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

114a this function is defined by a nd i ts t ime i

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Unformatted text preview: l t reatment. Using t he u nit s tep function, we c an describe such functions by a single expression t hat is valid for all t. f (-t) = { 1.3-4 ~ - '/2 - 12 - t > - 5 o therwise or 1:::; t < 5 • Combined Operations C ertain c omplex operations require simultaneous use of more t han one of t he above operations. T he m ost general operation involving all t he t hree o perations is f (at - b), which is realized in two possible sequences of operation: u (t) = 1 . T ime-shift f ( t ) b y b t o obtain f (t - b). Now time-scale the shifted signal f (t - b) by a ( that is, replace t w ith at) t o o btain f (at - b). 2 . Time-scale f (t) by a t o obtain f (at). Now time-shift f (at) by ~ ( that is, replace t w ith (t - ~) t o o btain f[a(t - ~)J = f (at - b). I n either case, if a is negative, time scaling involves time inversion. For instance, t he signal f (2t - 6) c an be obtained in two ways: first, delay f (t) by 6 t o o btain f (t - 6) a nd t hen time-compress this signal by factor 2 (replace t w ith 2t) t o o btain f (2t - 6). Alternately, we first time-compress f (t) by factor 2 to obtain f (2t), t hen delay this signal by 3 (replace t w ith t - 3) t o o btain f (2t - 6). 1.4 Some Useful Signal Models I n t he a rea o f signals a nd systems, t he s tep, t he impulse, a nd t he e xponential functions are very useful. They n ot only serve as a basis for representing other signals, b ut t heir u se c an simplify many aspects of t he signals a nd systems. :1 D 2 4 {~ t- t20 (1.20) t <O t- O -\ ( b) (a) F ig. 1 .15 R epresentation of a r ectangular p ulse by s tep functions. Consider, for example, t he r ectangular pulse depicted in Fig. 1.15a. We can express such a pulse in terms of familiar s tep functions by observing t hat t he pulse f (t) c an be expressed as t he s um of t he two delayed u nit s tep functions as shown in Fig. 1.15b. T he u nit s tep function u (t) delayed by T seconds is u (t - T). From Fig. 1.15b, i t is clear t hat f (t) = u (t - 2) - u(t - 4) 68 1 I ntroduction t o Signals a nd S ystems f")~ 023 1.4 S ome U seful Signal M odels 69 u ( -t) (0) t- f l ( t) 0 t- 2 0 t(b) (a) F ig. 1 .17 T he Signal for Exercise E1. 7. t- 2 f'] L (b) f2 ( t) 2 2 2 0 t- 0 t- t-- Fig. 1 .18 The signal for Exercise E l.8. (c) F ig. 1 .16 Representation of a signal defined interval by interval. 4 Compare this expression with the expression for t he same function found in Eq. 1.17. • b. E xercise E l. 7 Show that the signals depicted in Figs. 1.17a and L ITh can be described as u (-t), and • E xample 1 .6 Describe the signal in Fig. 1.16a. T he signal illustrated in Fig. l.16a can be conveniently handled by breaking it up into the two components f t(t) a nd h (t), depicted in Figs. 1.16b and 1.16c respectively. Here, f t(t) c an be obtained by multiplying the ramp t by the gate pulse u (t) - u(t - 2), as shown in Fig. 1. 16b. Therefore f t(t) = t [u(t) - = f t(t) + h (t) = t [u(t) =t u(t) - - u(t - 2)]- 2(t - 3) [u(t - 2) - u(t - 3)] 3(t - 2)u(t - 2) + 2 (t - 3)u(t - 3) • •...
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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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