signalsandsystems(2)

Chapter 5The Discrete-TimeFourier Transform崔琳莉ContentsThe Discrete-Time FourierTransform (DTFT)The Fourier Transform forPeriodic SignalsProperties of the DTFTNotation:CFS: ( the continuous Fourier series,FS)DFS: ( the discrete Fourier series)CTFT: ( the continuous time FourierTransform,FT)DTFT: ( the discrete time FourierTransform)5.1 Representation of AperiodicSignals: DTFT(1) Fourier series (periodic signal)⎪⎩⎪⎨⎧==∑∑>=<−>=<NnnNjkkNknNjkkenxNaeanx)/2()/2(][1][ππ5.1.1 Development of DTFT (from DFS to DTFT)Fourier series (periodic signal)x[n]akFourier transform (aperiodic signal )(2) Fourier transform (aperiodic signal )x[n]][~n x 01N −2N 01N −2N N−NSince over ][~][n x n x =21N n N ≤≤−∑∑+∞−∞=−>=<−==n n N jk N n n N jk k e n x N e n x N a )/2()/2(][1][~1ππ∑+∞−∞=−=n nj j en x eX ωω][)(When ∞→N)(2ωωπj k e X Na k N→→SoDTFT)(10ωjk k e X Na =The coefficients a k are proportional toX(e j ω) samples ofWhere is the spacing of the samples in the frequency domain.N /20πω=0)/2(0000)(21)(1][~ωπωωωωπ∑∑∑>=<>=<>=<===N k njk jk N k n jk jk N k nN jk keeX e e X Nea n x When ∞→N ∫∑→→→→,,],[][~00ωωωωd k n x n x ∫=πωωωπ2)(21][d e e X n x n j j ∑∫∞+−∞=−==n nj j n j j en x e X dwe e X n x ωωπωωπ][)()(21][2The discrete-time Fourier transform pairSynthesis equation Analysis equation)(][ωj DTFTe X n x ⎯⎯→←¾That means, x[n] is composed of e j ωn at different frequencies ωwith “amplitude ”X(e j ω)d ω/2π; X(e j ω)is referred to the spectrum of x[n].Note:1. Relation between of DFS and DTFTkDFS a n x ⎯⎯→←][~Discrete<-->periodic Periodic<-->discrete)(][ωj DTFTeX n x ⎯⎯→←)(1)(1)(000ωωωωωωωjk k j k k kj e X Ne X N a a N e X ==⋅===Discrete <-->periodicAperiodic <-->continuous2. Comparing DTFT to FSkFS a t x ⎯→←)(~Continuous-time 、Discrete-frequency)(][ωj DTFTe X n x ⎯⎯→←Discrete-time 、Continuous-frequencyDualityEq.(5.9)can be regarded to FS in frequency-domain;Eq.(5.8) is just that x[n] is the FS coefficients of X(e j ω).3. Relation between FT and DTFTLet ∑∑∑∞−∞=−∞−∞=∞−∞==⎯→←−=−=n nTj p FTn n p enT x j X nT t nT x nT t t x t x ωωδδ)()()()()()()(and ∑∞−∞=Ω−Ω=⎯⎯→←=n nj j DTFTen x e X nT x n x ][)(][][soTj X e X p j ωω=Ω=Ω)()(Example Poisson formula If then)()(ωj X t x FT⎯→←∑∑∑∑∞−∞=−∞−∞=∞−∞=∞−∞=⋅=−==−=n Tjn k k tjk k T enT x T k j X j X e T jk X nT t x t x ωωωωωω)())(()(~)2()()()(~)1(000Proof :∑∑∞−∞=∞−∞=−⋅=−⋅=⎯→←∗=k k T FTT T k Tjk X k j X j X t t x t x )(2)()()()(~)()()(~)1(0000ωωπδωωωδωωωδ)(200ωωπδωk e FTt jk −⎯→←Q ∑∞−∞==∴k tjk T e Tjk X t x 0)()(~0ωω)(~1)(2)(21)()()()()()2(0ωωωδπωπδδj X Tk Tj X nT t nT x t t x t x k FTn T s ⋅=−∗⎯→←−=⋅=∑∑∞−∞=∞−∞=nTj FTe nT t ωδ−⎯→←−)(Q )(~1)()()(ωδωj X TenT x nT t nT x n nTj FTn ⋅=⎯→←−∴∑∑∞−∞=−∞−∞=Note:∑∑∑∞−∞=−∞−∞=∞−∞==⎯→←−⎯→←−=n nTj FTk FTn T ek nT t t ωωωδωδδ)()()(0That implies∑∑∞−∞=−∞−∞==−n nTj k ek ωωωδω)(00It is a Fourier Series in Frequency-Domain.5.1.2 Examples of DTFTExample 5.11||],[][<=a n u a n x n ()ωωωωj n nj n nj nj ae ae en u a e X −∞=−∞−∞=−−===∑∑11][)(0ωωωωωcos 1sin )(,cos 211)(12a a tg e X a a e X j j −−=∠−+=−Example 5.21||,][<=a a n x n()()22110cos 211111)(a a a ae ae ae ae aeeaea e ae