A filter is appropriately described by the transfer function. It is a ratio between two polynomials

Transcription

1 Imaginary Part Matlab examples Filter description A filter is appropriately described by the transfer function. It is a ratio between two polynomials H(s) = N(s) D(s) = b ns n + b n s n + + b s a m s m + a m s m + + a s. In Matlab the filter (or system) is described by setting the coefficents in the numerator and denominator polynomials. The coefficients must be set in the correct order. The following example demonstrates this for an analog second order high pass filter H(s) = s 2 s s +. Clear all num = [ ] den = [ sqrt(2) ] tf(num,den) The transfer function appears in the command window: Transfer function: s^ s^ s + 2 Poles and zeros If the transfer function is known, then it is possible to find the poles and zeros. In this case there are two zeros in origo and two complex conjugated poles. Clear all num = [ ] den = [ sqrt(2) ] [z,p,k] = tf2zp(num,den) zplane(z,p) The pole-zero diagram becomes: The result in the command window:.5 z = 2 p = i i -.5 k = Real Part

2 Phase (degrees) Magnitude Imaginary Part If the poles and zeros are known, then it is possible to find the transfer function. Assume that a system has two zeros, one at -2 and one in the origin, and two poles at -2 ± i. clear all k = ; z = [-2; ]; p = [-2+i;-2-i]; zplane(z,p) axis([ ]) [num,den] = zp2tf(z,p,k) tf(num,den) The result in the command window: num = 2 den = 4 5 Transfer function: s^2 + 2 s s^2 + 4 s + 5 The pole-zero diagram Real Part 3 The transfer function in a diagram - The frequency response The magnitude and phase of the frequency response can be obtained by the following commands. num = [ 2 ]; den = [ 4 5]; freqs(num,den) Another way to do this: H = tf(num,den); bode(h) Note that with the last command, we obtain the magnitude in db Frequency (rad/s) Frequency (rad/s) 2

3 Magnitude (db) 4 The magnitude response in decibel and different plot-commands We can also obtain the magnitude in db by using the freqs-function and controlling the appearance of the graph. num = [ 2 ]; den = [ 4 5]; [h,w] = freqs(num,den) semilogx(w,2*log(abs(h))) ylabel('magnitude (db)') xlabel('frequency (rad/s)') Try the following commands in separate figures (use figure to create a new figure-window): figure plot(w,abs(h)) figure plot(log(w),2*log(abs(h))) figure loglog(w,abs(h)) figure semilogy(log(w),abs(h)) Result from semilogx-command: Frequency (rad/s) We can also control the frequency axis by creating frequency vectors to be used in freqs: w = linspace(,) or w = logspace(-2,2) freqs(num,den,w) 5 The phase in degrees We can also create a phase diagram using a new frequency vector. num = [ 2 ]; den = [ 4 5]; w = logspace(-2,2,) h = freqs(num,den,w) semilogx(w,8/pi*angle(h),'.') ylabel('phase (degrees)') xlabel('frequency (rad/s)') The resulting figure: 3

4 Phase (degrees) Frequency (rad/s) 6 Numerical values In the command window numerical values can be shown in columns instead of being arranged as rows. num = [ ]; den = [ sqrt(2) ]; w = logspace(-,,); h = freqs(num,den,w); h_db = 2*log(abs(h)); q = [w' h_db'] For a linear frequency scale: w = ::; h = freqs(num,den,w); h_db = 2*log(abs(h)); q = [w' h_db'] The result in the command window: q = For the linear frequency vector: q =

5 7 Fifth order Butterworth filter [z,p,k] = buttap(5); ; zplane(z,p) title('5:th order Butterworth'); ; w=logspace(-,,5); semilogx(w,2.*log(abs(h))); title('5:th order Butterworth'); axis([.,,-8,+5]); 8 Fifth order Chebyshev filter, type, 3 db ripple in the passband clear all [z,p,k]=chebap(5,3); zplane(z,p); title('5:th order Chebyshev-'); w=logspace(-,,5); semilogx(w,2.*log(abs(h))); title('5:th order Chebyshev-'); axis([.,,-8,+5]); 9 Fifth order Chebyshev filter, type 2, 4 db ripple in the stopband [z,p,k]=cheb2ap(5,4); figure(3) zplane(z,p); title('5:th order Chebyshev-2'); figure(4) w=logspace(-,,5); semilogx(w,2.*log(abs(h))); title('5:th order Chebyshev-2'); axis([.,,-8,+5]); Fifth order Bessel filter [z,p,k]=besselap(5); zplane(z,p); title('5:th order Bessel filter'); w=logspace(-,,5); semilogx(w,2.*log(abs(h))); 5

