FIR Filter Design by Windowing

Size: px

Start display at page:

Download "FIR Filter Design by Windowing"

Harriet Atkins
5 years ago
Views:

1 FIR Filter Desig by Widowig Take the low-pass filter as a eample of filter desig. FIR filters are almost etirely restricted to discretetime implemetatios.

2 Passbad ad stopbad Magitude respose of a ideal low-pass aalog filter showig the toleraces: passbad: [0, w p ], trasitio regio:[w p, w s ], stopbad: [w s, ], passbad tolerace: p, stopbad tolerace: s.

3 FIR Filter Desig by Widowig The widowig method: cosider a ideal desired frequecy respose (eg. ideal low-pass filter) that ca be represeted as jw H ( e ) h [ ] e d d jw

4 Recall of DTFT Pairs Time domai rectagular widow DTFT domai Dirichlet kerel Time domai syc fuctio DTFT domai rectagular widow

5 FIR Filter Desig by Widowig Therefore, ideally we hope that h d [ ] is a syc fuctio i the time domai, so that its DTFT respose is ideal lowpass. However, we caot compute from - to A practical way to obtai a FIR filter is to use oly a fiite portio (i.e., trucatio) of the syc fuctio. That is, multiplyig the syc fuctio by a rectagle widow i time domai.

6 otherwise M h h d 0 0 ] [ ] [ i.e. where w[] is a rectagular widow: ] [ ] [ ] [ w h h d otherwise M w ] [

7 Time domai multiplicatio implies frequecy domai covolutio: H( e jw ) (1/ 2 j j( w ) ) H d ( e ) W ( e ) d where W(e jw ) is the frequecy respose of the rectagular widow w[], which is a Dirichlet kerel as follows. jw W ( e ) e jwm /2 M 0 e jw 1 e 1 e si[ wm ( 1) / 2] si( w / 2) jw( M 1) jw

8 As M icreases, the mai lobe width of the Dirichlet kerel decreases, ad so the shape approimates more to impulses. (The mai lobe is usually defied as the regio betwee the first zero-crossigs o either side of the origi. )

9 Covolutio of a rectagular widow ad a Dirichlet kerel i the DTFT domai: A illustratio The above shows a typical approimatio of the ideal lowpass filter by usig rectagular-widow.

10 Gibbs pheomeo of rectagular widow: as M icreases, the maimum amplitude of the oscillatio does ot approach zero whe the rectagular widow is used. Fuctioal approimatio of square wave usig 5 harmoics usig 25 harmoics usig 125 harmoics

11 Trasitio regio: the iterval betwee a passbad ad a stopbad. For the rectagular widow, the width of the mai lobe decreases as M icreases. Hece, the trasitio regio gets smaller but the ripple remais due to the Gibbs pheomeo. The Gibbs pheomeo always occurs ad results i sharp discotiuity i the frequecy respose whe the rectagular widow is used, which is udesired.

12 Solutio to Sharp Discotiuity of Rectagular Widow: Use widows with o abrupt discotiuity i their time domai respose ad cosequetly low side-lobes i their frequecy respose. Hece, like the case of spectrogram costructio, other widow fuctios with differet mai lobes ad side lobe gais are used for w[] to avoid Gibbs pheomeo. Whe usig other widow fuctios, the reduced ripple comes at the epese of a wider trasitio regio (remember that rectagular widow has the smallest mai lobe amog all widows of the same legth). However, this ca be compesated for by icreasig the legth of the filter.

13 Hece, i FIR filter desig, we usually use other widow fuctios w[] : Some commoly used widows: Bartlett (triagular): Haig: Hammig: otherwise M M M M M w 0 2 / / / 0 / 2 ] [ otherwise M M w 0 0 ) / 0.5cos(2 0.5 ] [ otherwise M M w 0 0 ) / 0.46 cos( ] [

14 These widows are all symmetric to M/2. w[ ] w[ M 0 ] 0 M otherwise

15 Kaiser widow Kaiser foud that a ear-optimal widow defied as I0[ (1 [( ) / ] w[ ] 0 M / 2 where ad I 0 () represets the zeroth-order modified Bessel fuctio of the first kid. I cotrast to the other widows, the Kaiser widow has two parameters: the legth (M+1) ad a shape parameter. 2 ) 1/ 2 ] 0 M otherwise I sum, like the case of spectrogram, usig differet widow fuctios allows us to trade off betwee the trasitio regio (iflueced by the mai lobe of the widow) ad maimum oscillatio (iflueced by the side-lobe gai of the widow).

