代码:
%% ------------------------------------------------------------------------ %% Output Info about this m-file fprintf('\n***********************************************************\n'); fprintf(' <DSP using MATLAB> Problem 8.11 \n\n'); banner(); %% ------------------------------------------------------------------------ %d = 0.10 %d = 0.05 d = 0.01 a1 = (2-d)/(1+d); a2 = (2-d)*(1-d)/((2+d)*(1+d)); % digital IIR 2nd-order allpass filter b = [a2 a1 1] a = [1 a1 a2] figure('NumberTitle', 'off', 'Name', 'Problem 8.11 Pole-Zero Plot') set(gcf,'Color','white'); zplane(b,a); title(sprintf('Pole-Zero Plot, d=%.2f',d)); %pzplotz(b,a); [db, mag, pha, grd, w] = freqz_m(b, a); % --------------------------------------------------------------------- % Choose the gain parameter of the filter, maximum gain is equal to 1 % --------------------------------------------------------------------- gain1=max(mag) % with poles K = 1 [db, mag, pha, grd, w] = freqz_m(K*b, a); figure('NumberTitle', 'off', 'Name', 'Problem 8.11 IIR allpass filter') set(gcf,'Color','white'); subplot(2,2,1); plot(w/pi, db); grid on; axis([0 2 -60 10]); set(gca,'YTickMode','manual','YTick',[-60,-30,0]) set(gca,'YTickLabelMode','manual','YTickLabel',['60';'30';' 0']); set(gca,'XTickMode','manual','XTick',[0,0.25,0.5,1,1.5,1.75,2]); xlabel('frequency in \pi units'); ylabel('Decibels'); title('Magnitude Response in dB'); subplot(2,2,3); plot(w/pi, mag); grid on; %axis([0 1 -100 10]); xlabel('frequency in \pi units'); ylabel('Absolute'); title('Magnitude Response in absolute'); set(gca,'XTickMode','manual','XTick',[0,0.25,0.5,1,1.5,1.75,2]); set(gca,'YTickMode','manual','YTick',[0,1.0]); subplot(2,2,2); plot(w/pi, pha); grid on; %axis([0 1 -100 10]); xlabel('frequency in \pi units'); ylabel('Rad'); title('Phase Response in Radians'); subplot(2,2,4); plot(w/pi, grd*pi/180); grid on; %axis([0 1 -100 10]); xlabel('frequency in \pi units'); ylabel('Rad'); title('Group Delay'); set(gca,'XTickMode','manual','XTick',[0,0.25,0.5,1,1.5,1.75,2]); %set(gca,'YTickMode','manual','YTick',[0,1.0]); figure('NumberTitle', 'off', 'Name', 'Problem 8.11 IIR allpass filter') set(gcf,'Color','white'); plot(w/pi, -pha/w); grid on; %axis([0 1 -100 10]); xlabel('frequency in \pi units'); ylabel('Rad'); title('Phase Delay in samples'); % Impulse Response fprintf('\n----------------------------------'); fprintf('\nPartial fraction expansion method: \n'); [R, p, c] = residuez(K*b,a) MR = (abs(R))' % Residue Magnitude AR = (angle(R))'/pi % Residue angles in pi units Mp = (abs(p))' % pole Magnitude Ap = (angle(p))'/pi % pole angles in pi units [delta, n] = impseq(0,0,20); h_chk = filter(K*b,a,delta); % check sequences % ------------------------------------------------------------------------------------------------ % gain parameter K % ------------------------------------------------------------------------------------------------ %h = 0.2202 * ((-0.9385) .^ n) + (-0.8308) * ((-0.7887) .^ n) + 1.3509 * delta; %d=0.1 %h = 0.1099 * ((-0.9688) .^ n) + (-0.4112) * ((-0.8884) .^ n) + 1.1619 * delta; %d=0.05 h = 0.0220 * ((-0.9937) .^ n) + (-0.0820) * ((-0.9766) .^ n) + 1.0305 * delta; %d=0.01 % ------------------------------------------------------------------------------------------------ figure('NumberTitle', 'off', 'Name', 'Problem 8.11 IIR allpass filter, h(n) by filter and Inv-Z ') set(gcf,'Color','white'); subplot(2,1,1); stem(n, h_chk); grid on; %axis([0 2 -60 10]); xlabel('n'); ylabel('h\_chk'); title('Impulse Response sequences by filter'); subplot(2,1,2); stem(n, h); grid on; %axis([0 1 -100 10]); xlabel('n'); ylabel('h'); title('Impulse Response sequences by Inv-Z'); [db, mag, pha, grd, w] = freqz_m(h, [1]); figure('NumberTitle', 'off', 'Name', 'Problem 8.11 IIR filter, h(n) by Inv-Z ') set(gcf,'Color','white'); subplot(2,2,1); plot(w/pi, db); grid on; axis([0 2 -60 10]); set(gca,'YTickMode','manual','YTick',[-60,-30,0]) set(gca,'YTickLabelMode','manual','YTickLabel',['60';'30';' 0']); set(gca,'XTickMode','manual','XTick',[0,0.25,1,1.75,2]); xlabel('frequency in \pi units'); ylabel('Decibels'); title('Magnitude Response in dB'); subplot(2,2,3); plot(w/pi, mag); grid on; %axis([0 1 -100 10]); xlabel('frequency in \pi units'); ylabel('Absolute'); title('Magnitude Response in absolute'); set(gca,'XTickMode','manual','XTick',[0,0.25,1,1.75,2]); set(gca,'YTickMode','manual','YTick',[0,1.0]); subplot(2,2,2); plot(w/pi, pha); grid on; %axis([0 1 -100 10]); xlabel('frequency in \pi units'); ylabel('Rad'); title('Phase Response in Radians'); subplot(2,2,4); plot(w/pi, grd*pi/180); grid on; %axis([0 1 -100 10]); xlabel('frequency in \pi units'); ylabel('Rad'); title('Group Delay'); set(gca,'XTickMode','manual','XTick',[0,0.25,1,1.75,2]); %set(gca,'YTickMode','manual','YTick',[0,1.0]);
运行结果:
这里放d=0.1的运行结果。
二阶全通滤波器的留数、极点
系统零极点图,可以看出,两个零点都在单位圆外,幅角为π
方法一,利用系统函数直接形式,将脉冲序列做输入,得到脉冲响应h,得到系统幅度谱、相位谱和群延迟,如下图
方法二,将二阶全通系统函数部分分式展开,然后查表求逆z变换,得到脉冲响应h_chk
幅度谱、相位谱和群延迟,可以看到,ω=π时,幅度有衰减
可见,两种方法得到的脉冲响应h有区别,我们将各自前21个元素列出来,方框处二者稍有区别。
但,为何有区别,没搞懂,欢迎各位博友不吝赐教。