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T herefore 1 H(s) R es>  5 = s +5 and the input
I (t) = e 'u(t) + e  2'u(t) The input I (t) is of the type depicted in Fig. 6.50g, and the region of convergence
es not exist. In this case, we must determine separately the system response
t
)
d I ( ) _  2' ( t)
to each of the two i nput components, / I(t) = e u (t an 2 t  e u
.
f or F () d0
s 1 R es>  1 FI(s) = ;+l
1 T ransporting all b ut t he first t erm o n t he r ighthand side yields S3 y (S) = b3S3F(s) Dividing t hroughout b y s3 + [ a2Y(s) + b2F(s)]s2 + [ aIY(s) + b IF(s)]s
+ [ aoY(s) + boF(s)]
yields R es <  2 F 2(S)=s+2 I f YI(t) and Y2(t) a re the system responses to / I(t) and f2(t), respectively, then YI(s) = (s + 1)(s + 5)
1 /4 R es>  1 1/4 = s+ls+5
so t hat and
1 Y2(S) = (s + 2)(s + 5)
_  1/3
 s+2 + 1 /3
s +5  5<Res<2 1
+ : daoY(s)
s + boF(s)] (6.112) Therefore, Y (s) c an be generated by a dding four signals a ppearing o n t he r ighthand
side of Eq. (6.112). We shall build Y (s) s tep by s tep, a dding one component a t a
t ime. Figure 6.53a shows only t he first component; t hat is, b3F(S). F igure 6.53b
shows Y (s) formed by t he first two components, b3F(S) a nd ~[a2Y(s) + b2F(S)].
Observe t hat t he t erm a 2Y(s) is o btained from Y (s) itself. We a dd  azY(s) t o
b2F(s) a nd p ass i t t hrough a n i ntegrator t o o btain ~[a2Y(s) + b2F(s)]. F igure
6.53c shows Y (s) b uilt u p from t he first t hree c omponents. Finally, Fig. 6.53d shows
Y (s) b uilt up from all t he c omponents. This is t he final form, which represents a n
a lternative realization of H (s) i n Eq. (6.111).
T his r ealization c an b e readily generalized for a n n thorder t ransfer f unction
in Eq. (6.70) using n i ntegrators. 6.10 S ummary 6 ContinuousTime System Analysis Using t he Laplace Transform 458 ( a) ( b) ( c) ( d) F (s) F ig. 6 .53 Second (observer) canonical realization of an nthorder transfer function. 6 .10 Summary c; T he F ourier transform cannot be used directly in analysis of u~stable,
ev~n
m ar inally s table systems. Moreover, t he i nputs must also ? e re~tncted t o o uner
tran~formable signals, which leaves o ut exponentially growIng slgnal~. B oth th~se
l imitations a re t he r esult o fthe fact t hat t he s pectral components used In t~~ Fo~~~r
e
t ransform to synthesize a signal f (t) a re ordinary sinusoi~s ~r e xponentlt s ° lane
form e jwt , whose frequencies are restricted t o t he JW a xiS In t he comp ex p
. 459 These spectral components are incapable o f synthesizing exponentially growing signals. We extend t he Fourier transform by generalizing t he frequency variable from
s = j w t o s = (T + j w. T he r esulting transform is t he Laplace transform, which
c an analyze all types o f LTIC systems. T he Laplace transform can also handle
exponentially growing signals.
T he s ystem response t o a n everlasting exponential est is also a n everlasting
exponential H (s)e st , where H (s) is t he s ystem transfer function. We c an view t he
Laplace transform as a too...
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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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