This preview shows page 1. Sign up to view the full content.
Unformatted text preview: E xample 1 .7
Describe the signal in Fig. 1.11a by a single expression valid for all t.
Over t he interval from  1.5 t o 0, t he signal can be described by a c onstant 2, a nd
over t he interval from 0 t o 3, i t can be described by 2 e  t / 2 • Therefore
J (t) = 2[u(t + 1.5)  u(t)] + 2 e t / 2 [u(t)  u(t  3l]
'v'
f l(t) = 2 u(t + 1.5)  2(1  ' b. E xercise E 1.8 Show that the signal shown in Fig. 1.18 can be described as
f (t) = ( t  l)u(t  1)  (t  2)u(t  2)  u(t  4) v 2. \l T he Unit Impulse Function 8(t) T he u nit i mpulse f unction 8(t) is one o f t he m ost i mportant f unctions i n t he
s tudy o f s ignals a nd s ystems. T his f unction w as first defined b y P. A. M D irac a s I: h (t) =  2(t  3) [u(t  2)  u(t  3)]
f (t) \l u(t  2)] T he signal h (t) c an be obtained by mUltiplying another ramp by t he gate pulse illustrated
in Fig. 1.16c. This r amp has a slope  2; hence it can be described by  2t+c. Now, because
the ramp has a zero value a t t = 3, t he constant c = 6, a nd the ramp can be described by
 2(t  3). Also, t he gate pulse in Fig. 1.16c is u (t  2)  u(t  3). Therefore and e atu(  t), respectively. ' h (t) e t / 2 )u(t)  2e t / 2u(t  3) 8(t) = 0
8(t) dt = 1 (1.21) We c an v isualize a n i mpulse as a t all, n arrow r ectangular p ulse of u nit a rea,
a s i llustrated in Fig. 1.19b. T he w idth o f t his r ectangular p ulse is a very s mall
v alue f > O. C onsequently, i ts h eight is a very l arge v alue 1 /f. T he u nit i mpulse
t herefore c an b e r egarded a s a r ectangular p ulse w ith a w idth t hat h as b ecome
i nfinitesimally small, a h eight t hat h as b ecome i nfinitely large, a nd a n o verall a rea
t hat h as b een m aintained a t u nity. T hus 8(t) = 0 e verywhere e xcept a t t = 0, w here
i t is u ndefined. F or t his r eason a u nit i mpulse is r epresented b y t he s pearlike s ymbol
i n F ig. 1. 19a.
O ther p ulses, such a s e xponential p ulse, t riangular p ulse, o r G aussian p ulse
m ay also b e u sed i n i mpulse a pproximation. T he i mportant f eature o f t he u nit
i mpulse f unction is n ot i ts s hape b ut t he f act t hat i ts effective d uration ( pulse w idth) 70 I ntroduction to Signals a nd Systems
1
"( O (t) 1.4 Some Useful Signal Models 71 Sampling Property o f the Unit Impulse Function From Eq. (1.23a) it follows t hat o t I: t (a) ¢ (t)6(t) d t = ¢(O) F ig. 1.19 A unit impulse and its approximation. t t( c) (b) F ig. 1 .20 Other possible approximations to a unit impulse.
approaches zero while its area remains a t unity. For example, the exponential pulse
at
u(t) in Fig. 1.20a becomes taller a nd narrower as a increases. In t he limit as
a ..... 0 0, t he pulse height ..... 0 0, a nd its width or duration ..... O. Yet, t he a rea under
the pulse is u nity regardless of the value of a because
O Ie 1 00
O leat dt = 1 (1.22) T he pulses in Figs. 1.20b a nd 1.20c behave in a similar fashion.
From Eq. ( l.21), i t follows t hat t he function k6(t) = 0 for all t f= 0, and its
area is k. T hus, k 8(t) is an impulse function whose area is k (in contrast t o t he unit
impulse function, whose area is 1).
Multiplication o f...
View
Full
Document
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

Click to edit the document details