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

# 1k clearly t his signal is a function of k a nd may

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

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: e = (7.389)t (iv) e - 2t • VI e . = (0.2231) = (4.4817)-k ( b) Show t hat (i) 2 k = eO.693k (ii) ( 0.5)k = . -O.693k (iii) ( 0.8)-k = eO.2231k 'V o C omputer E xample C S.l S ketch t he d iscrete-time s ignals ( a) ( -0.5)k ( b) ( 2)-k ( c) ( _2)k ( a) ( b) ( c) 4. f::,. 2 456 k =0:5j k =k'j f kl=(-0.5). - kj s tem(k,fk) k =0:5j k =k'j f k=2. - (-k)j s tem(k,fk) k =0:5j k =k'jfk=(-2). - kj s tem(k,fk3) Discrete-Time Exponential 0 e jrlk A general discrete-time exponential e jflk (also called h ). valued f t' fk d h p a sor IS a complex p art ~~c 1O~ 0 a n t erefore its graphical description requires two plots (real t~: v~%:~I~~r~gka~: ~~e~:~~I~~ep~:! ~~;Ie) plot~, plot . . To avloid twfo we shall in F ' 8 7Th f v anous va ues 0 k as Illustrated Ig. &quot; e u nction j[kJ = e jflk takes on values ejO e jrl j 2rl j3rl k- 0123 t' I ' ,e ,e , . .. at - .' , , , . .. , respec Ive y. For t he sake of simplicity we shall ignore t h negative values of k for t he t ime being. Note t hat e 8 Discrete-time Signals a nd Systems 546 Locus of e JOk 8.2 Some Useful Discrete-time Signal models 547 Locus of e- jQk k=3 F ig. 8 .8 o (a) A d iscrete-time sinusoid c ost Uk C omputer E xample C 8.2 Sketch t he d iscrete-time sinusoid cos + f ). (Uk + f) k =-36:30j k =k'j f k=cos(k*pi/12+pi/4); s tem(k,fk) 0 F ig. 8 .7 Locus of (a) e jOk (b) e - jOk • e jnk = r ejB, r = 1, S ampled C ontinuous-Time Sinusoid Yields a D iscrete-Time Sinusoid a nd 0 = kO T his fact shows t hat t he m agnitude a nd angle of e jrlk are 1 a nd kO, respectively. Therefore, t he p oints e jO , e jn , e j2n , e j3n , . .. , e jkn , . .. lie on a circle o f u nit radius (unit circle) a t angles 0, 0 , 2 0, 3 0, . .. , kO, . .. respectively, as shown in Fig. 8.7a. For e ach u nit increase in k, t he function I[k] = e jrlk moves along t he u nit circle counterclockwise by a n angle O. Therefore, t he locus of ejnk m ay be viewed as a p hasor r otating counterclockwise a t a uniform speed of 0 r adians p er u nit s ample interval. T he e xponential e - jnk , on t he o ther hand, takes on values e jO = 1, e - j O, e - j2rlk , e - j3n , . .. a t k = 0 ,1,2,3, . .. , as depicted in Fig. 8.7b. Therefore, e - jnk may be viewed as a phasor rotating clockwise a t a uniform speed of 0 r adians p er u nit sample interval. Using E uler's formula, we can express a n exponential e jnk in terms of sinusoids of t he form cos (Ok + 0), a nd vice versa e e- +j jrlk = (cos Ok sin Ok) (8.5a) jrlk = (cos Ok - j sin Ok) (8.5b) These e quations show t hat t he f requency o f b oth e jnk a nd e - jrlk i s 0 (radians/sample). Therefore, t he frequency of e jnk is 101. Because of Eqs. (8.5), exponentials a nd sinusoids have similar properties and peculiarities. T he discretetime sinusoids will be considered next. 5. Discrete- Time Sinusoid c os (Ok + 0) A g eneral discrete-time sinusoid can be expressed as C cos (Ok + 0), where C is t he a mplitude, 0 is t he f requency (in radians per sample), a nd 0 is t he p hase (in k + 1)· radians). F igure 8.8 shows a discrete-time sinu...
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