1、most periodic signals have the Fourier series representations:,How can do the aperodic signals?,4 The continuous time Fourier transform,4 The continuous time Fourier transform,Emphases in this chapter: The Fourier transform of the continuous-time aperiodic signal.The Fourier transform of the continu
2、ous-time periodic signal. The properties of Fourier transform. Basic Fourier transform pairs.Fourier inverse transform.analyze LTI systems in frequency domain.,4.1 The continuous time Fourier transform,4.1 Representation of Aperiodic signals:The Continuous-time Fourier Transform,4.1.1 Development of
3、 the Fourier transform representation of the continuous-time Fourier transform,(1) Example,( From Fourier series to Fourier transform ),4.1 The continuous time Fourier transform,(2) Fourier transform representation of Aperiodic signal,For aperiodic signal x(t) :,For periodic signal :,4.1 The continu
4、ous time Fourier transform,T,4.1 The continuous time Fourier transform,When T ,So,4.1 The continuous time Fourier transform,4.1 The continuous time Fourier transform,The Fourier transform equation of continuous-time aperiodic signal x(t):,The inverse Fourier transform equation of continuous-time ape
5、riodic signal:,The spectral of x(t).,4.1 The continuous time Fourier transform,The Fourier transform pair:,4.1 The continuous time Fourier transform,Relation between Fourier series ak and Fourier transform X(j) :,4.1 The continuous time Fourier transform,4.1 The continuous time Fourier transform,4.1
6、 The continuous time Fourier transform,According to the Fourier transform pairs, we can obtain special formula:,Applications:,4.1 The continuous time Fourier transform,Example: The spectrum X(j) of the signal x(t) is depicted in figure:,-2 -1 0 1 2,1,X(j),(1) Find,(2) Find,4.1 The continuous time Fo
7、urier transform,Solution:,(1),(2),Odd function,4.1.2 Convergence of Fourier transform,Dirichlet conditions:(1) x(t) is absolutely integrable.(2) x(t) have a finite number of maxima and minima within any finite interval.(3) x(t) have a finite number of discontinuity within any finite interval. Furthe
8、rmore, each of these discontinuities must be finite.,4.1 The continuous time Fourier transform,4.1.3 Examples of Continuous time Fourier Transform,Example 4.1 4.2 Compute the Fourier transform of the following signals.,(a),(b),4.1 The continuous time Fourier transform,Example 4.3,4.1 The continuous
9、time Fourier transform,Example 4.4,4.1 The continuous time Fourier transform,Example 4.5 Compute the inverse Fourier series of the following signal.,4.1 The continuous time Fourier transform,the sinc function :,The properties of sinc function: 1) sinc() is a even function; 2) When is integer, sinc()
10、=0，and,Figure 4.10,4.1 The continuous time Fourier transform,3) The total area of sinc() is 1.,4.1 The continuous time Fourier transform,The other form of the function is called sample function Sa(x):,The total area of Sa(x) is .,Compare example 4.4 with example 4.5.,The duality property of Fourier
11、transform,4.1 The continuous time Fourier transform,4.1 The continuous time Fourier transform,Example :,4.1 The continuous time Fourier transform,(a),(b),4.1 The continuous time Fourier transform,Solution:,(a),(b),4.1 The continuous time Fourier transform,Note :1）绝对可积条件，仅是付利叶变换存在的充分条件，而不是必要条件。2）在允许付
12、利叶变换采用冲激函数的前提下，使许多不满足绝对可积条件的信号存在付利叶变换，这样就可以把周期信号和非周期信号的分析方法统一起来，使付利叶变换获得更加广泛的应用。3）不满足绝对可积条件的付利叶变换一般都存在冲激函数。,4.2 The Fourier Transform for Periodic Signal,Periodic signal:,thus,4.2 The Fourier Transform for Periodic Signal,Example 4.6 4.7 4.8,(a) the periodic square wave signal,(b),(c),4.2 The Fourie
13、r Transform for Periodic Signal,Note: The Fourier series coefficient ak of periodic signal figure is bar lineThe Fourier transform X(j) of periodic signal figure is impulse train.,Compare figure 4.2 with figure 4.12,4.2 The Fourier Transform for Periodic Signal,注意：当要求一个信号的傅利叶变换时，首先要分清该 信号是周期信号还是非周期信
