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Non-ParametricNon-ParametricPower Spectrum Power Spectrum

EstimationEstimation Methods Methods

Eric HuiEric Hui

SYDE 770 Course ProjectSYDE 770 Course Project

November 28, 2002November 28, 2002

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IntroductionIntroduction

Applications of Power Spectrum Applications of Power Spectrum Estimation (PSE):Estimation (PSE): Wiener FilterWiener Filter Feature ExtractionFeature Extraction

Non-parametricNon-parametric PSE does NOT PSE does NOT assume any data-generating process assume any data-generating process or model (e.g. autoregressive or model (e.g. autoregressive model).model).

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MotivationMotivation

Ideal autocorrelation:Ideal autocorrelation:

Actual autocorrelation:Actual autocorrelation:

Limited (finite length of) data due to:Limited (finite length of) data due to: Availability of dataAvailability of data Assumption of stationaryAssumption of stationary

N

NnN

x nxknxN

kr )()(12

1lim)(

kN

nx nxknx

Nkr

1

0

)()(1

)(ˆ

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Periodogram MethodPeriodogram Method

)()(1

)()(1

)()(1

)(ˆ1

0

kxkxN

nxknxN

nxknxN

kr

NN

nNN

kN

nx

n

x(n)

N0

2)(

1)( j

Nj

per eXN

eP

DT

FT

redefinedas

n

xN(n)

N0

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Periodogram MethodPeriodogram Method

)()(1

)()(1

)()(1

)(ˆ

1

0

1

0

1

0

krN

kNkr

N

nxknxEN

nxknxN

EkrE

x

kN

nx

kN

n

kN

nx

)()sin(

)sin(1

2

1)(ˆ

2

21

21

j

xj

per ePN

NePE

DT

FT

N0-Nk

DT

FT

N

kNkw

)(

0

2

21

21

)sin(

)sin(1)(

N

NeW j

k

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““Good” Method?Good” Method?

Necessary conditions for Necessary conditions for mean-mean-square convergencesquare convergence:: Asymptotically UnbiasedAsymptotically Unbiased

Zero VarianceZero Variance

)()(ˆlim jwjw

NePePE

0)(ˆlim

jw

NePVar

k

PSD

k

PSD

as N ↑

k

as N ↑

k

PSDPSD

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Evaluation of MethodsEvaluation of Methods

ResolutionResolution How much “blurring” effect is there on the How much “blurring” effect is there on the

power spectrum?power spectrum? Bias (Asymptotic)Bias (Asymptotic)

Does the estimation approach the true Does the estimation approach the true value with more data (i.e. as N increases)?value with more data (i.e. as N increases)?

VarianceVariance Does the amount of deviation from the true Does the amount of deviation from the true

value depend on the data length (i.e. N)?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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Different PSE MethodsDifferent PSE Methods Periodogram MethodPeriodogram Method

Apply Apply rectangular windowrectangular window to x(n) to get x to x(n) to get xNN(n).(n).

Modified Periodogram MethodModified Periodogram Method Apply Apply non-rectangular windownon-rectangular window to x(n) to get x to x(n) to get xNN(n).(n).

Bartlett’s MethodBartlett’s Method AverageAverage the Periodogram estimate of the Periodogram estimate of non-overlapping non-overlapping

sub-intervals of x(n).sub-intervals of x(n).

Welch’s MethodWelch’s Method AverageAverage the Modified Periodogram estimate of the Modified Periodogram estimate of

overlappingoverlapping sub-intervals of x(n).sub-intervals of x(n).

Blackman-Turkey MethodBlackman-Turkey Method Apply Apply non-triangular windownon-triangular window to r(x). to r(x).

kN0-N

DT

FT

N

kNkw

)(

k0

2

21

21

)sin(

)sin(1)(

N

NeW j

k

as N ↑

k

PSDPSD

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Application: Feature Application: Feature ExtractionExtraction

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

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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

Linearized PSD Slope (Horizontal)

PSD

linearize

repeat for whole image

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Questions or Comments?Questions or Comments?

……