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Unformatted text preview: 9) is y(t) = IH(jw)1 cos [wt + 9 + LH(jw)] (7.5b) T his r esult, w here we have let s = jw, is valid only for asymptotically s table s ystems
because t he r elationship (7.1) applies only for t he values of 8 lying in t he r egion of
convergence f or H (s). For t he case of u nstable a nd m arginally s table s ystems, t his
region does n ot i nclude t he i maginary axis s = jw (see also t he f ootnote o n p. 243).
E quation ( 7.5) shows t hat for a sinusoidal i nput of r adian frequency w, t he
s ystem r esponse is also a sinusoid o f t he s ame frequency w. T he a mplitude o f t he
o utput s inusoid i s IH(jw)1 t imes t he i nput a mplitude, a nd t he p hase o f t he o utput
s inusoid is s hifted b y L H(jw) w ith respect t o t he i nput p hase (see Fig. 7.1). For
instance, if a c ertain s ystem has IH(j10)1 = 3 a nd LH(jIO) =  30°, t hen t he s ystem
amplifies a s inusoid o f frequency w = 10 b y a factor of 3 a nd delays i ts p hase by 30°.
T he s ystem r esponse t o a n i nput 5 cos (lOt + 50°) is 3 x 5 cos (lOt + 50°  30°) =
15 cos (lOt + 2 0°).
C learly I H (jw) I is t he s ystem g ain, a nd a p lot of IH (jw) I versus w shows t he
s ystem g ain as a function of frequency w. T his f unction is more commonly known as
t he a mplitude r esponse. Similarly, L H(jw) is t he p hase r esponse a nd a p lot of
LH (jw) v ersus w shows how t he s ystem modifies or changes t he p hase of t he i nput
sinusoid. T hese two plots together, as functions of w, a re called t he f requency
r esponse o f t he s ystem. O bserve t hat H (jw) h as t he i nformation of IH(jw)1 a nd
L H(jw). For t his reason, H (jw) is also called t he f requency r esponse o f t he
s ystem. T he f requency response plots show a t a glance how a system responds t o
s inusoids of v arious frequencies. T hus, t he frequency response of a s ystem r epresents
i ts filtering c haracteristic.
E xample 7 .1
Find the frequency response (amplitude and phase response) of a system whose transfer function is
H (s) = s + 0.1
s +5
Also, find the system response y(t) if the input f (t) is ( a) cos 2t ( b) cos (lOt  50°). o 2 10 H ( 'w)
J = j w + 0.1
jw + 5 Recall t hat t he magnitude of a complex number is equal to the square root of the sum of
the squares of i ts real and imaginary parts. Hence
and L H(jw) = t anI (0~1)  tanI (~) 2 10 (a) y ( t ) ; 0.372 cos (21 + 65.30) (b) F ig. 7 .1 Frequency response of an LTIC system in Example 7.1. Both the amplitUde a nd the phase response are de ic
.
.
t·
b
p ted m Fig. 7.1a as functions of w
These plots furnish the complete inf
.
.
o rma Ion a o ut the freq
t o smusoidal inputs.
uency response of the system
( a) For the input f (t) = cos 2t, w = 2, a nd IH(j2)1 = J (2)2 + 0.01 _
J (2)2 + 25  0.372 • In this case o (0_ L H(j2) = t an  1 (0~1)  tan  I G) = 87.1 °  21.8° = 65.30 . We also could have read these values directl fro
Fig. 7 .la corresponding t o w = 2 T h'
u l Y m t he frequency response plots in
frequency w = 2, the amplitude gain of ~h:es t me.ans t hat for a sinusoidal input with
In other words, the o ut...
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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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