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amplit~des c an t ake o n M v alues is a n M ary s ignal of which b inary ( M = 2) is
a special case. T he t erms c ontinuoustime a nd d iscretetime qualify t he n ature o f
a signal along t he t ime ( horizontal) axis. T he t erms a nalog a nd digital, o n t he
o ther h and, qualify t he n ature o f t he signal a mplitude (vertical axis). Figure 1.5
shows examples of various t ypes o f signals. I t is clear t hat a nalog is n ot necessarily
continuoustime a nd d igital need n ot b e d iscretetime. Figure 1.5c shows a n e xample
of a n a nalog discretetime signal. A n a nalog signal c an b e converted i nto a d igital
signa~ [ analogtodigital ( A/D) conversion] t hrough q uantization ( rounding off), as
explamed in Sec. 5.13. 1.23 Periodic and Aperiodic Signals
A s ignal j (t) is said t o b e p eriodic if for some positive c onstant To j (t) = j (t + To) for all t t __ (1.6) T he s mallest v alue of To t hat satisfies t he p eriodicity condition (1.6) is t he p eriod
of j (t). T he signals in Figs. 1.2b a nd 1.3e are periodic signals w ith p eriods 2 a nd 1,
respectively. A signal is a periodic if i t is n ot p eriodic. Signals in Figs. 1.2a, 1.3a,
1.3b, 1.3c, a nd 1.3d are all aperiodic.
B y definition, a periodic signal j (t) r emains unchanged w hen t imeshifted by
one period. For t his r eason a periodic signal m ust s tart a t t =  00 b ecause if i t
s tarts a t s ome finite i nstant, say t = 0, t he t imeshifted signal j (t + To) will s tart a t
t =  To a nd j (t + To) would n ot b e t he s ame as f (t). T herefore a periodic signal,
b y definition, m ust s tart a t t =  00 a nd c ontinuing forever, as i llustrated i n Fig. 1.6.
A nother i mportant p roperty o f a periodic signal j (t) is t hat j (t) c an b e generated b y p eriodic e xtension o f any segment of j (t) of d uration To ( the p eriod).
As a result we c an g enerate j (t) from a ny s egment of j (t) w ith a d uration of one
p eriod b y placing t his s egment a nd t he r eproduction t hereof e nd t o e nd a d infinitum o n e ither side. Figure 1.7 shows a periodic signal j (t) o f p eriod To = 6. T he
s haded p ortion o f Fig. 1.7a shows a segment of j (t) s tarting a t t =  1 a nd h aving
a d uration of one period (6 s econds). T his s egment, w hen r epeated forever in e ither
d irection, results in t he p eriodic signal j (t). F igure 1.7b shows a nother s haded
s egment of j (t) o f d uration To s tarting a t t = O. A gain we see t hat t his s egment,
w hen r epeated forever o n e ither side, results in j (t). T he r eader c an verify t hat t his
c onstruction is possible w ith a ny s egment of j (t) s tarting a t a ny i nstant a s long as
t he s egment d uration is one period.
I t is helpful t o l abel signals t hat s tart a t t =  00 a nd c ontinue for ever as
everlasting signals. Thus, an everlasting signal exists over t he e ntire interval  00 <
t < 0 0. T he signals in Figs. L Ib a nd 1.2b are examples of everlasting signals.
Clearly, a periodic signal, by definition, is a n e verlasting signal.
A signal t hat does not s 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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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