = sos2tf([B A]); %form A(z) and B(z)
freqz(B,A,n);
%compute and plot the frequency response with
quantized coefficients
title('Frequency Response for Desired Wordlength');
==================================================
function Stability(b,a,maxbits)
%function Stability computes the pole radius and angle of each second
%order section of the transfer function before(r1,angle1) and after
%(r2,angle2) the coefficient quantization for different wordlengthes
1
format long;
fprintf('\n\nnbits\tstage\tA1\tA2\tradius1\tangle1\tradius2\tangle2\n
');
for nbits=2:maxbits
[B,A]=QuantizeCoeff(b,a,nbits);
for i=1:size(b,1)
r1 = sqrt(abs(A(i,3)));
angle1 = 180/pi*acos(A(i,2)/(-2.0*r1));
r2 = sqrt(abs(a(i,3)));
angle2 = 180/pi*acos(a(i,2)/(-2.0*r2));
fprintf('%d\t%d\t%-7.4f\t%-7.4f\t%-7.4f\t%-7.2f\t%7.4f\t%-7.2f\n',nbits,i,A(i,2),A(i,3),r1,angle1,r2,angle2);
end
end
format;
=====================================
function ScaleFactor(b,a,nstep,size,structure)
% Obtain scale factors, using L and cheb norms, for transfer function
% either of canonic or of direct structure with single or multiple
sections
if structure
norm0 = DirectScale(b,a,0,nstep); % obtain L1 norms
norm1 = DirectScale(b,a,1,nstep); % obtain L2 norms
norm2 = DirectScale(b,a,2,size); % obtain Loo norms
else
norm0 = CanonicScale(b,a,0,nstep); % obtain L1 norms
norm1 = CanonicScale(b,a,1,nstep); % obtain L2 norms
norm2 = CanonicScale(b,a,2,size); % obtain Loo norms
end
disp('L1-norms of the second order sections:'); disp(norm0);
disp('L2-norms of the second order sections:'); disp(norm1);
disp('Loo-norms of the second order sections:');disp(norm2);
==============================================
function s = DirectScale(b,a,iopt,n)
% function DirectScale computes the scale factors for a transfer
% function realized by cascaded second order sections in direct
structure
if(iopt>=3) % scaling not required
s = ones(1,size(b,1));
return;
else
A = 1; B = 1;
for i=1:size(b,1)
%loop for each stage
A = conv(A,a(i,:));
B = conv(B,b(i,:));
s(i) = GetScaleFactor(B,A,iopt,n);
end
end
=============================
2
function s = CanonicScale(b,a,iopt,n)
% function CanonicScale computes the scale factors of a transfer
% function realized by cascaded second order canonic sections
if(iopt>=3)
% scaling not required
s = ones(1,size(b,1));
return;
else
A = 1; B = 1;
for i=1:size(b,1)
%loop for each stage
A = conv(A,a(i,:));
if i>1
B = conv(B,b(i-1,:));
end
s(i) = GetScaleFactor(B,A,iopt,n);
end
end
====================================
function FWA
% Program for a limited finite
IIR
% filters in cascade, parallel
% Analysis performed are for:
% (1) ADC quantization noise
% (2) Coefficient quantization
% (3) Product Roundoff noise
% (4) Scaling
%
% For details see:
% Digital Signal Processing: A
% E C Ifeachor and B W Jervis,
wordlength analysis on user-specified
or direct form
noise
Practical Approach
Addison Wesley, 1993
clear all;
isel=3;
%0
for ADC noise analysis
%1
for coefficient quantization error analysis
%2 for product roundoff noise analysis
%3 for scale factors computations
FM = 1 ;
%filter multiplier (set to 1 if there is none)
b1 = [1 0.0339 1]; %numerators of the second order sections
b2 = [1 -0.7563 1];
b3 = [1 0.5331 1];
b4 = [1 -1.1489 1];
a1 = [1 -0.1743 0.9662]; %denominators of the second order sections
a2 = [1 -0.5588 0.9675];
a3 = [1 -0.2711 0.9028];
a4 = [1 -0.4441 0.9045];
b = [b1; b2; b3; b4]; a = [a1; a2; a3; a4];
%above coeff are for example 7.16 (quantization error analysis)
b=[1.2916 -0.08407 0];a=[1 -0.131 0.3355];
b=[7.5268 0 0];a=[1 -0.049 0];
%above coeff are for example 7.19 (scaling factor)
b1 = [1 0.2189 1]; %numerators of the second order sections
b2 = [1 -0.5291 1];
b3 = [1 1.5947 1];
a1 = [1 -0.0127 0.9443]; %denominators of the second order sections
a2 = [1 -0.1731 0.7252];
3
a3 = [1 -0.6152 0.2581];
b = [b1; b2; b3]; a = [a1; a2; a3];
%above coeff are for example 7.18 (scaling factor)
b1=[1 -2 1]; b2=[1 -0.707 1];
a1=[1 0.777 0.3434]; a2=[1 0.01877 0.801];
b=[b1 ; b2]; a = [a1 ; a2];
%above coeff are in user guide: example 7.21
b=[0.1436 0.2872 0.1436];a=[1 -1.8353 0.9747];
