BMIE_DSP_Chap3New

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Completeness of space 空间完备性
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Function H
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Function H
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Additional noise such as thermal noise,1/f noise or shot noise, which have an additional relationship with signal and additional noise does exist whether there is signal or not.
Multiplicative noise is usually caused by non-ideal characteristics of signal channel. It has a multiplicative relationship with signal, and it co-exists with the signal.
In general communication, additional randomicity is regarded as background noise of the system, while multiplicative randomicity is regarded as time varying volatility of the system, e.g. fading, Doppler, or caused by non-linearity.
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Chapter 3
Fourier Transform of Discrete-time signals
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Chapter 3 Discrete Fourier Transform
Fourier transform
is a basic tool for
analyzing and
processing signals
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White light passing through
triangular prism
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《实用Fourier变换及C++实现》1CD. 作者:孙鹤泉编著.
科学出版社
www.1-
/Book.197182.aspx
Besides Spectral analysis, convolution, correlation and so on can be
calculated using DFT
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3.1 Fourier transform of continuous signals 1. Fourier series
FS
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Fourier coefficient is the coefficient of
th harmonic wave, therefore are discrete on
frequency axis, and an interval of exits.
0()X k W k 0()X k W 0W
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L
L
()
x t A
T
2
t 2t -0
L
L
T
X(k Ω0)X(j Ω)
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dt
dt
and vice versa?
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Periodic signal :Non-periodic signal :
Then ,can Fourier transform be implemented in periodic signals?
W
-dt
e
t
x t j
)(
0 =-¥
å
¥Periodic signal
FS
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e.g.
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0()
X k W 1/2
1/2
1
1
-0
()
X j W 0
p
p
W -W FS
FT
Line spectrum
3.2 Fourier transform for discrete time series 1.Definition
Relationships between DTFT and Z transform!
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2. Properties
x n
()
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m
n m n ¹=
Four kinds of Fourier transforms
1.Cont. Non-period
3.Discrete Non-period.
Understand the relationships among the 4 FTs!
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4 types of Fourier transform
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3.Properties
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Time-domain convolution theorem Frequency domain convolution theorem!
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Cross correlation:
* Autocorrelation:
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Attention:
2
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()()
2
j
n
x n X e d
p w
p
w
p-
=
åò
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2
2()j N X e w
Remarks:
valuation theory’is formed.
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)
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4.Applications:
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Attention:All finite signals can be seen as a product of infinite signals and a rectangle window. The key is its effect in frequency domain.
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f f
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w
Low-pass high-pass phase-
2010-10-1142
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•Sampling theorem
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s
nT t t x n x t x ==Þ|)()()(D
A
/
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w
ÞW s
T j j X j
P e
X W =W *W =w w
|)()()(Property of DTFT
2010-10-11
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k s
T =-¥
Periodic extension Infinite superposition
W
()
X j W c
W c
-W W
()
P j W s
W s
-W L L w ()
j X e w
L
L
Possible
effect after superposition
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(~)
p p -Request :Nyquist/Shannon sampling theorem
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()
x t ()
x n ()
a H s /A D
How to reconstruct from
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