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

1 17 t he signal for exercise e1 7 t 2 f l b f2

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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 pear-like 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 le-at 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...
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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.

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