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 imeintegration 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 timedifferentiation 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 ContinuousTime 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 piecewiselinear signal using the timedifferentiation property. • E xample 4 .15
Find the zerostate response of a stable LTIC system with transfer functiont
1 (4.53) H (s}=s+2 From the timedifferentiation 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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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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