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

# Would negative feedback make a system stable a nd

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Unformatted text preview: rm table. Changmg the sign of s in F2( - s) yields F2(S). + eatu(t) (6.106) We a lready know the Laplace transform of the causal component R es> a For the anticausal component, h (t) = e btu(_t), we have (6.107) 452 6 C ontinuous-Time S ystem A nalysis Using t he L aplace T ransform _ h ( - t) = e b tu(t) 1 6.8 s+b s o t hat R es < b f- .;@, T herefore b -1 ¢ =} - - s- b ' 08 R es < b (6.108) - --+-0-- , ,-- 1 t:l 0" ·So an ec ~ o o f - .;@, . .,-- z8 0 ,,-- a nd t he L aplace t ransform o f f It) i n E q. (6.106) is 1 i cc . .,-- -1 F2(S)=--=-s + b s- b e t u( - t) 453 R es> - b ¢ =} - - 1 T he B ilateral L aplace T ransform s F(s)=--+R es> a a nd s-b s -a a -b a < R es < b ( s-b)(s-a) R es < b (6.109) F igure 6 .50 s hows f (t) a nd t he r egion of convergence o f F (s) for v arious v alues of a a nd b. E quation (6.109) i ndicates t hat t he r egion of convergence o f F (s) d oes n ot e xist if a > b, w hich is precisely t he c ase i n F ig. 6.50g. O bserve t hat t he p oles o f F (s) a re o utside ( on t he e dges) o f t he r egion of convergence. T he p oles of F (s) b ecause o f t he a nticausal c omponent o f f (t) lie t o t he r ight o f t he r egion of convergence, a nd t hose d ue t o t he c ausal c omponent of f (t) lie t o i ts l eft . • E xample 6 .22 F ind t he inverse Laplace transform of -3 F (s) = (s if t he region o f convergence is ( a) + 2)(s _ 1) - 2 < R es < 1 ( b) i R es> 1 ( e) t:l R es < - 2. , ,-1 1 F(s)=--s+2 s -1 ( a) f-.;@, ~--+o-- f-.;@, - --+-0-- f-.;@, - ---1-- 0 f-.;@, - --+-0-- ,,-- Now, F (s) h as poles a t - 2 a nd 1. T he strip of convergence is - 2 < R es < 1. T he pole a t - 2, being t o t he left of t he s trip of convergence, corresponds t o t he causal signal. T he pole a t 1, being t o t he right of t he s trip of convergence, corresponds t o t he anticausal signal. Equations (6.107) a nd (6.108) yield f (t) = e - 2t u(t) + e tu(-t) ( b) B oth poles lie to t he left of t he region of convergence, so b oth poles correspond t o causal signals. Therefore f (t) = ( e- 2t - et)u(t) ( c ) Both poles lie t o t he right of t he region of convergence, so b oth poles correspond t o a nticausal signals, and f It) = ( -e - 2t + e t)u( - t) o 4 54 6 C ontinuous-Time S ystem A nalysis U sing t he L aplace T ransform 6.8 T he B ilateral L aplace T ransform 455 f {t) I In f (t) IF (a) o -4 (a) f (l) f(l) -4 4 o 2 -2 1_ (b) ( b) T he response y (t) is t he inverse transform of F (s)H(s) Figure 6.51 shows the three inverse transforms corresponding t o t he s ame F (s) b ut w ith different regions of convergence. • y (t) = C Linear System Analysis Using t he Bilateral Transform y (t) = C - 1 [ F(s)H(s)] E xample 6 .23 F ind t he c urrent y(t) for t he R C circuit in Fig. 6.52a if the voltage f (t) is + e2t u( - t) -8 (8 + 1)(8 - 1)(s - 2) [~_1_ + ~_1_ _ ~ 6 s+1 2 8-1 1] 3 8-2 y (t) = ~e-tu(t) + ~etu(t) + ~e2tu(_t) Figure 6.52c shows y(t). Note t hat in this example, if f (t) = e - 4t u(...
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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.

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