Filter Design Problem

Transcription

1 Filter Design Problem Design of frequency-selective filters usually starts with a specification of their frequency response function. Practical filters have passband and stopband ripples, while exhibiting a gradual roll-off in the transition band. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 1/40

2 Filter specifications Absolute specifications In the passband: 1 δ p H(e jω ) 1 + δ p, for 0 ω ω p, with δ p 1. In the stopband: H(e jω ) δ s, for ω s ω π, with δ s 1. Relative specifications 1 δ p 1 + δ p H(e jω ) 1, 0 ω ω p, H(e jω ) δ s 1 + δ p, ω s ω π. The passband A p and stopband A s ripples are defined as ( ) 1 + δp A p 20 log 10 1 δ p and A s 20 log 10 ( 1 + δp δ s ). Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 2/40

3 Continuous-time filter specification In practical applications, the passband F p and stopband F s cutoff frequencies are specified in Hz. The values of ω p and ω s are calculated according to ω p = 2π F pass F s, ω s = 2π F stop F s. IIR filters are typically derived from analog filters, which are traditionally specified using the quantities ɛ and A as follows. ( ) 20 log ɛ 2 = A p and 20 log 10 A = A s, which gives ɛ = Ap 1 and A = As. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 3/40

4 Optimality criteria for filter design Let H(z) be a rational system function that approximates some ideal frequency response H d (e jω ). The coefficients of H(z) depends on the approximation criterion. Mean-squared-error approximation: E 2 [ 1/2 1 H(e jω ) H d (e jω ) dω] 2, 2π B with B being a frequency interval of interest. Minimax approximation: E max ω B H(ejω ) H d (e jω ). The solution minimizing this error function is called a minimax or Chebyshev approximation. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 4/40

5 FIR filters with linear phase Without loss of generality, we focus on the ideal lowpass filter (LPF) with its impulse and frequency responses given by { h lp [n] = sin ω c(n α), H lp (e jω e jαω, ω ω c, ) = π(n α) 0, ω c < ω π. Depending on the choice of α, there are three possible cases 1 If α Z, h lp [n] is symmetric about its sample at n = α. 2 If 2α Z, h lp [n] is symmetric about the middle point between the samples at α 1/2 and α + 1/2. 3 If 2α is not an integer, there is no symmetry at all. A causal FIR filter is created by setting h[n] = h lp [n] for n = 0,..., M and zero elsewhere. There are four types of FIR filter with linear phase. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 5/40

6 Type-I FIR linear-phase filters A Type-I FIR system has a symmetric impulse response with even order M, that is, h[n] = h[m n], 0 n M. The frequency response of a Type-I FIR filter can be expressed as M/2 H(e jω ) = a[k] cos ωk e j(m/2)ω = A(e jω )e j(m/2)ω, k=0 } {{ } A(e jω ) where A(e jω ) is a real, even, and periodic function of ω with a[0] = h[m/2], a[k] = 2h[(M/2) k], k = 1,..., M/2. This type of filters is the most general one. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 6/40

7 Type-II FIR linear-phase filters A Type-I FIR system has a symmetric impulse response with odd order M. The frequency response of a Type-II FIR filter can be expressed as (M+1)/2 H(e jω ) = b[k] cos [ω(k 1/2)] e j(m/2)ω = k=1 } {{ } A(e jω ) where the delay M/2 is fractional and = A(e jω )e j(m/2)ω, b[k] = 2h[(M + 1)/2 k], k = 1,..., (M + 1)/2. We note that at ω = π, A(e jω ) = 0, independently of h[k]. This implies that Type-II filters cannot be high-pass. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 7/40

8 Type-III FIR linear-phase filters A Type-III FIR system has an antisymmetric impulse response with even order M, i.e. h[n] = h[m n], 0 n M. In this case, the frequency response is given by M/2 H(e jω ) = c[k] sin ωk j e j(m/2)ω = j A(e jω )e j(m/2)ω, where k=1 } {{ } A(e jω ) c[k] = 2h[(M/2) k], k = 1,..., M/2. The condition h[m/2] = h[m/2] requires that h[m/2] = 0 for every Type-III filter. Note that A(e jω ) = 0, at ω {0, π}. Such filters are most suitable for designing differentiators and Hilbert transformers. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 8/40

9 Type-IV FIR linear-phase filters If the impulse response is antisymmetric and M is odd, then we have (M+1)/2 H(e jω ) = d[k] sin [ω(k 1/2)] j e j(m/2)ω = k=1 } {{ } A(e jω ) where the delay M/2 is fractional and = ja(e jω )e j(m/2)ω, d[k] = 2h[(M + 1)/2 k], k = 1,..., (M + 1)/2. For Type-IV filters, ones has A(e jω ) = 0 at ω = 0, independently of the choice of h[k]. Thus, this class of filters is most suitable for approximating differentiators and Hilbert transformers. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 9/40

