1 / 21

IIR-filters Design

(1). Difference equation:. IIR-filters Design. (2). Canonical Form:. u k. y k. Im. R=1. z 1. Re. Restrictions:. (1). General form:. (2). Amplitude response:. |W(z)| 2 =W(z)W(z -1 ). Phase response:. (3). (4). (5). (6). (7). Design of Analog Prototypes. General form:. (8).

zariel
Download Presentation

IIR-filters Design

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. (1) Difference equation: IIR-filters Design (2) Canonical Form: uk yk

  2. Im R=1 z1 Re Restrictions: (1) General form: (2) Amplitude response: |W(z)|2 =W(z)W(z-1). Phase response: (3) (4) (5) (6) (7)

  3. Design of Analog Prototypes General form: (8) General form for Butterworth, Chebyshev, Bessel, Caurer (elliptic) filters: (9) Butterworth LP- filter: (10) (11) (12) (12) After Wiener factorization of (11):

  4. (13) (14) Normalized transfer function: (15) Non-normalized transfer function: (16) For case (14,15): (17) For case (16):

  5.  Zero-pole Map of Butterworth Filter (18)

  6. [n, Wn]=buttord(0.25,0.4,1,60) N=14, Wn=0.2658 [b,a]=butter(n,Wn); fvtool(b,a) For analog prototype: [n, Wn]=buttord(0.25,0.4,1,60,’s’) Converting analog prototype to digital filter: - Bilinear transform (19) (20) [numd,dend]= bilinear(num,den,fs); or: [numd,dend]= bilinear(num,den,fs,fp); - Invariance of impulse response (21) (22) (24)

  7. [bz,az]=impinvar(b,a,fs); [bz,az]=impinvar(b,a,fs,tol)); Design of Butterworth Filter in MATLAB %Butterworth filter design Fs=1000; %sampling frequence tm=0:1/Fs:0.5;%time vector %Data vector: k=2;am=3; z=am*cos(2*pi*50*tm); x=z+k*randn(size(tm)); %low-pass filter design specification objectives: d=fdesign.lowpass('N,F3db',5,100,Fs); %order=5, edge frequency=100 %Invoke Butterworth filter design method Hd=design(d,'butter'); y=filter(Hd,x); %results of filtering figure(1); subplot(2,1,1),plot(tm,x,'b'), grid on subplot(2,1,2),plot(tm,y,'k',tm,z,'m'), grid on %design results visualization fvtool(Hd)

  8. Results of Design

  9. x z y Results of Filtering Maxflat filters n = 10; % order of polynomial in denominator m= 2; % order of polynomial in numerator Wn = 0.25; % normalized cutoff frequency [b,a]=maxflat(n,m,Wn); fvtool(b,a)

  10. Bessel Filters (1) Bessel Functions (2) Cut off frequency: (3) n=5; %order Wn=1000; [b,a]= besself(5,1000) % Wn – edge frequency of band, within which

  11. Chebyshev Filters (4) Chebyshev polynomials: (5) Amplitude of ripples in the pass band: (6) [b,a] = cheby1(n,Rp,Wn,options) n=4; Rp=1 (dB); Ω=0.4 Roots of denominator are located on the ellipse:

  12. Chebyshev 1 Chebyshev 2 [b,a]=cheby2(n,R,Wst,’ftype’) n=4, R=30, Wst=0.4. [A,B,C,D]=cheby2(n,R,Wst,’ftype’) [z,p,k]=cheby2(n,R,Wst,’ftype’)

  13. Elliptic filter [b,a] = ellip(n,Rp,Rs,Wn,options) [z,p,k] = ellip(n,Rp,Rs,Wn,options) [A,B,C,D] = ellip(n,Rp,Rs,Wn,options) n=6, Rp=1, Rs=40, Wn=0.25

  14. Conversion of filters Basic filter in MATLAB is lp-filter with normalized edge frequency Wn=1 rad/sec 2) [bt,at] = lp2hp(b,a,Wo) [At,Bt,Ct,Dt] = lp2hp(A,B,C,D,Wo) 1) [bt,at] = lp2lp(b,a,Wo) [At,Bt,Ct,Dt] = lp2lp(A,B,C,D,Wo) 3) [bt,at] = lp2bs(b,a,Wo,Bw) [At,Bt,Ct,Dt] = lp2bs(A,B,C,D,Wo,Bw) 4) [bt,at] = lp2bp(b,a,Wo,Bw) [At,Bt,Ct,Dt] = lp2bp(A,B,C,D,Wo,Bw) Example (case 2):

  15. Integrating IIR-filters 1. Rectangular formula: (1) (2) (3)

  16. 2. Trapezoid formula: (4) (5) (6)

  17. 3. Simpson’s formula: (7) (8) (9)

  18. Fast Convolution Based on FFT (10) (11) (12)

  19. Block-diagram of the Fast Convolution with Overlapping and Storage Algorithm H(n) y(n) x(k) Output control& storage FFT IFFT Frames & Overlap Y(n)=H(n)X(n) x(k) Alg. Ov & St L L h(k) 2N-1 3N-1 kT N-1 L L N N 1st frame y(n) n N N 2nd frame 3rd frame

  20. ADAPTATION TO THE VARIATIONS OF THE PLANT’S MAGNITUDE FREQUENCY RESPONSE

More Related