s-plane to z-plane transformation (2024)

clear all

close all

%%Define G-filter in s-plane according to ISO 7196:1995E

%poles

p=[ -0.707+j* 0.707

-0.707-j* 0.707

-19.27+j* 5.16

-19.27-j* 5.16

-14.11+j*14.11

-14.11-j*14.11

-5.16+j*19.27

-5.16-j*19.27];

%zeros

z=[0 0 0 0]';

%scalar gain

k=10^(116/20);

G_0 = zpk(z,p,k)

fs = 400;

G_1 = c2d(G_0,1/fs,'tustin');

bode(G_0);

hold all;

bode(G_1);

legend({'continuous' 'discrete filter'})

% Extract the filter coefficients

G_1_tf = tf(G_1);

B = G_1_tf.num{1}

A = G_1_tf.den{1}

% Compare the time series output using a random input signal

figure;

r = randn(1,10000);

t = 1/fs * (0:9999);

y1 = lsim(G_0,r,t);

y2 = filter(B,A,r);

plot(t,y1,t,y2,'r:');

legend({'continuous' 'discrete filter'})

[bz,az]=bilinear(bG,aG,fs/(2*pi));

[bz,az]=bilinear(bG,aG,fs);

clear all

close all

%%G-filter in s-plane according to ISO 7196:1995E

%poles

p=[ -0.707+j* 0.707

-0.707-j* 0.707

-19.27+j* 5.16

-19.27-j* 5.16

-14.11+j*14.11

-14.11-j*14.11

-5.16+j*19.27

-5.16-j*19.27];

%zeros

z=[0 0 0 0]';

%scalar gain

k=10^(116/20);

%get analog filter coefficients

[bG,aG] = zp2tf(z,p,k);

%get filter response

[hG,fG] = freqs(bG,aG,[0 logspace(log10(0.2),log10(200),1000)]);

%%Try to get digital coefficients from yule-walker method

fs=200;

[b,a] = yulewalk(8,fG/max(fG),abs(hG));

%get digital filter response

[h,f] = freqz(b,a,100*fs,fs);

%%Try to get digital coefficients from bilinear transform

[bz,az]=bilinear(bG,aG,fs/(2*pi));

%get digital filter response

[Hz, fz]= freqz(bz,az,100*fs,fs);

%%plot

figure

semilogx(fG,20*log10(abs(hG)),'--'...

,f,20*log10(abs(h))...

,fz,20*log10(abs(Hz)))

xlim([0.2 fs])

xlabel('frequency (Hz)'), ylabel('response (dB)')

legend('G in s-plane','G in z-plane, yule-walker','G in z-plane, bilinear');

set(gca,'XTick',[0.2 0.5 1 2 5 10 20 50 100 200])

set(gca,'XTickLabel',{'0,2','0,5','1','2','5','10','20','50','100','200'})