apter 8 The discrete Fourier Transform K8:0 Introduction K8.1 Representation of Periodic Sequence: the Discrete fourier series 98.2 Properties of the Discrete Fourier Series 8.3 The Fourier transform of Periodic signal 8.4 Sampling the Fourier Transform 98.5 Fourier Representation of Finite-Duration Sequence: the discrete Fourier Transform < 8.6 Properties of the Discrete Fourier Transform 18.7 Linear Convolution using the Discrete Fourier transform 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 ◆8.2 Properties of the Discrete Fourier Series ◆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 ◆8.6 Properties of the Discrete Fourier Transform ◆8.7 Linear Convolution using the Discrete Fourier Transform ◆8.8 the discrete cosine transform (DCT)
Filter Design Techniques 8.0 Introduction
3 Filter Design Techniques 8.0 Introduction
8.0 Introduction Discrete Fourier Transform(DFT)for finite duration sequence DFT is a sequence rather than a function of a continuous variable DFT corresponds to samples equally spaced in frequency of the Discrete-time Fourier transform(dTFT) of the signal
4 8.0 Introduction ◆Discrete Fourier Transform (DFT) for finite duration sequence ◆DFT is a sequence rather than a function of a continuous variable ◆DFT corresponds to samples, equally spaced in frequency, of the Discrete-time Fourier transform (DTFT) of the signal
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 义]=Xn+rN The fourier series representation can be written as 2丌/Nk 小=∑[k]e Q Fourier series representation of continuous-time periodic signals require infinitely many complex exponentials, for discrete-time periodic signals 2丌 2丌 k+mN)n j (2rmn 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 ◆Given a periodic sequence with period N so that x[n] ~ x[n rN] ~ x[n] ~ = + 1 (2 / ) [ ] k j N kn x n X k N e = ( ) 2 j k N N m n e + ◆The Fourier series representation can be written as , 0,1,2, , 1 k N = − 2 j kn N e = ( ) 2 2 j kn N j mn e e =