16 Correlatio I additio to covolutio, there is aother operatio called correlatio Give a pair of sequeces [] ad y[], their cross correlatio sequece is r y [l] is defied as for all iteger l. r y l y l The cross correlatio sequece ca help measure similarities betwee two sigals.

17 Cross correlatio is very similar to covolutio, uless the idices chages from l to l. Relatio betwee cross correlatio ad covolutio: Autocorrelatio r l l

18 Cosider the followig o-egative epressio: a That is, Thus, the matri Properties of correlatio a y l a a y l y l 2 r 0 2ar l r 0 0 a y Its determiate is oegative. yy 0 l l 0 r r a y 1 0 ry r yy 1 r r y 0 l r r y yy l 0 for all a is positive semidefiite.

19 The determiat is r [0]r yy [0] r y2 [l] 0. Properties r [0]r yy [0] r y2 [l] r 2 [0] r 2 [l] Normalized cross correlatio ad autocorrelatio: l r r l The properties imply that [0] 1 ad yy [0] 1. This property ca also be eplaied by Schwartz iequality. Cosider ad y to be two ifiite-log vectors. The r [0] ad r yy [0] are the squared legth of ad y, respectively. r y [l] is the ier product betwee [] ad y[-l], where y[l] ad y[] have the same squared legths. l y l 0r y 0 r yy r 0

20 Autocorrelatio is quite ofte be used for fidig the period of a periodical sigal. By defiitio, autocorrelatio peaks at the iteger multiples of the period. Property i the frequecy domai Give a real sequece [l], The DTFT of the autocorrelatio sigal r [l] is the squared magitude of the DTFT of [l], i.e., X(e jw ) 2. That is Proof by the relatioship betwee correlatio ad covolutio, r [l] = [l] [-l]. Time domai covolutio implies frequecy domai multiplicatio. I additio, [l] is real, so [l]=*[l]; its DTFT is X*(e jw ) accordig to the symmetry property of DTFT. Takig DTFT for both sides, we have DTFT(r ) = X(e jw )X*(e jw ) = X(e jw ) 2

21 Hece, sigals with the same autocorrelatio share the same magitude spectrum i the frequecy domai, albeit their phase resposes are differet. A eample of describig the property of a class of sigals. Correlatio is useful i radom sigal modelig ad processig

22 Radom (Stochastic) Processes Radom (or Stochastic) Process (or Sigal) A radom process is radom fuctio, ot oly a radom variable. A sequece [], < <. Each idividual sample [] is assumed to be a outcome of some uderlyig radom variable X. Differece from a sigle radom variable: for a radom variable the outcome of a radom-samplig eperimet is a umber, whereas for a radom process the outcome is a sequece.

23 Radom (Stochastic) Processes Cosider a radom process [], < <, where each [] is draw from the radom variable X. Hece, there are ifiite radom variables, X, < <. Strictly speakig, its joit distributio, p(, X -2 =[-2], X -1 =[-1],, X 0 =[0],, X 1 =[1], X 2 =[2], ) is a probability distributio i a ifiite-dimesioal space.

24 Radom (Stochastic) Processes However, it is impractical ad ifeasible to represet the radom (or stochastic) process as a distributio i a ifiite-dimesioal space. The most commo way to describe a radom process is to characterize the distributios for some fiite samples, say, { 1, 2,, k }, ad specify their probability distributios i fiite-dimesioal spaces: p(x 1 =[ 1 ], X 2 =[ 2 ],, X k =[ k ])

25 Eample: Gaussia Process For eample, the Gaussia process is defied as follows: If for ay set of samples 1, 2,, k ( i Z, kn + ), the radom process satisfies that the joit distributio of these samples p(x 1 =[ 1 ], X 2 =[ 2 ],, X k =[ k ]) is a multivariate Gaussia distributio, the this process is called a Gaussia process. Gaussia process is ofte used i machie learig for oliear regressio.

26 Radom process i sigal processig I sigal processig, the radom process cosidered focused more o the joit distributios of two samples, p(x 1 =[ 1 ], X 2 =[ 2 ]). I additio, the shift-ivariat property is further imposed (called statioary process). Details will be give below.

