Signal Processing. Lecture 10: FIR Filter Design. Ahmet Taha Koru, Ph. D. Yildiz Technical University Fall

Transcription

1 Signal Processing Lecture 10: FIR Filter Design Ahmet Taha Koru, Ph. D. Yildiz Technical University Fall ATK (YTU) Signal Processing Fall 1 / 47

2 Introduction Introduction ATK (YTU) Signal Processing Fall 2 / 47

3 Introduction What is FIR filter? FIR = Finite impulse response Impulse response has finite duration Settles to zero in finite time. x[n] n Figure: A FIR filter impulse response. ATK (YTU) Signal Processing Fall 3 / 47

4 Introduction Definition For a causal discrete-time FIR filter of order N, the output is y[n] = a 0 x[n] + a 1 x[n 1] + a 2 x[n 2] + + a N x[n N] N y[n] = a k x[n k] k=0 The impulse response of the filter is N h[n] = a k δ[n k] k=0 ATK (YTU) Signal Processing Fall 4 / 47

5 Introduction Structure of FIR Filter x[n] z 1 z 1 z 1 h[0] h[1] h[2] h[3] y[n] Figure: Block diagram of a FIR filter ATK (YTU) Signal Processing Fall 5 / 47

6 Introduction FIR Properties Require no feedback of the output. Simpler implementation. Always stable. For x B x y N a k B x k=0 Can easily be designed to be linear phase by making the coefficient sequence symmetric. Disadvantage: Requires more taps and computational power is required compared to an IIR filter with similar sharpness or selectivity, especially for low frequency cut-offs. ATK (YTU) Signal Processing Fall 6 / 47

7 Introduction Design Methods Window design method Frequency sampling method Weighted least squares design Linear programming Parks-McClellan method Equiripple FIR filters design Etc... ATK (YTU) Signal Processing Fall 7 / 47

8 Introduction Filtering Specifications H(e jω ) 1 + δ 1 1 δ ω Transition band δ Ripple [ω s, π] Stop band [0, ω p] Pass band ω δ ω p ω c ω s π ω ATK (YTU) Signal Processing Fall 8 / 47

9 Window Design Method Window Design Method ATK (YTU) Signal Processing Fall 9 / 47

10 Window Design Method What is window design method? Ideal low-pass filters have infinite length of duration. Windowing is truncating the ideal low pass impulse response. Windowed impulse response is not ideal but an approximation There are different types of windows. ATK (YTU) Signal Processing Fall 10 / 47

11 Window Design Method An example: Low pass filter h i [n] n w[n] n h FIR [n] = h i [n] w[n] n ATK (YTU) Signal Processing Fall 11 / 47

12 Window Design Method Ideal Filters Window Design Method: Ideal Filters ATK (YTU) Signal Processing Fall 12 / 47

13 Window Design Method Ideal Filters Low-pass filter H LP (e jω ) H LP (e jω ) = { 1, ω ωc 0, ω c < ω π π ω c ω c π n h LP [n] h LP [n] = sin(ω cn), πn Matlab implementation: >> hlp = sinc(wc/pi*n)*wc/pi; n ATK (YTU) Signal Processing Fall 13 / 47

14 Window Design Method Ideal Filters High-pass filter H HP (e jω ) H HP (e jω ) = { 0, ω ωc 1, ω c < ω π π ω c ω c π n h HP [n] h HP [n] = δ[n] sin(ω cn) πn Matlab implementation: >> hhp = [dirac(n2)>1] - sinc(wc/pi*n2)*wc/pi; n ATK (YTU) Signal Processing Fall 14 / 47

15 Window Design Method Ideal Filters Band-pass filter 0, ω ω c1 H BP (e jω ) = 1, ω c1 < ω ω c2 0, ω c2 < ω π H BP (e jω ) h BP [n] = sin(ω c 2 n) πn sin(ω c 1 n) πn ω c2 -ω c1 ω c1 ω c2 π n Matlab implementation: >> hbp = sinc(wc2/pi*n)*wc2/pi - sinc(wc1/pi*n)*wc1/pi; ATK (YTU) Signal Processing Fall 15 / 47

