FIR Filter Design: Part I

Transcription

1 EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I. Itroductio FIR Filter Desig: Part I I this set o otes, we cotiue our exploratio o the requecy respose o FIR ilters. First, we cosider some ituitive examples o low-pass ad high-pass FIR ilters; we had previously cosidered these examples i our course itroductio (see /6 lecture otes. Secod, we cosider the desig o FIR ilters with more desirable requecy respose characteristics. We cosider two approaches. I the irst approach, we derive low-pass FIR ilters rom the impulse respose o ad ideal low-pass ilter; that is, a ilter with sharp cut-o requecies betwee the passbad (requecy rage or which requecies are allowed through the system, ad the rejectiobad (requecy rage or which requecies are zeroed. As we will see, this approach leads to FIR ilters with overshoot characteristics ear the cuto requecy. To mitigate this problem, we the cosider a secod approach, where we allow a iite-legth trasitio betwee the passbad ad rejectio-bad, ad observe that the resultig FIR ilters more closely approximate a idealized requecy respose.. Simple low-pass FIR ilters A. Itroductio Previously (/6 lecture otes, we had cosidered low-pass FIR ilters o the ollowig orm: y [ ] L --x[ k] L k or which the output y [ ] is a average o the curret iput x [ ] ad the ( L previous iputs x [ k], k {,,, L }. This type o FIR ilter is kow as a ruig-average ilter, ad teds to atteuate high requecies i a iput sigal, while leavig lower requecies relatively utouched (hece, the label lowpass ilter. For these ilters, the requecy respose uctio He ( j is give by, ( He ( j L --e. ( L j B. Examples Here we cosider the requecy respose o ilters o type (, or L, L 3 ad L. Figure below plots the magitude ad phase part o the requecy respose or each o these ilters as a uctio o the requecy variable [ π, π]. Note that the larger the value o L, the more arrow the cetral peak about ; that is, loger ruig averages will ted to suppress requecy compoets i the iput sigal o lower ad lower requecy. The zero-requecy compoet (, however, remais uaected o matter how large the value o L, sice averagig a costat iput sequece will ot chage that costat value. To veriy the requecy respose characteristics o these ilters, we ow coduct the ollowig experimet. We sample a cotiuous-time, oisy 4Hz cosie wave at s Hz ad the pass the resultig discretetime sequece x [ ] through the three ilters above. That is, xt ( cos( 8πt (3 x [ ] x ( s + oise, < <, (4 where, or each sample, the oise compoet o the sigal cosists o a uiormly distributed radom umber i the iterval [, ]. I Figures, 3 ad 4, or each o the three ilters, we plot a -legth (secod part o the iput sequece x [ ] ad a -legth part o the output sequece y [ ]. Moreover, we plot the magitude FFT o each -legth discrete-time sequece as a uctio o requecy. Note that or. You may wat to review the /6 lecture otes to complemet the materials i this sectio. - -

2 EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I L He ( j He ( j L 3 He ( j 3 He ( j L He ( j 3 He ( j Figure this samplig process, the relatioship betwee the requecy variable i Figure ad the real requecy (i Hertz is give by, π s ( so that the requecy variable rage, [ π, π] (6 correspods to the requecy rage, [ s, s ] [ Hz, Hz]. (7 A ew observatios: irst, ote that the uiorm oise added to the 4Hz cosie waveorm at the iput shows up as o-zero requecies throughout the requecy rage [ s, s ] ; secod, ote that each ilter teds to smooth out the oisy cosie waveorm by suppressig most o the oise-iduced requecy compoets the loger the ilter, the more the oise-iduced requecies are suppressed, ad, cosequetly, the smoother the resultig output waveorm y [ ]. I the ollowig sectio, we recosider some high-pass ilters that we derived ituitively i a earlier lecture (/6. - -

3 EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I x [ ] L y [ ] FFT ( x[ ] 4 3 FFT ( y[ ] x [ ] Figure L y [ ] FFT ( x[ ] 4 3 FFT ( y[ ] Figure 3 3. Simple high-pass FIR ilters A. Itroductio Previously (/6 lecture otes, we had cosidered the ollowig high-pass FIR ilters:. You may wat to review the /6 lecture otes to complemet the materials i this sectio

