信号与系统 第五章课件

Ch5 The Discrete-Time Fourier Transform 第5章离散时间傅立叶变换Abbreviations（缩写）：1. CFS :The Continuous-Time Fourier Series ——连续时间傅立叶级数2. DFS :The Discrete-Time Fourier Series ——离散时间傅立叶级数3. CTFT :The Continuous-Time Fourier Transform ——连续时间傅立叶变换4. DTFT :The Discrete-Time Fourier Transform ——离散时间傅立叶变换Focus on itin thisMain content1.Development of the Discrete-Time Fourier Transform (离散时间傅立叶变换的导出);2.Basic Fourier Transform Pairs (常用信号的离散时间傅立叶变换对);3. The Fourier Transform for Periodic Signals (离散时间周期信号的傅立叶变换);4. Properties of the Discrete-Time Fourier Transform (傅立叶变换的性质);5. The frequency response and frequency-domain methods for discrete-time signals and systems (离散系统的频率响应与频域分析方法);5.0 IntroductionAnalytical objects : aperiodic discrete-time signals and systemsAnalytical methods: (similar to CTFT)1)An aperiodic d-t signal can be viewed a periodic d-t signals with an infinite period.2)As the period becomes infinite, the discrete-time Fourier series represen-tation becomes the discrete-time Fourier transform .3)Using the property of the Fourier transform to examine frequency-domain analysis of signals and LTI systems.note that the similarities and differences between continuous-time and discrete-time Fourier transform analysis.5.1 Representation of Aperiodic Signals: The Discrete-Time Fourier Transform(p359)（非周期信号的表达：离散傅里叶变换）5.1.1Development of the Discrete-Time Fourier Transform(离散傅里叶变换的导出)Consider a discrete-time periodic signal:N→∞periodic aperiodic[][]221,j kn j kn N N k k k N n N x n a e a x n e N ππ-=<>=<>==∑∑for a d-t periodic signal , we have the discrete-time Fourier series pair:[]x n [][]22/2/2N j kn j kn N N k n N n Na x n e x n e ππ+∞--=-=-∞==∑∑so discrete-time Fourier transform[]j j n n X e x n e ωω∞-=-∞=∑（） , 2N k N πω→∞→　as lim j k N Na X e ω→∞（）Define:so []00000012(),1()2jk jk n k N jk jk n k N x n X e e N N X e e ωωωωπωωπ=<>=<>=⋅==⋅⋅∑∑0211()()jk j k k N a X e X e N N ωωπω===Compare with ,k a j X e ω（）we have []212j j n x n X(e )e d ωωπωπ∴=⎰N →∞asan aperiodic sequence can be thought of as a linear combination of complex exponentials infinitesimally close infrequency and with amplitudes 12j X(e )d ωωπ[]212j j n x n X(e )e d ωωπωπ=⎰[]j j nX(e )x n e ωω∞--∞=∑结论：discrete-time Fourier transform pairDifferences between the c-t and d-t Fourier transform :2) the finite interval of integration in the synthesis equation.1) periodicity of the discrete-timetransform j X(e )ω(2)()()j j X eX e ωπω+=In discrete-time ,Low frequencies are the values of near evenmultiple of ;high frequencies are the values of near oddmultiples of .