27 Discrete-time Radom Sigals Why radom process or radom sigals? Util ow, we have assumed that the sigals are determiistic, i.e., each value of a sequece is uiquely determied. I may situatios, the processes geeratig the sigals are so comple as to make precise descriptio of a sigal etremely difficult or udesirable. A radom or stochastic sigal is cosidered to be characterized by a set of probability desity fuctios. Discrete-time radom process: A sequece [], < <. Each idividual sample [] is assumed to be a outcome of some uderlyig radom variable X.

28 Cotiuous-time Radom Sigals A cotiuous-time radom sigal is a sigal (t) whose value at each time is a radom variable. Radom sigals appear ofte i real life. Eamples iclude: 1. The oise heard from a radio receiver that is ot tued to a operatig chael 2. The oise heard from a helicopter rotor. 3. Electrical sigals recorded from a huma brai through electrodes put i cotact with the skull (these are called electroecephalograms, or EEGs).

29 4. Mechaical vibratios sesed i a vehicle movig o a rough terrai. 5. Agular motio of a boat i the sea caused by waves ad wid. 6. Televisio sigal 7. Radar sigal

30 Defiitios (Oppeheim, Appedi) Probability desity fuctio of []: p(, ) The pdf is varyig with the time ide. Joit distributio of two samples [] ad [m]: p,, m, m The joit pdf of time idices ad m Eg., 1 [] = A cos(w + φ ), where A ad φ are radom variables for all < <, the 1 [] is a radom process.

31 [] ad [m] are idepedet iff is a statioary process iff for all k. That is, the joit distributio of [] ad [m] depeds oly o the time differece m. Idepedece ad Statioary m p p m p m m,,,,, m p k m k p m k m k,,,,,,

32 Statioary (cotiue) I particular, the above defiitio of statioary process is also applicable to the situatio of m=. Hece, a statioary radom process should also satisfy p, k p k, That is, the pdf of a statioary process is ot varyig with the time ide. Implies that [] is shift ivariat.

33 Stochastic Processes vs. Determiistic Sigal I may applicatios of DSP, radom processes serve as sigal-source models i the sese that a particular sigal ca be cosidered a sample sequece of a radom process. Although such a sigals are upredictable makig a determiistic approach to sigal represetatio is iappropriate certai average properties of the esemble ca be determied, give the probability law of the process.

34 Average Esembles: Epectatio Mea (or average) p m, deotes the epectatio operator d Defied i associatio with the time ide g g p, For idepedet radom variables y y m m d

35 Statistics: Mea Square Value ad Variace Mea squared value (also called power of the radom sigal) 2 } { p, 2 d Variace: var m 2

36 Autocorrelatio ad Autocovariace Autocorrelatio correlatio betwee time idices m ad of a process {,m} Autocovariace m covariace betwee time idices m ad of a process {,m} {,m} m p m,, m, m d m m m m m m d * m

37 Statioary Process Accordig to the defiitio of statioary process, the autocorrelatio of a statioary process is depedet oly o the time differece m. Hece, for statioary process, we have m 2 m m If we deote the time differece by k, we have k k, 2 k Mea ad variace are idepedet to Autocorrelatio is depedet oly to the time differece k

38 Wide-sese Statioary I the above, the statioary property is defied i the strict sese that the pdf should remai the same for all time. However, for may istaces, we ecouter radom processes that are ot statioary i the strict sese. Istead, oly the statistics up to the 2-d order are ivariat with time. To rela the defiitio, if the followig equatios hold, we call the process wide-sese statioary (w. s. s.). m 2 m m 2 k k, k

39 Eample of statioary As we see, ituitively, a WSS sigal looks more or less the same at differet time itervals. Although its detailed form varies, its overall (or macroscopic) shape does ot.

40 Eample of ostatioary A eample of a radom sigal that is ot WSS is a seismic wave durig a earthquake. As we see, the amplitude of the wave shortly before the begiig of the earthquake is small. At the start of the earthquake the amplitude grows suddely, sustais its amplitude for a certai time, the decays.