10 Frequency responses of the FIR filters The frequency responses of all Type I-IV FIR filters with linear phase can be expressed in the form H(e jω ) = M h[n]e jωn = A(e jω )e jψ(ejω), n=0 where the amplitude response A(e jω ) is real and Ψ(e jω ) αω + β. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 10/40

11 Frequency responses of the FIR filters (cont.) The constant β results from the presence of j in H(e jω ), and it does not affect the group delay τ gd (ω) = dψ(ejω ) dω = α. In contrast to H(e jω, A(e jω ) is analytic, that is, its derivative exists for all ω. As a result, in contrast to H(e jω ), Ψ(e jω ) a continuous function of ω. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 11/40

12 Additional insights Ψ(e jω ) = αω + β is called a generalized linear phase. Recall the following Fourier transform relations: h e [n] A e (e jω ) = h e [n n 0 ] A e (e jω )e jωn0. h o [n] ja o (e jω ) = h o [n n 0 ] A o (e jω )e jωn0+jπ/2. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 12/40

13 Design of FIR filters by windowing Suppose we wish to approximate an ideal frequency response H d (e jω ) = h d [n]e jωn n= with an FIR filter h[n], n = 0,..., M by minimizing the MSE ɛ 2 = 1 2π π π H(e jω ) H d (e jω ) 2 dω. Using Parsevals identity, one can express ɛ 2 as ɛ 2 = M (h[n] h d [n]) 2 + n=0 1 n= (h d [n]) 2 + Thus, the optimal solution is { h d [n], n = 0,..., M h[n] =, 0, otherwise which is nothing else but a truncation of h d [n]. n=m+1 (h d [n]) 2 Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 13/40

14 Truncation in the DTFT domain The truncated impulse response is the product of h d [n] with which suggests w[n] = u[n] u[n M 1], H(e jω ) = 1 2π π π H d (e jθ )W (e j(ω θ) )dθ, where W (e jω ) = sin [ω(m + 1)/2] e jωm/2. sin ω/2 The convolution produces passband and stopband ripples which is a result of the Gibbs phenomenon. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 14/40

15 Truncation in the DTFT domain (cont.) Truncating h d [n] yields passband ripples about 20 log 10 (1.0895) = = 0.75 db, as well as a minimum stopband attenuation of about 20 log 10 (1/0.0895) = 21 db, irrespective of the filter length M + 1. Both are not sufficient for practical applications. The design can be improved by using smoother windows. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 15/40

16 Filter design by windowing Below are the most commonly used windows. Bartlett: Hann: Hamming: w[n] = w[n] = 2n/M, 0 n M/2, w[n] = 2 2n/M, M/2 < n M,. 0, otherwise { cos(2πn/m), n = 0,..., M, 0, otherwise. { cos(2πn/m), n = 0,..., M, 0, otherwise. Blackman: { cos(2πn/m) cos(4πn/m), n = 0,..., M, w[n] = 0, otherwise. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 16/40

17 Fixed windows Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 17/40

18 Fixed windows (cont.) Windows are always symmetric, that is { w[m n], 0 n M, w[n] = 0, otherwise. Therefore, windowing an FIR filter with linear phase does not change its type. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 18/40

19 Kaiser window We see that there is a trade-off between main-lobe width and side-lobe height; we cannot reduce both quantities at the same time. Kaiser window maximizes the ratio of the energy in a frequency band about ω = 0 over the total energy of the window. Formally, [ ] I 0 β 1 [(n α)/α] 2 w[n] = I 0(β), 0 n M, 0, otherwise, where α = M/2 and I 0 is the zeroth-order modifier Bessel function of the first kind. The shape of the Kaiser window is controlled by the shape parameter β (in MATLAB: kaiser(l,beta)), which controls the aforementioned trade-off. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 19/40

20 Kaiser window (cont.) Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 20/40

21 Kaiser window (cont.) A prescribed value A for A s is related to β according to 0, A < 21, β = (A 21) (A 21), 21 A 50, (A 8.7), A > 50, where A = 20 log 10 δ, with δ = max{δ p, δ s } (since the windowing method produces filters with δ p δ s, irrespective of the design specifications.) The range 0 β 8 provides useful windows. The order M required to achieve prescribed values of A and ω is estimated as M = A ω. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 21/40

