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- 1. By Imtiyaz Mohiuddin 10361A0434 In charge Dr.M.Narayana Electronics and Communications Engineering
- 2. SECTION OUTLINE: Introduction to Digital filter design Introduction to FIR Filter Design of FIR Filter using WINDOW Techniques Introduction to IIR Filter Design of IIR Filter using Approximation Method MATLAB Code of Designed Filters
- 3. Introduction: A digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. In digital signal processing, there are two important types of systems: Digital filters: perform signal filtering in the time domain Spectrum analyzers: provide signal representation in the frequency domain
- 4. Digital Filter: xn yn Digital Filter Sampling frequency fS A D C D A C x(t) y(t) Analog anti- aliasing filter Analog smoothing filter
- 5. Preliminaries: The design of a digital filter is carried out in three steps: Specifications: they are determined by the applications Approximations: once the specification are defined, we use various concepts and mathematics that we studied so far to come up with a filter description that approximates the given set of specifications. (in detail) Implementation: The product of the above step is a filter description in the form of either a difference equation, or a system function H(z), or an impulse response h(n). From this description we implement the filter in hardware or through software on a computer.
- 6. Classification: Digital filters are classified into one of two basic forms, according to how they respond to a unit impulse: Finite impulse response Infinite impulse response
- 7. Finite Impulse Response: In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response is of finite duration, because it settles to zero in finite time. FIR digital filters use only current and past input samples, and none of the filter's previous output samples, to obtain a current output sample value
- 8. The transfer function is given by The length of Impulse Response is N All poles are at Z=0. . Zeros can be placed anywhere on the z-plane 1 0 ).()( N n n znhzH
- 9. Filter Design by Windowing Simplest way of designing FIR filters Start with ideal frequency response Choose ideal frequency response as desired response Most ideal impulse responses are of infinite length n nj d j d enheH deeH 2 1 nh njj dd else0 Mn0nh nh d
- 10. Rectangular: Bartlett: Hamming: Blackman: Kaiser: 2 1 N nN n2 1 N n2 cos1 N n2 cos46.054.0 N n N n 4 cos08.0 2 cos5.042.0 )( 1 2 1 0 2 0 J N n J Commonly used windows
- 11. Kaiser window Kaiser window β Transition width (Hz) Min. stop attn dB 2.12 1.5/N 30 4.54 2.9/N 50 6.76 4.3/N 70 8.96 5.7/N 90
- 12. Rectangular Window else0 Mn01 nw Narrowest main lob – 4/(M+1) – Sharpest transitions at discontinuities in frequency Large side lobs – Large oscillation around discontinuities – -13 dB Simplest window possible
- 13. Hamming Window else0 Mn0 M n2 cos46.054.0 nw Medium main lob – 8/M Good side lobs – -41 dB – Simpler than Blackman
- 14. Kaiser Window Parameterized equation forming a set of windows Parameter to change main-lob width and side-lob area trade-off I0(.) represents zeroth-order modified Bessel function of 1st kind else0 Mn0 I 2/M 2/Mn 1I nw 0 2 0
- 15. MATLAB CODE: %Design of LPF&HPF using rectangular,hamming and kaiser windows clc;clear all;close all; rp=input('enter attenuation in pass band'); rs=input('enter attenuation in stop band'); fp=input('enter pass band frequency'); fs=input('enter stop band frequency'); Fs=input('enter sampling frequency'); wp=2*pi*fp/Fs; ws=2*pi*fs/Fs; %formula for FIR filter num=-20*log10(sqrt(rp*rs))-13; den=14.6*(fs-fp)/Fs; n=ceil(num/den); disp('order of filter is n'); disp(n); disp('press any key to continue'); pause; n1=n+1; %For even order if(rem(n,2)~=0) n1=n; end %LPF
- 16. %LPF s1=input('enter the value for window 0-rectangularLPF 1-kaiserLPF 2-hammingLPF 3- rectangularHPF 4-kaiserHPF 5-hammingHPF'); switch(s1); case 0 y=rectwin(n1); [b,a]=fir1(n,wp,'low',y); freqz(b,a,512); case 1 y=kaiser(n1); [b,a]=fir1(n,wp,'low',y); freqz(b,a,512); case 2 y=hamming(n1); [b,a]=fir1(n,wp,'low',y); freqz(b,a,512); case 3 y=rectwin(n1); [b,a]=fir1(n,wp,'high',y); freqz(b,a,512); case 4 y=kaiser(n1); [b,a]=fir1(n,wp,'high',y); freqz(b,a,512); case 5 y=hamming(n1); [b,a]=fir1(n,wp,'high',y); freqz(b,a,512); end
- 17. Pros & Cons: FIR filters have the following advantages: Exactly linear phase is possible Always stable, even when quantized Design methods are generally linear Efficient hardware realizations Startup transients have finite duration FIR filters have the following disadvantages: • Higher filter order than IIR filters • Corresponding greater delays
- 18. Infinite Impulse Response Filter: IIR systems have an impulse response function that is non-zero over an infinite length of time. This is in contrast to finite impulse response (FIR) filters, which have fixed-duration impulse responses IIR filters may be implemented as either analog or digital filters
- 19. Cont.. While designing a digital IIR filter , an analog filter (e.g. Chebyshev filter, Butterworth filter) is first designed and then is converted to a digital filter by applying discretization techniques such as Bilinear transform or Impulse invariance.
