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

# T he f requency response of a system is d etermined

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Unformatted text preview: gnal, depicted in Fig. 8.1, is t herefore a sequence of numbers. This signal may be denoted by f (kT) a nd viewed a s a function of time t where signal values are specified a t t = kT. I t m ay also b e d enoted by f [k] a nd viewed as a function of k (k, integer). For instance, a continuous-time exponential f (t) = e - t , when sampled every T = 0.1 second, r esults in a discrete-time signal f (kT) given by D iscrete-time signals arise naturally in situations which are inherently discretetime, such as population studies, amortization problems, national income models, a nd r adar tracking. T hey may also arise as a result of sampling continuous-time signals in sampled d ata systems, digital filtering, and so on. Digital filtering is a p articularly interesting application in which continuous-time signals are processed by discrete-time systems, using appropriate interfaces a t t he i nput a nd o utput, as illustrated in Fig. 8.2. A continuous-time signal f (t) is first sampled to convert it into a discrete-time signal f [k], which is t hen processed by a discrete-time system t o yield t he o utput y[k]. A c ontinuous-time signal y(t) is finally constructed from y[k]. We shall use t he n otations C /D a nd D /C for continuous-to-discrete-time a nd discrete-to-continuous-time conversion. Using the interfaces in this manner, we can process a continuous-time signal with a n a ppropriate discrete-time system. As we shall see l ater in our discussion, discrete-time systems have several advantages over continuous-time systems. For this reason, t here is a n accelerating t rend toward processing continuous-time signals with discrete-time systems. ~ \ It.. ~ 1 k~ jrrlrrrlIl! t. Discrete to / y ( I) Continuous ole f (kT) = e - kT = e - O. 1k Clearly, t his signal is a function of k a nd may be expressed as J[k]. We c an plot this signal as a function of t o r as a function of k (k, integer). T he r epresentation f [k] is m ore convenient and will be followed throughout this book. A discrete-time signal therefore m ay b e viewed as a sequence of numbers, and a discrete-time system may be seen as processing a sequence of numbers f [k] a nd yielding as o utput a nother sequence of numbers y[k]. 540 F ig. 8 .2 Processing a continuous-time signal by a discrete-time system. 8.2 Some Useful Discrete-Time Signal Models We now discuss some i mportant d iscrete-time signal models which are encountered frequently in t he s tudy of discrete-time signals a nd systems. 8 Discrete-time Signals a nd Systems 542 543 8.2 Some Useful Discrete-time Signal models (a) (b) k- m (a) Fig. 8.3 Discrete-time impulse function. 1. D iscrete- Time Impulse Function 8[k] T he d iscrete-time counterpart of the continuous-time impulse function 8(t) is 8[kJ, defined by 8[k] = g k =O k l-O 3. Discrete- Time Exponential -yk A continuous-time exponential {~ I f we w ant a signal t o s tart a t k = 0 (so t hat i t has a zero value for all k < 0), we need o nly multiply the signal with u[k]. 4 6 k- Fig. 8 .4 A discrete-time unit step functi...
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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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