Discrete-Time Filters

Transcription

1 Discrete-Time Filters

2 Puttig LTI Systems to Work x h y y[] = x[] h[] = h[ m] x[m] m= Y (z) = H(z) X(z), Y (ω) = H(ω) X(ω) Goal: Desig a LTI system to perform a certai task i some applicatio Key questios: What is the rage of tasks that a LTI system ca perform? What are the parameters uder our cotrol for desig purposes?

3 What Do LTI Systems Do? Recall Eigeaalysis LTI system eigevectors: s ω [] = e jω LTI system eigevalues: λ ω = H(ω) = = h[] e jω (frequecy 4 respose) λ ω cos(ω) cos(ω) s ω h λω s ω λ ω si(ω) si(ω)

4 LTI Systems Filter Sigals x h y Importat iterpretatio of Y (ω) = H(ω) X(ω) x[] = π dω X(ω) ejω π 2π h y[] = π dω H(ω) X(ω) ejω π 2π A LTI system processes a sigal x[] by amplifyig or atteuatig the siusoids i its Fourier represetatio (DTFT) X(ω) by the complex factor H(ω) Ispires the termiology that X(ω) is filtered by H(ω) to produce Y (ω)

5 Desig Parameters of Discrete-Time Filters (LTI Systems) x H y Impulse respose: h[] Trasfer fuctio: poles ad zeros H(z) Frequecy respose: H(ω) + + Movig/recursive average parameters: {a i }, {b i }

6 Filters Archetypes: Low-Pass Ideal low-pass filter Example low-pass impulse respose h[].5 h[] H (ω) Example frequecy respose H(ω) π π/2 ω π/2 π

7 Filters Archetypes: High-Pass Ideal high-pass filter Example high-pass impulse respose h[] h[] H (ω) Example frequecy respose H(ω) π π/2 ω π/2 π

8 Filters Archetypes: Bad-Pass Ideal bad-pass filter Example bad-pass impulse respose h[] h[] H (ω) Example frequecy respose H(ω) 2 π π/2 ω π/2 π

9 Filters Archetypes: Bad-Stop Ideal bad-stop filter Example bad-stop impulse respose h[] h[] H (ω) Example frequecy respose H(ω) π π/2 ω π/2 π

10 Summary Now that we uderstad what LTI systems do, we ca desig them to accomplish certai tasks A LTI system processes a sigal x[] by amplifyig or atteuatig the siusoids i its Fourier represetatio (DTFT) Equivalet desig parameters of a discrete-time filter Impulse respose: h[] z-trasform: H(z) (poles ad zeros) Frequecy respose: H(ω) Movig/recursive average parameters: {a i}, {b i} Archetype filters: Low-pass, high-pass, bad-pass, bad-stop We will emphasize ifiite-legth sigals, but the situatio is similar for fiite-legth sigals

11 Discrete-Time Filter Desig

12 Recall Discrete-Time Filter x h y y[] = x[] h[] = h[ m] x[m] m= Y (z) = H(z) X(z), Y (ω) = H(ω) X(ω) A discrete-time filter fiddles with a sigal s Fourier represetatio Recall the filter archetypes: ideal low-pass, high-pass, bad-pass, bad-stop filters

13 Ideal Lowpass Filter Ideal low-pass filter frequecy respose H(ω) = { ω c ω ω c otherwise Impulse respose is the ifamous sic fuctio h[] = 2ω c si(ω c) ω c.5 h[] Problems: System is ot BIBO stable! ( h[] = ) Ifiite computatioal complexity (H(z) is ot a ratioal fuctio)

14 Filter Specificatio Fid a filter of miimum complexity that meets a give specificatio Example: Low-pass filter Pass-bad edge frequecy: ω p Stop-bad edge frequecy: ω s Betwee pass- ad stop-bads: trasitio bad Pass-bad ripple ɛ p (ofte expressed i db) Stop-bad ripple ɛ s (ofte expressed i db) Clearly, the tighter the specs, the more complex the filter

15 Two Classes of Discrete-Time Filters Ifiite impulse respose (IIR) filters Uses both movig ad recursive averages H(z) has both poles ad zeros Related to aalog filter desig usig resistors, capacitors, ad iductors Geerally have the lowest complexity to meet a give spec + + Fiite impulse respose (FIR) filters Uses oly movig average H(z) has oly zeros Uachievable i aalog usig resistors, capacitors, ad iductors Geerally higher complexity (tha IIR) to meet a give spec But ca have liear phase (big plus)