ωπωπLow-frequency signalhigh-frequency signal5.1.2 examples of discrete-time Fourier transform (p362)（离散傅里叶变换的例子）[]()11nj nj nj j n n X(e)a u n eaeaeωωωω∞∞---=-∞====-∑∑Where is a complex function j X(e )ω11j a sin X(e)tga cos ωωω-=--Example 5.1(p362)[][]1nx n a u n ,a =<2112j X(e )a acos ωω=+-Magnitude:Phase:All of values are of theDecaying same signLow-frequencysignal<<01ahigh-frequency signal10a -<<alternate in valueDecayingExample 5.2 (p364)[]1nx n a ,a =<[][][]1nnx n a u n a u n -=--+1122111112j n j nn j nn n nj nn j nn n j j j X (e)aea ea ea eae a ae ae a a cos ωωωωωωωωω-∞---=-∞=∞∞-==-=+=+-=+=--+-∑∑∑∑01a <<Real and even sequenceReal and even functionLow-frequency signalExample 5.3(p365)[]1110n N ,xn ,n N ≤⎧=⎨>⎩1112122N j j nn N sin(N )X (e)esinωωωω-=-+==∑12N Real and even sequenceReal and even function1211k sin k(N )N a ,N sin k Nππ+=①Compare with the corresponding periodic square wave signal21j k k Na X (e )N ωπω==so 12122j sin(N )X(e )sin ωωω+=②Compare with the corresponding c-t aperiodic square wave signal1112T sin T X(j )T ωωω=12122j sin(N )X(e )sin ωωω+=Example 5.4(p367)[][]x n n δ=[]n δ0n1[]()1j j nn X e x n eωω∞-=-∞==∑()j X e ω1π-πω5.1.3Convergence Issues associated with the Discrete-Time Fourier Transform （p366)(离散傅里叶变换的收敛问题)Question:[]j j nX(e )x n e ωω∞--∞=∑①What is the conditions on thatguarantee the convergence of thissum []x n Answer 1:Condition:[]n x n +∞=-∞<∞∑( i.e. is absolutely summable)[]x nAnswer 2:Condition:[]2n x n +∞=-∞<∞∑( i.e. the sequence has finite energy)The two conditions above can guarantee the convergence of []j j nX(e )x n eωω∞--∞=∑②Are there convergence issuesassociated with ?[]212j j n x n X(e )e d ωωπωπ=⎰NO!Because the integral in this equation is over a finite interval of integration.We approximate by[]x n[]()12Wj j nWˆx n X e e dωωωπ+-=⎰[]nδ5.2 The Fourier Transform For Periodic Signals(p367)（周期信号的离散傅里叶变换）consider the Fourier transform of the sequence0[]j nx n eω=In c-t time, we saw ()002j teωπδωω↔-(Note the d-t Fourier transform must be periodic in with )ω2π0022j nk ek ωπδωωπ∞=-∞↔--∑（）Therefore, we expect[]002,jk nk k N x n a eNωπω=<>==∑[]0()2(2)j kk N l x n X e a k l ωπδωωπ∞=<>=-∞↔=--∑∑02021()(-)2j nj j nj nX e ed ed eπωωωωπωδωωωπ==⎰⎰To check the validity of the expression above,consider an arbitrary periodic sequence []x n so0022j nk ek ωπδωωπ∞=-∞↔--∑（）()22()j k k X e a k N ωππδω∞=-∞=-∑Example 5.5 :（P371）00011[]cos 22j n j n x n n e e ωωω-==+025πω=with 22()(2)(2)55j k X e k k ωπππδωπδωπ∞=-∞⎡⎤←−→=--++-⎢⎥⎣⎦∑F22()()()55j X e ωπππδωπδωπωπ=-++-≤<That isExample 5.6 :（P371）[][]k x n n kN δ∞=-∞=-∑periodic impulse train [][]001111N jk njk nk n N n a x n en eNNNωωδ---=<>====∑∑22()j k X e k NN ωππδω∞=-∞⎛⎫=- ⎪⎝⎭∑5.3 Properties of The Discrete-Time Fourier Transform(P372)（离散时间傅立叶变换的性质）Adopting notation below to indicate the pairing of a signal and its transform[]()j x n X eω←−→F5.3.1Periodicity (周期性)(2)()()j j X eX e ωπω+=5.3.2Linearity (线性)5.3.3Time Shifting and Frequency Shifting(时移和频移)[][]1212()()j j ax n bx n aX e bX e ωω+↔+[]()j x n X e ω←−→Fthen[]()j x n X e ω↔00()j n j x n n X e e ωω--←−→⎡⎤⎣⎦F Time Shifting []00()()j nj x n eX eωωω-←−→FFrequencyShiftingSolution:Determine the relationship of the impulseresponses of an ideal lowpass filter and an ideal highpass filter .