%above coeff are for example 7.20 (scaling factor)
b1 = [1 0.2189 1]; %numerators of the second order sections
b2 = [1 -0.5291 1];
b3 = [1 1.5947 1];
a1 = [1 -0.0127 0.9443]; %denominators of the second order sections
a2 = [1 -0.1731 0.7252];
a3 = [1 -0.6152 0.2581];
b = [b1; b2; b3]; a = [a1; a2; a3];
%b=[1 2 1];a=[1 -1.0581359 0.338544];
isel = 3;
if isel==0
n=300;
% length of impulse response
FM = 1;
ADCNoise(b,a,n,FM);
elseif
isel==1
Fs = 1000; %sampling frequency in Hz
maxbits = 8; %maximum wordlength
ftype = 2; %filter type: 0 - LPF, 1 - HPF, 2 - BPF, 3 - BSF
f = [0 100 120 300 320 500]; %bandedge frequencies(Hz) (4 or 6)
of filters
CoeffQuantizeErr(b,a,maxbits,ftype,f,Fs); %coefficient
quantization error analysis
Stability(b,a,maxbits);
elseif isel==2
iopt= 2;
% scaling type: 0 - L1, 1 - L2, 2 - Loo, 3 - no
scaling
n = 300;
% time points(300) for L1 and L2, frequency
points(256) for Loo
nbits = 30; % word length of the coefficients
RoundoffNoise(b,a,n,nbits,iopt);
else
structure = 0 ; % 0 - canonic, 1 - direct
nstep = 256;
size = 300;
ScaleFactor(b,a,nstep,size,structure);
end
===========================================
function s = GetScaleFactor(b,a,iopt,n)
% function GetScaleFactor computes the scale factors for L1, L2, and
% Loo forms from a transfer function described by coefficients a and
b
if (iopt==0|iopt==1)
[h,t] = impz(b,a,n); %impulse response
if (iopt)
s = sqrt(sum(h.^2));
else
s = sum(abs(h));
end
elseif (iopt==2)
4
[h,w] = freqz(b,a,n); % frequency response
s = max(abs(h));
else
s = 1;
end
% otherwise no scale
=======================================
function [B,A]=QuantizeCoeff(b,a,nbits)
if(nbits==32)
B = b; A = a;
return;
end
ip2=2^(nbits-1);
B = floor(sign(b).*(abs(b)*ip2+0.5))/ip2;
A = floor(sign(a).*(abs(a)*ip2+0.5))/ip2;
==================================
function RoundoffNoise(b,a,n,nbits,iopt)
% function RoundoffNoise computes the scale factors and roundoff
% noise of the transfer function realized by the cascaded second
% order canonic sections
if(nbits<2)
nbits=2;
elseif(nbits>28)
nbits=32;
end
norm = CanonicScale(b,a,iopt,n);
% obtain scale factors
temp = norm./[norm(2:size(b,1)) 1];
for i=1:size(b,1) % scale the coefficients
B(i,:)=b(i,:)*temp(i);
end
[B,A] = QuantizeCoeff(B,a,nbits);
norm_square = NormSquare(B,A);
% norm_square(i)=||Hi(z)||2 squared
roundoff_noise_gain = 3*norm_square(1);
for i=2:size(b,1)
roundoff_noise_gain = roundoff_noise_gain+5*norm_square(i);
end
roundoff_noise_gain = (roundoff_noise_gain + 3)/12;
fprintf('Scale factors: '); disp(norm);
fprintf('Norm squared values for Fi(z): '); disp(norm_square);
================================================
function norm_square = NormSquare(b,a)
% function NormSquare computes the scale norm squares of a transfer
% function realized by cascaded second order canonic sections
A = 1; B = 1;
for i=1:size(b,1) %loop for each stage
A = conv(A,a(i,:));
B = conv(B,b(i,:));
end
for i=1:size(b,1) %loop for each stage
if i>1
A = deconv(A,a(i-1,:));
5
B = deconv(B,b(i-1,:));
end
s(i) = GetScaleFactor(B,A,1,300); % get L2 norm from 300 points
end
norm_square = s.*s;
==================================
function dev = RippleAtten(ftype,f,amag,n,Fs)
m=floor(2*n*f/Fs);
if ftype==0
% LPF: obtain passband ripple and stopband
attenuation
dev(1)=20.0*log10(1.0+max(amag(m(1)+2:m(2)))min(amag(m(1)+2:m(2))));
dev(2)=-20.0*log10(max(amag(m(3)+2:m(4))));
dev(3)=0.0;
elseif ftype==1
% HPF: obtain passband and stopband
attenuation
dev(1)=-20.0*log10(max(amag(m(1)+2:m(2))));
dev(2)=20.0*log10(1.0+max(amag(m(3)+2:m(4)))min(amag(m(3)+2:m(4))));
dev3=0.0;
elseif ftype==2
% bandpass filter
dev(1)=-20.0*log10(max(amag(m(1)+2:m(2))));
dev(2)=20.0*log10(1.0+max(amag(m(3)+2:m(4)))min(amag(m(3)+2:m(4))));
dev(3)=-20.0*log10(max(amag(m(5)+2:m(6))));
elseif ftype==3
%BSF: obtain passband and stopband
attenuation
dev(1)=20.0*log10(1.0+max(amag(m(1)+2:m(2)))min(amag(m(1)+2:m(2))));
dev(2)=-20.0*log10(max(amag(m(3)+2:m(4))));
dev(3)=20.0*log10(1.0+max(amag(m(5)+2:m(6)))min(amag(m(5)+2:m(6))));
end
==============================================
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