16 Window Design Method Ideal Filters Band-stop filter 1, ω ω c1 H BS (e jω ) = 0, ω c1 < ω ω c2 1, ω c2 < ω π H BS (e jω ) ω c2 -ω c1 ω c1 ω c2 π n h BS [n] = sin(ω c 1 n) +δ[n] sin(ω c 2 n) πn πn Matlab implementation: >> hbs = sinc(wc1/pi*n)*wc1/pi + [dirac(n)>1]... - sinc(wc2/pi*n)*wc2/pi; ATK (YTU) Signal Processing Fall 16 / 47

17 Window Design Method Ideal Filters Multi-level Frequency Filter H ML (e jω ) = A k, for ω k 1 ω ω k H ML (e jω ) h ML [n] = L l=1 with A L+1 = 0. [ (A l A l+1 ) sin(ω ] ln) πn ω 1 ω 2 ω 3 π n Matlab implementation: >> n = -100:100; % Length of filter >> A = [3, 1, 0, 2, 0]; % List magnitudes >> w = [0.3*pi, 0.55*pi, 0.8*pi,1*pi]; % List frequencies >> h = zeros(size(n)); >> for k = 1:4 >> h = h + (A(k) - A(k+1))*sinc(w(k)/pi*n)*w(k)/pi; >> end ATK (YTU) Signal Processing Fall 17 / 47

18 Window Design Method Ideal Filters Hilbert Transformer { H HT (e jω j, π ω < 0 ) = j, 0 ω π { 0, for n even h HT [n] = for n odd 2 πn, Matlab implementation: >> h = mod(n,2)*2/pi./n; ATK (YTU) Signal Processing Fall 18 / 47

19 Window Design Method Ideal Filters Differantiator H DIFF (e jω ) = jω { 0, n = 0 h DIFF [n] = cos(πn) n, n > 0 Matlab implementation: >> h = cos(pi*n)./n; >> h(find(isinf(h))) = 0; ATK (YTU) Signal Processing Fall 19 / 47

20 Window Design Method Window Types Window Design Method: Window Types ATK (YTU) Signal Processing Fall 20 / 47

21 Window Design Method Window Types Window Types: Rectangular w[n] = u[n + M] u[n M 1] w[n] 1 M M n ATK (YTU) Signal Processing Fall 21 / 47

22 Window Design Method Window Types Window Types: Bartlett w[n] = 1 n M + 1, w[n] 1 M n M M 1 M + 1 n ATK (YTU) Signal Processing Fall 22 / 47

23 Window Design Method Window Types Window Types: Hann w[n] = 1 2 [ ( )] 2πn 1 + cos, M n M 2M + 1 w[n] 1 M M n ATK (YTU) Signal Processing Fall 23 / 47

24 Window Design Method Window Types Window Types: Hamming ( ) 2πn w[n] = cos, M n M 2M + 1 w[n] 1 M M n ATK (YTU) Signal Processing Fall 24 / 47

25 Window Design Method Window Types Window Types: Blackman ( ) ( ) 2πn 4πn w[n] = cos cos, M n M 2M + 1 2M + 1 w[n] 1 M M n ATK (YTU) Signal Processing Fall 25 / 47

26 Window Design Method Window Types Properties of some fixed window functions Type of Window Main Lobe Width Relative Sideslobe Level Minimum Stopband Attenuation Transition Bandwidth ω Rectangular 4π/(2M + 1) 13.3 db 20.9 db 0.92 π/m Bartlett 4π/(2M + 1) 26.5 db - - Hann 8π/(2M + 1) 31.5 db 43.9 db 3.11 π/m Hamming 8π/(2M + 1) 42.7 db 54.5 db 3.32 π/m Blackman 12π/(2M + 1) 58.1 db 75.3 db 5.56 π/m ATK (YTU) Signal Processing Fall 26 / 47

