Frequency Response of FIR Filters

Transcription

1 EEL335: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we itroduce the idea of the frequecy respose of LTI systems, ad focus specifically o the frequecy respose of FIR filters.. Steady-state frequecy respose of LTI systems A. Itroductio Let us cosider a discrete-time, LTI system with impulse respose h [ ]. Oe questio of great sigificace i aalyzig systems is how such a system will modify siusoidal iputs of various frequecies. For example, a low-pass filter might allow low-frequecy compoets of a sigal through relatively uchaged, while dampeig or atteuatig higher frequecies. I cotiuous time, we represet frequecies as cosie fuctios: xt () cos( πft) where f deotes the frequecy (i Hz) of xt (). The discrete-time equivalet is, of course, just a sampled versio of xt (): x [ ] x ( f s ) cos[ πf( f s )] () where f s deotes the samplig frequecy i Hz. Note that i equatio (), we ca group all the costat terms iside the cosie fuctio together: θ such that, x [ ] cos( θ), < <. (4) Note that θ deotes the ormalized frequecy variable that we have see before i our discussio of the discrete-time Fourier trasform (DTFT), ad if we kow f s for a particular discrete-time sigal x [ ] we ca use equatio (3) to covert betwee the frequecy variable θ ad correspodig real frequecies f. So, for a discrete-time LTI system with impulse respose h [ ], we will ow derive the output for a discrete-time siusoidal iput x [ ] as give by equatio (4). B. Derivatio πf f s Recall from the iverse Euler relatios, that we ca express equatio (4) i terms of two complex expoetials: e x [ ] cos( θ) jθ + e jθ (5) Therefore, by liearity, we ca compute the output by computig the outputs y [ ] ad y [ ] for the followig two complex expoetials: x [ ] e jθ ad x [ ] e jθ (6) such that, x [ ] --x, ad, [ ] + --x [ ] (7) --y. (8) [ ] + --y [ ] For a LTI system the output correspodig to a iput x [ ] ad impulse respose h [ ] ca be writte as: () (3) - -

2 EEL335: Discrete-Time Sigals ad Systems h [ ] * x[ ] hk [ ]x [ k] k Hece, y [ ] h [ ] * x [ ] hk [ ]x [ k] k y [ ] hk [ ]e j( k)θ k Now, recall our defiitio of the DTFT for a sequece x [ ]: y [ ] e jθ hk [ ]e jkθ k Xe ( jθ ) x [ ]e jθ Therefore, we ca rewrite equatio () as: y [ ] e jθ He ( jθ ) (9) () () () (3) (4) where He ( jθ ) deotes the DTFT of the impulse respose h [ ]. We ca pursue a similar derivatio for x [ ] : y [ ] h [ ] * x [ ] hk [ ]x [ k] k y [ ] hk [ ]e j( k)θ k y [ ] e jθ hk [ ]e jkθ k Note that we ca rewrite equatio (7) as: y [ ] e jθ He ( jθ ) (5) (6) (7) (8) where, He ( jθ ) hk [ ]e jkθ. (9) k Usig equatio (8), we ow ca compute the output for the siusoidal iput x [ ]: --y [ ] + --y [ ] jθ He ( jθ ) + jθ He ( jθ ) Note from the defiitios of He ( jθ ) ad He ( jθ ), that the followig relatioships hold true: () () - -

