Chapter 8 The Discrete Fourier Transform ◆8.0 Introduction 8.1 Representation of Periodic sequence: the Discrete Fourier Series(DFS) 8.2 Properties of the dFs 8. 3 The Fourier Transform of Periodic signal 98.4 Sampling the Fourier Transform 98.5 Fourier Representation of Finite-Duration Sequence: the Discrete Fourier Transform FT 8.6 Properties of the dt 98.7 Linear Convolution using the DFT 8. 8 the discrete cosine transform DCT
2 Chapter 8 The Discrete Fourier Transform ◆8.0 Introduction ◆8.1 Representation of Periodic Sequence: the Discrete Fourier Series (DFS) ◆8.2 Properties of the DFS ◆8.3 The Fourier Transform of Periodic Signal ◆8.4 Sampling the Fourier Transform ◆8.5 Fourier Representation of Finite-Duration Sequence: the Discrete Fourier Transform(DFT) ◆8.6 Properties of the DFT ◆8.7 Linear Convolution using the DFT ◆8.8 the discrete cosine transform (DCT)
Chapter 8 The Discrete Fourier transform 8.0 Introduction
3 Chapter 8 The Discrete Fourier Transform 8.0 Introduction
8.0 Introduction Discrete Fourier Transform(DFT) is Transform of finite duration sequence DFT corresponds to samples equally spaced in frequency, of the Discrete-time Fourier transform(DTFT) of the signal DFT is a sequence rather than a function of a continuous variable o
4 8.0 Introduction ◆Discrete Fourier Transform (DFT) is Transform of finite duration sequence. ◆DFT corresponds to samples, equally spaced in frequency, of the Discrete-time Fourier transform (DTFT) of the signal. ◆DFT is a sequence rather than a function of a continuous variable ω
8.0 Introduction Derivation and interpretation of dft is based on relationship between periodic sequence and finite-length sequences The Fourier series representation of the periodic sequence corresponds to the DFT of the finite-length sequence
5 8.0 Introduction ◆Derivation and interpretation of DFT is based on relationship between periodic sequence and finite-length sequences： ◆The Fourier series representation of the periodic sequence corresponds to the DFT of the finite-length sequence
8.1 Representation of Periodic Sequence: the discrete fourier series Given a periodic sequence x[n] with period N so that xn]=xntrN The fourier series representation can be written as x[n]= Xk]ej(2T/N)kn Fourier series representation of continuous-time periodic signals require infinitely many complex exponentials, for discrete-time periodic signals 2丌 (k+mn)n ,(2zm) k=0,1,2,…,N-1
6 ◆Fourier series representation of continuous-time periodic signals require infinitely many complex exponentials, ◆for discrete-time periodic signals: 8.1 Representation of Periodic Sequence: the Discrete Fourier Series ( ) 2 j k N N m n e + ◆The Fourier series representation can be written as 2 j kn N e = ( ) 2 2 j kn N j mn e e =