4 EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I x [ ] L y [ ] FFT ( x[ ] 4 3 FFT ( y[ ] Figure 4 y [ ] 4x [ ] x [ ] + 4x [ ] (secod-order (8 y [ ] 8x [ ] 3 8x [ ] + 3 8x [ ] 8x [ 3] (third-order (9 y 3 [ ] 6x [ ] 4x [ ] + 3 8x [ ] 4x [ 3] + 6x [ 4] (ourth-order ( which we derived by takig cosecutive discrete-time derivatives o the irst-order high-pass ilter: y [ ] x [ ] x [ ]. ( These ilters will ted to atteuate low requecies (zeroig ay DC or zero-requecy compoet, while passig through higher requecies relatively utouched (hece, the label high-pass ilter. For these ilters, the requecy respose uctios are straightorward to compute: H ( e j e j + --e 4 j ( H ( e j e 8 j e 8 j --e 8 j3 (3 H 3 e j ( e. (4 6 4 j 3 --e 8 j --e 4 j e 6 j4 Figure below plots the magitude ad phase part o the requecy respose or each o these ilters as a uctio o the requecy variable [ π, π]. Note that as the legth o the ilter is icreased, more ad more low requecies are virtually zeroed out etirely, while the highest requecies remai utouched. B. Experimets To veriy the requecy respose characteristics o the high-pass ilters i equatios ( through (4, we ow coduct the same experimet as or the low-pass ilters i the previous sectio. We sample a cotiuoustime, oisy 4Hz cosie wave at s Hz ad the pass the resultig discrete-time sequece x [ ] through the three high-pass ilters above. That is, - 4 -

5 EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I H e j H 3 e j H ( e j H ( e j ( H ( e j ( H 3 ( e j - - Figure xt ( cos( 8πt ( x [ ] x ( s + oise, < <, (6 where, or each sample, the oise compoet o the sigal cosists o a uiormly distributed radom umber i the iterval [, ]. I Figures 6, 7 ad 8, or each o the three high-pass ilters, we plot a - legth (secod part o the iput sequece x [ ] ad a -legth part o the output sequece y [ ]. Moreover, we plot the magitude FFT o each -legth discrete-time sequece as a uctio o requecy. Note that or this samplig process, the relatioship betwee the requecy variable i Figure ad the real requecy (i Hertz is agai give by, π s (7 so that the requecy variable rage, [ π, π] (8 correspods to the requecy rage, [ s, s ] [ Hz, Hz]. (9 - -

6 EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I x [ ] y [ ] FFT ( x[ ] 4 3 FFT ( y [ ] Figure x [ ] y [ ] FFT ( x[ ] 4 3 FFT ( y [ ] - - Figure Note that each ilter removes the low-requecy cosie wave rom the sigal ad preserves oly high-requecy compoets (to varyig degrees. Also ote that these experimetal results closely match the magitude requecy respose uctios i Figure. The Mathematica otebook ir_ilter_desig_parta.b was used to geerate the examples i the previous two sectios. Additioal speech ad image processig examples or the low-pass ad high-pass ilters discussed so ar ca be oud i the otes correspodig to the /6 lecture. I the ext two sectios, we illustrate how oe might go about desigig low-pass FIR ilters, where you, as the desiger, have much more cotrol over the ilter s resultig requecy respose

7 EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I x [ ] y 3 [ ] FFT ( x[ ] 4 3 FFT ( y 3 [ ] Figure 8 4. Desig o a low-pass FIR ilter: approach # A. Itroductio I the previous two sectios, we saw examples o two types o simple FIR ilters: ( low-pass ad ( highpass. Although these ilters were relatively simple to derive, we had little cotrol over the precise shape o the requecy respose uctio correspodig to each ilter. Here, we cosider the desig o a low-pass ilter, begiig with a ideal low-pass ilter with zero-width trasitios betwee the passbad ad rejectio-bad. B. Ideal low-pass ilter Figure 9 illustrates the requecy respose o a ideal low-pass ilter with ormalized cuto requecy at c. Note that a ilter with these characteristics will ot chage the magitude o ay requecies i the iput sigal or the rage [ c, c ], but will zero out all requecies outside that rage. Moreover, the liear-phase property o such a ilter will shit iput requecies i the rage [ c, c ] by a amout proportioal to each requecy ad, thereore, will ot itroduce ay phase distortio ito the output o the system. The slope o the lie ( a i the phase plot will determie the extet o the shit; a slope o zero will correspod to o phase shit (i.e. o delay. Below, we will use the iverse DTFT to determie the time-domai impulse respose with a (i.e. o delay. Recall that the iverse DTFT is give by, h [ ] o a ideal ilter He ( j He ( j π c c π Figure 9 slope a π c c π - 7 -