3 EEL335: Discrete-Time Sigals ad Systems He ( jθ ) He ( jθ ) He ( jθ ) We ow substitute, He ( jθ ) () (3) He ( jθ ) He ( jθ ) e j Hejθ ( ) ad He ( jθ ) He ( jθ ) ej He jθ ( ) (4) ito equatio (), ad the use properties () ad (3) to simplify the expressio for : jθ He ( jθ ) e j Hejθ ( ) jθ He ( jθ ) ej He ( jθ + ) (5) -- He ( jθ ) ejθ j Hejθ [ + ( ) + He ( jθ ) e jθ j Hejθ ( )] (6) He ( jθ e ) j [ θ + Hejθ ( )] e j θ Hejθ + [ + ( )] (7) He ( jθ ) cos[ θ + He ( jθ )]. (8) To summarize, the output of a LTI system with impulse respose h [ ] for a siusoidal iput x [ ], x [ ] cos( θ), < <, (9) is give by, He ( jθ ) cos[ θ + He ( jθ )], (3) where, He ( jθ ) hk [ ]e jkθ DTFT of the impulse respose h [ ]. (3) k The fuctio He ( jθ ) is kow as the frequecy respose fuctio, ad gives us the amplitude ad phase at the output of the system for siusoids of differet frequecies. That is for all discrete-time frequecies θ, we ca use equatio (3) to compute how differet frequecy compoets at the iput are modified (both i amplitude ad phase). C. Geeralizatio to arbitrary siusoidal iputs Now, we wat the geeralize the result i equatios (9) through (3) for geeral siusoidal iputs of the form, x [ ] Acos( θ+ α), < <. (3) will just be a scaled ad time-shifted versio of equa- Due to liearity ad time ivariace, the output tio (3): He ( jθ ) Acos[ θ+ α+ He ( jθ )]. (33) D. Frequecy respose of FIR filters FIR LTI systems are give by the followig geeral equatio: M b k x [ k] k The impulse respose h [ ] for such systems is give by, (34) - 3 -

4 EEL335: Discrete-Time Sigals ad Systems h [ ] b {,,, M} elsewhere (35) Therefore, we ca rewrite the frequecy respose fuctio He ( jθ ) i terms of the filter coefficiets b k for FIR systems: M He ( jθ ) b k e jkθ. (36) k Note that the limits i the summatio above are ow o loger ifiite. Below, we explore properties of a simple FIR filter. 3. Simple FIR filter example A. Itroductio Cosider the followig simple FIR system: x [ ] + x [ ] I a previous lecture, we have already see that the filter i equatio (37) is a example of a low-pass filter (see /6 lecture otes). Ituitively, we ca see this is the case if we cosider the output of the system for two differet iputs x [ ] ad x [ ] : x [ ] cos( θ), < <, (38) θ x [ ] cos( θ) cos( π),. (39) θ π eve odd < < Note that the first iput sequece correspods to zero frequecy ( θ ), while the secod iput sequece correspods to the highest possible ormalized frequecy ( θ π). For these iputs, the correspodig outputs are give by, y [ ] ( ) + ( ) (37) ad (4) y [ ] ( ) + ( ). (4) These iput ad output sequeces are plotted i Figure below. Thus, it appears that this filter passes through the lowest frequecy uchaged, while completely zeroig out the highest possible discrete-time frequecy; that s why we would call this filter a low-pass filter. I the ext sectio, we will derive the frequecy respose fuctio He ( jθ ) for the filter i equatio (37), ad explore some of its properties. B. Frequecy respose fuctio From the defiitio i equatio (36), He ( jθ ) for the filter i equatio (37) is give by, He jθ ( ). (4) jkθ -- + jθ k We ca rewrite equatio (4) as: He ( jθ ) e jθ jθ + jθ e jθ cos( θ ) (43) From equatio (43), He ( jθ ) ad He ( jθ ) are straightforward to compute: He ( jθ ) e jθ cos( θ ) e jθ cos( θ ) cos( θ ) (44) - 4 -

5 EEL335: Discrete-Time Sigals ad Systems x [ ] x [ ] y [ ] y [ ] Figure He ( jθ ) e jθ cos( θ ) θ θ ± π cos( θ ) > cos( θ ) < I Figure, we plot He ( jθ ) ad He ( jθ ) as a fuctio of θ [ π, (recall that outside the plotted iterval, He ( jθ ) (i.e. the DTFT) is periodic) θ He ( jθ ).5 He ( jθ ) Figure Note that the magitude plot of the frequecy respose fuctio cofirms our specific results for θ ad θ π. Also, ote that He ( jθ ) gives us the scaled output magitudes for iput frequecies betwee θ ad θ π, while the phase plot He ( jθ ) tells us the phase delay for every frequecy. Note that the phase profile i Figure is kow as a liear phase respose, because the phase He ( jθ ) is a liear fuctio of θ i the iterval θ [ π,. It might seem strage that differet frequecies appear to get shifted by differet amouts; however, the liear phase property of this system is exactly what is desirable if we do t wat to create phase distortio i a sigal. This cocept of phase distortio is cosidered i greater detail i the followig sectio θ (45) - 5 -