16 Summary A discrete-time filter fiddles with a sigal s Fourier represetatio Ideal filters are ot practical System is ot BIBO stable! Ifiite computatioal complexity (H(z) is ot a ratioal fuctio) Filter desig: Fid a filter of miimum complexity that meets a give spec Two differet types of filters (IIR, FIR) mea two differet types of filter desig

17 IIR Filter Desig

18 IIR Filters Use both movig ad recursive averages + + Trasfer fuctio has both poles ad zeros H(z) = Y (z) X(z) = b + b z + b 2 z 2 + b M z M + a z + a 2 z a N z N = z N M (z ζ )(z ζ 2 ) (z ζ M ) (z p )(z p 2 ) (z p N ) We desig a IIR filter by specifyig the locatios of its poles ad zeros i the z-plae Geerally ca satisfy a spec with lower complexity tha FIR filters

19 IIR Filters from Aalog Filters I cotrast to FIR filter desig, IIR filters are typically desiged by a two-step procedure that is slightly ad hoc Step : Desig a aalog filter (for resistors, capacitors, ad iductors) usig the Laplace trasform H L (s) (this theory is well established but well beyod the scope of this course) Step 2: Trasform the aalog filter ito a discrete-time filter usig the biliear trasform (a coformal map from complex aalysis) s = c z z + The discrete-time filter s trasfer fuctio is give by H(z) = H L (s) s=c z z+

20 Three Importat Classes of IIR Filters Butterworth filters butter commad i Matlab No ripples (oscillatios) i H(ω) Getlest trasitio from pass-bad to stop-bad for a give order Chebyshev filters cheby ad cheby2 commads i Matlab Ripples i either pass-bad or stop-bad Elliptic filters ellip commad i Matlab Ripples i both pass-bad ad stop-bad Sharpest trasitio from pass-bad to stop-bad for a give order (use with cautio!).5 H (ω) π π/2 ω π/2 π

21 Butterworth IIR Filter Maximally flat frequecy respose Largest umber of derivatives of H(ω) equal to at ω = ad π N zeros ad N poles Zeros are all at z = Poles are located o a circle iside the uit circle Example: N = 6 usig butter commad i Matlab h[] H (ω).8 Im(z ) π π/2 ω π/2 π.5.5 Re(z)

22 Chebyshev Type IIR Filter Ripples/oscillatios (of equal amplitude) i the pass-bad ad ot i the stop-bad N zeros ad N poles Zeros are all at z = Poles are located o a ellipse iside the uit circle Example: N = 6 usig cheby commad i Matlab h[] H (ω).8 Im(z ) π π/2 ω π/2 π Re(z)

23 Chebyshev Type 2 IIR Filter Ripples/oscillatios (of equal amplitude) i the stop-bad ad ot i the pass-bad N zeros ad N poles Zeros are distributed o uit circle Poles are located o a ellipse iside the uit circle Example: N = 6 usig cheby2 commad i Matlab h[] H (ω).8 Im(z ) π π/2 ω π/2 π Re(z)

24 Elliptic IIR Filter Ripples/oscillatios i both the stop-bad ad pass-bad N zeros ad N poles Zeros are clustered o uit circle ear ω p Poles are clustered close to uit circle ear ω p Example: N = 6 usig ellip commad i Matlab h[] H (ω).8 Im(z ) π π/2 ω π/2 π Re(z)

25 IIR Filter Compariso Butterworth (black), Chebyshev (blue), Chebyshev 2 (red), Elliptic (gree) H (ω).5 π π/2 π/2 π ω

26 Summary IIR filters use use both movig ad recursive averages ad have both poles ad zeros Typically desiged by trasformig a aalog filter desig (for use with resistors, capacitors, ad iductors) ito discrete-time via the biliear trasform Four families of IIR filters: Butterworth, Chebyshev (,2), Elliptic Useful Matlab commads for choosig the filter order N that meets a give spec: butterord, chebyord, cheby2ord, ellipord

27 FIR Filter Desig

28 FIR Filters Use oly a movig average Trasfer fuctio has oly zeros (ad trivial poles at z = ) H(z) = Y (z) X(z) = b + b z + b 2 z 2 + b M z M = z M (z ζ )(z ζ 2 ) (z ζ M ) We desig a FIR filter by specifyig the values of the taps b, b,..., b M (this is equivalet to specifyig the locatios of the zeros i the z-plae) Geerally require a higher complexity to meet a give spec tha a IIR filter