- 20. Discretization techniques
- 21. Chebyshev Filter: Chebyshev filters are analog or digital filters having a steeper roll-off and more passband ripple (type I) or stopband ripple (type II) Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter, but with ripples in the passband
- 22. Cont.. Type-1 Chebyshev Filter Type-2 Chebyshev Filter:
- 23. Butterworth filter The Butterworth filter is a type of signal processing filter designed to have as flat a frequency response as possible in the pass band. It is also referred to as a maximally flat magnitude filter
- 24. MATLAB Prototype Filter Design Commands [B,A] = BUTTER(N,Wn) [B,A] = CHEBY1(N,R,Wn) [B,A] = CHEBY2(N,R,Wn) [B,A] = ELLIP(N,Rp,Rs,Wn) – N = filter order – R = pass band ripple (cheby1) or stop-band ripple (cheby2) in dB. (Rp and Rs respectively for the elliptic filter) – Wn = cut-off frequency (radians/sec for analog filters or normalized digital frequencies for digital filters) – [B,A] = filter coefficients, s-domain (analog filter) or z-domain (digital filter)
- 25. Design Example Filter Specifications: Butterworth response Pass-band edges = 400 Hz and 600 Hz Stop-band edges = 300 Hz and 700 Hz Pass-band ripple = 1 dB Stop-band attenuation = -20 dB Sampling Frequency = 2000 Hz
- 26. Design Example Results 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Frequency (kHz) Magnitude Magnitude Response Band Edges (-1dB and -20 dB)
- 27. Design Example Chebyshev II High-Pass Filter Filter specifications: Chebyshev II response (stop-band ripple) Pass-band edge = 1000 Hz Stop-band edge = 900 Hz Pass-band ripple = 1 dB Stop-band attenuation = -40 dB Sampling frequency = 8 kHz
- 28. MATLAB Code for Design Example >> fs=8000; >> Wp=[2*1000/fs]; % Pass-band edge normalized digital frequency >> Ws=[2*900/fs]; % Stop-band edge normalized digital frequency >> [N,Wn]=cheb2ord(Wp,Ws,1,40); % The “order” command >> [B,A]=cheby2(N,40,Wn,'high'); % cheby2 is the “filter” command. In this command % the syntax requires the stop-band attenuation % as the second parameter >> fvtool(B,A)
- 29. Design Example Results 0 0.5 1 1.5 2 2.5 3 3.5 -100 -80 -60 -40 -20 0 20 Frequency (kHz) Magnitude(dB) Magnitude Response (dB)
- 30. MATLAB CODE: %Design of IIR filters fp1=input('enter pass band frequency'); fs1=input('enter stop band frequency'); Fs1=input('enter sampling frequency'); wp1=fp1/Fs1; ws1=fs1/Fs1; [n1,wn1]=buttord(wp1,ws1,2,60); [x,y]=butter(n1,wn1,'low'); figure; freqz(x,y,512); [n1,wn1]=buttord(wp1,ws1,2,60); [x,y]=butter(n1,wn1,'high'); figure; freqz(x,y,512); [n1,wn1]=cheb1ord(wp1,ws1,2,60); [x,y]=cheby1(n1,3,wn1,'low'); figure; freqz(x,y,512); [n1,wn1]=cheb1ord(wp1,ws1,2,60); [x,y]=cheby1(n1,3,wn1,'high'); figure; freqz(x,y,512);
- 31. Summary of IIR Filter: IIR filters can be design by pole-zero location – Digital oscillators: poles on the unit circle – Notch filters: zeros on the unit circle with nearby poles to control notch width Classic analog filters can be designed using the bilinear transformation IIR filters have the advantage of smaller filter order for a given frequency response. IIR filters have the disadvantages of possible instability due to coefficient quantization effects and non-linear phase response.
- 32. References: “Design of IIR Filter” by K.S Chandra, M.Tech, IIT-Bombay, Jan-2006 “Digital Filter Design” by Prof. A.G. Constantinides, University of Auckland, 2006 “FIR Filter Design”, Gao Xinbo,School of E.E., Xidian Univ. xbgao@ieee.org “FIR Filter by Windowing”- The lab Book Pages.com “Digital Signal Processing”, Prof.Ramesh Babu, Pondicherry Govt. College, TataMcgraw-Hill publication. Wikipedia.org

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