Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.341:DISCRETE-TIME SIGNAL PROCESSING OpenCourse Ware 2006 Lecture 15 The Discrete Fourier Transform (DFT) Reading:Sections 8.1-8.6 in Oppenheim,Schafer Buck (OSB). Here are some basic points about the discrete Fourier Transform (DFT),the discrete-time Fourier Transform (DTFT),and the fast Fourier transform (FFT). 1.The DTFT can't be computed 2.The DFT can be computed 3.The DFT isn't the DTFT 4.The FFT isn't the DFT 5.The FFT is not necessarily the best/most effient way to compute whatever it is that it computes In this lecture,we will cover the first three points,and discuss the FFT in lecture 19. Sampling in Frequency The DTFT of xIn]is defined as follows: x(e)=∑rmle-om n Since w is a continuous variable,there are an infinite number of possible values of w from 0 to 2m or from-to Thus,X(ej)can be computed only at a finite set of frequencies: X(eiae)=∑zmle-un n As a special case,we use N samples equally spaced around the unit circle: 2πk Wk= N, k=0,1,,N-1 and define the N samples of X(ew): 1Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.341: Discrete-Time Signal Processing OpenCourseWare 2006 Lecture 15 The Discrete Fourier Transform (DFT) Reading: Sections 8.1 - 8.6 in Oppenheim, Schafer & Buck (OSB). Here are some basic points about the discrete Fourier Transform (DFT), the discrete-time Fourier Transform (DTFT), and the fast Fourier transform (FFT). 1. The DTFT can’t be computed 2. The DFT can be computed 3. The DFT isn’t the DTFT 4. The FFT isn’t the DFT 5. The FFT is not necessarily the best/most effient way to compute whatever it is that it computes In this lecture, we will cover the first three points, and discuss the FFT in lecture 19. Sampling in Frequency The DTFT of x[n] is defined as follows: e X −jωn (ejω) = �x[n] n Since ω is a continuous variable, there are an infinite number of possible values of ω from 0 to 2π or from −π to π. Thus, X(ejω) can be computed only at a finite set of frequencies: e X −jωkn (ejωk ) = �x[n] n As a special case, we use N samples equally spaced around the unit circle: 2πk ωk = N , k = 0, 1, . . . , N − 1 and define the N samples of X(ejω): 1