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

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online