29 FIR Filters Are Iterestig FIR filters are specific to discrete-time; they caot be built i aalog usig R, L, C FIR filters are always BIBO stable FIR filters ca be desiged to optimally meet a give spec Ulike IIR filters ad all aalog filters, FIR filters ca have (geeralized) liear phase A oliear phase respose H(ω) distorts sigals as they pass through the filter Recall that a liear phase shift i the DTFT is equivalet to a simple time shift i the time domai

30 Impulse Respose of a FIR Filter Easy to see by iputtig x[] = δ[] that the impulse respose of a FIR filter cosists of the taps weights 2 2 M M + M + 2 h[] b b b 2 b M Note: Filter order = M; filter legth = M +

31 Symmetric FIR Filters Ulike IIR filters, FIR filters ca be causal ad have (geeralized) liear phase Liear phase filters must have a symmetric impulse respose Four cases: eve/odd legth, eve/odd symmetry Differet symmetries ca be useful for differet filter types (low-pass, high-pass, etc.) We will focus here o low-pass filters with odd-legth, eve-symmetric impulse respose Odd legth: M + is odd (M is eve) Eve symmetric (aroud the ceter of the filter): h[] = h[m ], =,,..., M Example: Legth M + = 2 h[]

32 Frequecy Respose of a Symmetric FIR Filter () Compute frequecy respose whe h[] is odd-legth ad eve-symmetric (h[] = h[m ]) H(ω) = M h[] e jω = = M/2 = h[] e jω + h[m/2] e jωm/2 + M =M/2+ h[] e jω = = M/2 = M/2 = h[] e jω + h[m/2] e jωm/2 + h[] e jω + h[m/2] e jωm/2 + M =M/2+ M/2 r= h[m ] e jω h[r] e jω(m r) = h[m/2] e jωm/2 + M/2 = h[] (e jω + e jω( M))

33 Frequecy Respose of a Symmetric FIR Filter (2) Compute frequecy respose whe h[] is odd-legth ad eve-symmetric (h[] = h[m ]) H(ω) = h[m/2] e jωm/2 + M/2 = h[] (e jω + e jω( M)) = h[m/2] e jωm/2 + = h[m/2] + = A(ω) e jωm/2 M/2 = M/2 = h[] e jωm/2 ( e jω( M/2) + e jω( M/2)) 2 h[] cos(ω( M/2)) e jωm/2

34 Geeralized Liear Phase FIR Filters Frequecy respose whe h[] is odd-legth ad eve-symmetric (h[] = h[m ]) H(ω) = A(ω) e jωm/2 with A(ω) = h[m/2] + M/2 = 2 h[] cos(ω( M/2)) A(ω) is called the amplitude of the filter; it plays a role like H(ω) sice However, A(ω) is ot ecessarily H(ω) = A(ω) e jωm/2 is a liear phase shift H(ω) has liear phase except whe A(ω) chages sig, i which case its phase jumps by π rad

35 FIR Filter Desig Frequecy respose whe h[] is odd-legth ad eve-symmetric (h[] = h[m ]) H(ω) = A(ω) e jωm/2 with A(ω) = h[m/2] + M/2 = 2 h[] cos(ω( M/2)) Desig of H(ω) is equivalet to the desig of A(ω); spec chages slightly Stop-bad spec ow allows egative values i A(ω) For simplicity, same ɛ i both pass- ad stop-bad (this is easy to geeralize)

36 Optimal FIR Filter Desig Goal: Fid the optimal A(ω) (i terms of shortest legth M + ) that meets the specs A(ω) = h[m/2] + M/2 = 2 h[] cos(ω( M/2)) Parameters uder our cotrol: The M/2 + filter taps h[], =,,..., M/2 Problem solved by James McClella ad Thomas Parks at Rice Uiversity (97) Parks-McClella Filter Desig

37 Key Igrediets of Optimal FIR Filter Desig Goal: Fid the optimal A(ω) (i terms of shortest legth M + ) that meets the specs A(ω) = h[m/2] + M/2 = 2 h[] cos(ω( M/2)) Ripples: A(ω) oscillates M/2 times i the iterval ω π Equiripple property: The oscillatios of the optimal A(ω) are all the same size Alteratio Theorem: The optimal A(ω) will touch the error bouds M/2 + 2 times i the iterval ω π

