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

2 w ith a m inor difference i n t he p resent e

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Unformatted text preview: 0 (1O.36) k =-oo T his condition is sufficient b ut n ot necessary for t he existence of F{!1). For instance, t he signal j [k] = sin k / k violates t he condition (1O.36), b ut does have D TFT (see Example 10.6). Physical Appreciation o f t he D iscrete-Time Fourier Transform In understanding any aspect of t he Fourier transform, we should remember t hat Fourier representation is a way of expressing a signal j [k] as a sum of everlasting F(O) =L -yke- jOk k=O = L he-jnl k=O This is a geometric progression with a common ratio -ye- jn . Therefore, [see Sec. (B.7-4)] F(O) _ 1 - 1- -ye- jn 10 Fourier Analysis of Discrete-Time Signals 630 p rovided t hat I ),e-inl Therefore 10.2 Aperiodic Signal Representation by Fourier Integral 631 < 1. B ut b ecause I e-in I = 1, t his c ondition implies 11'1 < 1. F (rl) = 1 1 - ),e 11'1 in I f 1)'1 > 1, F (rl) d oes n ot converge. T his r esult is i n c onformity w ith c ondition (10.36). F rom E q. (10.37) F (rl) = 1 F (rl) = n J ) ,e- (10.37) <1 b cosrl-l)-j)'sinrl IF(rl)1 = so t hat IF(rl)1 = J(1- 1 J l +),2 - LF(rl) = t an- 1 (1O.39a) 1 )'cos rl)2 + b sin rl)2 (10.40) T herefore (10.38) 1.. 1 -)'cosrl+nsmrl h i> 1 1 - [ 2)'cos r l ) 'sin n J ) 'cosrl-l (10.41) T he F ourier t ransform ( and t he f requency s pectra) for t his s ignal i s identical t o t hat o f j [k] = ) 'ku[k]. Y et t here i s n o a mbiguity i n d etermining t he I DTFT o f F (rl) = ' Y.- I"-1 b ecause o f t he r estrictions o n t he v alue o f )' in each case. I f 1)'1 < 1, t hen t he i nverse t ransform is J [k] = ) 'ku[k]. I f 1)'1 > 1, i t is j [k] = ) 'k[_(k + 1)]. • L F(rl) = _ tan- 1 [ 1 ) 'sin r l r lJ (1O.39b) - ),cos flkl] F igure 10.4 shows j [k] = ) 'ku[k] a nd i ts s pectra f or), = 0.8. O bserve t hat t he frequency s pectra a re c ontinuous a nd p eriodic functions o f r l w ith t he p eriod 211". As explained earlier, we n eed t o u se t he s pectrum o nly o ver t he f requency interval o f 211". We often select this interval t o b e t he f undamental f requency r ange (-11",11"). T he a mplitude s pectrum IF(rl)1 is a n e ven function a nd t he p hase s pectrum L F(rl) is a n o dd f unction o f rl. • ""III i'l1II "" (a) flkl o -10 k- F ig. 1 0.6 Fig. 1 0.5 E xponential ) 'ku[_(k + 1)]. • • E xample 1 0.5 F ind t he D TFT o f t he d iscrete-time r ectangular p ulse i llustrated i n Fig. 1O.6a. T his p ulse is also known a s t he 9 -point r ectangular w indow function. E xample 1 0.4 F ind t he D TFT o f ) 'ku[-(k + 1)] d epicted i n Fig. 10.5. F (rl) = f ) 'ku[_(k + 1 )]e- iOk = k=-oo f b e-in)k = k =-1 f (~ein)-k k =-1 F (n) = I : J[k]e- ink k =-oo S etting k = - m yields j [k] = D iscrete-time g ate p ulse a nd i ts F ourier s pectrum. 1 'n ~(;/) 00 m l 'n =:/ + ( l 'n ; :/ T his is a geometric series w ith a c ommon r atio e jn )2 ( )3 1 jn + ::ye + . .. / )'. T herefore, from Sec. B.7-4, M =9 T his is a g eometric p rogression w ith a c ommon r atio e - in...
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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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