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Unformatted text preview: onse of t he m ain filter. 7 .9 Problems 1 . Wai-Kai C hen, Passive and active Filters, Wiley, New York, 1986. 2 . Van Valkenberg, M.E., Analog Filter Design, Holt, R inehart a nd W inston, New
3 . C hristian E., a nd E. E isenmann, F ilter Design Tables and Graphs, Transmission Networks International, Inc., Knightdale, N.C., 1977. Problems S ummary T he r esponse of a n LTIC s ystem with transfer function H (s) t o a n everlasting
sinusoid of frequency w is also a n everlasting sinusoid of t he same frequency. T he
o utput a mplitude is IH(jw)1 t imes t he i nput amplitude, a nd t he o utput sinusoid is
shifted in p hase w ith respect t o t he i nput sinusoid by ! .H ( jw) radians. T he p lot of
I H(jw)1 vs w i ndicates t he a mplitude gain o f sinusoids of various frequencies a nd
is called t he amplitude response of t he system. T he plot of ! .H ( jw) vs w i ndicates
t he p hase s hift of sinusoids of various frequencies a nd is called t he phase response.
P lotting of t he frequency response is r emarkably simplified by using logarithmic
units for a mplitude as well as frequency. Such plots are known as t he Bode plots.
T he use of logarithmic units makes i t possible t o add ( rather t han multiply) t he
a mpiitude response of four basic types of factors t hat occur in transfer functions:
(1) a c onstant (2) a pole or a zero a t t he origin (3) a first order pole or a zero,
a nd (4) complex conjugate poles or zeros. For phase plots, we use linear units for
phase and logarithmic units for t he frequency. T he phases corresponding to t he
t hree basic t ypes of factors mentioned above add. T he a symptotic properties of t he
a mplitude a nd phase responses allow their plotting with remarkable ease even for
transfer functions of high orders.
T he f requency response of a system is d etermined by t he locations in t he complex plane o f poles a nd zeros of its transfer function. We can design frequency
selective filters by proper placement of its transfer function poles a nd zeros. P lacing a pole ( a zero) near a frequency j wo in t he complex plane enhances (suppresses)
t he f requency response a t t he frequency w = woo W ith t his concept, a proper combination of poles a nd zeros a t s uitable locations can yield desired filter characteristics.
Two families of analog filters are considered: Butterworth a nd Chebyshev.
T he B utterworth filter has a maximally flat amplitude response over t he p assband. 7.1-1 For an LTIC system described by the transfer function
H (s) _
- S2 8 +2
+ 5s + 4 find the response to the following everlasting sinusoidal inputs: (a) 5 cos (2t +
30°) ( b) 10 sin (2t + 45°) (c) lOcos (3t + 40°). Observe t hat these are everlasting
7.1-2 For an LTIC system described by the transfer function
H (s) = (8 + 2)2 find the steady-state system response to the following inputs:
(a) lOu(t) ( b) cos (2t + 60 0 )u(t) (c) sin (3t - 45°)u(t) ( d) ej3t u(t)
7.1-3 For an allpass filter specified by the transfer function
H(8) = - (s...
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