Signal Processing and Linear Systems-B.P.Lathi copy

5 7 7 3 bandstop filters 1 s tep 1 5 d etermine t

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Unformatted text preview: sensitive to amplitude distortion. Therefore, in video applications we c annot ignore phase distortion. I n pulse communication, b oth t he amplitude a nd t he p hase d istortion a re i mportant for correct information transmission. Thus, in practice, w e also need t o design filters primarily for p hase linearity in video applications. In pulse communication applications, it is i mportant t o have filters with c onstant a mplitude response a nd a linear phase response. We shall briefly discuss some aspects and approaches t o t he design o f such filters. More discussion a ppears in t he l iterature. 2 We s howed [see Eq. (4.59)] t hat t he t ime delay t d r esulting from t he signal transmission t hrough a filter is t he negative of t he slope o f t he filter phase response LH(jw); t hat is, t d(W) d = - -LH(jw) 7.8 Filters to Satisfy Distortionless Transmission Conditions 535 14 12 ~ c 0 u ~ 10 8 ~..,. E C:l 6 4 2 0 36 0.1 _ ______0~.2~,0._3_ __ _.5~~0'r8~_ ___________~_ ________ -.10 .~ 0 8 w --+ 32 ~ c 0 28 24 " ~ 20 -..,. ~ 16 12 (7.64) dw I f t he slope o f LH (jw) is c onstant over t he desired band ( that is, if L H(jw) is linear with w), all t he c omponents are delayed by t he same time interval t d. I n t his case t he o utput is a replica of t he i nput, assuming t hat all components are a ttenuated equally; t hat is, IH(jw)1 = c onstant over t he p assband. I f t he s lope of t he p hase response is n ot constant, t d, t he t ime delay, varies with frequency. This variation means t hat different frequency components undergo different a mounts of time delay, a nd consequently the o utput waveform will not be a replica o f t he i nput waveform even if t he a mplitude response is c onstant over t he p assband. A good way of judging phase linearity is t o p lot t d as a function of frequency. F or a distortionless system, t d ( the negative slope of L H(jw)) should b e c onstant over t he b and of interest. This is in addition t o t he r equirement of constancy of t he a mplitude response. Generally speaking, t he two requirements of distortionless transmission conflict. T he more we a pproach t he ideal amplitude response, t he f urther we d eviate from t he ideal phase response. T he s harper t he cutoff characteristic (smaller t he t ransition b and), t he m ore nonlinear is t he p hase response near t he t ransition band. We c an verify t his fact from Fig. 7.34, which shows t he delay characteristic of the B utterworth a nd t he Chebyshev family of filters. T he Chebyshev filter, which has a s harper c utoff t han t hat of t he B utterworth, shows considerably more variation in time delay of various frequency components as compared t o t hat of t he B utterworth. For t he a pplications where t he phase linearity is also i mportant, t here are two possible approaches: 1 I f t d = c onstant (phase linearity) is t he p rimary requirement, we design a filter for which t d is maximally flat around w = 0 a nd a ccept t he r esulting amplitude r espons...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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