Unformatted text preview: t) + e  2t u(  t) the. n t he region of convergence of F(8) is  4 < Re 8 <  2 .
()
Here no region of convergence
t£
eXlS s or F s H(s), a nd t he response y (t) goes to infinity.
•
• E xample 6 .24 F ind the response y(t) o f a noncausal system with t he t ransfer function T he t ransfer function H (s) of t he circuit is given by H(8)=~ H(s) = _s_
s +1 8 1 Because h(t) is a causal function, t he region of convergence of H(s) is R es>  1. Next,
t he b ilateral Laplace transform of f (t) is given by
1
1
1
F ( 8) = 81  82 = :(81:7):(8,2"'") [ T he region of convergence of F (8)H( ) . h
.
F(8) a nd H(s) T h' . 1 R
8 IS t a t regIon of convergence common to both
.
IS IS < e 8 < 2. T he poles s  ±1 r
h
~~nver~ence and, therefore, correspond t o causal sign;;]s; theI~:I~ : : ~e~~e~ft:ht~:er~i~~ o~
e regIon of convergence a nd t hus represents an anticausal signal. Hence
g0 (6.110) T his e xpression is valid o nly i f F (s)H (s) e xists. T he r egion of convergence of
F (s)H(s) is t he r egion w here b oth F (s) a nd H (s) e xist. I n o ther w ords, t he r egion
o f c onvergence o f F (8) H (8) is t he r egion c ommon t o t he r egions of convergence of
b oth F (8) a nd H (8). T hese i deas a re c larified in t he following examples. f (t) = e tu(t) l = c .I S ince t he b ilateral L aplace t ransform c an h andle n oncausal s ignals, we c an
a nalyze n oncausal L TIC s ystems u sing t he b ilateral L aplace t ransform. W e h ave
s hown t hat t he ( zerostate) o utput y (t) is given b y • ( c) F ig. 6 .52 Response of a circuit t o a noncausal i nput (Example 6.23). (c) F ig. 6 .51 T hree possible inverse transforms of ( 8+2)/.1)' 6 .81 I <Res<2 2 R es < 1 to t he i nput f (t) = e  2t u(t).
We have F(8) = _ 1_
s +2
and R e8>  2 6 C ontinuousTime S ystem Analysis Using t he L aplace Transform l56 6.9 457 A ppendix 6.1: Second C anonical realization so t hat Y (s) 1 = F(s)H(s) = (s _ 1)(s + 2)
Therefore f he region of convergence of F (s)H(s) is, therefore, the region  2 < R es < 1. By partial
'raction expansion y (t)  1/3
1 /3
Y (s)=s_l+s+2
and
y (t) =~  2<Res<1 [e'u(  t) = ~e2'u(t)+ (~et+ l2 e 5t )U(t) • + e  2t u(t)] Note t hat the pole of H (s) lies in the RHP a t 1. Yet the system is not uns.table. ~he
pole(s) in the RHP may indicate instability or noncausality, dep~nding o~ Its lo:a~l~n
with respect to the region of convergence of H(s). For example, If H(s)  1/(
.)
with R es> 1, the system is causal and unstable, with h (t) = _etu~t). In c~nt~as~t
H (s) =  1/(sl) w ith R es < 1, the system is noncausal and stable, with h (t)  e u( ). 6 .9 Appendix 6.1: Second Canonical Realization A n n thorder t ransfer f unction can also b e r ealized by a second canonical ( observer c anonical) form. As in t he case o f t he first canonical, we b egin w ith a
r ealization o f a t hirdorder t ransfer function in E q. (6.71) •
• = YI(t) + Y2(t) E xample 6 .25 H (s) = Y (s) = b 3S3+bzsz+bIS+bo
F (s)
s3 + a2s2 + a ls + ao . (6.111) Find the response y (t) of a system with the transfe...
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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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