Lecture 19 The Discrete Fourier Series Embedded Signal 讲座19离散傅立叶系列嵌式信号.pptVIP

Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing The Discrete Fourier Series Quote of the Day Whoever despises the high wisdom of mathematics nourishes himself on delusion. Leonardo da Vinci? Discrete Fourier Series Given a periodic sequence with period N so that The Fourier series representation can be written as The Fourier series representation of continuous-time periodic signals require infinite many complex exponentials Not that for discrete-time periodic signals we have Due to the periodicity of the complex exponential we only need N exponentials for discrete time Fourier series Discrete Fourier Series Pair A periodic sequence in terms of Fourier series coefficients The Fourier series coefficients can be obtained via For convenience we sometimes use Analysis equation Synthesis equation Example 1 DFS of a periodic impulse train Since the period of the signal is N We can represent the signal with the DFS coefficients as Example 2 DFS of an periodic rectangular pulse train The DFS coefficients Properties of DFS Linearity Shift of a Sequence Duality Symmetry Properties Symmetry Properties Cont’d Periodic Convolution Take two periodic sequences Let’s form the product The periodic sequence with given DFS can be written as  Periodic convolution is commutative Periodic Convolution Cont’d Substitute periodic convolution into the DFS equation Interchange summations The inner sum is the DFS of shifted sequence Substituting Graphical Periodic Convolution The Fourier Transform of Periodic Signals Periodic sequences are not absolute or square summable Hence they don’t have a Fourier Transform We can represent them as sums of complex exponentials: DFS We can combine DFS and Fourier transform Fourier transform of periodic sequences Periodic impulse train with values proportional to DFS coefficients This is periodic with 2? since DFS is periodic The inverse transform can be written as Example