41 Time Averages For ay sigle sample sequece [], defie their time average to be 1 lim L 2L 1 L Similarly, time-average autocorrelatio is m lim L L 1 2L 1 L L Defied by averagig all time idices for a arbitrary istace of the radom process m

42 Ergodic Process Note that the above time average is defied for a determiistic sigal sampled from the radom process. A statioary radom process for which time averages equal esemble averages is called a ergodic process: m m m

43 Ergodic Process (cotiue) It is commo to assume that a give sequece is a sample sequece of a ergodic radom process, so that averages ca be computed from oly a sigle sequece. I practice, we caot compute with the limits, but istead the fiite-sum quatities for approimatio mˆ 2 1 L 1 L L1 0 L1 0 mˆ 1 m m L L 2 L1 0

44 W. S. S. correlatio sequeces Defiitio: Autocorrelatio ad cross-correlatio sequeces (or fuctios) of W. S. S. radom process m m m y y m Note that, due to the statioary property, the above defiitios eist ad are idepedet to the time idices. Autocorrelatio sequece (or fuctio) is a determiistic sigal (ot a radom sigal), which caot be well defied for a radom process that is ot W. S. S.

45 Properties of correlatio ad covariace sequeces (cotiue) Property (similar properties have already bee show i the determiistic case) The above implies 2 m y yy 0 0 m 0

46 Properties of correlatio ad covariace sequeces (cotiue) Property: (shift ivariace) If y 0 yy m m Property 2 0 Mea Squared Value

47 Fourier Trasform Represetatio of Radom Sigals Sice the autocorrelatio sequece of a radom process is a determiistic sigal, its DTFT eists ad are bouded i w π. Let the DTFT of the autocorrelatio sequece be jw m e By doig so, we ca view a W. S. S. radom process i the spectral domai.

48 Fourier Trasform Represetatio of Radom Sigals (cotiue) Applyig the iverse DTFT: Recall that Cosequetly, 1 jwm m e e dw 2 jw 0 0 jw 2 1 e dw 2 also called the power of the radom sigal. 2

49 Deote Fourier Trasform Represetatio of P to be the power spectral desity; power desity spectrum (or power spectrum) of the radom process. Hece, we have Radom Sigals (cotiue) jw w e 2 1 P wdw 2 That is, the total area uder power desity i [ π, π] is the total power of the sigal.

50 Power Spectral Desity I sum, from 2 1 P wdw 2 P (w) ca be treated as the desity at the frequecy w of the total power. Itegratig all the desities from π to π the costitutes the total eergy of a w.s.s. radom sigal. This is why P (w) is called power spectral desity.

51 Power Desity Spectrum Property of the power desity spectrum: P (w) is always real-valued sice () is cojugate symmetric. For real-valued radom processes, P (w) = φ (e jω ) is both real ad eve. Whe the ergodic property is available, we ca realize more ature about the power desity spectrum.

52 Power Spectral Desity Estimatio from Determiistic Sigal Suppose we sample a determiistic sigal y from the radom process. Remember the autocorrelatio sequece defied for a determiistic sigal is yy r l y y l

53 Power Spectral Desity Estimatio from Determiistic Sigal Applyig the ergodic property: By the ergodic property, we ca use the autocorrelatio of a arbitrary sampled sigal y to estimate the autocorrelatio of the radom process. So, we ca also use the DTFT of r yy [l] to estimate P (w), where P (w) is the DTFT of the autocorrelatio of the radom process. Remember we have see that the DTFT of the determiistic sigal r yy [l] is equal to the squared magitude of the DTFT of y. DTFT(r ) = Y(e jw ) 2

54 Power Spectral Desity ad Squared Magitude of DTFT Hece, the power spectral desity P (w) of a radom process is equal to the squared magitude spectrum of ay of its istace y whe the ergodic assumptio is hold. So, we ca use a sample sequece (or a set of sample sequeces) to estimate the power spectrum of a radom sigal. Computig the DTFT magitude square of the sample sequece(s) the estimates the power spectrum.

55 Power Spectral Desity ad Squared Magitude of DTFT Like the determiistic case, we caot perform itegratio i [, ], ad so ca use oly a fiite rage [-T, T] istead. Widow fuctios (such as Hammig, Kaiser) are also used. Hece, the power spectrum estimatio process is the same as the spectrogram costructio process of a determiistic sigal. Note that the power spectral desity is the squared magitude frequecy respose, which does ot cotai phase iformatio ad is always positive.

56 Iput W.S.S. radom process to a LTI system We have see that a radom process is actually a collectio of sigals, istead of a sigle or uique sigal. To apply a radom process as iput to a LTI system, we mea that each sigal i this collectio serves as iput, ad we obtai a collectio of output sigals. We wat to characterize the output collectio of sigals. What are their esemble properties?