14、号。,非周期信号的傅利叶变换对：,周期信号的傅利叶变换对：,例如，习题4.21(g) (h),4.2 The Fourier Transform for Periodic Signal,4.3 Properties of the Continuous time Fourier Transform,then,If,4.3.1 Linearity,4.3 Properties of the Continuous time Fourier Transform,then,If,4.3.2 Time Shifting,4.3 Properties of the Continuous time Fouri
15、er Transform,时域里时移，对应频域里相移。,4.3 Properties of the Continuous time Fourier Transform,Example 4.9 Use the time shifting property to compute the Fourier transform of x(t).,4.3 Properties of the Continuous time Fourier Transform,4.3.3 Conjugation and Conjugate Symmetry,then,(1) If,4.3 Properties of the
16、Continuous time Fourier Transform,then,(2) If x(t) is real signal,4.3 Properties of the Continuous time Fourier Transform,(3) If,then,4.3 Properties of the Continuous time Fourier Transform,4.3.4 Differentiation and Integration,then,(1) If,4.3 Properties of the Continuous time Fourier Transform,补充：,
17、4.3 Properties of the Continuous time Fourier Transform,then,(2) If,4.3 Properties of the Continuous time Fourier Transform,Example 4.11 Determine the Fourier transform of x(t)=u(t).,4.3 Properties of the Continuous time Fourier Transform,Solution:,Example 4.12 Calculate the Fourier transform of x(t
18、) as figure:,4.3 Properties of the Continuous time Fourier Transform,Solution:,4.3 Properties of the Continuous time Fourier Transform,Example:,Calculate the Fourier transform of triangular function q(t):,t,-/2,q(t),1,4.3 Properties of the Continuous time Fourier Transform,Solution:,/2,Solution:,4.3
19、 Properties of the Continuous time Fourier Transform,4.3.5 Time and Frequency Scaling,then,If,A signal is stretched |a| times in time, corresponding to compressed |a| times in frequcy. A signal is compressed |a| times in time, corresponding to stretched |a| times in frequcy.,4.3 Properties of the Co
20、ntinuous time Fourier Transform,时域扩展，对应频域压缩,4.3 Properties of the Continuous time Fourier Transform,Especially,reversal in time domain reversal in frequency domain.,补充,4.3 Properties of the Continuous time Fourier Transform,Example:,4.3 Properties of the Continuous time Fourier Transform,4.3.6 Duali
21、ty(对称性）,then,If,（即：与原信号x(t) 的频谱函数X(j)具有相同形式 的信号X(jt)的傅立叶变换为2x(-).）,If x(t) is even signal,补充,4.3 Properties of the Continuous time Fourier Transform,4.3 Properties of the Continuous time Fourier Transform,4.3 Properties of the Continuous time Fourier Transform,Example 4.13,4.3 Properties of the Cont
22、inuous time Fourier Transform,Example :,4.3 Properties of the Continuous time Fourier Transform,(符号函数),Example:,补充,4.3 Properties of the Continuous time Fourier Transform,Defining:,Prove: We look Sgn(t) as the following signal,4.3 Properties of the Continuous time Fourier Transform,1. The frequency
23、shifting property:,if,4.3 Properties of the Continuous time Fourier Transform,x(t) 在时域里乘以ej0t，对应于频谱X(j) 在频域里产生频移（右移0)。,Prove:,4.3 Properties of the Continuous time Fourier Transform,Example:,4.3 Properties of the Continuous time Fourier Transform,2. Differentiation property in frequency :,if,4.3 Pro
24、perties of the Continuous time Fourier Transform,Prove:,4.3 Properties of the Continuous time Fourier Transform,3. Integration property in frequency :,if,4.3 Properties of the Continuous time Fourier Transform,Example:,补充,4.3 Properties of the Continuous time Fourier Transform,注：求傅立叶反变换时，需要这些公式。,4.3
25、 Properties of the Continuous time Fourier Transform,Example:,4.3.7 Parsevals Relation,then,If,4.3 Properties of the Continuous time Fourier Transform,: The energy-density spectrum of x(t).,4.3 Properties of the Continuous time Fourier Transform,Example 4.14 The Fourier transform X(j) of x(t) isdepi
