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

# T his s egment w hen r epeated forever in e ither d

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Unformatted text preview: art b efore t = 0 is a c ausal signal. I n o ther words, j (t) is a causal signal if j (t) = 0 t &lt;0 (1.7) Signals in Figs. 1.3a, b, c, a s well as in Figs. 1.9a a nd 1 .9b are causal signals. A signal t hat s tarts before t = 0 is a n oncausal signal. All t he signals i n Figs. 1.1 a nd 1.2 are noncausal. O bserve t hat a n e verlasting signal is always noncausal b ut a n oncausal signal is n ot necessarily everlasting. T he e verlasting signal in Fig. 1.2b is noncausalj however, t he n oncausal signal in Fig. 1.2a is n ot e verlasting. A signal t hat is zero for all t :2: 0 is called a n a nticausal signal. 60 1 I ntroduction t o Signals a nd S ystems 61 1.3 Some Useful Signal O perations (a) (a) ~ F ig. 1 . 7 G eneration o f a p eriodic s ignal by p eriodic extension o f i ts s egment o f o ne-period d uration. cf== Comment: A t rue e verlasting s ignal c annot b e g enerated in practice for obvious reasons. W hy s hould w e b other t o p ostulate such a signal? In later c hapters we s hall see t hat c ertain s ignals (including a n e verlasting sinusoid) which c annot b e g enerated in practice do s erve a very useful p urpose i n t he s tudy o f signals a nd s ystems. 1.2-4 Energy and Power Signals A s ignal w ith finite energy is a n e nergy s ignal, a nd a s ignal w ith finite a nd n onzero power is a p ower s ignal. Signals in Fig. 1.2a a nd 1.2b a re e xamples of energy a nd p ower signals, respectively. O bserve t hat power is t he t ime average of energy. Since t he averaging is over a n infinitely large interval, a signal w ith finite energy h as z ero power, a nd a s ignal w ith finite power has infinite energy. Therefore, a signal c annot b oth b e a n e nergy a nd a power signal. I f i t is one, i t c annot b e t he o ther. O n t he o ther h and, t here a re signals t hat a re neither energy nor power signals. T he r amp s ignal is such a n e xample. Comments All p ractical signals have finite energies a nd a re therefore energy signals. A power signal m ust necessarily have infinite duration; otherwise i ts power, which is its energy a veraged over a n infinitely large interval, will n ot a pproach a (nonzero) limit. Clearly, i t is impossible t o g enerate a t rue power signal in practice because such a signal h as infinite d uration a nd infinite energy. Also, b ecause o f p eriodic repetition, periodic signals for which t he a rea u nder If(t)1 2 over o ne p eriod is finite are power signals; however, n ot all power signals are periodic. C::. E xercise E 1.4 Show t hat a.n everlasting exponential e - at is neither an energy nor a power signal for any real value of a. However, if a is imaginary, it is a power signal with power P f = 1 regardless of '7 the value of a. 1.2-5 Deterministic and Random Signals A s ignal w hose physical description is known completely, e ither i n a m athematical form or a graphical form, is a d eterministic s ignal. A s ignal whose values ( c) F ig. 1 .8 T ime s hifting a s ignal. c annot b e p redicted pre...
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