10. The Discrete-Time Fourier Transform (DTFT)
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1 Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w considr discrt signals and dvlop a Fourir transform for ths signals calld th discrt-tim Fourir transform, abbrviatd DTFT Th discrt-tim Fourir transform of a discrt squnc ( m is dfind as follows: ( ( m m m ( whr is calld th normalizd frquncy Th notation ( is ustifid by th obsrvation that th frquncy dpndncy is in ponntial form In ordr for th DTFT of a squnc to ist, th summation in ( must convrg It will hold if m is absolutly summabl, that is ( m ( m < ( Not that th DTFT of a discrt-tim squnc is a function of a continuous variabl
2 Sinc ( + m( + ( ( m ( m m m m m ( m ( m m thn th DTFT is priodic in with a priod of Sinc ( is priodic in with th priod qual to, w can prss it by an ponntial Fourir sris in variabl Thrfor m m ( c m c m m m (3 whr m ( d c m (4 Comparison of quations ( and (3 shows that th discrt signal ( m corrsponding to th spctrum ( is ( m c m Thrfor, from quation (4 th invrs discrt-tim Fourir transform is ( m ( m d (5 To shd mor light on th subct w will driv th DTFT using an altrnativ approach Lt us considr th Fourir transform of a continuous-tim signal ( t ( ( t t dt W approimat th intgral as follows:
3 3 ( ( mt mt s s Ts m (6 and rplac th product T s by th normalizd frquncy If w ignor th scal factor summation, dnotd by ( Eampl Lt us considr th signal Th DTFT of this signal is T s f T s and rplac ( mt s, is th DTFT s by ( m, th rsulting ( ( m m (7 m m ( m a u( m a < m m ( a ( a m m Sinc th prssion on th right hand sid is th gomtric sris, w obtain providd a < Eampl Lt ( m b th unit sampl δ ( m ( a Th DTFT of this signal is ( ( m m δ m m
4 4 Som proprtis of th DTFT In this sction w formulat som proprtis of th discrt tim Fourir transform Priodicity This proprty has alrady bn considrd and it can b writtn as follows Linarity ( + ( ( (8 Th DTFT is a linar oprator, i th discrt-tim Fourir transform of a signal is whr ( Shifting k ( m a ( m a ( m + ( a ( + a ( is th DTFT of ( m ( k, Lt us considr a shiftd signal Th DTFT of this signal is (s (7 k ˆ ( m ( m m ( m m m ( ( ( m m m m ˆ m m m Lt k m m, thn ˆ m k m ( ( k ( k Thus, w conclud that shifting in tim rsults in th multiplication of th DTFT m by a compl ponntial
5 5 Eampl 3 Lt us considr th shiftd unit sampl ( m δ( m m proprty and knowing that th DTFT of δ ( m is, w obtain Frquncy shifting Lt us considr a signal ( m Th DTFT of this signal is ˆ ( m m multiplid by m ( m ( m ˆ Using th shifting m m m( ( ( ( m ( m ( m m Thus, multiplying a squnc by a compl ponntial in frquncy of th DTFT Convolution thorm Th convolution of signals ( m and ( m y is givn by m rsults in shifting Th DTFT of th convolution is k ( m y( m ( k y ( m k k m k m ( k y( m k ( k y( m k ( m k k ( ( k k y m k ( k y( p m ( Y ( k k m p m p
6 6 whr p m k Thus, th DTFT of a convolution of signals ( m y m th product of th DTFTs of ( m Parsval s thorm and ( Lt us considr a discrt signal ( m Parsval s thorm stats that and ( m y is m ( m ( d (9 In Sction it will b shown that this quation givs signal nrgy in th tim domain and in th frquncy domain 3 Comparing of th DTFT to th DFT Rcall that th DFT of a squnc { ( m } m whr m,,, N is n N m N mn mw m m mn N n,,, N ( On th othr hand, th DTFT of th sam squnc is ( N m Comparing ( to ( w find m m ( n n,,, N n N ( ( Equation ( stats that th cofficints of th DFT ar sampls of th continuous spctrum givn by th DTFT at n N Not that th DFT cofficints corrspond to N sampls of th ( z takn at N qually spacd points around th unit circl
7 7 n z N n,,, N 4 Gnralizd DTFT Som discrt-tim signals do not hav a DTFT but thy hav a gnralizd DTFT as plaind blow Lt th DTFT of a signal ( m b ( δ( To find this signal, w us th invrs DTFT: m ( m δ( d This rsult stats that th constant signal ( m has th DTFT qual to ( Hnc, th constant signal ( m has th DTFT qual to ( ( m ( δ( δ, or δ (3 Not that th signal bcaus th sris ( m dos not hav th DTFT in th ordinary sns m m is not convrgnt Thrfor w say that ( signal ( m Now w considr a discrt signal ( m ( δ( + Thn m d m ( m δ( + δ is a gnralizd DTFT of th having th DTFT or holds m ( m ( δ( + (4
8 8 Likwis, w find m ( m ( δ( (5 Sinc cos ( m+α ( m+α ( m + α + α m + α m thn using (4 and (5 w dtrmin th DTFT of th signal m cos m + α as follows: ( ( α ( δ( α + δ( + α ( δ( α + δ( + or α α ( m ( m + α ( ( δ( + δ( + cos (6 A similar approach lads to th DTFT of th signal ( m sin ( m + α ( m+α ( m+α ( m sin( m + α α m α m To dtrmin th DTFT of ( m, w apply (4 and (5 α ( δ( α δ( + α ( δ( α + δ( or α α ( m ( m + α ( ( δ( + δ( sin (7
9 9 In th spcial cas whn α w hav: ( m m ( ( δ( + δ( + cos (8 ( m m ( ( δ( + δ( sin (9 5 Frquncy rspons of LTI discrt systms Lt an LTI discrt systm b rprsntd by its unit sampl rspons h ( m (s Fig Th rspons of th systm du to th input ( m is givn by convolution y k ( m h( m ( m h( k ( m k (m h(m y(m Convolution thorm stats that whr Y ( ( m and H ( Fig An LTI discrt systm ( ( ( H is th DTFT of th output ( m Y ( y, ( is th DTFT of th input is calld th frquncy rspons function of th discrt systm Not that H ( is, in gnral, a compl-valud function of th frquncy and can b writtn in th polar rprsntation φ ( ( ( H H H (
10 3 Thus, th frquncy rspons function of th discrt systm is th DTFT of th h m and is a continuous function of Having th unit sampl rspons ( magnitud H ( Eampl 4 and th phas φ ( Lt us considr th input signal H w find: ( ( ( H Y ( φ Y ( φ ( + φ ( (3 H ( m Acos ( m + α To find th output rspons of a systm spcifid by a frquncy rspons function H (, w apply ( Th DTFT of th signal ( m is givn by (6 rpatd blow Hnc, w obtain Y α α ( A( δ( + δ( + ( AH ( α α δ( AH ( + δ( + α A( H α ( δ( + H δ ( ( + ( ( A H ( ( φh +α H ( ( φh +α δ + δ + ( To find y ( m, w us th invrs DTFT y A ( ( ( ( ( ( φh +α m H ( φh +α ( δ + δ + A A H ( m+φh +α ( m+φh +α ( ( + ( cos( m + φ + α H H m d
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