X j j j m nj n nj n nj n n nj n n nj nj +−−=−+−=+=+==−∞=∞=−−−∞=−−∞=−∞−∞=−∑∑∑∑∑ωωωωωωωωωωWe can conclude:real and even real and evenExample 5.3⎪⎩⎪⎨⎧>≤=11,0,1][N n N n n x 2sin)21sin()()(1)(1222)21()21(2)1(111111ωωωωωωωωωωωωω+=−−=−−==−−+−+−−+−−=−∑N e e eee e e e e ee X j j j N j N j j j N j N j N N n nj j Similar to a sinc ,but periodic with a period 2π.Note:If the periodization of x[n] which periodN is made,∑∞−∞=−=r rN n x n x ][][~kNkN N eX Na k Nj k πππsin 2)21sin()(112+==We can get DFS coefficients:Just same as (3.104) in P218Example 5.4][][n n xδ=1][)(==∑∞−∞=−n nj j en e X ωωδ][n δnω0)(ωj e X 1nWn dw e dw e e X n x WWnj n j j πππωππωωsin 21)(21][===∫∫−−Example 5.5Ideal DT LPFA discrete sinc in time domain5.1.3 Convergence of DTFTConvergence conditions:∑∫∞+−∞=−==n nj j n j j en x e X d e eX n x ωωπωωωπ][)()(21][2∞<∞<∑∑∞−∞=∞−∞=n n n x n x 2][][or––Absolutely summable —Finite energyGibbs phenomenon in DTFT?][][n n x δ=1][)(==∑∞−∞=−n nj j en e X ωωδnWn d e n x d e eX n x WWnj n j j πωπωπωπωωsin 21][~)(21][2==⇒=∫∫−W=π/4W=3π/8W=π/2W=3π/4W=7π/8W=πThere are not any behaviors like the Gibbs phenomenon in evaluation the discrete-time Fourier transform synthesis equation.(see page 368 fig5.7)5.2 DTFT for periodic signalsRecall CT result:)(2)()(00ωωπδωω−=⎯→←=j X e t x t j What about DT:?)(][0=⎯→←=ωωj n j e X e n x a) We expect an impulse (of area 2π) at ω=ωo b) But X(e j ω) must be periodic with period 2π,In fact)2(2)(0l e X l j πωωπδω−−=∑∞−∞=)(ωj e X 0ω02ωπ+02ωπ+−ωIfω0=0)2(21l l DTFT πωπδ−⎯⎯→←∑∞−∞=0][n x n10ω)(ωj e X π2π2−π4nj nj eX l ed el n x j 0)(0)2(221][ωππωωπωωπδπω=−−=∫∑−∞−∞=444344421Now consider a periodic sequenceNea n x n x N n x N k njk kπωω2,][][][00===+∑>=<)2(200l el nj πωωπδω−−⎯→←∑∞−∞=Q )(2)2(2)(0∑∑∑∞−∞=∞−∞=>=<−=−−=⇒k kl N k k j k a l k a e X ωωδππωωπδωSee page 370Fig5.9Example[]∑∞−∞=−−+−+=⎯⎯→←=l j DTFT l l e X n n x )2()2()(cos ][000πωωδπωωδπωω[]∑∞−∞=−−−−+=⎯⎯→←=l j DTFTl l j e X n n x )2()2()(sin ][000πωωδπωωδπωω1. 2.∑∑∞−∞=∞−∞=−=⎯⎯→←−=l j DTFTr l NNe X rN n n x )2(2)()(][πωδπδω3.5.3 Properties of DTFT5.3.1 Periodicity)()()2(ωπωj j e X e x =+5.3.3 Time Shifting and Frequency Shifting)()(][][2121ωωj j DTFTe bX e aX n bx n ax +⎯⎯→←+5.3.2 Linearity)(][00ωωj e X e n n x n j DTFT −⎯⎯→←−)(][)0(0ωωω−⎯⎯→←j e X n x e DTFT n j ][)1(][][,0n x n x e n y n n j −===ππωExample5.3.4 Conjugation and Conjugate Symmetry)(][ωj e X n x −∗∗=5.3.5 Differencing and Accumulation)()1(]1[][ωωj e X e n x n x j DTFT−−⎯⎯→←−−∑∑∞−∞=−−∞=−+−⎯⎯→←k j DTFTnm k e X e X e m x j j ]2[)()(11][0πωδπωωIf x[n] is real)()(ωωj j e X e X −∗= 5.3.6 Time Reversal)(][ωjk DTFTk e X n x ⎯⎯→←5.3.7 Time Expansion)(][ωj e X n x DTFT−⎯⎯→←−⎩⎨⎧=,],/[][o k n x n x k if n is a multiple of kif n is not a multiple of k.5.3.8 Differentiation in Frequency∫∑=∞−∞=πωωπ222)(21][d eX n x jk n 5.3.7 Parseval’s Relationωωd e dX jn nx j DTFT)(][⎯⎯→←5.4 The Convolution Property)()()(][][][ωωωj j j DTFTe H e X e Y n h n x n y =⎯⎯→←∗=P386 Ex5.165.5 The Multiplication Property∫−=⎯⎯→←⋅=πθωθωθπ2)()()(21)(][][][d e H e X e Y n h n x n y j j j DTFTPeriodic convolutionHomework: P400-5.2* 5.10*。