8 EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I π h [ ] ( π He ( j e j d π For the ideal ilter i Figure 9 (with o delay, equatio ( reduces to, h [ ] c e π j d c ( h [ ] c ej ej c e j c ej c e j c π j π j j π j c si( c π ( si( c h ideal [ ] , < <. (3 π Note that the impulse respose or a ideal ilter with o delay is iiite i legth, both orwards ad backwards i time. For example, i Figure, we plot h [ ],, or c Figure h ideal [ ] C. Approximatig a ideal ilter with a FIR ilter Now, we will try to approximate the ideal ilter with a causal FIR ilter usig the ollowig procedure: First we will retai values or h ideal [ ] oly or a limited rage o, max max (4 sice values o h [ ] approach zero as. Let us deote this iite impulse respose as h [ ], such that, h [ ] h ideal [ ] max max elsewhere ( Secod, we will shit h [ ] to be causal; let us deote the resultig impulse respose as h [ ] such that, h [ ] h [ max ]. (6 Note that this shit will ot chage the steady-state requecy respose o the resultig system, except to itroduce a delay at the output. Thus, the dierece equatio correspodig to h [ ] ca be writte as: y [ ] b k x [ k] where, max k (7-8 -

9 EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I b k h ideal [ max + k] si( ( max + k c ( max + kπ (8 I Figure, we plot the impulse respose h [ ] ad the correspodig magitude requecy respose H ( e j or various values o max (, ad ad a umeric cuto requecy o c. For each requecy respose plot, the blue lie idicates the ideal magitude requecy respose. Note that as we icrease the legth o the FIR ilter (i.e. max the trasitio regio at the cuto requecy becomes more well-deied; however, also ote the overshoot pheomeo (Gibbs pheomeo at the cuto requecy, which is similar to what we observed earlier i the course or the iite-sum Fourier series approximatio o periodic waveorms with discotiuities (such as a square wave. While icreasig the FIR ilter legth will make the trasitio regio i the requecy respose uctio icreasigly arrow, the overshoot at the cuto requecy will remai. Thereore, this approach or desigig low-pass FIR ilters will yield ilters that approximate the ideal ilter characteristic arbitrarily well, except ear ± c, where the overshoot problem will remai. This problem motivates our secod approach to ilter desig i the ollowig sectio, where we explicitly allow some arrow, ozero trasitio regio betwee the passbad ad rejectio bad o the ilter max max max h [ ] h [ ] h [ ] H ( e j H ( e j H ( e j Figure - 9 -

10 EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I. Desig o a low-pass FIR ilter: approach # A. Itroductio I this sectio, we will ollow exactly the same approach to desigig a low-pass FIR ilter as i the previous sectio, except that we will start o with a slightly dieret magitude requecy respose, as illustrated i Figure below. Note that ulike beore, we ow allow a iite-width trasitio regio o width rom oe to zero i the magitude requecy respose. He ( j π ( c c + c c + π Figure Below, we will use the iverse DTFT to determie the time-domai impulse respose assumig o delay. For this ilter, the iverse DTFT is give by, h [ ] o this ilter, h [ ] Note that, π π c -- ( + c + e j d e ( c + π j d c ( c + -- ( c + e j d c c (9 He ( j -- ( +, (3 c + [ ( c +, c ] He j ( -- (, (3 c + [ c, ( c + ] are the equatios o the two lies i the trasitio regios. We solve the itegratio i equatio (9 usig Mathematica (see ir_ilter_desig_partb.b ad arrive at the ollowig solutio or h [ ]: cos( c cos( ( c + h [ ] ,. (3 < < π Usig L Hopital s rule, we ca show that this result is, i act, idetical to the oe previously derived or the ideal ilter [equatio (3], as : lim h [ ] lim cos( c cos( ( c π (33 lim h [ ] lim d { cos( d c cos( ( c + } d { d π } (34 si( ( c + si( lim h [ ] c lim π π. ( - -

