2 ObjectivesDescribe the general approach to filter design and the design equation for FIR filters.Define phase distortion and show how it affects signals.Demonstrate how linear phase response eliminates phase distortion.Demonstrate the symmetry condition on FIR impulse response that results in linear phase.Derive the impulse responses of ideal low-pass and high-pass linear phase filters.Describe and demonstrate the window design method for linear phase FIR filters.Demonstrate the design of low-pass, high-pass, band-pass, and band-reject FIR filters.Describe and demonstrate the sampling method of linear phase FIR filter design.Demonstrate the tools in MATLAB for optimized FIR filter design using the Parks-McClellan algorithm.
3 Design of Frequency Selective FIR Filters For frequency selective filters the design problem is to find the impulse response, h[n], for a desired frequency response, H(Ω)In principle, this can be done using the inverse discrete-time Fourier transform (DTFT):
4 Phase DistortionPhase distortion results from a variable time delay (phase delay) for different frequency components of a signal.If the phase response of a filter is a linear function of frequency, then the phase delay is constant for all frequencies and no phase distortion occurs.
5 Linear PhaseAn input signal (blue sinusoid) processed in the pass-band of a filter (output red sinusoid) will, in general, experience a phase change (θ1 to θ2) and a time delay (t1 to t2).
6 Linear Phase (Analog Filter) The output amplitude equals the input amplitude in the pass-bandThis is the phase change condition for constant phase delay independent of frequency: the phase change is a linear function of frequency
7 Linear Phase (Digital Filter) The output amplitude equals the input amplitude in the pass-bandΔn is a constant if Δt is constantThe condition for Δn being constant is that Δθ be a linear function of ΩA constant group delay is a signature of linear phase
8 Constant Phase vs. Linear Phase >> t=0:.04/1000:.04; % The small increment of t approximates an analog signal>> f0=50;>> term1=(2/pi)*sin(2*pi*(f0)*t);>> term2=(2/(3*pi))*sin(2*pi*(3*f0)*t);>> term3=(2/(5*pi))*sin(2*pi*(5*f0)*t);>> term4=(2/(7*pi))*sin(2*pi*(7*f0)*t);>> s=term1+term2+term3+term4;>> plot(t*1000,s);title('4 Term Square Wave');xlabel('milliseconds');Zero phase change for each sinusoid
9 Constant Phase vs. Linear Phase >> term1=(2/pi)*sin(2*pi*(f0)*t+pi/6);>> term2=(2/(3*pi))*sin(2*pi*(3*f0)*t+pi/6);>> term3=(2/(5*pi))*sin(2*pi*(5*f0)*t+pi/6);>> term4=(2/(7*pi))*sin(2*pi*(7*f0)*t+pi/6);>> s=term1+term2+term3+term4;>> plot(t*1000,s);>> title('4 Term Square Wave with Constant 30 deg Phase');xlabel('milliseconds');A constant phase change of +30 degrees for each sinusoid results in phase distortion.
10 Constant Phase vs. Linear Phase >> term1=(2/pi)*sin(2*pi*(f0)*t-.01*pi*(f0));>> term2=(2/(3*pi))*sin(2*pi*(3*f0)*t-.01*pi*(3*f0));>> term3=(2/(5*pi))*sin(2*pi*(5*f0)*t-.01*pi*(5*f0));>> term4=(2/(7*pi))*sin(2*pi*(7*f0)*t-.01*pi*(7*f0));>> s=term1+term2+term3+term4;>> plot(t*1000,s);title('4 Term Square Wave with 5ms Phase Delay');xlabel('milliseconds')Computed with a constant phase (time) delay of 5 ms and linear phase. Linear phase results in no phase distortion.
11 Sufficient Condition for Linear Phase If a FIR filter consists of an odd number of coefficients, M + 1, where M is even, and is symmetrical about the M/2 term, the filter has a linear phase response of Δθ = −(M/2)Ω. The filter will have a group delay of M/2This is termed a “type I” filter (odd length and positive symmetry). Other types have different permutations of length and symmetry. This type is the most common and easiest to design.