27 Parks-McClellan Method Parks-McClellan Method ATK (YTU) Signal Processing Fall 27 / 47

28 Parks-McClellan Method What is Parks-McClellan Algorithm Is an iterative algorithm for finding the optimal Chebyshev FIR filters. Powerful tool for design and implement efficient and optimal FIR filters. Can easily be calculated with help of a computer. ATK (YTU) Signal Processing Fall 28 / 47

29 Parks-McClellan Method What does algorithm do? A desired filter frequency response Tries to minimize error the difference with desired response Matlab Code: firpm ATK (YTU) Signal Processing Fall 29 / 47

30 Parks-McClellan Method Example: Low-pass Filter H d (e jω ) 1 Desired frequency response H(e jω ) ω p ω s π Parks-McClellan algorithm design n ω s π n ATK (YTU) Signal Processing Fall 30 / 47

31 Parks-McClellan Method Matlab Implementation N = 5; wc = pi/2; dw = 0.1*pi; % Length of filter % Cut-off Frequency % Transition Band % Parks-McClellan Algorithm h = firpm(n*2, [0 (wc-dw)/pi (wc+dw)/pi 1], [ ]); %Plot the response [H,W] = freqz(h); plot(w/pi, abs(h)) ATK (YTU) Signal Processing Fall 31 / 47

32 Linear Programming Method Linear Programming Method ATK (YTU) Signal Processing Fall 32 / 47

33 Linear Programming Method What is Linear Programming? Linear programming is optimization Linear cost function Linear constraints General form of a linear program is minimize subject to c T x Ax b where x is the vector to be designed. ATK (YTU) Signal Processing Fall 33 / 47

34 Linear Programming Method The General Structure of a FIR Filter Let us consider a symmetric FIR filter such that h[ n] = h[n], for all n Suppose the coefficients of the filters are a k such that N h[n] = a k δ[n k] k= N Then, the frequency response of the filter is N H(e jω ) = a a k cos(kω) k=1 We would like H(e jω ) to approximate an desired filter H d (e jω ). ATK (YTU) Signal Processing Fall 34 / 47

35 Linear Programming Method Low-pass Filter Linear Programming Method: Low-pass Filter ATK (YTU) Signal Processing Fall 35 / 47

36 Linear Programming Method Low-pass Filter Filtering Specifications H(e jω ) 1 + δ 1 1 δ ω Transition band δ Ripple [ω s, π] Stop band [0, ω p] Pass band ω δ ω p ω c ω s π ω ATK (YTU) Signal Processing Fall 36 / 47

37 Linear Programming Method Low-pass Filter Constraints: Pass-band (1/2) In the pass-band: N a a k cos(kω) 1 δ k=1 for all 0 ω ω c. Equivalently, ( ) N a a k cos(kω) δ 1 k=1 ( ) N a a k cos(kω) δ 1 k=1 ATK (YTU) Signal Processing Fall 37 / 47

38 Linear Programming Method Low-pass Filter Constraints: Pass-band (2/2) If we re-write the conditions in vector form [ 1 2 cos(ω) 2 cos(2ω) 2 cos(nω) cos(ω) 2 cos(2ω) 2 cos(nω) 1 ] a 0 a 1. a N δ [ 1 1 ] for any 0 ω ω p. ATK (YTU) Signal Processing Fall 38 / 47

39 Linear Programming Method Low-pass Filter Constraints: Transition-band In the transition-band: No constraints ATK (YTU) Signal Processing Fall 39 / 47

40 Linear Programming Method Low-pass Filter Constraints: Stop-band (1/2) In the stop-band: a N a k cos(kω) δ k=1 Equivalently, ( ) N a a k cos(kω) δ 0 k=1 ( ) N a a k cos(kω) δ 0 k=1 ATK (YTU) Signal Processing Fall 40 / 47