6 EEL335: Discrete-Time Sigals ad Systems C. Liear phase ad phase distortio Let us see how the liear phase characteristic of the filter i equatio (37) affects the output of the system for differet frequecies. From equatio (33), we ca write the output as a fuctio of the frequecy variable θ : He ( jθ ) cos[ θ + He ( jθ )] cos( θ ) cos[ θ θ ] cos( θ ) cos[ θ( ) ] (46) Note that the liear phase characteristic of the filter results i exactly the same shift of for all ormalized frequecies θ. Therefore, a ideal filter should have a phase respose of the followig form: He ( jθ ) aθ. (47) For the filter i equatio (37) a ; i fact, causal filters will always have the property that a. Also ote that more egative values of a itroduce a greater time delay ito the overall system respose. Let us ow compare two hypothetical filters with the followig frequecy respose fuctios: H ad. ( e jθ ) e j5θ H ( e jθ ) e j5 (48) Both filters have the same magitude resposes, H ( e jθ ) H ( e jθ ) (49) but differet phase resposes: H ( e jθ ) 5θ ad H ( e jθ ) 5. (5) Note that the first filter has a liear-phase respose, while the secod filter has a costat phase respose. We ow test this system with the followig iput sequece: x [ ] π π cos ,. (5) 5 cos 5 < < The correspodig outputs of the two systems will be give by, y [ ] x [ ] π ( 5) 4π cos ( 5) 5 cos 5 (5) y [ ] π π cos (53) 5 cos 5 The iput ad two output sequeces are plotted i Figure 3 below. Note that the first system results i a output that is idetical to the iput except for a shift to the right of five time uits; the secod system, however, results i a output that is a distortio of the iput, because the two iput frequecies get shifted by costat amouts, ot amouts proportioal to each frequecy compoet. Thus, y [ ] exhibits phase distortio; that is although both frequecies i the iput sigal are preset at the output with the same amplitude as the iput, the sigal becomes distorted by costat phase shifts to each frequecy compoet.. Note that H correspods to the system x [ 5], while the secod correspods to a filter with complex filter coefficiets

7 EEL335: Discrete-Time Sigals ad Systems 3 x [ ] D. Compariso to time-domai covolutio For a FIR system, we ow have two ways of computig the output for a give siusoidal iput. We ca either apply equatio (33) above, or compute i the time-domai directly through the covolutio sum. Cosider, for example, the followig siusoidal iput x [ ] applied to the system i equatio (37): x [ ] cos( θ), < <, θ [ π,. (54) Usig the covolutio sum, the output y [ ] y [ ] Figure 3 is give by, h [ ] * x[ ] hk [ ]x [ k] b k x [ k] k k --x[ ] + --x[ ] (55) (56) -- cos( θ) + -- cos( θ( ) ) (57) From equatio (46), we ca also compute the output from the frequecy respose fuctio: cos( θ ) cos[ θ( ) ] (58) We ca show that the outputs i (57) ad (58) are equivalet by applyig the followig trigoometric idetity: cos( α) cos( β) -- cos( α β) + -- cos( α+ β) (59) Sice cos( θ ), θ [ π,, we ca first rewrite equatio (58) as: cos( θ ) cos[ θ( ) ] (6) Now we apply trigoometric idetity (59) to equatio (6) by lettig: - 7 -

