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same sinusoid compressed by a factor 3 and expanded by a factor 2. \ l 1.33 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 imeinvert 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 imeexpanded 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 imeinvert 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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 Spring '13
 Bayliss
 Signal Processing, The Land

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