ch3 Fourier Series Representation of Periodic Signals.ppt

INTRODUCTION
Representation of continuous-time and discrete-time periodic signals — Fourier series.
Use Fourier methods to analyze and understand signals and LTI systems.
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1. The Response of LTI Systems plex Exponentials
1) Important concept — signal position
(1) basic signals: possess two properties
A.  The set of basic signals can be used to construct a broad and useful class of signals.
B. It should be convenient for us to represent the response of an LTI system to any signal constructed as a bination of the basic signals.
(2) complex exponential signals
in discrete time:
in continuous time:
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2) The Response of an LTI System to plex Exponential input is the plex Exponential with only a change in amplitude
in discrete time:
in continuous time:
Complex amplitude facts H(s) or H(z) is in general a function of plex variable s or z .
Why?
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in a continuous–time LTI system with impulse response h(t) and an input
So the output
in a discrete–time LTI system with impulse response h(n) and an input
So the output
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3) Eigenfunction(特征函数)
Eigenfunction and Eigenvalue( of the system):
A signal for which the system output is a (plex) constant times the input is referred to as an eigenfunction of the system,and the amplitude factor is referred to as the system’s eigenvalue(特征值).
For a specific value of sk or zk , or : eigenvalue .
is the eigenfunction of continuous–time systems
is the eigenfunction of discrete–time systems
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If the input to a continuous (discrete) time LTI system is represented as a bination plex exponentials:
then from the eigenfunction property and the superposition property, the output will be:
4) posing general signals in terms of eigenfunctions
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2. Fourier Series Representation of Continuous-Time Periodic Signals
1) Complex Exponential Fourier Series
Given periodic x(t) with fundamental period T , plex exponential Fourier series is :
The signals in the set are harmonically plex exponentials.
: ponents or the first p