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Unformatted text preview: cisely b ut a re known only in t erms o f probabilistic description, such as m ean value, m ean s quared value, a nd so o n is a r andom s ignal. I n
t his b ook we shall exclusively deal w ith d eterministic signals. R andom signals are
beyond t he s cope of t his s tudy. 1 .3 Some Useful Signal Operations We discuss here t hree useful signal operations: shifting, scaling, a nd inversion.
Since t he i ndependent variable in o ur s ignal description is time, these o perations a re
discussed as t ime s hifting, t ime scaling, a nd t ime inversion (or folding). However,
t his discussion is valid for functions having i ndependent variables o ther t han t ime
(e.g., frequency or distance). 1.31 Time Shifting C onsider a signal f (t) (Fig. 1.8a) a nd t he s ame signal delayed by T seconds
(Fig. 1.8b), which we shall denote by ¢ (t). W hatever h appens in f (t) (Fig. 1.8a) a t
some i nstant t also h appens in ¢ (t) (Fig. 1.8b) T s econds later a t t he i nstant t + T .
T herefore
(1.8)
¢ (t + T ) = f (t)
a nd
(1.9)
¢ (t) = f (t  T)
Therefore, t o t imeshift a signal by T , we replace t w ith t  T. T hus j (t  T)
r epresents j (t) t imeshifted by T seconds. I f T is positive, t he s hift is t o t he r ight
(delay). I f T is negative, t he shift is t o t he left (advance). Thus, j (t  2) is j (t)
delayed (rightshifted) by 2 seconds, a nd j (t + 2) is f (t) a dvanced (leftshifted) by
2 seconds. 62 1 I ntroduction t o Signals a nd S ystems 1.3 Some Useful Signal O perations 63 1 (a) 1 (b) 1 (b) t f(t+ 1 )
(c) 1 (c) 1 F ig. 1 .10 Time scaling a signal.
Fig. 1 .9 (a) signal i tt) (b) i tt) delayed by 1 second (c) i tt) advanced by 1 second. • E xample 1 .3
An exponential function i tt) = e  2t shown in Fig. 1.9a is delayed by 1 second. Sketch
and mathematically describe the delayed function. Repeat the problem if i tt) is advanced
by 1 second.
The function i tt) can be described mathematically as
i tt) = { ~  2t t;:::O
(1.10) { ~  2(tl) t l;:::O or t;:::1 t l<O or t <1 (1.11) Let i a(t) represent the function i tt) advanced (leftshifted) by 1 second as depicted in Fig.
1.9c. This function is i (t + 1); its mathematical description can be obtained from i tt) by
replacing t with t + 1 in Eq. (1.10). Thus
e 2(t+l) iOo(t) = i (t + 1) = { 0 t +l;:::O or t ;:::1
(1.12) t +l<O or t <l • Exercise E 1.5
Write a mathematical description of the signal h (t) in Fig. 1.3c. This signal is delayed
by 2 seconds. Sketch the delayed signal. Show that this delayed signal fd(t) can be described
mathematically as fd(t) = 2(t  2) for 2 ::; t ::; 3, and equal to 0 otherwise. Now repeat the
b, 1.32 Time Scaling T he c ompression or expansion of a signal in t ime is known as t ime s caling.
C onsider t he s ignal f (t) o f Fig. 1.10a. T he s ignal 4>(t) i n Fig. 1.10b is f (t) compressed in t ime b y a factor of 2. T herefore, whatever h appens in f (t) a t some
i nstant t also h appens t o 4>(t) a t t he i nstant t /2, so t hat t...
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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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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