Digital filter design 1 FREQUENCY RESPONSE OF FILTERS x(t) - G - y(t) For input x(n) = cos(!n) the output is y(n) = AG(!) cos(!n + 'G(!)) AG(!): filter gain or magnitude 'G(!): filter phase 2 Reason: the output can always be written as y(n) = h(0) cos(!n) + h(1) cos(!n − !) + ··· h(k) cos(!n − !k) + ··· = AG(!) cos(!n + 'G(!)) The functions AG(!), 'G(!) form the frequency response of a filter Matlab routine: freqz(B,A,W) where B= [b0 a2 : : : bM ], B= [1 a1 a2 : : : aN ], W: vector with frequencies (if specified) 3 Relation between transfer function and frequency response Let x(n) = ej!n = cos(!n) + j sin(!n) Then: y(n) = h(0)ej!n + h(1)ej!(n−1) + ··· h(k)ej!(n−k) + ··· ( ) = h(0) + h(1)e−j! + ··· h(k)e−j!k + ··· ej!n = H(ej!) ej!n where H(z) is the transfer function of the system. 4 H(ej!) is the frequency response: determines how a frequency component ! is affected in magnitude and phase. H(ej!) is a complex number which can be written in polar form: j! H(ej!) = jH(ej!)j ej arg(H(e )) Then for input x(n) = ej!n = cos(!n) + j sin(!n) we have: y(n) = H(ej!) ej!n j! = jH(ej!)j ej arg(H(e )) ej!n j! = jH(ej!)j ej(!n+arg(H(e ))) ( ) = jH(ej!)j cos(!n + arg(H(ej!))) + j sin(!n + arg(H(ej!))) or y(n) = AH(!) cos(!n + 'H(!)) + jAH(!) sin(!n + 'H(!)) where j! AH(!) = jH(e )j (filter gain) and j! 'H(!) = arg(H(e )) (phase shift of filter) 5 FREQUENCY RESPONSE OF SERIES CONNECTION y1(n) - - - x(n) H1 H2 y(n) For x[n] = cos(!n) we have: y1(n) = A1(!) cos(!n + '1(!)) y(n) = A2(!)A1(!) cos(!n + '1(!) + '2(!)) ) Magnitude of series connection is multiplicative: A(!) = A2(!) · A1(!) Phase of series connection is additive: '(!) = '2(!) + '1(!) 6 FILTER DESIGN Example: Low-pass filter magnitude specs 1.2 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 - maximum deviation in passband, 1 − δp ≤ A(f) ≤ 1 + δp - damping in stopband, A(f) ≤ δs - width of transition band (normalized) 7 Filter magnitude specs often defined in dB (decibel) units: - maximum deviation in passband: Ap = 20 log(1 + δp) (dB) Note: Ap ≈ 8:7δp for small δp - damping in stopband: As = −20 log δs (dB) 8 FUNDAMENTAL RELATIONSHIP BETWEEN AMPLITUDE AND PHASE Z 1 π d log(A(!0)) sin((!0 + !)=2) d!0 ≤ · '(!) 0 log 0 − 0 π 0 d log(! ) sin((! !)=2) ! 0 sin((! +!)=2) where log sin((!0−!)=2) > 0 Implication: d log(A(!)) Low-pass filter with large magnitude slope, d log(!) << 0 results in large negative phase shifts, '(!) << 0! 9 EFFECT OF PHASE SHIFT - LINEAR PHASE FILTERS Observation: Phase shift in passband may cause signal distortion. Illustrative Example Consider a signal consisting of two sinusoidal components s(n) = s1(n) + s2(n) where s1(n) = sin(!1n) s2(n) = sin(!2n) 10 1 0.5 s 1 0 −0.5 −1 1 0.5 s 2 0 −0.5 −1 2 1 s 0 −1 −2 Top: s1(n) Middle: s2(n) Below: s(n) 11 Add a high-frequency disturbance e(n), x(n) = s(n) + e(n) where e[n] = sin(!en);!e > !1;!2 2 1 s 0 −1 −2 1 0.5 e 0 −0.5 −1 2 x 0 −2 12 Use low-pass filtering to suppress the disturbance 1 1 0.5 0.5 s ,y s ,y 1 1 0 1 1 0 −0.5 −0.5 −1 −1 1 1 0.5 0.5 s ,y s ,y 2 2 0 2 2 0 −0.5 −0.5 −1 −1 2 2 1 1 s,y s,y 0 0 −1 −1 −2 −2 Left: Nonlinear phase filter gives passband signal distortion Right: Linear phase filter reconstructs delayed version of passband signal 13 Linear phase property: x(t) - G - y(t) For input x(n) = cos(!n) we have the output y(n) = A(!) cos(!n + '(!)) To avoid phase distortion we require that all frequencies in passband delayed are by the same time, i.e., y(n) = A(!) cos(!(n − τ)) in passband. This is achieved if the filter gives phase shift '(!) = −! · τ 14 PHASE SHOULD BE A LINEAR FUNCTION OF FREQUENCY IN PASSBAND! Implication for frequency response: j! We should have 'H(!) = arg(H(e )) = −!τ or H(ej!) = jH(ej!)je−j!τ in passband. • Linear phase property can be achieved with symmetric FIR filters • Linear property can be achieved only approximately with IIR filters 15 LINEAR PHASE FIR FILTERS Linear phase property is achieved by symmetric FIR filters: y(n) = h(0)x(n) + h(1)x(n − 1) + ··· +h(1)x(n + 2 − N) + h(0)x(n + 1 − N); n = 0; 1; 2;:::;M where h(k) = h(N − 1 − k); k = 0; 1;:::;N − 1 For input x(n) = cos(!n) the output is y(n) = A(!) cos(!n − !