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

S tep 1 f ind j ips t he owpass p rototype f ilter t

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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 . j w-a-jb - a+j(w-b) 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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