22 FIR filter design using fixed windows 1 Given the design specifications {ω p, ω s, A p, A s }, determine the ripples δ p and δ s and set δ = min{δ p, δ s }. 2 Determine the cutoff frequency of the ideal LPF as ω c = ωp+ωs 2. 3 Determine the parameters A = 20 log 10 δ and ω = ω s ω p. 4 From the table, choose the window function that provides the smallest stopband attenuation greater than A. 5 For this window function, determine the required value of M by selecting the corresponding value of ω. 6 Determine the impulse response of the ideal lowpass filter by h d [n] = sin [ω c(n M/2)]. π(n M/2) 7 Compute the impulse response h[n] = h d [n] w[n] using the chosen window. 8 Check whether the designed filter satisfies the prescribed specifications; if not, increase M and go back to step 6. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 22/40

23 Example: M = 40, ω c = π/4 Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 23/40

24 Multiband filter design The above approach is straightforward to generalize to the design of highpass, bandpass, bandstop, and other multiband filters. For example, the magnitude response of a 5-band ideal multiband filter can look like In general, the impulse response of a multiband filter is given by h mb [n] = K (A k A k+1 ) sin [ω k(n M/2)], π(n M/2) k=1 where K is the number of bands. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 24/40

25 Design of FIR filters by frequency sampling Suppose we are given L samples of a desired frequency response H d [k] = H d (e j2πk/l ), k = 0, 1,..., L 1. The associated impulse response is given by h d [n] = 1 L which is L periodic. L 1 H d [k]w nk L = h d [n ml], m Z k=0 Multiplying h d with a window of length L results in with its DTFT given by h[n] = h d [n]w[n], H(e jω ) = 1 L 1 H d [k]w (e j(ω 2πk/L) ). L k=0 Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 25/40

26 Design by frequency sampling (cont.) Thus, the frequency response of the designed filter is obtained by interpolating between the samples H d [k] using W (e jω ) as an interpolation function. The windowing method creates an FIR filter by truncating h d [n]; the frequency sampling method creates an FIR filter by aliasing or folding h d [n]. If w[n] is an L-point rectangular window then h[n] is the primary period of h[n]. In this case the approximation error is zero at the sampling frequencies, and finite between them. For nonrectangular windows, the approximation error is not zero at the sampling frequencies. In practice, to minimize the time-domain aliasing distortion, we start with a large value for L, and then we choose a much smaller value for the length of the window w[n]. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 26/40

27 Design by frequency sampling (cont.) To guarantee the resulting filter has linear phase, we use the following formulas H d [k] = A d [k]e jψ d[k], A d [k] = Ψ d [k] = { A d (e j0 ), k = 0, A d (e j2π(l k)/l ), k = 1,..., L, { ± π 2 q L 1 2 π 2 q + L 1 2 where Q = (L 1)/2. 2π L 2π L k, k = 0, 1,..., Q, (L k), k = Q + 1,..., L 1, Note that the parameter q = 0 for type I-II FIR filters, and q = 1 for type III-IV FIR filters. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 27/40

28 Example: w c = 0.45π, M = 19 Note that the samples of the frequency response transition sharply from passband to stopband. Also, the approximation error is larger near the sharp transition and is smaller away from it. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 28/40

29 Better design approaches Since the transition band is typically unspecified, we can improve the quality of the approximation by enforcing a smoother transition band. For many applications, it is sufficient to use either a straight-line roll-off A(e jω ) = (ω s ω)/(ω s ω p ), ω p < ω < ω s, or a raised-cosine roll-off A(e jω ) = cos(π(ω s ω)/(ω s ω p )), ω p < ω < ω s. The main drawback is to create a wider transition bandwidth in the resulting design which can be reduced by increasing the order M. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 29/40

30 Example: w c = 0.45π, M = 19 Notice a substantial reduction in passband and stopband ripples. The ripples obtained using the raised-cosine approach are almost non-noticeable. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 30/40

31 Example: w c = 0.45π, M = 19 Using non-rectangular windows can also be used to reduce passband and stopband ripple. The resulting frequency response is similar to the one obtained using the raised-cosine smooth transition. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 31/40

32 Design procedure To design a lowpass filter with specifications ω p, ω s, A p, and A s, the following steps can be used: 1 Choose the order of the filter M by placing at least two samples in the transition band. 2 For a window design approach obtain samples of the desired frequency response H d [k]. For a smooth transition band approach, add transition band samples. 3 Compute the (M + 1)-point IDFT of H d [k] to obtain h[n]. For a window design approach multiply h[n] by the appropriate w[n]. 4 Compute the log-magnitude response and verify the design over passband and stopband. 5 If the specifications aren t met, increase M and go back to step 1. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 32/40