(N − 1)=2) = A(!) cos(!(n − (N − 1)=2)) ) Each frequency component delayed by (N − 1)=2 time steps. 16 REASON: Pairwise combination of terms k and N − 1 − k gives ( ) h(k) x(n − k) + x(n + 1 − N + k) = ( ) = h(k) cos(!(n − k)) + cos(!(n + 1 − N + k)) ( N−1 N−1 = h(k) cos(!(n − ) + !( − k)) 2 2 ) N−1 N−1 + cos(!(n − ) − !( − k)) 2 2 N−1 = h(k)2 cos(!( − k)) · cos(!(n − (N − 1)=2)) 2 where we have used cos(α ± β) = cos α · cos β ∓ sin α · sin β ) All frequency components x(n) = cos(!n) are delayed by (N −1)=2 samples 17 From above, we have (for N even) y(n) = AH(!) cos(!n + 'H(!)) N=X2−1 N−1 = h(k)2 cos(!( − k)) · cos(!(n − (N − 1)=2)) 2 k=0 ) Filter has frequency response with: • Linear phase: 'H(!) = −!(N − 1)=2 • Gain: N=X2−1 N−1 A (!) = 2 h(k) cos(!( − k)) H 2 k=0 Similarly if N is odd-valued (see lecture notes) 18 FIR FILTER DESIGN We consider two important FIR filter design methods: • Windowed-Sinc filters • Optimization-based FIR filters 19 Windowed-Sinc filters The idea is to modify the ideal filter. The ideal low-pass filter with bandwidth fc has impulse response hD(0) = 2fc sin(2πf k) h (k) = c ; k = ±1; ±2;::: D πk Cannot be implemented, because: • impulse response hD(k) cannot be realized by any finite-dimensional difference equation • not causal (output depends on future inputs) 20 Practical low-pass FIR filter: • cut the ideal filter response to finite length N = 2M + 1 (hD(k); k = −M; −M + 1;:::;M), y(n) = hD(−M)x(n + M) + ··· + hD(−1)x(n + 1) + +hD(0)x(n) + hD(1)x(n − 1) + ··· + hD(M)x(n − M) • make it causal by introducing a delay M: y(n) = hD(−M)x(n) + ··· + hD(−1)x(n − M + 1) + +hD(0)x(n − M) + ··· + hD(M)x(n − 2M − 1) • multiply impulse response coefficients by a window function w(k) in order to smooth out end effects: y(n) = hw(−M)x(n) + ··· + hw(0)x(n − M) + ··· + hw(M)x(n − N); where hw(k) = w(k)hD(k), N = 2M + 1. 21 Some common window functions: • Rectangular window (trivial), w(k) = 1; jkj ≤ (N − 1)=2 • Hanning window, w(k) = 0:5 + 0:5 cos(2πk)=N); jkj ≤ (N − 1)=2 • Hamming window, w(k) = 0:54 + 0:46 cos(2πk=N); jkj ≤ (N − 1)=2 • Blackman window, w(k) = 0:42 + 0:5 cos(2πk=N) + 0:08 cos(4πk=N); jkj ≤ (N − 1)=2 22 Observe that: • Length of windowed filter is N = 2M + 1 (odd) • Windowed filters are linear phase, as they satisfy symmetry property: hD(−k) = hD(k) and w(−k) = w(k) ) hw(−k) = hw(k) Selection of window function is a trade-off between performance and complexity: • Performance (deviation in passband and damping in stopband) is determined by the window function. • Filter length N is determined by width ∆f of transition band: for each window function, ∆f = cw=N, where cw = window-specific constant. • Better passband and stopband performance implies increased filter length for a given width of transtion band. 23 Summary of windowed filter properties: Window Width of Maximal passband Damping in function transition band deviation stopband (Hz, normed) (dB) (dB) Rectangular 0:9=N 0.7416 21 Hanning 3:1=N 0.0546 44 Hamming 3:3=N 0.0194 53 Blackman 5:5=N 0.0017 74 Kaiser 2:93=N (β = 4:54) 0.0274 50 4:32=N (β = 6:76) 0.00275 70 5:71=N (β = 8:96) 0.000275 90 24 IIR FILTERS Good filter performance requires a sufficiently long memory in order to allow distinct response to different frequency components. In FIR filters this implies a large number of taps. IIR filters have infinite memory due to recursiveness ) simpler filter for given magnitude performance specifications Standard IIR filters: - Butterworth - Chebyshev - Elliptic 25 Butterworth filters Digital low-pass Butterworth filter with cut-off frequency !c = 2πfc has frequency response magnitude [ ]1=2 j j! j 1 BN (e ) = 2N (1) 1 + [tan(!=2)= tan(!c=2)] - monotonically varying gain in both passband and stopband Matlab routines: [B,A]=butter(N,Wn,'ftype') designs filter [N,Wn]=buttord(Wp,Ws,Rp,Rs) computes required minimum order 26 Chebyshev filters • Chebyshev filter type I has equiripple gain in passband and monotonically varying gain in stopband.