《DSP using MATLAB》Problem 8.31
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代码：
%% ------------------------------------------------------------------------ %% Output Info about this m-file fprintf(‘\\n***********************************************************\\n‘); fprintf(‘ <DSP using MATLAB> Problem 8.31 \\n\\n‘); banner(); %% ------------------------------------------------------------------------ Fp = 3.2; % analog passband freq in kHz 6.4 kpi Fs = 3.8; % analog stopband freq in kHz 7.6 kpi fs = 8; % sampling rate in kHz 16.0 kpi % ------------------------------- % Ω=(2/T)tan(ω/2) % ω=2*[atan(ΩT/2)] % Digital Filter Specifications: % ------------------------------- wp = 2*pi*Fp/fs % digital passband freq in rad 0.8pi %wp = Fp; ws = 2*pi*Fs/fs % digital stopband freq in rad 0.95pi %ws = Fs; Rp = 0.5; % passband ripple in dB As = 45; % stopband attenuation in dB Ripple = 10 ^ (-Rp/20) % passband ripple in absolute Attn = 10 ^ (-As/20) % stopband attenuation in absolute % Analog prototype specifications: Inverse Mapping for frequencies T = 1/8000; % set T = 1 %fs = 1/T; OmegaP = (2/T)*tan(wp/2) % prototype passband freq 1.9593pi 15675pi OmegaS = (2/T)*tan(ws/2) % prototype stopband freq 8.089pi 64712pi % Analog Chebyshev-1 Prototype Filter Calculation: [cs, ds] = afd_chb1(OmegaP, OmegaS, Rp, As); % Calculation of second-order sections: fprintf(‘\\n***** Cascade-form in s-plane: START *****\\n‘); [CS, BS, AS] = sdir2cas(cs, ds) fprintf(‘\\n***** Cascade-form in s-plane: END *****\\n‘); % Calculation of Frequency Response: [db_s, mag_s, pha_s, ww_s] = freqs_m(cs, ds, 8*pi/T); % -------------------------------------------------------------------- % find exact band-edge frequencies for the given dB specifications % -------------------------------------------------------------------- [diff_to_45dB, ind] = min(abs(db_s+45)) db_s(ind-3 : ind+3) % magnitude response, dB ww_s(ind)/(pi) % analog frequency in kpi units %ww_s(ind)/(2*pi) % analog frequency in Hz units [sA,index] = sort(abs(db_s+45)); AA_dB = db_s(index(1:8)) AB_rad = ww_s(index(1:8))/(pi) AC_Hz = ww_s(index(1:8))/(2*pi) % ------------------------------------------------------------------- % Calculation of Impulse Response: [ha, x, t] = impulse(cs, ds); % Impulse Invariance Transformation: %[b, a] = imp_invr(cs, ds, T); % Bilinear Transformation [b, a] = bilinear(cs, ds, 1/T) [C, B, A] = dir2cas(b, a) % Calculation of Frequency Response: [db, mag, pha, grd, ww] = freqz_m(b, a); % -------------------------------------------------------------------- % find exact band-edge frequencies for the given dB specifications % -------------------------------------------------------------------- [diff_to_45dB, ind] = min(abs(db+45)) db(ind-3 : ind+3) % magnitude response, dB ww(ind)/(pi) (2/T)*tan(ww(ind)/2)/pi [sA,index] = sort(abs(db+45)); AA_dB = db(index(1:8))‘ AB_rad = ww(index(1:8))‘/pi AC_Hz = (2/T)*tan(ww(index(1:8))‘/2)/pi % ------------------------------------------------------------------- %% ----------------------------------------------------------------- %% Plot %% ----------------------------------------------------------------- figure(‘NumberTitle‘, ‘off‘, ‘Name‘, ‘Problem 8.31 Analog Chebyshev-I lowpass‘) set(gcf,‘Color‘,‘white‘); M = 1.0; % Omega max subplot(2,2,1); plot(ww_s/pi, mag_s); grid on; %axis([-10, 10, 0, 1.2]); xlabel(‘ Analog frequency in \\piHz units‘); ylabel(‘|H|‘); title(‘Magnitude in Absolute‘); % set(gca, ‘XTickMode‘, ‘manual‘, ‘XTick‘, [-8.089, -1.9593, 0, 1.9593, 8.089]); % T = 1 set(gca, ‘XTickMode‘, ‘manual‘, ‘XTick‘, [-80000, -64712, -15675, 0, 15675, 64712, 80000]); % T = 1/8000 set(gca, ‘YTickMode‘, ‘manual‘, ‘YTick‘, [0, 0.006, 0.94, 1.0, 1.5]); subplot(2,2,2); plot(ww_s/pi, db_s); grid on; %axis([0, M, -50, 10]); xlabel(‘Analog frequency in \\piHz units‘); ylabel(‘Decibels‘); title(‘Magnitude in dB ‘); % set(gca, ‘XTickMode‘, ‘manual‘, ‘XTick‘, [-8.089, -1.9593, 0, 1.9593, 5.7, 8.089]); % T = 1 set(gca, ‘XTickMode‘, ‘manual‘, ‘XTick‘, [-80000, -64712, -15675, 0, 15675, 45696, 64712, 80000]); % T = 1/8000 set(gca, ‘YTickMode‘, ‘manual‘, ‘YTick‘, [-45, -1, 0]); set(gca,‘YTickLabelMode‘,‘manual‘,‘YTickLabel‘,[‘45‘;‘ 1‘;‘ 0‘]); subplot(2,2,3); plot(ww_s/pi, pha_s/pi); grid on; %axis([-10, 10, -1.2, 1.2]); xlabel(‘Analog frequency in \\piHz nuits‘); ylabel(‘radians‘); title(‘Phase Response‘); % set(gca, ‘XTickMode‘, ‘manual‘, ‘XTick‘, [-8.089, -1.9593, 0, 1.9593, 8.089]); % T = 1 set(gca, ‘XTickMode‘, ‘manual‘, ‘XTick‘, [-80000, -64712, -15675, 0, 15675, 45696, 64712, 80000]); % T = 1/8000 set(gca, ‘YTickMode‘, ‘manual‘, ‘YTick‘, [-1:0.5:1]); subplot(2,2,4); plot(t, ha); grid on; %axis([0, 30, -0.05, 0.25]); xlabel(‘time in seconds‘); ylabel(‘ha(t)‘); title(‘Impulse Response‘); figure(‘NumberTitle‘, ‘off‘, ‘Name‘, ‘Problem 8.31 Digital Chebyshev-I lowpass‘) set(gcf,‘Color‘,‘white‘); M = 2; % Omega max subplot(2,2,1); plot(ww/pi, mag); axis([0, M, 0, 1.2]); grid on; xlabel(‘ Digital frequency in \\pi units‘); ylabel(‘|H|‘); title(‘Magnitude Response‘); set(gca, ‘XTickMode‘, ‘manual‘, ‘XTick‘, [0, 0.8, 0.95, M]); set(gca, ‘YTickMode‘, ‘manual‘, ‘YTick‘, [0, 0.0056, 0.9441, 1]); subplot(2,2,2); plot(ww/pi, pha/pi); axis([0, M, -1.1, 1.1]); grid on; xlabel(‘Digital frequency in \\pi nuits‘); ylabel(‘radians in \\pi units‘); title(‘Phase Response‘); set(gca, ‘XTickMode‘, ‘manual‘, ‘XTick‘, [0, 0.8, 0.95, M]); set(gca, ‘YTickMode‘, ‘manual‘, ‘YTick‘, [-1:1:1]); subplot(2,2,3); plot(ww/pi, db); axis([0, M, -80, 10]); grid on; xlabel(‘Digital frequency in \\pi units‘); ylabel(‘Decibels‘); title(‘Magnitude in dB ‘); set(gca, ‘XTickMode‘, ‘manual‘, ‘XTick‘, [0, 0.8, 0.93, 0.95, M]); set(gca, ‘YTickMode‘, ‘manual‘, ‘YTick‘, [-70, -45, -1, 0]); set(gca,‘YTickLabelMode‘,‘manual‘,‘YTickLabel‘,[‘70‘;‘45‘;‘ 1‘;‘ 0‘]); subplot(2,2,4); plot(ww/pi, grd); grid on; %axis([0, M, 0, 35]); xlabel(‘Digital frequency in \\pi units‘); ylabel(‘Samples‘); title(‘Group Delay‘); set(gca, ‘XTickMode‘, ‘manual‘, ‘XTick‘, [0, 0.8, 0.95, M]); %set(gca, ‘YTickMode‘, ‘manual‘, ‘YTick‘, [0:5:35]); figure(‘NumberTitle‘, ‘off‘, ‘Name‘, ‘Problem 8.31 Pole-Zero Plot‘) set(gcf,‘Color‘,‘white‘); zplane(b,a); title(sprintf(‘Pole-Zero Plot‘)); %pzplotz(b,a); % ---------------------------------------------- % Calculation of Impulse Response % ---------------------------------------------- figure(‘NumberTitle‘, ‘off‘, ‘Name‘, ‘Problem 8.31 Imp & Freq Response‘) set(gcf,‘Color‘,‘white‘); t = [0: 0.000005 : 8*0.0001]; subplot(2,1,1); impulse(cs,ds,t); grid on; % Impulse response of the analog filter axis([0, 8*0.0001, -1.5*10000, 2.0*10000]);hold on n = [0:1:7*0.0001/T]; hn = filter(b,a,impseq(0,0,7*0.0001/T)); % Impulse response of the digital filter stem(n*T,hn); xlabel(‘time in sec‘); title (sprintf(‘Impulse Responses T=%2d‘,T)); hold off % Calculation of Frequency Response: [dbs, mags, phas, wws] = freqs_m(cs, ds, 8*pi/T); % Analog frequency s-domain [dbz, magz, phaz, grdz, wwz] = freqz_m(b, a); % Digital z-domain %% ----------------------------------------------------------------- %% Plot %% ----------------------------------------------------------------- subplot(2,1,2); plot(wws/(2*pi), mags/T, ‘b+‘, wwz/(2*pi*T), magz, ‘r‘); grid on; xlabel(‘frequency in Hz‘); title(‘Magnitude Responses‘); ylabel(‘Magnitude‘); text(-0.8,0.15,‘Analog filter‘, ‘Color‘, ‘b‘); text(0.6,1.05,‘Digital filter‘, ‘Color‘, ‘r‘); %% ----------------------------------------------------------------------- %% MATLAB cheby1 function %% ----------------------------------------------------------------------- % Analog Prototype Order Calculations: ep = sqrt(10^(Rp/10)-1); % Passband Ripple Factor A = 10^(As/20); % Stopband Attenuation Factor OmegaC = OmegaP; % Analog Chebyshev-1 prototype cutoff freq OmegaR = OmegaS/OmegaP; % Analog prototype Transition ratio g = sqrt(A*A-1)/ep; % Analog prototype Intermediate cal N = ceil(log10(g+sqrt(g*g-1))/log10(OmegaR+sqrt(OmegaR*OmegaR-1))); fprintf(‘\\n\\n ********** Chebyshev-I Filter Order = %3.0f \\n‘, N) % Digital Chebyshev-1 Filter Design: wn = wp/pi; % Digital Chebyshev-1 cutoff freq in pi units [b, a] = cheby1(N, Rp, wn) [C, B, A] = dir2cas(b, a) % Calculation of Frequency Response: [db, mag, pha, grd, ww] = freqz_m(b, a); % -------------------------------------------------------------------- % find exact band-edge frequencies for the given dB specifications % -------------------------------------------------------------------- [diff_to_45dB, ind] = min(abs(db+45)) db(ind-3 : ind+3) % magnitude response, dB ww(ind)/(pi) (2/T)*tan(ww(ind)/2)/pi [sA,index] = sort(abs(db+45)); AA_dB = db(index(1:8))‘ AB_rad = ww(index(1:8))‘/pi AC_Hz = (2/T)*tan(ww(index(1:8))‘/2)/pi % ------------------------------------------------------------------- %% ----------------------------------------------------------------- %% Plot %% ----------------------------------------------------------------- figure(‘NumberTitle‘, ‘off‘, ‘Name‘, ‘Problem 8.31 Digital Chebyshev-I lowpass by cheby1 function‘) set(gcf,‘Color‘,‘white‘); M = 2; % Omega max subplot(2,2,1); plot(ww/pi, mag); axis([0, M, 0, 1.2]); grid on; xlabel(‘Digital frequency in \\pi units‘); ylabel(‘|H|‘); title(‘Magnitude Response‘); set(gca, ‘XTickMode‘, ‘manual‘, ‘XTick‘, [0, 0.8, 0.95, M]); set(gca, ‘YTickMode‘, ‘manual‘, ‘YTick‘, [0, 0.0056, 0.9441, 1]); subplot(2,2,2); plot(ww/pi, pha/pi); axis([0, M, -1.1, 1.1]); grid on; xlabel(‘Digital frequency in \\pi nuits‘); ylabel(‘radians in \\pi units‘); title(‘Phase Response‘); set(gca, ‘XTickMode‘, ‘manual‘, ‘XTick‘, [0, 0.8, 0.95, M]); set(gca, ‘YTickMode‘, ‘manual‘, ‘YTick‘, [-1:1:1]); subplot(2,2,3); plot(ww/pi, db); axis([0, M, -100, 10]); grid on; xlabel(‘Digital frequency in \\pi units‘); ylabel(‘Decibels‘); title(‘Magnitude in dB ‘); set(gca, ‘XTickMode‘, ‘manual‘, ‘XTick‘, [0, 0.8, 0.93, 0.95, M]); set(gca, ‘YTickMode‘, ‘manual‘, ‘YTick‘, [-60, -45, -1, 0]); set(gca,‘YTickLabelMode‘,‘manual‘,‘YTickLabel‘,[‘60‘;‘45‘;‘ 1‘;‘ 0‘]); subplot(2,2,4); plot(ww/pi, grd); grid on; %axis([0, M, 0, 35]); xlabel(‘Digital frequency in \\pi units‘); ylabel(‘Samples‘); title(‘Group Delay‘); set(gca, ‘XTickMode‘, ‘manual‘, ‘XTick‘, [0, 0.8, 0.95, M]); %set(gca, ‘YTickMode‘, ‘manual‘, ‘YTick‘, [0:5:35]); figure(‘NumberTitle‘, ‘off‘, ‘Name‘, ‘Problem 8.31 Pole-Zero Plot‘) set(gcf,‘Color‘,‘white‘); zplane(b,a); title(sprintf(‘Pole-Zero Plot‘)); %pzplotz(b,a); % ---------------------------------------------- % Calculation of Impulse Response % ---------------------------------------------- figure(‘NumberTitle‘, ‘off‘, ‘Name‘, ‘Problem 8.31 Imp & Freq Response‘) set(gcf,‘Color‘,‘white‘); t = [0: 0.000005 : 8*0.0001]; subplot(2,1,1); impulse(cs,ds,t); grid on; % Impulse response of the analog filter axis([0, 8*0.0001, -1.5*10000, 2.0*10000]);hold on n = [0:1:7*0.0001/T]; hn = filter(b,a,impseq(0,0,7*0.0001/T)); % Impulse response of the digital filter stem(n*T,hn); xlabel(‘time in sec‘); title (sprintf(‘Impulse Responses T=%2d‘,T)); hold off % Calculation of Frequency Response: [dbs, mags, phas, wws] = freqs_m(cs, ds, 8*pi/T); % Analog frequency s-domain [dbz, magz, phaz, grdz, wwz] = freqz_m(b, a); % Digital z-domain %% ----------------------------------------------------------------- %% Plot %% ----------------------------------------------------------------- subplot(2,1,2); plot(wws/(2*pi), mags/T, ‘b+‘, wwz/(2*pi*T), magz, ‘r‘); grid on; xlabel(‘frequency in Hz‘); title(‘Magnitude Responses‘); ylabel(‘Magnitude‘); text(-0.8,0.15,‘Analog filter‘, ‘Color‘, ‘b‘); text(0.6,1.05,‘Digital filter‘, ‘Color‘, ‘r‘);
运行结果：
这里放上T=1/8000sec的结果。
模拟chebyshev-1型低通，幅度谱、相位谱和脉冲响应
采用双线性变换法，得到数字chebyshev-1型低通滤波器，幅度谱、相位谱和群延迟响应
采用MATLAB自带cheby1函数得到的数字低通，其幅度谱、相位谱和群延迟
cheby1函数得到的数字低通，和相应的模拟原型的脉冲响应，二者形态不同。
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