msp15 the fourier transform (cont’) lim, 1990. msp16 the fourier series expansion suppose g(t) is...
Post on 20-Dec-2015
219 views
TRANSCRIPT
MSP1
The FourierTransform (cont’)
Lim, 1990Lim, 1990
MSP2
The Fourier Series Expansion
• Suppose g(t) is a transient function that is zero outside the interval [-T/2,T/2] (e.g., a cycle of a periodic function). We can obtain a sequence of coefficients by making s a discrete variable and integrating over the interval (with period T), so that
MSP3
The Fourier Series Expansion (cont’)
0
)(2
0
)(2
2
2
)(2
1)()(
)()()(
n
T
ntj
nn
stnj
T
T
stnj
eGT
sesnGtg
dtetgsnGnG
Ts 1where
MSP4
The Discrete Fourier Transform (DFT)
• If we discretize both time and frequency the Fourier transform pair of a series become
n
nN
ij
nn
tisnji
N
Nn
iN
nj
i
N
Nn
tisnj
eGT
sesnGtigtg
egN
TtetigsnGnG
)(2)(2
2
2
)(22
2
)(2
1)()()(
)()()(
MSP5
The DFT (cont’)
• If {fi} is a sequence of length N (by taking samples of a continuous function at equal intervals) then its discrete Fourier transform pair is given by
1,0 ,1
1
1
0
2
1
0
2
NnieFN
f
efN
F
N
n
nN
ij
ni
N
i
iN
nj
in
MSP6
Properties of the Fourier Transform
• The addition theorem (addition in time/spatial domain corresponds to addition in frequency)
• The shift theorem (shifting a function causes to only phase shift)
• The convolution theorem (convolution is equivalent to multiplication in the other domain)
• …
MSP7
The Addition Theorem
)()(
)()(
)()(
)()(
22
2
sGsF
dtetgdtetf
dtetgtf
tgtf
stjstj
stj
Castleman, 1996Castleman, 1996
MSP8
The Fourier Transform of a 2D Sequence x(m,n)
• The Fourier Transform Pair
2121212
212121
)}(exp{),()2(
1),(
,- )},(exp{),(),(
1 2
ddnmjXnmx
nmjnmxXm n
MSP9
A 2D Fourier Transform
Castleman, 1996Castleman, 1996
MSP10
Properties of 2D Fourier Transform
Castleman, 1996Castleman, 1996
MSP11
The Fourier Transform (cont’)
• Example 1 (xh)
)cos(3
1)cos(
3
1
3
1
6
1
6
1
6
1
6
1
3
1
)}(exp{),(),(
21
22112121
2121
1 2
jjjj
n n
eeee
nnjnnhH
Lim, 1990Lim, 1990
MSP12
The Fourier Transform (cont’)
Lim, 1990Lim, 1990
MSP13
The Fourier Transform (cont’)
Lim, 1990Lim, 1990
MSP14
The Fourier Transform (cont’)
)2cos-(3)2cos-(3
)()(),(
21
221121
HHH
Lim, 1990Lim, 1990