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Unformatted text preview: (x = 7r) is t he s ame as t he
fre~uency 11 ~ 0 (c~nstant signal). These conclusions can be verified from Fig. 8.14,
which shows smusOids offrequencies 11 = (a) 0 or 27r (b) i or 1~" (c) ~ or ~ (d) 7r.
6. E xponentially V arying D iscrete-Time S inusoid " '(k cos (l1k + 8) ! his is a sin~soi~ cos (l1k + 8) w ith a n exponentially varying amplitude "'(k. I t is
o btamed by mUltlplymg t he sinusoid cos (l1k + 8) by a n e xponential "'(k. F igure 8.15 8 Discrete-time Signals a nd Systems 556
_ __ (0.9)* 8.3 Sampled Continuous-Time Sinusoids a nd Aliasing 557 called a p ower s ignal. As in t he continuous-time case, a discrete-time signal can
either be a n energy signal or a power signal, b ut c annot be b oth a t t he same time.
Some signals are neither energy nor power signals. 5
( a) /:;,. E xercise E 8.7 ( a) Show t hat t he s ignal aku[kJ is a n e nergy signal o f e nergy I -tal 2 i f lal < 1. I t is a p ower
signal of power P f = 0 .5 if lal = 1. I t is n either a n e nergy signal n or a p ower signal if lal > 1. 'V 8 .3 O n t he surface, t he fact t hat d iscrete-time sinusoids of frequencies differing b y
27rm a re identical may a ppear innocuous. B ut in reality it creates a serious problem
for processing continuous-time signals b y digital filters. A continuous-time sinusoid
f (t) = cos wt s ampled every T seconds ( t = kT) results in a discrete-time sinusoid
f [k] = cos wkT. T hus, t he s ampled signal f [k] is given b y 5 f [k] = cos wkT
= cos O k - 13 ( b) F ig. 8 .15 Examples of exponentially varying discrete-time sinusoids. shows signals (O.9)kcos(%k - ~), a nd ( l.l)kcos(%k - ~). Observe t hat if
t he a mplitude decays, a nd if hi > 1, t he a mplitude grows exponentially. 8.2-1 Sampling Continuous-Time Sinusoid and Aliasing hi < 1, Size of a Discrete-Time Signal A rguing along t he lines similar t o those used in continuous-time signals, t he
size of a discrete-time signal f [k] will be measured by its energy E I defined by
00 EI = L (8.15) I f[kW k =-oo T his definition is valid for real or complex f [k]. For this measure t o b e meaningful,
t he energy o f a signal must be finite. A necessary condition for t he energy t o be
finite is t hat t he signal amplitude must - > 0 as Ikl - > 0 0. Otherwise t he s um in Eq.
(8.15) will n ot converge. I f E I is finite, t he Signal is called a n e nergy s ignal.
I n some cases, for instance, when t he a mplitude of f [k] does n ot - > 0 as Ikl - >
0 0, t hen t he s ignal energy is infinite, a nd a m ore meaningful measure of t he signal
in such a c ase would be t he t ime average of t he energy (if it exists), which is t he
signal power P I defined by
1 PI L If[k]1
2N + 1 = lim - N~oo N 2 (8.16) -N F or periodic signals, the time averaging need be performed only over one period in
view of t he p eriodic repetition of t he signal. I f P I is finite a nd nonzero, t he signal is where 0 = wT Recall t hat t he discrete-time sinusoids cos O k have unique waveforms only for t he
values of frequencies in t he r ange 0 ~ 7r or wT ~ 7r ( f...
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