26、cted as following figure,to find,4.3 Properties of the Continuous time Fourier Transform,(a),(b),4.3 Properties of the Continuous time Fourier Transform,Solution:,(a),4.3 Properties of the Continuous time Fourier Transform,(b),4.4 The Convolution Property,Consider a LTI system:,4.3 Properties of the
27、 Continuous time Fourier Transform,时域里卷积，对应频域里相乘。,4.3 Properties of the Continuous time Fourier Transform,Example 4.15 Use the Fourier transform to compute y(t).,4.4.1 Examples,4.3 Properties of the Continuous time Fourier Transform,Example 4.16For a differentiator,4.3 Properties of the Continuous t
28、ime Fourier Transform,Example 4.17For a integrator,4.3 Properties of the Continuous time Fourier Transform,Example 4.19,4.3 Properties of the Continuous time Fourier Transform,4.5 The Multiplication ( modulation ) Property,The multiplication(modulation) property:,4.3 Properties of the Continuous tim
29、e Fourier Transform,时域里相乘，对应频域卷积（再除以2）。,Example 4.21,We have known,and S(j) of signal s(t) as the following figure:,-1 1 ,A,S(j),4.3 Properties of the Continuous time Fourier Transform,4.3 Properties of the Continuous time Fourier Transform,The multiplication property is called the modulation proper
30、ty. In example 4.21, and s(t) is the modulated signal,p(t) is carrier wave signal.,4.3 Properties of the Continuous time Fourier Transform,4.3 Properties of the Continuous time Fourier Transform,Example 4.22 The demodulation process of example 4.21.,The frequency response of low pass filter H(j) is:
31、,4.3 Properties of the Continuous time Fourier Transform,filter,Conclusion:,4.3 Properties of the Continuous time Fourier Transform,Example 4.23,4.3 Properties of the Continuous time Fourier Transform,4.6 Tables of Fourier Properties and of Basic Fourier Transform Pairs,4.3 Properties of the Continu
32、ous time Fourier Transform,Table 4.1,补充,4.3 Properties of the Continuous time Fourier Transform,时域微分：,对偶性：,频域微分：,频域积分：,4.3 Properties of the Continuous time Fourier Transform,Modulation property:,Table 4.2,补充,(1),(2),(3),(4),4 The continuous time Fourier transform,以下补充练习: 请同学们自己做,Compute the Fourier
33、 transform of the following signals.,(1),(2),(3),4 The continuous time Fourier transform,(4),4 The continuous time Fourier transform,(5),(6),4 The continuous time Fourier transform,2. Compute the inverse Fourier transform of the following signals,(1),(2),4 The continuous time Fourier transform,3. We
34、 have known,and,Question:,4 The continuous time Fourier transform,4. We have known,Question:,4.7 System Characterized by Linear Constant-Coefficient Differential Equation,Constant-coefficient differential equation:,Fourier transform:,4.7 Analyze LTI system in frequency domain,Input x(t),then,1) The
35、unit impulse response of the system,2) The output y(t) of the system with rest initial condition.,Frequency response,4.7 Analyze LTI system in frequency domain,Example 4.24,4.7 Analyze LTI system in frequency domain,Example 4.26,4.7 Analyze LTI system in frequency domain,补充：求傅立叶反变换方法,求傅立叶反变换有三种方法： 方
36、法一：用傅立叶反变换公式求解,方法二：利用傅立叶变换性质和常用函数傅立叶变换对方法三：利用部分分式展开的方法求,方法一：利用傅立叶反变换公式求,例题：求傅立叶反变换,解：,补充：求傅立叶反变换方法,方法二： 利用傅立叶变换性质和常用函数傅立叶变换对表求,参见表4.1和表4.2,Example:,Solution:,补充：求傅立叶反变换方法,If X(j) is the Fourier transform of the signal x(t),把上式的分母按j因式分解后，再把 X(j)写成部分分式和的形式，分以下几种情况讨论。,方法三：利用部分分式展开法求傅立叶反变换,补充：求傅立叶反变换方法,
37、情况1：p1,p2pN为X(j)分母的不同实根时，X(j)可写成如下形式：,补充：求傅立叶反变换方法,式中，,补充：求傅立叶反变换方法,Example 1,补充：求傅立叶反变换方法,Solution:,补充：求傅立叶反变换方法,情况2：X(j)分母的有共轭复根p1=p2*时，把共 轭复根看作两个不同根处理，方法同情况1。,补充：求傅立叶反变换方法,suppose,then,对共轭根部分有：,Example 2,补充：求傅立叶反变换方法,Solution:,补充：求傅立叶反变换方法,补充：求傅立叶反变换方法,情况3：X(j)的分母有r重实根p1，而其它根为不同实根时，X(j) 可写成如下形式：,
38、补充：求傅立叶反变换方法,重根p1,其他单根,补充：求傅立叶反变换方法,对重根p1的系数A1k：,对单根pi的系数Ai：方法同情况1,再利用,可以求出x(t).,补充：求傅立叶反变换方法,对重根p1:,对单根pi:,Example 3,补充：求傅立叶反变换方法,Solution:,补充：求傅立叶反变换方法,补充：求傅立叶反变换方法,补充：求傅立叶反变换方法,Problems: 4.1 4.3 4.4(a) 4.8 4.14 4.21(a) (b)(g) (h) (i) 4.32(a)(b) 4.33(a) (b) 4.35 4.36,4 The continuous time Fourier transform,