41 Linear Programming Method Low-pass Filter Constraints: Stop-band (2/2) If we re-write the conditions in vector form [ 1 2 cos(ω) 2 cos(2ω) 2 cos(nω) cos(ω) 2 cos(2ω) 2 cos(nω) 1 ] a 0 a 1. a N δ [ 0 0 ] for any ω s ω π. ATK (YTU) Signal Processing Fall 41 / 47

42 Linear Programming Method Low-pass Filter Optimization Variables and Other Parameters Optimization variables to be designed are listed in the vector a 0 a 1 x =. a N δ Given parameters are Number of taps: N Cut-off frequency: ω c Transition bandwidth: ω From ω, pass-band freq: ω p = ω c ω/2 From ω, stop-band freq: ω s = ω c + ω/2 ATK (YTU) Signal Processing Fall 42 / 47

43 Linear Programming Method Low-pass Filter Cost Function and the Optimization Problem The cost function to be minimized is the amount of ripple δ and δ = [ ] x Then, the optimization problem is minimize subject to [ ] x a 0 a [ ] cos(ω) 2 cos(nω) cos(ω) 2 cos(nω) 1. a Nδ [ a 0 a ] cos(ω) 2 cos(nω) cos(ω) 2 cos(nω) 1. a Nδ [ 1 1 [ 0 0 ], ω [ 0, ω p ] ], ω [ω s, π] There are infinitely many constraints. Problem is not solvable. ATK (YTU) Signal Processing Fall 43 / 47

44 Linear Programming Method Low-pass Filter Discretization We can not evaluate at every ω point but instead in M samples among ω [0, π]. Let s choose M = 15N The samples are chosen as ω k = π k M, Then, the constraints become cos(ω 1 ) 2 cos(nω 1 ) cos(ω 1 ) 2 cos(nω 1 ) 1 k = 0, 1,, M. 1 2 cos(ω M ) 2 cos(nω M ) cos(ω M ) 2 cos(nω M ) 1. a N δ a 0 a ATK (YTU) Signal Processing Fall 44 / 47

45 Linear Programming Method Low-pass Filter Matlab Implementation (1/3) % % Set the given parameters L = 20; % Length of the filter N = 2L + 1 wc = pi/2; % Cut-off frequency dw = 0.1*pi; % Transition bandwidth wp = wc - dw/2; ws = wc + dw/2; % Pass-band and stop-band freqs. % % Discretize M = 15*L; wl = linspace(0, pi, M + 1); R = sum(wl <= wp) + sum(wl >=ws) - 2; % Number of constraints A = zeros(r, L+2); b = zeros(r,1); cnt = 1; ATK (YTU) Signal Processing Fall 45 / 47

46 Linear Programming Method Low-pass Filter Matlab Implementation (2/3) for k = 1:length(wl) w = wl(k); if w <= wp A(cnt,:) = [1, 2*cos([1:L]*w), -1]; A(cnt+1,:) = [-1, -2*cos([1:L]*w), -1]; b(cnt) = 1; b(cnt+1) = -1; cnt = cnt+2; elseif w >= ws A(cnt,:) = [1, 2*cos([1:L]*w), -1]; A(cnt+1,:) = [-1, -2*cos([1:L]*w), -1]; b(cnt) = 0; b(cnt+1) = 0; cnt = cnt+2; end end ATK (YTU) Signal Processing Fall 46 / 47

47 Linear Programming Method Low-pass Filter Matlab Implementation (3/3) c = [zeros(1,l+1), 1]; x = linprog(c, A, b); hhalf = x(1:l+1); h = [hhalf(end:-1:2); hhalf]; % Cost function to minimize c'*x % Minimize c'*x subject to Ax <= b % Construct the filter [H, W] = freqz(h); plot(w/pi, abs(h)) ATK (YTU) Signal Processing Fall 47 / 47