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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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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