38 Remez Exchage Algorithm for Optimal FIR Filter Desig Goal: Fid the optimal A(ω) (i terms of shortest legth M + ) that meets the specs A(ω) = h[m/2] + M/2 = 2 h[] cos(ω( M/2)) Alteratio Theorem: iterval ω < π The optimal A(ω) will touch the error bouds M/2 + 2 times i the Parks ad McClella proposed the Remez Exchage Algorithm to fid the h[] such that A(ω) satisfies the alteratio theorem Matlab commad firpm ad firpmord (be careful with the parameters)

39 Example : Optimal FIR Filter Desig () Optimal low-pass filter of legth M + = 2 with ω p =.3π, ω s =.35π Note the M/2 + 2 = 2 alteratios H (ω) π π/2 ω π/2 π h[] Im(z ) Re(z)

40 Example : Optimal FIR Filter Desig (2) Optimal low-pass filter of legth M + = 2 with ω p =.3π, ω s =.35π Note the M/2 + 2 = 2 alteratios log H (ω) Im(z ).5 π π/2 π/2 π ω h[] Re(z)

41 Example 2: Optimal FIR Filter Desig () Optimal low-pass filter of legth M + = with ω p =.3π, ω s =.35π Note the M/2 + 2 = 52 alteratios H (ω) π π/2 ω π/2 π h[] Im(z ) Re(z)

42 Example 2: Optimal FIR Filter Desig (2) Optimal low-pass filter of legth M + = with ω p =.3π, ω s =.35π Note the M/2 + 2 = 52 alteratios log H (ω) Im(z ) 5 π π/2 ω π/2 π h[] Re(z)

43 Matlab Example: Optimal FIR Filter Desig Process a chirp sigal through a optimal low-pass filter with Legth M + = ω p = π/3 ω s = π/2

44 Summary FIR filters correspod to a movig average ad have oly zeros (o poles) FIR filters are specific to discrete-time; they caot be built i aalog usig R, L, C Symmetrical FIR filters have (geeralized) liear phase, which is impossible with IIR or aalog filters Desig optimal FIR filters usig the Parks-McClella algorithm (Remez exchage algorithm) FIR filters are always BIBO stable ad very umerically stable (to coefficiet quatizatio, etc.) Geerally require a higher complexity to meet a give spec tha a IIR filter, but the beefits ca outweigh the computatioal cost

45 Iverse Filter ad Decovolutio

46 LTI Sigal Degradatios I may importat applicatios, we do ot observe the sigal of iterest x but rather a versio y processed by a LTI system with impulse respose g x g y Examples: Digital subscriber lie (DSL) commuicatio (log wires) Echos i audio sigals Camera blur due to misfocus or motio (2D) Medical imagig (CT scas),... Goal: Ameliorate the degradatio by passig y through a secod LTI system i the hopes that we ca cacel out the effect of the first such that x = x x g y h x

47 LTI Sigal Degradatios i the z-trasform Domai Goal: Ameliorate the degradatio by passig y through a secod LTI system i the hopes that we ca cacel out the effect of the first such that x = x x g y h x Easy to uderstad usig z-trasform X(z) = H(z) Y (z) = H(z) G(z) X(z) Therefore, i order to have x = x, ad thus X(z) = X(z), we eed H(z) = G(z) H(z) G(z) = or H(z) = G(z) is called the iverse filter, ad this process is called decovolutio

48 Iverse Filter Poles ad Zeros If the degradatio filter G(z) is a ratioal fuctio with zeros {ζ i } ad poles {p j } G(z) = z N M (z ζ )(z ζ 2 ) (z ζ M ) (z p )(z p 2 ) (z p N ) the the iverse filter H(z) is a ratioal fuctio with zeros {p j } ad poles {ζ i } H(z) = G(z) = (z p )(z p 2 ) (z p N ) zm N (z ζ )(z ζ 2 ) (z ζ M ) Assumig that G(z) ad H(z) are causal, if ay of the zeros of G(z) are outside the uit circle, the H(z) is ot BIBO stable, which meas that the iverse filter does ot exist Whe G(z) is causal ad all of its zeros are iside the uit circle, we say that it has miimum phase; i this case a exact iverse filter H(z) exists

49 Example: Exact Iverse Filter.5 x[] G 4 2 y[] y[] H x[] G(ω) (blue) ad H(ω) = G(ω) (red) G(z) H(z) π π/2 π/2 π ω Im(z ) Im(z ) Re(z).5.5 Re(z)

