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

Clearly i t is impossible t o g enerate a t rue power

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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). 1.3-1 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 ime-shift a signal by T , we replace t w ith t - T. T hus j (t - T) r epresents j (t) t ime-shifted 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 (right-shifted) by 2 seconds, a nd j (t + 2) is f (t) a dvanced (left-shifted) 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(t-l) t -l;:::O or t;:::1 t -l<O or t <1 (1.11) Let i a(t) represent the function i tt) advanced (left-shifted) 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, 1.3-2 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.

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