57 Mea of the output process Cosider a liear system with the impulse respose h[]. If [] is a statioary radom sigal with mea m, the the output y[] is also a statioary radom sigal with mea m y equalig to m y y hk k hk m k k k Sice the iput is statioary, m [ k] = m, ad cosequetly, m y m k h j0 k H e m

58 Statioary ad LTI System If [] is a real ad statioary radom sigal, the autocorrelatio fuctio of the output process is, yy m yy m k k r h h k hr k m r k hr k m r r Sice [] is statioary, ε k + m r depeds oly o the time differece m + k r.

59 Statioary ad LTI System (cotiue) Therefore, yy k yy m h k h r m k r, m r Hece, the output power desity is also statioary. Geerally, for a LTI system havig a wide-sese statioary iput, the output is also wide-sese statioary.

60 Power Desity Spectrum ad LTI System Furthermore, by substitutig l = r k i the above, where yy m m l hkh l k l c hh l m lc l k A sequece of the form of c hh [l] is called a determiistic autocorrelatio sequece of the system. hh l hk hl k k autocorrelatio of the impulse respose

61 Power Desity Spectrum ad LTI System Hece, m m lc l yy hh l That is, the autocorrelatio sequece of the output radom process is the covolutio of that of the iput radom process ad c hh [l]. So, i the DTFT domai, yy jw jw jw e C e e hh where C hh (e jw ) is the Fourier trasform of c hh [l].

62 What is C? hh e jw For real c hh [l], c C hh hh l hl h l jw jw jw e H e H e Correlatio of a[] ad b[] is the covolutio of a[[ ad b[-] So C hh e jw H e jw 2

63 Power Desity Spectrum ad LTI System (cotiue) We have the relatio of the iput ad the output power spectra as follows: yy 2 jw jw jw e H e e 2 1 jw 0 e jw jw y 0 H e e yy total average 2 2 power of the output dw total average dw power of the iput

64 Power Desity Spectrum ad LTI System (cotiue) I sum: whe iput a W.S.S. radom process with the autocorrelatio φ [] to a LTI system of impulse respose h[]: m m lc l yy hh l Time domai respose yy 2 jw jw jw e H e e Frequecy domai respose

65 Power Desity Property We have see that P ω = Φ (e jw ) ca be viewed as desity. Property: The area over a bad of frequecies, w a < w <w b, is proportioal to the power i the sigal i that bad. We ca eplai this agai from the liear-system property above. Cosider a ideal bad-pass filter. Let H bp (e jw ) be the frequecy respose of the ideal bad pass filter for the bad w a < w <w b. Ideal bad pass filter is a LTI system. H bp e jw 1 wa w w 0 otherwise b

66 Power Desity Property Cosider the power of the output radom sigal y whe the ideal bad-pass filter is applied: 1 jw yy 0 yy e dw 2 1 wa 1 w jw b jw e dw e dw 2 wb 2 wa is just equivalet to the power of the radom sigal i the bad w a < w <w b.

67 White Noise (or White Gaussia Noise) A white oise sigal is a sigal for which m 2 m delta fuctio Because its autocorrelatio is a delta fuctio, its samples at differet istats of time are ucorrelated. The power spectrum of a white oise is a costat jw 2 e

68 White Noise (or White Gaussia Noise) White oise is very useful i quatizatio error aalysis. White Gaussia oise: if a radom process is both white oise ad Gaussia process it is called a white Gaussia oise.

69 White Noise (cotiue) The average power of a white-oise is therefore 1 0 jw 2 2 e dw dw 2 2 White oise is also useful i the represetatio of radom sigals whose power spectra are ot costat i the frequecy domai. A radom sigal y[] with the power spectrum φ yy (e jw ) below ca be modeled as the output of a LTI system with a white-oise iput. yy 1 2 jw jw 2 e H e

Practical Spectral Anaysis (continue) (from Boaz Porat s book) Frequency Measurement

Practical Spectral Anaysis (continue) (from Boaz Porat s book) Frequency Measurement Practical Spectral Aaysis (cotiue) (from Boaz Porat s book) Frequecy Measuremet Oe of the most importat applicatios of the DFT is the measuremet of frequecies of periodic sigals (eg., siusoidal sigals),