50 Approximate Iverse Filter Whe G(z) is o-miimum phase, a exact iverse filter does ot exist, because or more poles outside the uit circle G(z) has oe We ca still fid a approximate iverse filter by regularizig G(z) ; for example H a (z) = G(z) + r where r is a costat (techically this is called Tikhoov regularizatio) Typically we try to choose the smallest r such that H a (z) is BIBO stable We o loger have x = x, but rather x x

51 Example: Approximate Iverse Filter.5 x[] G 4 2 y[] y[] H a x[] G(ω) (blue) ad H a (ω) = G(ω)+ 6 (red) G(z) H a (z) Im(z ) Im(z ) π π/2 π/2 π ω Re(z).5.5 Re(z)

52 Summary Decovolutio: Ameliorate the degradatio from a LTI system G by passig the degraded sigal through a secod LTI system H i the hopes that we ca cacel out the effect of the first such that x = x x g y h x Iverse filter: Poles/zeros of G(z) become zeros/poles of H(z) H(z) = G(z) Best case: Whe G(z) is causal ad all of its zeros are iside the uit circle, we say that it has miimum phase; i this case a exact iverse filter H(z) exists Puzzler: What do we do whe N M i G(z)? Advaced topics beyod the scope of this course: blid decovolutio, adaptive filters (LMS alg.)

53 Matched Filter

54 Ier Product ad Cauchy Schwarz Iequality Recall the ier product (or dot product) betwee two sigals x, y (whether fiite- or ifiite-legth) y, x = y[] x[] Recall the Cauchy-Schwarz Iequality (CSI) y, x y 2 x 2 Iterpretatio: The ier product y, x measures the similarity of y ad x Large value of y, x y ad x very similar Small value of y, x y ad x very disimilar

55 Sigal Detectio Usig Ier Product We ca determie if a target sigal x of iterest is preset withi a give sigal y simply by computig the ier product ad comparig it to a threshold t > { t sigal is preset d = y, x < t sigal is ot preset (Aside: I certai useful cases, this is the optimal way to detect a sigal) Example: Detect the square pulse x i a oisy versio y = x + e; we calculate d = y, x = 4.92 x[] y[] = x[] + e[]

56 Sigal Detectio With Ukow Shift I may importat applicatios, the target is time-shifted by some ukow amout l Example: Square pulse with shift l = 5 y[] = x[ l] + e[] x[] y[] = x[ 5] + e[]

57 Solutio: Sigal Detectio With Ukow Shift I may importat applicatios, the target is time-shifted by some ukow amout l y[] = x[ l] + e[] Solutio: Compute ier product betwee y ad shifted target sigal x[ m] for all m Z d[m] = y[], x[ m] I statistics, d[m] is called the cross-correlatio; it provides both The detectio statistic to compare agaist the threshold t for each value of shift m A estimate for l (the m the maximizes d[]) Example: Square pulse with shift l = 5 ad m =, 7, 27 y[] = x[ 5] + e[]

58 Matched Filter Useful iterpretatio of the cross correlatio: Let x[] = x [ ]; the d[m] = y[], x[ m] = y[] x [ m] = y[] x[m ] I words, the cross-correlatio d[m] equals the covolutio of y[] with the time-reversed ad cojugated target sigal x[] = x [ ] x[] = x [ ] is the impulse respose of the matched filter y x [ ] d

59 Example: Matched Filter Square pulse shifted by l = 5: y[] = x[ 5] + e[] x[] y[] = x[ 5] + e[] d[]

60 Applicatio: Radar Imagig () I a radar system, the time delay l is liearly proportioal to 2 the distace betwee the atea ad the target x[] y[] = x[ 5] + e[] d[]

61 Applicatio: Radar Imagig (2) I a radar system, the time delay l is liearly proportioal to 2 the distace betwee the atea ad the target x[] y[] = x[ 5] + e[] d[]

62 Summary Ier product ad Cauchy Schwarz Iequality provide a atural way to detect a target sigal x embedded i aother sigal y Compare magitude of ier product to a threshold Whe the target sigal is time-shifted by a ukow time-shift l, compute the cross-correlatio: ier products at all possible time shifts Cross-correlatio ca be iterpreted as the covolutio of the sigal y with a time-reversed ad cojugated versio of x: the matched filter Matched filter is ubiquitous i sigal processig: radar, soar, commuicatios, patter recogitio ( Where s Waldo? ),...

63 Ackowledgemets c 24 Richard Baraiuk, All Rights Reserved