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

648b a nd 648c 1 48 as long as k r emains below 48

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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) = -8---1 - -8---2 = -:-(8---1:-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 ( zero-state) 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 .8-1 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 ontinuous-Time 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) = ~e-2'u(-t)+ (~e-t+ -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/(s-l) 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 th-order 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 hird-order 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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