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# ch3 Fourier Series Representation of Periodic Signals

3 Fourier Series Representation of Periodic Signals

jhhu@phy.ccnu.edu.cn

3 Fourier Series Representation of Periodic Signals
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3.1 A Historical Perspective 3.2 The Response of LTI Systems to Complex Exponentials 3.3 Fourier Series Representation of ContinuousTime Periodic Signals 3.4 Convergence of the Fourier Series 3.5 Properties of Continuous-Time Fourier Series 3.8 Fourier Series and LTI Systems 3.9 Filtering 3.10 Examples of Continuous-Time Filters Described by Differential Equations 3.12 Summary

3.1 A Historical Perspective
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Jean Baptiste Joseph Fourier, born in 1768, in France.
1807,periodic signal could be represented by sinusoidal series. 1829,Dirichlet provided precise conditions. 1960s,Cooley and Tukey discovered fast Fourier transform.

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3.1 A Historical Perspective
Time domain Basic signal Frequency domain

? (t )
+?

cos?t , e j?t
Linear combination of the basic signals

Input signal

x(t ) =

x(t )
Output signal

??

? x(τ )δ(t - τ )dτ

y(t )

y(t ) ? h(t ) * x(t )

?e j?t ? H ( j? ) X ? e j?t Y

3 Fourier Series Representation of Periodic Signals
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3.1 A Historical Perspective 3.2 The Response of LTI Systems to Complex Exponentials 3.3 Fourier Series Representation of ContinuousTime Periodic Signals 3.4 Convergence of the Fourier Series 3.5 Properties of Continuous-Time Fourier Series 3.8 Fourier Series and LTI Systems 3.9 Filtering 3.10 Examples of Continuous-Time Filters Described by Differential Equations 3.12 Summary

3.2 The Response of LTI Systems to Complex Exponentials
Basic signals
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The set of basic signals can be used to construct a broad and useful class of signals.
It should be convenient for us to represent the response of an LTI system to any signal constructed as a linear combination of the basic signals.

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Complex exponential signals
in continuous time:

e

st

3.2 The Response of LTI Systems to Complex Exponentials
The Response of an LTI System to a Complex Exponential input is the same Complex Exponential with only a change in amplitude in continuous time: e st ? H ( s )e st

x(t ) ? e

H ( s ) ? ? h(? )e ? s? d?
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st

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Why?
?? ??

h(t)

y( t ) ? x( t ) * h( t ) ? ? ?? e
?? ?? s ( t ?? )

x( t ? ? )h(? )d?
st

h(? )d? ? e

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??

h(? )e ? s? d?

H ( s) ?

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? e st H ( s )

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h(? )e ? s? d?

( system function )

3.2 The Response of LTI Systems to Complex Exponentials
Eigenfunction and eigenvalue ?A signal for which the system output is a (possibly complex) constant times the input is referred to as an eigenfunction of the system. ?The amplitude factor is referred to as the system’s eigenvalue.
Eigenfunction

Continuous-time system

e st

Eigenvalue H ( s)

3.2 The Response of LTI Systems to Complex Exponentials
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Decomposing general signals in