Unformatted text preview:  00 where Y (w), t he Fourier transform of y(t), is given b yt
h (t)eiwtdt (4.17) T he r ighthand side is t he Fourier transform of h (t), and according t o o ur n otation
introduced i n Eq. (4.3) this is H (w), whereas t he same entity is d enoted by H (jw) t The r elationship (4.19) applies only t o t he a symptotically s table s ystems. T he r eason is t hat
when s = jw, t he i ntegral o n t he r ighthand side o f Eq. (2.48) does n ot converge for u nstable
systems. Moreover, even for marginally s table s ystems, t hat i ntegral does n ot converges in t he
o rdinary sense, a nd H(jw) [or H(w)) c annot b e o btained b y r eplacing s i n H(s) w ith jw. As shown
in Eq. (4.44b), Eq. (4.19) c an b e applied t o m arginally s table s ystems p rovided H(w) is i nterpreted
as t he F ourier t ransform o f h(t) r ather t han a s H(s) w ith s r eplaced b y jw. 244 4 ContinuousTime Signal Analysis: T he Fourier Transform Y (w) = F (w)H(w) 4.2 Transforms of Some Useful Functions
1 (4.19) 245 r eet( x ) I n c onclusion, we showed t hat for a n LTIC s ystem with transfer function H (s), if
t he i nput a nd t he o utput a re f it) a nd y(t), respectively, a nd if f it) = F(w) y(t) = I Y (w) 1: 1 "2 "1 2" F ig. 4 .7 A gate pulse. t hen for asymptotically stable systems Y (w) = F(w)H(w)
We shall derive this result again l ater in a more formal way.
T he p rocedure o f t he frequencydomain m ethod is identical t o t hat o f t he timedomain m ethod. I n t he t imedomain case we express t he i nput f (t) a s a s um o f
its impulse components; in t he frequencydomain case, t he i nput is expressed as
a s um o f everlasting exponentials (or sinusoids). I n t he former case, t he response
y(t) o btained by summing t he s ystem's responses t o impulse components results
in t he c onvolution integral; in t he l atter case, t he response o btained b y summing
t he s ystem's response to everlasting exponential components results in t he Fourier
integral. T hese ideas can be expressed mathematically as follows: 4 .2 Transforms o f S ome Useful Functions
F or convenience, we now introduce a compact n otation for some useful functions
such a s g ate, triangle, a nd i nterpolation functions.
U nit Gate Function We define a u nit g ate f unction reet (x) as a gate pulse of u nit height a nd u nit
w idth, centered a t t he origin, a s i llustrated in Fig. 4.7a:t Ixl > ~
Ixl = ~
Ixl < ~ 1 For t he t imedomain case tilt) ===> h it)
f it) = 1: the impulse response of the system is h (t) f (x)ti(t  x)dx T he g ate pulse in Fig. 4.7b is t he u nit g ate p ulse rect (x) e xpanded by a factor
a nd t herefore can be expressed as reet(~) (see Sec. 1.32). Observe t hat T , t he
d enominator o f t he a rgument o f rect(~), i ndicates t he w idth o f t he pulse. expresses J (t) as a sum of impulse components U nit Triangle Function expresses yet) as a sum of responses to impulse components y(t) = [ : f (x)h(t  x)dx We define a u nit t riangle function t .(x) as a t riangular pulse of unit height a nd
u nit width, cent...
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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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