8 EEL335: Discrete-Time Sigals ad Systems α θ ad β θ( ) (6) -- cos( θ θ( ) ) + -- cos( θ + θ( ) ) -- cos( θ + θ) + -- cos( θ) (6) (63) -- cos( θ) + -- cos( θ( ) ). (64) Note that equatio (64) is idetical to the output derived directly from the covolutio sum [equatio (57)]. I the ext sectio, we will exted our result for siusoidal iputs (i.e. relatioship betwee time-domai covolutio ad the frequecy respose fuctio) to arbitrary iput sequeces x [ ]. 4. Time-domai covolutio i the frequecy domai A. Itroductio So far, we have show that the aalytic output of a LTI system to a siusoidal iput x [ ], x [ ] Acos( θ+ α), < <, θ [ π,, (65) is give by, He ( jθ ) Acos[ θ+ α+ He ( jθ )] where He ( jθ ) is the frequecy respose of the system (i.e. the DTFT of the impulse respose h [ ]). Below we develop the system respose of a LTI system for arbitrary iput sequeces x [ ] i the frequecy domai ad derive a importat result that relates covolutio i the time domai with multiplicatio i the frequecy domai. B. Frequecy-domai computatio of system output Let us cosider the output of a system for a arbitrary iput sequece x [ ]. The DTFT Xe ( jθ ), Xe ( jθ ) x [ ]e jθ gives us the frequecy cotet of x [ ] as a fuctio of the ormalized frequecy variable θ. That is, Xe ( jθ ) tells us the magitude ad phase of each frequecy θ that makes up the time-domai sigal x [ ]. The frequecy respose fuctio He ( jθ ), o the other had, tells us how each frequecy compoet of x [ ] will be modified by the system. Therefore, the frequecy represetatio of the output Ye ( jθ ) is give by, Ye ( jθ ) Xe ( jθ )He ( jθ ) where Ye ( jθ ) represets the DTFT of the output. If we compare equatio (68) to the covolutio represetatio of the output i the time domai, x [ ] * h[ ] we see that covolutio i the time domai correspods to multiplicatio i the frequecy domai. This is a extremely importat result, that applies ot just to discrete-time systems, but to cotiuous-time systems as well. Below, we show aalytically that equatios (68) ad (69) are equivalet. The right-had side of equatio (68) ca be writte as: (66) (67) (68) (69) Ye ( jθ ) Xe ( jθ )He ( jθ ) xm [ ]e jmθ hp [ ]e jpθ m p (7) - 8 -

9 EEL335: Discrete-Time Sigals ad Systems Ye ( jθ ) xm [ ]e jmθ hp [ ]e jpθ m p (7) Ye ( jθ ) xm [ ]hp [ ]e j( m + p)θ m p From equatio (69) ad the defiitio of the DTFT, we ca write the followig: (7) Ye ( jθ ) e jθ xk [ ]h [ k] e jθ k (73) Ye ( jθ ) xk [ ]h [ k]e (74) k We ow have to show that equatios (7) [derived from (68)] ad (74) [derived from (69)] are equivalet. To do this, let us make the substitutios, k m (75) k p ito equatio (74): (76) Ye ( jθ ) xm [ ]hp [ ]e j( m p)θ (77) p m Note that equatio (77) is equivalet to (7) (except for the order of summatio, which ca readily be iterchaged). Thus, we have show the followig importat correspodece: x [ ] * h[ ] Ye ( jθ ) Xe ( jθ )He ( jθ ). (78) 5. Coclusio The Mathematica otebook fir_frequecy_respose.b was used to geerate the example o phase distortio, ad shows the equivalece of time-domai covolutio with frequecy-domai multiplicatio for a simple FIR system. I this set of otes, we itroduced the cocept of frequecy respose for LTI systems, gave the formula for the frequecy respose of a FIR LTI system i terms of the coefficiets of the system s differece equatio [equatio (36)], ad explored some of the properties of the frequecy respose fuctio through a simple FIR filter example. Fially, we showed that covolutio i the time domai correspods to multiplicatio i the frequecy domai