()j hpH e ω()j lp H e ωExample 5.7(p374)Shifting by ()()()j j h p l p H e H eωωπ-=()j lp H e ωπBy frequency shifting property,[][]()[]1nj nh p l p l p h n eh n h n π==-5.3.4Conjugation and Conjugate Symmetry (共轭及共轭对称)[]()j x n X e ω←−→F[]**()j x n X eω-←−→F*()()j j X e X eωω-=When is real[]x n Re ()Re ()Im ()Im ()j j j j X e X e X e X e ωωωω--⎧⎡⎤⎡⎤=⎪⎣⎦⎣⎦⎨⎡⎤⎡⎤=-⎪⎣⎦⎣⎦⎩()()()()j j j j X e X e X e X e ωωωω--⎧=⎪∴⎨=-⎪⎩{}{}[]()j x n X e ω←−→FEv Re {}{}[]()j x n j X e ω←−→FOd Im5.3.5Differencing and Accumulation (差分与求和)[]1(2)1j k u n k e ωπδωπ∞-=-∞∴←−→+--∑F[][]nk u n k δ=-∞=∑[]1n δ←−→F[][][]1(1)()1()(2)1j j nj j k k x n x n eX e x k X e k e ωωωωπδωπ-∞-=-∞=-∞--←−→-⎡⎤←−→+-⎢⎥-⎣⎦∑∑FFExample 5.8 :（P376）Determine []u n ←−→F？[]()j x n X eω--←−→F[]()j x n X e ω←−→F5.3.6Time Reversal (时域反转)If is real and even ，then[]x n ()()()j j j X e X eX e ωωω-*==i.e. is real and even.()j X e ωIf is real and odd ，then[]x n i.e. is purely imaginary and odd.()()()j j j X e X e X e ωωω-*=-=-()j X e ω5.3.7Time Expansion (时域扩展)()[/],[]0,k x n k if n is a multiple of kx n if n is not a multiple of k ⎧=⎨⎩[][]()j j nj rkk kkn r X e x n ex rk eωωω∞∞--=-∞=-∞==∑∑[]()j rkjk r x r eX eωω∞-=-∞==∑[]()jk k x n X eω∴↔inverse relationship[]x n Example 5.9(p379)Determine of ()j X eω(2)(2)[][]2[1]x n y n y n =+-Solution:2sin(5/2)()sin(/2)j j Y e eωωωω-=4(2)sin(5)[]sin()F j y n eωωω-←−→5(2)sin(5)2[1]2sin()F j y n eωωω--←−→()4()sin(5)12sin()j j j X e eeωωωωω--⎛⎫=+ ⎪⎝⎭power-density spectrumEnergy-density spectrum5.3.8Differentiation in Frequency (频域微分)[]()j dX e jnx n d ωω-↔5.3.9Parseval’s Relation (帕斯瓦尔定理)[]22212()j n x n X e d ωπωπ∞=-∞=∑⎰[]221kn N k N x n a N=<>=<>=∑∑Compare with DFS①The convolution property maps (映射）the convolution of two signals to the simple algebraic operation of multiplying their Fourier transforms.5.4 The Convolution Property (P382)（卷积性质）[][][]*()()()j j j y n x n h n Y e X e H e ωωω==if thenImpulse response of LTI systemInput of LIT systemoutput of LITsystemFrequency response of LTI system()j H eω✓②The frequency response captures the change in complex amplitude of the Fourier transform of the input at each frequency ω✓③The convolution property is the basis of the frequency domain analysis of signals and systems.Consider an LTI system with 0[][]h n n n δ=-Analyze that how the system impacts on the input in frequency domain?Solution:0()[]j n j j nn H e n n eeωωωδ+∞--=-∞=-=∑The frequency response is()()j n j j Y e eX e ωωω-=Therefore0[][]y n x n n =-Using time-shifting property, A pure time shift system()j n j H e eωω-=Magnitude = 1 at all frequenciesphase =that is linear with frequency .0n ω-Consider the d-t ideal lowpass filter[]?h n =Solution:[]()()1122sin ccj j nj j nc h n H e ed He e d nnωπωωωωπωωωππωπ++--===⎰⎰Not causaloscillatory5. 5 The Multiplication Property (P388)（相乘性质）[][][]12()122121()()()21()()2j j j j j y n x n x n Y e X e X e d X e X e ωθωθπωωθππ-=⋅==⊗⎰ifthen periodic convolution5. 6 TABLES OF FOURIER TRANSFORM PROPERTIES AND BASIC FOURIER TRANSFORM PAIRS(P390)（傅立叶变换的性质及基本变换对列表）5. 7 duality(P390)（对偶）5.7 对偶性（Duality ）∑∑>=<->=<==N n nNjk k N k n Njk ken x Na ea n x ππ22)(1,)(由于本身也是以N 为周期的序列，当然也可以将其展开成DFS 形式。