syde 770 a presentation
TRANSCRIPT
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Non-Parametric
Power Spectrum EstimationMethods
Eric Hui
SYDE 770 Course ProjectNovember 28, 2002
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Introduction
Applications of Power Spectrum Estimation
(PSE):
Wiener Filter
Feature Extraction
Non-parametricPSE does NOT assume any
data-generating process or model (e.g.
autoregressive model).
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Motivation
Ideal autocorrelation:
Actual autocorrelation:
Limited (finite length of) data due to:
Availability of data
Assumption of stationary
N
NnN
x nxknxN
kr )()(12
1lim)(
kN
n
x nxknxN
kr1
0
)()(1
)(
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Periodogram Method
)()(1
)()(1
)()(1
)(1
0
kxkxN
nxknx
N
nxknxN
kr
NN
n
NN
kN
n
x
n
x(n)
N0
2
)(1
)( jNj
per eXN
eP
DTF
T
redefined
as
n
xN(n)
N0
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Periodogram Method
)()(1
)()(1
)()(1)(
1
0
1
0
1
0
krN
kNkrN
nxknxEN
nxknxN
EkrE
x
kN
n
x
kN
n
kN
n
x
)()sin(
)sin(1
2
1)(
2
21
21
j
x
j
per ePN
NePE
DTFT
N0-Nk
DTFT
N
kN
kw
)(
0
2
21
21
)sin(
)sin(1)(
N
NeW j
k
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Good Method?
Necessary conditions for mean-square
convergence:
Asymptotically Unbiased
Zero Variance
)()(lim jwjwN
ePePE
0)(lim
jw
NePVar
k
PSD
k
PSD
as N
k
as N
k
PSDPSD
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Evaluation of Methods
Resolution
How much blurring effect is there on the powerspectrum?
Bias (Asymptotic) Does the estimation approach the true value with
more data (i.e. as N increases)?
Variance Does the amount of deviation from the true value
depend on the data length (i.e. N)?
k
PSD
k
PSD
as N
k
as N
k
PSDPSD
k
PSD
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Application: Feature Extraction
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5x 10
4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-250
-200
-150
-100
-50
0
50
LinearizedPSD Slope(Horizontal)
PSD
linearize
repeat for whole image
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Questions or Comments?