信号与系统英文版教学课件：ch5 The Discrete-Time Fourier Transform

1、1 INTRODUCTIONDiscrete-time Fourier transform and inverse Fourier transform.Similarities and differences between continuous-time and discrete-time Fourier transforms. 21. Representation of Aperiodic Signals: The Discrete-Time Fourier Transform As , N N1 0 N2 Nn N1 0 N2xnn1) Development of the Discre

2、te-Time Fourier Transform3Defining a function:Thus,Consequently,As , discrete-time Fourier transform discrete-time inverse Fourier transform 4analysis equation: (分析公式)synthesis equation: (综合公式)-spectrum-density function(频谱密度函数)The aperiodic Discrete-Time signals can still be represented as a linear

3、combination of complex exponentials. The magnitude of component with frequence isAn useful relationship:5CTFT:DTFT:-角频率(弧度/秒)-角度(弧度)，是 对 归一化的结果6Differences between the continuous-time and discrete-time Fourier transform are: periodicity of the discrete-time transform and the finite interval of integ

4、ration in the synthesis equation. In discrete time,Low frequencies are the values of near even multiple of ;high frequencies are those values of near odd multiples of . 2 0 2 2 0 2 0 nx1n 0 nx2n7Example 5.1 Consider the signal1/(1)1/(1+)2 0 2 2 0 2 1 0 0 -1 1/(1+)1/(1)2 0 2 2 0 2 8Example 5.2 Consid

5、er the signal 0 xnn (1+)/(1) (1)/(1+) 2 0 2 for 0 2M1/N s 2 spectrum of sampled signal with s 2MIn discrete-time case, the result of sampling theorem also exist.42Example 5.15 Consider a sequence xn whose Fourier transform Determine the lowest rate at which xn may be sampled without aliasing. Since

6、the corresponding sampling frequency is 2/4 = /2. So thatThusFrom the sampling theorem, we know430 xn n0 xpn n0 xbn nIn the time-domain: xbn = xpnN (1)Decimation In the frequency-domain:2) Discrete-Time Decimation and Interpolation44the effect of decimation is to spread thespectrum of the original s

7、equence over a larger portion of the frequency band. M M 2 11/N M M 2 1/N NM NM 2 Frequency-domain illustration of the relationship between sampling and decimation45Down-Sampling If the original sequence xn is obtained by sampling a continuous-time signal, the process of decimation can be viewed as

8、reducing the sampling rate on the signal by a factor of N. To avoid aliasing, cannot occupy the full frequency band. In other words, if the signal can be decimated without introducing aliasing, then the original continuous-time signal was over-sampled, and thus, the sampling rate can be reduced with

9、out aliasing. With the interpretation of the sequence xn as samples of a continuous-time signal, the process of decimation is often referred to as down-sampling.46 M M 2 11/N M M 2 1/N NM NM 2 Frequency-domain illustration of the relationship between sampling and decimation47(2)Interpolation (Up-Sam

10、pling ) 48Example 5.16 2 16/9 2/9 2/9 2210/9 8/9 8/9 10/9 2 2 2 2 /9/9 2 17/9 17/949SUMMARY 2. The Fourier transform for periodic discrete-time signals; 1. The Fourier transform for aperiodic discrete-time signals; 6. Convenient way to obtain the frequency response and sample response of a discrete-

11、time LTI system; 5. Fourier analysis method (also referred to frequency-domain analysis)(which is to used to change a linear constant-coefficient difference equation to an algebraic form);3. The difference between the continuous-time Fourier transform and the discrete-time Fourier transform (the discrete-time Fourier transform of an aperiodic signal is always periodic with period 2); 4. The properties of the Fourier transform (how differen