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

Repeat t he problem for t he same signal time

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Unformatted text preview: and the same sinusoid compressed by a factor 3 and expanded by a factor 2. \ l 1.3-3 Time Inversion (Time Reversal) C onsider t he s ignal f (t) i n F ig. 1.12a. W e c an v iew f (t) as a r igid w ire f rame h inged a t t he v ertical a xis. T o t ime-invert f (t), we r otate t his f rame 1 80 0 a bout t he v ertical a xis. T his t ime i nversion o r f olding [ the r eflection o f f (t) a bout t he v ertical axis] gives u s t he s ignal </>(t) ( Fig. 1 .12b). O bserve t hat w hatever h appens i n F ig. 1 .12a a t s ome i nstant t a lso h appens i n F ig. 1 .12b a t t he i nstant - to T herefore - 0.5 :::; t < 0 or 0 :::; t < 1 (1.18a) otherwise Observe t hat t he i nstants t = - 1.5 a nd 3 in f (t) c orrespond t o t he i nstants t = - 0.5, a nd 1 in t he c ompressed signal f (3t). F igure 1.11c shows f .(t), which is f (t) t ime-expanded by factor 2; consequently, i t c an b e described mathematically as f (t/2), which is obtained by replacing t w ith t /2 in f (t). T hus cfJ(-t) = f (t) a nd </>(t) = f (-t) (1.19) T herefore, t o t ime-invert a s ignal we r eplace t w ith - to T hus, t he t ime i nversion o f s ignal f (t) y ields f (-t). C onsequently, t he m irror i mage o f f (t) a bout t he v ertical a xis is f (-t). R ecall a lso t hat t he m irror i mage o f f (t) a bout t he h orizontal a xis is - f(t). 1 I ntroduction to Signals a nd Systems 36 s::::1, I 67 1.4 Some Useful Signal Models U ( I) (a) f Ul -7 1- {HII t S I 3 ( b) o 5 7 1- 1- I- w ~ F ig. 1 .14 (a) U nit s tep f unction u (t) (b) exponential e -a'u(t). F ig. 1 .13 A n e xample o f t ime inversion. 1. • E xample 1 .5 Unit Step Function u(t) F or t he s ignal I (t) i llustrated i n Fig. l .13a, s ketch f (-t), which is time inverted f (t). T he i nstants - 1 a nd - 5 i n f (t) a re m apped i nto i nstants 1 a nd 5 i n f (-t). B ecause f (t) = e '/2, we h ave f (-t) = e -'/2. T he signal f (-t) is d epicted in Fig. U 3b. We c an describe f (t) a nd f ( - t) as e '/2 - 12t>-5 f (t) = { 0 o therwise I n much of our discussion, t he signals begin a t t = 0 (causal signals). Such signals can be conveniently described in terms of u nit s tep function u (t) shown in Fig. 1.14a. This function is defined by a nd i ts t ime i nverted version f ( - t) is o btained b y replacing t w ith - t i n f (t) as I f we want a signal t o s tart a t t = 0 (so t hat i t h as a value o f zero for t < 0), we only need t o m ultiply t he signal with u (t). For instance, t he signal e - a ' represents a n everlasting exponential t hat s tarts a t t = - 00. T he causal form of this exponential illustrated in Fig. 1.14b can be described as e -atu(t) . T he u nit step function also proves very useful in specifying a function with different mathematical descriptions over different intervals. Examples of such functions appear in Fig. 1.11. These functions have different mathematical descriptions over different segments of time as seen from Eqs. (1.17), (1.1Sa), and (1.1Sb). Such a description often proves clumsy a nd inconvenient in mathematica...
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