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Unformatted text preview: generalized function definition
of a n impulse. Because t he u nit s tep function u (t) is discontinuous a t t = 0, i ts
derivative d u/dt does not exist a t t = 0 in the ordinary sense. We now show t hat 72 1 I ntroduction t o Signals a nd Systems 73 1.4 Some Useful Signal Models t his derivative does e xist in the generalized sense, a nd i t is, in fact, o(t). AB a proof,
l et us evaluate t he integral of (du/dt)<{I(t), using integration by parts: oo du <{I(t) dt = u(t)<{I(t)
dt J  00 00
 00  °1 = <{I(oo)  1 (a) 00 1 00 u(t)¢(t) dt (1.25) 00 ¢(t) dt = <{I(oo)  <{I(t)l~
= <{I(O) (1.26) T his r esult shows t hat d u/dt satisfies t he s ampling property of o(t). Therefore i t is
a n impulse o(t) i n t he generalized s ensethat is, ~~ = (1.27) o(t) F ig. 1 .21 Sinusoids of complex frequency Consequently 3. loo o(r)dr = u(t) (1.28) T hese r esults c an also b e o btained graphically from Fig. 1.19b. We observe
t hat t he a rea f rom  00 t o t u nder t he l imiting form of o(t) in Fig. 1.19b is zero if
t < 0 a nd u nity i f t 2: O. Consequently 1
t T he Exponential Function (7 + j w. e st One of the most important functions in t he a rea of signals and systems is the
exponential signal e st, where s is complex in general, given by
s= (7 + jw Therefore
(1.30a) O (r)dr={O If s· = 1  00 = u(t) (7  jw ( the conjugate of s), t hen (1.29) (1.30b) Derivatives o f impulse function can also b e defined as generalized functions (see
Prob. 1.410). a nd ;:,. Comparison of this equation with Euler's formula shows t hat e st is a generalization
of the function eiwt , where t he frequency variable j w is generalized t o a complex
variable s = (7 + j w. For this reason we designate t he variable s as the c omplex
f requency. From Eqs. (1.30) i t follows t hat t he function e st encompasses a large
class of functions. T he following functions are special cases of e st: E xercise E 1.9
Show t hat + 3)6(t) = ( a) (t3 ( c) e  2t 6(t) = 6(t) Hint: Use Eqs. (1.23).
;:,. 3 6(t) ( b)
( d) [sin ( t 2  ~)16(t) =  6(t)
w2 +1 1 w2 +9 6 (wl) = 5 '6(wl) \l 1
2
3
4 E xercise E 1.l0
Show t hat ( a) ( c) 1:
1: 6 (t)e j ",t dt = 1 e  2(xt)6(2  t) d t = e  2(x2) Hint: In p art c r ecall t hat 6 (x) is located a t x
is at t = 2.
\l (1.30c) = O. T herefore 6(2  t) is located a t 2  t = 0; t hat A c onstant k = k e Dt (s = 0)
A monotonic exponential eut (w = 0, s = ( 7)
A sinusoid cos wt (7 = 0 ,8 = ± jw)
An exponentially varying sinusoid eut cos wt (8 = (7 ± j w) These functions are illustrated in Fig. 1.21.
The complex frequency 8 can be conveniently represented on a c omplex f requency p lane (8 plane) as depicted in Fig. 1.22. T he horizontal axis is the real axis
(7 axis), and the vertical axis is t he imaginary axis (jw axis). The absolute value of
t he imaginary p art of s is Iwl ( the radian frequency), which indicates t he frequency 74 1 I ntroduction t o Signals a nd Systems 75 1.5 Even a nd O dd F unctions
fo ( t) Left h alf plane t 1 1 Fig. 1.23 An even and an odd function of t. >. ~ I 1....
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 Spring '13
 Bayliss
 Signal Processing, The Land

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