11 EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I As or the case o the ideal ilter, the impulse respose h [ ] i equatio (3 is iiite i legth, both orwards ad backwards i time. For example, i Figure 3, we plot h [ ],, or c ad.however, whe comparig Figures ad 3, we otice that the impulse respose or o-zero trasitio regios (i.e., Figure 3, decays to zero values much quicker tha or (Figure as. This is due to the act that the h [ ] values vary as or the ideal ilter i the previous sectio, while they vary as or the ilter with ozero-width trasitio regios i this sectio. Thereore, the eects o approximatig the iiite impulse respose i Figure 3 by retaiig oly values o h [ ] or max max should be less severe..3 h [ ] Figure 3 B. Approximatio with a FIR ilter Here we will approximate the iiite impulse respose i Figure 3, correspodig to the requecy respose i Figure (with o delay, with a causal FIR ilter, ollowig the same procedure as or the ideal ilter i the previous sectio. First we will retai values or h [ ] [see equatio (3] oly or a limited rage o, max max (36 sice values o h [ ] approach zero as. Let us deote this iite impulse respose as h [ ], such that, h [ ] h [ ] max max elsewhere (37 Secod, we will shit h [ ] to be causal; let us deote the resultig impulse respose as h [ ] such that, h [ ] h [ max ]. (38 Note that this shit will ot chage the steady-state requecy respose o the resultig system, except to itroduce a delay at the output. Thus, the dierece equatio correspodig to h [ ] ca be writte as: y [ ] b k x [ k] where, max k (39 b k h[ max + k] cos( ( max + k c cos( ( max + k ( c ( max + k π (4 I Figure 4, we plot the impulse respose h [ ] ad the correspodig magitude requecy respose H ( e j or various values o max (, ad ad umeric values, c ad. For each - -

12 EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I.3 max h [ ] H ( e j max h [ ]. H ( e j max h [ ]. H ( e j Figure 4 requecy respose plot, the blue lie idicates the iiite impulse magitude requecy respose.note that ulike the previous examples i Figure, we o loger have the same overshoot problem. I act, or max, we see that the FIR ilter is a very good approximatio o the IIR ilter. Agai, this is due to the act that we o loger require the trasitio rom oe to zero i the requecy respose to occur istataeously, but rather that we allow or that trasitio regio to be o some small, o-zero width (i.e.. C. Filterig examples I this sectio, we illustrate the output y [ ] or a FIR ilter o type (4 o some sample iput sigals x [ ]. For these examples, we assume a samplig rate o s Hz, a desired cuto requecy c Hz, ad a trasitio width o Hz. We also, limit our FIR ilter to a legth ( max +, where max. Usig the coversio rule betwee ad, π s (4 we ca compute our ormalized cuto requecy c ad trasitio-regio width : c π( c s π( π (4 π( s π( π. (43 - -

13 EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I Usig equatios (3, (37 ad (38, we ca ow compute the impulse respose h [ ] or the values o c ad i (4 ad (43 above. I Figure below, we plot that impulse respose ad the correspodig magitude requecy respose as a uctio o requecy. Note how well the FIR requecy respose approximates our desired cuto requecy ad trasitio width..4 h [ ] H ( e j π s Figure We ow apply the ollowig our test iputs to this system: x [ ] π cos s cos(.π (i.e. sampled Hz cosie wave (44 x [ ] x 3 [ ] x 4 [ ] π π 7 cos s cos s cos(.π + 3cos(.3π π π cos s cos s cos(.π + 3cos(.6π π π cos s cos s cos(.π + 3cos(.4π (4 (46 (47 For each test iput, we compute the output as the covolutio betwee x i [ ] ad h [ ] : y i [ ] x i [ ] * h [ ], i { 34,,, }, (48 ad compute the magitude FFT o -legth segmets o the steady-state iput ad correspodig output (see Mathematica otebook ir_ilter_desig_partb.b or time-domai plots with trasiet resposes as well.these FFTs are plotted i Figures 6 through 9 below. Note that or the irst three test sigals, the ilter leaves the magitude o the iput requecies less tha Hz uchaged, while zeroig out requecies above Hz. For the ourth sigal, we see that the Hz requecy compoet is approximately halved, sice it lies exactly halway iside the trasitio regio rom Hz to Hz. Thus, this low-pass FIR ilter appears to perorm exactly as desiged

14 EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I FFT ( x [ ] - - FFT ( y [ ] - - Figure FFT ( x [ ] FFT ( y [ ] - - Figure 7-4 -

15 EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I FFT ( x 3 [ ] FFT ( y 3 [ ] - - Figure FFT ( x 4 [ ] FFT ( y 4 [ ] - - Figure 9 - -

16 EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I 6. Coclusio The Mathematica otebooks ir_ilter_desig_parta.b ad ir_ilter_desig_partb.b were used to geerate all the examples i this set o otes. I a compaio set o otes, we coclude our discussio o FIR-ilter desig