6 title('5:th order Bessel filter'); axis([.,,-8,+5]); Fifth order Cauer/Elliptic filter [z,p,k]=ellipap(5,3,4); zplane(z,p); title('5:th order Cauer filter'); w=logspace(-,,5); semilogx(w,2.*log(abs(h))); title('5:th order Cauer filter'); axis([.,,-8,+5]); 2 Step response [z,p,k]=besselap(5); dt=.; t=:dt:3; h=step(tf(num,den),t); plot(t,h,'b'); hold on [z,p,k]=chebap(5,3); dt=.; t=:dt:3; h=step(tf(num,den),t); plot(t,h,'r'); title('red=chebyshev, blue=bessel'); 3 Impulse response [z,p,k]=besselap(5); dt=.; t=:dt:3; h=impulse(tf(num,den),t); plot(t,h,'b'); hold on [z,p,k]=chebap(5,3); dt=.; t=:dt:3; h=impulse(tf(num,den),t); plot(t,h,'r'); title('red=chebyshev, blue=bessel'); 6

7 4 Output signal from the system dt=.; t=:dt:3; uin=[zeros(,),sin(t),zeros(,)]; [zc,pc,kc]=chebap(5,3); [numc,denc]=zp2tf(zc,pc,kc); hc=impulse(numc,denc,t); uutc=conv(hc,uin).*dt; [zb,pb,kb]=besselap(5); [numb,denb]=zp2tf(zb,pb,kb); hb=impulse(numb,denb,t); uutb=conv(hb,uin).*dt; T=:dt:(length(uutb)-.)*dt; UIN=[zeros(,), sin(t),zeros(,length(t)- length(uin)+)]; plot(t,uutb,'r',t, uutc,'b', T,UIN,'k') axis([,6,-.2,.2]) A short duration sinusoid with frequency rad/s is filtered by Chebyshev and Bessel filter respectively. 5 Low pass-to-low pass (not using lp2lp) A system has the poles - ± 3j. We wish to double the cutoff frequency. The Matlab code to do this is: z=[]; p=[-+3.*j;--3.*j]; k=p().*p(2); dw=.; w=:dw:; H=freqs(num,den,w); semilogx(w,2.*log(abs(h)),'r'); hold on; %transformed system is marked by z=[]; p=2.*p; k=2.^2.*k; [num,den]=zp2tf(z,p,k); H=freqs(num,den,w); semilogx(w,2.*log(abs(h)),'b'); Low pass-to-high pass Determine the poles and zeros for a 5:th order high pass Chebyshev type filter with 3 db ripple and a cutoff frequency of khz. 7

8 Imaginary Part %original system is marked with [z, p,k]=chebap(5,3); %We move the cutoff frequency to % 2*pi* and transform to HP. % Transformed filter is marked by [num,den]=lp2hp(num,den,2.*pi.*); dw=; w=:dw:3; H=freqs(num,den,w); plot(w./2./pi,2.*log(abs(h)),'b'); axis([,5,-2,+5]) title('magnitude of H as a function of frequency') [z,p,k]=tf2zp(num,den) zplane(z,p) magnitude of H as a function of frequency x 4 In the command window the zeros and poles are: z = i i p =.e+4 * i i i i Real Part x 4 7 Low pass-to-band pass Starting from a 5:th order Chebyshev (type ) filter with 3 db ripple, we wish to transform it to a band pass filter with cutoff frequencies 8 Hz and 2 Hz, which gives a bandwidth of 4 Hz and a center frequency approximately khz (98 Hz). We determine the pole and zeros of the system. %original system is marked with [z, p,k]=chebap(5,3); % Transformed filter is marked by [num,den]=lp2bp(num,den,2.*pi.*98,2.*pi.* 4); dw=; w=:dw:3; H=freqs(num,den,w); plot(w./2./pi,2.*log(abs(h)),'b'); axis([,2,-2,+5]) magnitude of H as a function of frequenc

9 Imaginary Part title('magnitude of H as a function of frequency') [z,p,k]=tf2zp(num,den) zplane(z,p) In the command window the zeros and poles are: z = i i i i p =.e+3 * i i i i i i i i i i Real Part x 4 9