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

3 some p roperties of the fourier transform 263 then

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Unformatted text preview: F2(W) F (w) JW t hen (time differentiation)t df . - ~ J wF(w) dt (4.46) R epeated a pplication o f t his p roperty y ields a nd ( time integration) ( 4.48) f (r)dr t ; ~ - 00 F (w) JW + 1TF(O)6(w) (4.47) T he t ime-integration p roperty [Eq. (4.47)] h as a lready b een p roved i n E xample 4.13. P roof: D ifferentiation o f b oth s ides o f E q. ( 4.8b) yields • d 1 . _ f = _ ;00 j wF(w)e Jwt dw dt 271"_00 T his r esult s hows t hat df -~jwF(w) dt tValid only if the transform of df /dt exists. E xample 4 .14 Using t he time-differentiation property, find t he Fourier transform of t he t riangle pulse b.( ~) i llustrated in Fig. 4.25a. To find t he Fourier transform of this pulse we differentiate the pulse successively, as illustrated in Fig. 4.25b and c. Because df / dt is c onstant everywhere, its derivative, d2 f / dt 2 , is zero everywhere. B ut df / dt has j ump discontinuities with a positive j ump of 2 /r a t t = ±~, a nd a negative j ump of 4 /r a t t = O. Recall t hat t he derivative of a signal a t a j ump discontinuity is a n impulse a t t hat point of strength equal t o t he a mount of jump. Hence, d2 f / dt 2 , t he derivative of df / dt, consists of a sequence of impulses, as depicted in Fig. 4.25c; t hat is, ~:; = ;[8(t+~) - 28(t) + 8(t- ~)J (4.49) 4 Continuous-Time Signal Analysis: T he F ourier Transform 266 4.4 Signal Transmission T hrough L TIC S ystems 267 4.4 Signal Transmission through LTIC Systems t- t I f J(t) a nd yet) a re t he i nput a nd o utput o f a n L TIC s ystem w ith t ransfer function H(w), t hen, a s d emonstrated i n E q. (4.44b) (a) T Yew) d! dt 2 t t /2 t- 0 -t - 2- 2 ( b) 2 d '! d t' T 0 -t - 2- t t- (4.52) T his r esult applies only t o a symptotically ( and m arginally) s table s ystems because o f t he r easons discussed in t he f ootnote o f p. 243. Moreover, J(t) h as t o b e F ourier transformable. Consequently, exponentially growing i nputs c annot b e h andled by t his m ethod. I n C hapter 6, we shall see t hat t he L aplace transform, which is a g eneralized Fourier transform, is m ore versatile a nd c apable o f a nalyzing all kinds o f L TIC s ystems w hether s table, unstable, o r m arginally stable. Laplace t ransform c an also handle exponentially growing inputs. C ompared t o t he L aplace transform, t he Fourier t ransform i n s ystem a nalysis is clumsier. Hence, t he L aplace t ransform is preferable t o t he F ourier t ransform in L TIC s ystem a nalysis, a nd we shall n ot b elabor t he a pplication o f t he F ourier t ransform t o L TIC s ystem analysis. We consider j ust o ne example here. -2 T t H(w)F(w) = (c) T -4 ~ F ig. 4.25 Finding the Fourier transform of a piecewise-linear signal using the timedifferentiation property. • E xample 4 .15 Find the zero-state response of a stable LTIC system with transfer functiont 1 (4.53) H (s}=s+2 From the time-differentiation property (4.48) and the input f (t} = e -tu(t}. I n this case, (4.50a) 1 F (w}=-- jw+ 1 { =} e - jwto and F(w} = ~[e1"f - 2+ e -j"fJ = * (...
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