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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) = { 1.34 ~  '/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 imeshift f ( t ) b y b t o obtain f (t  b). Now timescale the shifted signal f (t  b)
by a ( that is, replace t w ith at) t o o btain f (at  b).
2 . Timescale f (t) by a t o obtain f (at). Now timeshift 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 timecompress this signal by factor 2 (replace t
w ith 2t) t o o btain f (2t  6). Alternately, we first timecompress 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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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