33 Example Consider the following specifications of a lowpass filter: ω p = 0.25π, ω s = 0.35π, A p = 0.1 db, A s = 40 db. Since the transition bandwidth is 0.1π, we will need more than 40 samples of the ideal frequency response. M = 44; L = M+1; alpha = M/2; Q = floor(alpha); Psi = -alpha*2*pi/l*[(0:q), -(L-(Q+1:M))]; Dw = 2*pi/L; k1 = floor(wp/dw); k2 = ceil(ws/dw); w = ((k2-1):-1:(k1+1))*dw; A = *cos(pi*(ws-w)/(ws-wp)); Ad = [ones(1,k1+1),a,zeros(1,l-2*k2+1),... flip(a),ones(1,k1)]; Hd = Ad.*exp(1j*Psi); h = real(ifft(hd)); %phase delay parameter %desired phase %transition band %trans. band samples Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 33/40

34 Example (cont.) The stopband attenuation (in db) of the designed filter can be measured as: H = freqz(h,1,linspace(0,2*pi,1001)); dw = 2*pi/1000; maxmag = max(abs(h)); Asd = min(-20*log10(abs(h(ceil(ws/dw):501))/maxmag)) Asd = The measured value of the stopband attenuation does not satisfy the given specifications. Clearly, we need to increase M or use another approach. Since a Hamming window provides more than 50 db of attenuation, one can use it instead of the rectangular window. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 34/40

35 Example (cont.) In this case, the script remains largely the same except for: omc = (wp+ws)/2; %cutoff frequency k1 = floor(omc/dw); k2 = ceil(omc/dw); Ad = [ones(1,k1+1),zeros(1,l-2*k2+1),ones(1,k1)]; Hd = Ad.*exp(1j*Psi); h = real(ifft(hd)).*hamming(l)'; %windowed impulse response In this case, the measured value of the stopband attenuation is 37.4 db, which is still insufficient. By increasing the value of M and checking for the resulting stopband attenuation, one can verify that M = 50 satisfies the given specifications for both approaches. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 35/40

36 Example (cont.) The above example shows that the frequency sampling technique is not well suited to standard filter design, because it is difficult, if not impossible, to determine a priori the number of samples needed for the correct design. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 36/40

37 Equiripple filter design Problem formulation Given a desired amplitude response A d (e jω ) and a weighting function W (ω) 0, the actual amplitude response A(e jω ) is characterized by the approximation error defined as E(ω) W (ω) [A d (e jω ) A(e jω )]. The design objective is to find the coefficients of a Type I-IV FIR filter minimizing the weighted Chebyshev error, defined by E = max ω B E(ω), where B is a union of disjoint closed subsets of [0, π]. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 37/40

38 Equiripple filter design (cont.) Since we focus on FIR filters with linear phase, A(e jω ) can be written as A(e jω ) = Q(e jω )P (e jω ) with 1, Type I R P (e jω ) = p[k] cos(ωk) and Q(e jω cos(ω/2), Type II ) = sin(ω), Type III k=0 sin(ω/2), Type IV where R = M/2, if M is even, and R = (M 1)/2, if M is odd. Consequently, the error E(ω) can be redefined as E(ω) = W (ω)q(e jω ) }{{} W (ω) [ Ad (e jω ) Q(e jω ) }{{} Ā d (e jω ) ] P (e jω ) = W ] (ω)[ād (e jω ) P (e jω ). The Chebyshev approximation problem {p[k]} R k=0 = arg min E. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 38/40

39 Alternation theorem for FIR filters The solution to the Chebyshev approximation problem is characterized by the alternation theorem. Theorem (Alternation theorem for FIR filters) A necessary and sufficient condition for P (e jω ) to be the unique solution of the Chebyshev approximation problem is that E(ω) exhibits at least R + 2 alternations in B. That is, there must exist R + 2 extremal frequencies ω 1 < ω 2 <... < ω R+2 such that for every k = 1,..., R + 2 E(ω k ) = E(ω k+1 ), E(ω k ) = max E(ω) δ. ω B The alternation theorem implies that the best Chebyshev approximation must have an equiripple error function. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 39/40

40 Alternation theorem for FIR filters (cont.) The conditions characterizing the optimum filter can be used to develop an efficient algorithm for obtaining filter coefficients. Parks and McClellan (1972) solved this problem by means of the Remez exchange algorithm. Modern methods of filter design are based on the tools of convex optimization. See, for example, FIR Filter Design Factorization and Convex Optimization, by S. P. Wu, S. Boyd, and L. Vandenberghe. Prof. O. Michailovich, Dept of ECE, Winter 2017 ECE 413: Digital Signal Processing / Section 8 40/40