12 Sufficient Condition for Linear Phase Odd number of impulse response valuesSymmetry point: n=2Equal values of the impulse response about the symmetry pointGroup delay = 2
13 Sufficient Condition for Linear Phase >> fvtool(h,1)
14 Sufficient Condition for Linear Phase The Running Average Filter >> raf7=(1/7)*ones(1,7);>> stem(n,raf7)>> xlabel('n');ylabel('h[n]');>> title('Impulse Response of a 7-Point Running Average Filter')>> axis([ ])>> fvtool(raf7,1)
17 Impulse Response of the Ideal LP Filter The impulse response of the ideal low pass filter is:It is easily seen that this impulse response has linear phase because it has the symmetry property of h[-n] = h[n]As given, however, this impulse response is not computable: it is infinite and anti-causalA finite set of M+1 h[n] values can be delayed by M/2 samples to create a finite and causal h[n]Creating the finite and causal impulse response will affect some filter properties, but not its linear phase response
18 Example: 21 Coefficient LP Filter with Ω0 = π/4 >> omega=pi/4; % This is the cut-off frequency>> h=(omega/pi)*sinc(omega*(n-10)/pi); % Note the 10 step shift>> stem(n,h)>> title('Sample-Shifted LP Impulse Response')>> xlabel('n')>> ylabel('h[n]')>> fvtool(h,1)Note the linear phase properties of the impulse response
19 Example: 201 Coefficient LP Filter with Ω0 = π/4 >> n=0:200; % This sets the order of the filter where length(n)=201>> omega=pi/4;>> h=(omega/pi)*sinc((n-100)*omega/pi); %Note the sample shift of 100 samples>> fvtool(h,1)Higher order = sharper transitionSide-lobe “ripple” (Gibbs phenomenon) due to abrupt truncation of the impulse response
20 Tapering WindowsUsed to taper the abrupt truncation of the impulse response towards zeroExample: Hamming window
21 Effect of Tapering Windows >> omega=pi/4;>> h=(omega/pi)*sinc((n-10)*omega/pi);>> w= *cos(2*pi*(n-10)/20);% This is the Hamming window. Note that% N-1 = 20 in this case>> hw=h.*w;% Note the use of “ .* “ to multiply h and w% sample-by-sample>> stem(n,w)>> title('Hamming Window')>> figure,stem(n,h,'ko')>> hold>> stem(n,hw,'bd')>> title('Comparison of Rectangular and Hamming Window Impulse Response')>> legend('Rectangular','Hamming')>> fvtool(hw,1)
29 Effect of Group Delay (Order 200 Filter) >> m=1:250; % The sinusoid will have a length of 250 samples>> f=2*pi*100/fs;>> x=sin(f*m);>> y=filter(hLPL,1,x); % The filter command executes the filter on the signal x>> subplot(2,1,1),plot(x),title('Input 100 Hz Sinusoid'),axis([0,250,-2,2]);>> subplot(2,1,2),plot(y),title('Low-Pass Filter Output'),axis([0,250,-2,2]);
30 Sampling Method of FIR Design Basic Theory The sampling method is based on the principle that the DFT is a sample of the DTFTTo see this, take the DFT of the frequency response of a filter:>> n=0:10; % Design a low pass filter by the window method>> omega=pi/4;>> h1=(omega/pi)*sinc(omega*(n-5)/pi);>> dtft_demo(h1,0,2*pi,512); % Display the DTFT of the filter>> hold>> [H1,f]=dft_demo(h1); % Take the DFT of the filter>> stem(f/pi,abs(H1),'k');>> legend('DTFT of h1','DFT of h1')>> title('Fourier Transforms of the Impulse Response h1')>> hold off
32 Sampling Method of FIR Design Basic Theory Evenly spaced samples of the frequency response in the frequency range Ω = 0 to 2π represents the discrete Fourier transform (DFT) of a finite impulse response of the same lengthTherefore, given the frequency response, the impulse response can be computed from the inverse DFT of the frequency response
33 Steps in a Sampling Design Using the Custom M-files Step 1: Determine the critical frequency Ω0Step 2: Determine the order M, where M is evenStep 3: Construct a vector of M+1 real-valued frequency response values evenly spaced from Ω = 0 to 2π. (Custom M-files help do this)[H,omega]=selectH_lp(Ω0,M+1) or[H,omega]=selectH_hp(Ω0,M+1)
34 Steps in a Sampling Design Step 4: Create a causal frequency response by delaying the response by M/2 stepsH_delay=exp(-j*omega*M/2).*HStep 5: Compute the impulse response of the delayed frequency response using the inverse discrete Fourier transformh=inv_dft_demo(H_delay)Step 6: Window the impulse response with a tapering window function; for example:hw=h.*hamming(length(h))'
35 Sampling Design Example 1 Design an order 24 HP filter with a cut-off frequency of π/2Generate and plot the sampled frequency response:>> [x,f]=selectH_hp(pi/2,25);>> stem(f/pi,x);>> title('Samples of the Frequency Response of a N=25 HP Filter with pi/2 Cutoff')>> xlabel('Digital Frequency in Units of Pi')>> axis([ ]);
