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

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

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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. " 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...
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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.

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