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Fourier Analysis(Ta3520)
Fourier – p. 1/16
Fourier AnalysisContinuous Fourier transformDiscrete Fourier Transform and Sampling TheoremLinear Time-Invariant (LTI) systems andConvolutionConvolution TheoremFiltersCorrelationDeconvolution
Fourier – p. 2/16
Continuous Fourier transformDefinition:
A(f) =
∫ +∞
−∞
a(t) exp(−2πift) dt
Inverse transform:
a(t) =
∫ +∞
−∞
A(f) exp(2πift) df
f = frequency (Hz); t = time (s)
Decomposition of signal into sines and cosinesFourier – p. 3/16
Decomposition into cosines
0
50
100
150
200
250
0 10 20 30
frequency
time
Fourier – p. 4/16
Decomposition into sines and cosines
Fourier – p. 5/16
Discretisation of signalDiscretisation in time makes spectrum periodic:
ADiscrete(f) =+∞∑
m=−∞
AContinuous(f +m
∆t)
At least two samples per frequency/wavelength.
Otherwise: aliasing.
Fourier – p. 6/16
Aliasing in time domain
0
0
0
(a)
(b)
(c)
Fourier – p. 7/16
Aliasing = overlapping spectraAcontinuous
A D
0
01∆t
A D
01∆t
1∆t
1∆t1 1
2 2
(a)
(b)
(c)
f
f
f
Fourier – p. 8/16
Discrete Fourier transform
An = ∆t
N−1∑
k=0
ak exp(−2πink/N) for n = 0, 1, ..., N − 1
k = index for time-domain coefficientn = index for Fourier-domain coefficient
ak = ∆f
N−1∑
n=0
An exp(2πink/N) for k = 0, 1, ..., N − 1
Basic relation:N∆t∆f = 1Nyquist criterion: fN = 1
2∆t Fourier – p. 9/16
Linear Time-Invariant (LTI) systemsand convolution
Output of linear time-invariant system = convolution ofinput with impulse response of system:
h(t)
h(t)
h(t)
h(t)
δ(t)
δ(t-τ)
s(τ)δ(t-τ)
s(τ)δ(t-τ) dτ
h(t)
h(t-τ)
s(τ)h(t-τ)
s(τ)h(t-τ) dτ
time-invariance :
linearity :
linearity :
= s(t)
Fourier – p. 10/16
Convolution theorem
Ft
(∫ +∞
−∞
h(t′)g(t − t′)dt′)
= Ft (h(t) ∗ g(t))
= H(f)G(f)
Fourier – p. 11/16
Filters
0 10 20 30 400
0.5
1.0
frequency
| A(f)
|
0 0.5 1.0 1.5-2
-1
0
1
2
time
a(t)
FT
0 10 20 30 400
0.5
1.0
| A(f)
|
frequency
0 0.5 1.0 1.5-2
-1
0
1
2
time
a(t)
-1FT
INPUT TO FILTER
0 10 20 30 400
1.0
frequency
| A(f)
|
OUTPUT OF FILTER
Fourier – p. 12/16
Filters: original Wassenaar data
0
0.5
1.0
1.5
time
(s)
100 200 300offset (m)
Fourier – p. 13/16
Filters: Wassenaar data0
0.5
1.0
1.5
time
(s)
100 300distance (m)
0
50
100
150
200
250
300
350
400
450
500
frequ
ency
(Hz)
100 300distance (m)
0
50
100
150
200
250
300
350
400
450
500
frequ
ency
(Hz)
100 300distance (m)
0
0.5
1.0
1.5
time
(s)
100 300distance (m)
Fourier – p. 14/16
Filters: filtered Wassenaar data
0
0.5
1.0
1.5
time
(s)
100 200 300offset (m)
Fourier – p. 15/16
CorrelationAuto-correlation:
Ft
(∫ +∞
−∞
a(τ)a∗(τ − t)dτ
)
= A(f)A∗(f) = |A(f)|2
Cross-correlation:
Ft
(∫ +∞
−∞
a(τ)b∗(τ − t)dτ
)
= A(f)B∗(f)
Fourier – p. 16/16