37 Design Example 1 Impulse Response Compute the causal frequency response:>> M=length(x)-1;>> H=exp(-j*f*M/2).*x;Compute the impulse response with the inverse DFT and plot the resulting frequency response:>> hhp1=inv_dft_demo(H);>> fvtool(hhp1,1)
39 Design Example 1 Compare the sampling design with an ideal window design >> omega=pi/2;>> hhp2=sinc(n-12)-(omega/pi)*sinc(omega*(n-12)/pi);>> [Hhp1,freq]=dtft_demo(hhp1,0,pi,512);>> [Hhp2,freq]=dtft_demo(hhp2,0,pi,512);>> plot(freq/pi,abs(Hhp1)); %This plots the magnitude of Hhp1 versus frequency>> hold>> plot(freq/pi,abs(Hhp2),'--k') %This plots the magnitude of Hhp2 versus frequency>> xlabel('Units of Pi')>> title('Comparison of Ideal Window and Sampling Method for HP Design')>> legend('Sampling Method – hhp1','Windowed Method –hhp2')
44 Using FIR2 for Sampling Designs >> B=fir2(N,F,A)N = orderF = Frequency break points, Ω in units of πA = Amplitudes corresponding to the break points in FF = [0, 0.3, 0.3, 1]A = [1, 1, 0, 0]0.3ππ
45 Sampling Design with FIR2 >> N=100; % Set filter order>> fs = 3000; % Set sampling frequency>> fc=500; % Set cutoff frequency>> F=[0,2*fc/fs,2*fc/fs,1]; % Vector of frequency break points (omega values in units of pi)>> A=[1,1,0,0]; % Amplitudes corresponding to F>> B=fir2(N,F,A); % Compute impulse response>> dtft_demof(B,0,1500,512,3000);>> title('Sampling Design Example with MATLAB FIR2')Note: By default FIR2 uses a Hamming window
47 Optimal Design in MATLAB The values for order, transition width, and ripple cannot be independently specified. Specifying two of the parameters forces a particular value for the third. For a specified ripple and transition width the order is “optimal” in a Parks-McClellan design.
48 A Parks-McClellan Design for a Low-Pass Filter Filter specifications:Order 20 low-passCritical frequency π/4Transition width 0.2πBy specifying the order and the transition width, the forced parameter is the pass-band and stop-band ripple>> f=[ ]; % This sets the band edges with normalized frequencies% The transition is between the .15 and .35 values = 0.20>> a=[ ]; % This sets the desired amplitude response in the pass-band and% stop-band by specifying amplitudes at the band edges.>> N=20; % This sets the filter order. The filter length will be N + 1.>> h=firpm(N,f,a);>> fvtool(h,1)
49 Parks-McClellan Low Pass Results To improve the ripple performance, either the order must be increased or the transition width must be increased
50 Comparison of the 3 Methods >> % Ideal Windowed Design>> n=0:20;>> omega=pi/4;>> hwin=(omega/pi)*sinc(omega*(n-10)/pi).*blackman(21)';>> % Sample Design>> [H,f]=selectH_lp(pi/4,21); %This is a custom M-file for generating response samples% and frequencies for a low-pass filter>> M=20;>> Hk=exp(-j*f*M/2).*H;>> hs=inv_dft_demo(Hk);>> hsamp=hs.*blackman(21)'; % Blackman window the sampled response>> % Optimal Design>> f=[0,.15,.35,1];>> a=[1,1,0,0];>> w=[1,1];>> N=20;>> hopt=firpm(N,f,a,w);
51 Comparison of the 3 Methods (order 20) Best transition performance for a given order
52 Optimal Design Example A Multiple Band-Pass Filter Filter specifications:Sampling frequency 8 kHzBand-pass at:300 to 400 Hz600 to 700 Hz2000 to 3000 HzTransition width 25 Hz-40 dB ripple in both the pass-bands and stop-bandsWith this design the forced parameter is the filter order; in this case it was found by experimentation to be 700.
53 Multiple Band-Pass MATLAB Code Optimal Parks-McClellan Filter >> fs=8000; % This is the sampling rate>> f1=2*300/fs; % The following lines define the frequency edges>> f2=2*400/fs;>> f3=2*600/fs;>> f4=2*700/fs;>> f5=2*2000/fs;>> f6=2*3000/fs;>> trans=2*25/fs; % This defines the transition width% The following vector defines the stop-band and pass-band edges>> f=[0,f1-trans,f1,f2,f2+trans,f3-trans,f3,f4,f4+trans,f5-trans,f5,f6,f6+trans,1];% The following vector gives the stop-band and pass-band amplitudes>> a=[0,0,1,1,0,0,1,1,0,0,1,1,0,0];>> N=700; % This is the filter order which was found by trial-and-error>> hmb=firpm(N,f,a); % The impulse response found by the Parks-McClellan algorithm>> fvtool(hmb,1)
54 Multiple Band-Pass Results Optimal Parks-McClellan Filter Stop-Band Response can be improved by either higher filter order or a relaxed transition specification
55 SummaryFIR filters allow the design of linear phase filters, which eliminate the possibility of signal phase distortion.Three methods of linear phase FIR design were discussed:The ideal window methodThe sampling methodThe optimal Parks-McClellan method