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Unformatted text preview: e, which may n ot b e so flat nor have a s harp cutoff characteristic. 4 0
0.1 0.2 0.3 0.5 0.8 I IO w - -+ F ig. 7 .34 D elay characteristics o f t he B utterworth a nd C hebyshev filters. C ontrast t his with t he B utterworth filter, which is designed t o yield t he maximally flat amplitude response a t w = 0 w ithout any a ttention t o t he p hase
response. A family of filters which yields a maximally fiat t d goes under the
name B essel-Thomson filters, which uses t he n th-order Bessel polynomial
in t he d enominator of nth-order H (s).
2 I f b oth a mplitude a nd p hase response are i mportant, we s tart w ith a filter t o
satisfy t he a mplitude response specifications, disregarding t he p hase response
specifications. We cascade this filter with another filter, a n e qualizer, whose
amplitude response is fiat for all frequencies ( the a Upass f ilter) a nd whose
t d c haracteristic is c omplementary t o t hat o f t he m ain filter in such a way t hat
t heir composite phase characteristic is a pproximately linear. T he cascade thus
has linear phase a nd t he amplitude response of t he main filter (as required).
Allpass Filters An allpass filter has equal number of poles a nd zeros. All the poles are in t he
L HP (left half plane) for stability. All t he zeros are mirror images of the poles a bout
t he i maginary axis. In other words, for every pole a t - a + j b, t here is a zero a t
a + j b. T hus, all t he zeros are in the RHP. Any filter with this kind of pole-zero
configuration is a n allpass filter; t hat is, its amplitude response is c onstant for all 536 7 Frequency Response a nd Analog Filters frequencies. W e c an verify this assertion by considering a transfer function with a
pole a t - a + j b a nd a zero a t a + jb:
s - a - jb
H (s) = - - - - ' s + a - jb a nd .
H (J w) =
= - ---::,"---:-:-'jw+a-jb
a +j(w-b) Therefore
(7.65) w- b]
[ ! 'H(jw) = t an- 1 ~
= 1r - - [w - w - b]
tan- 1 [- a- - tan- 1 [w a- =
- - b] 537 T he Chebyshev a mplitude response has ripples in t he p assband. O n t he o ther
hand, t he behavior of t he Chebyshev filter in t he s topband is s uperior t o t hat of
t he B utterworth filter. T he design procedure for lowpass filters c an be readily
applied to highpass, bandpass, a nd b andstop filters by using appropriate frequency
transformations discussed in Sec. 7.7.
Allpass filters have a constant gain b ut a variable phase with respect t o frequency. Therefore, placing a n allpass filter in cascade with a system leaves its
amplitude response unchanged, b ut a lters its phase response. Thus, a n allpass filter
can be used to modify t he p hase response of a system. b]
tan- 1 - a- References
1r - 2 t an-l [w a- (7.66)
- - b] Observe t hat a lthough t he a mplitude response is u nity regardless o f pole-zero
locations, t he p hase response depends on t he locations of poles (or zeros). By
placing poles in proper locations, we c an obtain a desirable phase response t hat is
c omplementary t o t he phase resp...
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