2-FFT-Based Power Spectrum Estimation

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Professor A G Constantinides© 1 AGC DSP Power Spectral Estimation The purpose of these methods is to obtain an approximate estimation of the power spectral density of a given real random process ) ( j yy e } { n y

Transcript of 2-FFT-Based Power Spectrum Estimation

Page 1: 2-FFT-Based Power Spectrum Estimation

Professor A G

Constantinides© 1

AGC

DSP

AGC

DSP

Power Spectral Estimation

The purpose of these methods is to obtain an approximate estimation of the power spectral density of a given real random process

)( jyy e

}{ ny

Page 2: 2-FFT-Based Power Spectrum Estimation

Professor A G

Constantinides© 2

AGC

DSP

AGC

DSP

Autocorrelation

The autocorrelation sequence is

The pivot of estimation is the Wiener-Khintchine formula (which is also known as the Einstein or the Rayleigh formula)

]}[][{][ nynmyEmyy

m

jmyy

jyy eme ][)(

Page 3: 2-FFT-Based Power Spectrum Estimation

Professor A G

Constantinides© 3

AGC

DSP

AGC

DSP

Classification The crux of PSD estimation is the determination of

the autocorrelation sequence from a given process. Methods that rely on the direct use of the given

finite duration signal to compute the autocorrelation to the maximum allowable length (beyond which it is assumed zero), are called Non-parametric methods

Methods that rely on a model for the signal generation are called Modern or Parametric methods.

Personally I prefer the names “Direct” and “Indirect Methods”

Page 4: 2-FFT-Based Power Spectrum Estimation

Professor A G

Constantinides© 4

AGC

DSP

AGC

DSP

Classification & Choice

The choice between the two options is made on a balance between “simple and fast computations but inaccurate PSD estimates” Vs “computationally involved procedures but enhanced PSD estimates”

Page 5: 2-FFT-Based Power Spectrum Estimation

Professor A G

Constantinides© 5

AGC

DSP

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DSP

Direct Methods & Limitations

Apart from the adverse effects of noise, there are two limitations in practice Only one manifestation ,

known as a realisation in stochastic processes, is available

Only a finite number of terms, say , is available

]}[{ ny

12 N

Page 6: 2-FFT-Based Power Spectrum Estimation

Professor A G

Constantinides© 6

AGC

DSP

AGC

DSP

Assumptions

Assume to be Ergodic so that statistical expectations can

be replaced by summation averages Stationary so that infinite averages can be

estimated from finite averages Both of these averages are to be derived

from

]}[{ ny

]}[{ ny

Page 7: 2-FFT-Based Power Spectrum Estimation

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DSP

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DSP

Windowing

Thus an approximation is necessary. In effect we have a new signal given by

where is a window of finite duration selecting a segment of signal from .

][][][ nynwnx

][nw

]}[{ ny

]}[{ nx

Page 8: 2-FFT-Based Power Spectrum Estimation

Professor A G

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DSP

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DSP

The Periodogram The Periodogram is defined as

Clearly evaluations at

are efficiently computable via the FFT.

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0][

1)(

N

n

jnjN enx

NeI

kNk 2

1

0

1

0][][

1)(

N

r

jrN

n

jnjN erxenx

NeI

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Professor A G

Constantinides© 9

AGC

DSP

AGC

DSP

Limited autocorrelations

Let

which we shall call the autocorrelation sequence of this shorter signal.

These are the parameters to be used for the PSD estimation.

mN

nxx nxmnxm

1

0][][][

Page 10: 2-FFT-Based Power Spectrum Estimation

Professor A G

Constantinides© 10

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DSP

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DSP

PSD Estimator It can be shown that

The above and the limited autocorrelation expression, are similar expressions to the PSD. However, the PSD estimates, as we shall see, can be bad.

Measures of “goodness” are the “bias” and the “variance” of the estimates?

1

)1(][

1)(

N

Nm

jmxx

jN em

NeI

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Professor A G

Constantinides© 11

AGC

DSP

AGC

DSP

The Bias

The Bias pertains to the question:

Does the estimate tend to the correct value as the number of terms taken tends to infinity?

If yes, then it is unbiased, else it is biased.

Page 12: 2-FFT-Based Power Spectrum Estimation

Professor A G

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DSP

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DSP

Analysis on Bias

For the unspecified window case considered thus far, the expected value of the autocorrelation sequence of the truncated signal is

]}[][][][{

}][][{]}[{

1

0

1

0

nynwmnymnwE

nxmnxEmE

mN

n

mN

nxx

Page 13: 2-FFT-Based Power Spectrum Estimation

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Constantinides© 13

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DSP

AGC

DSP

Analysis on Bias

or

Thus

2

1

)1(

)(*)(21

]}[{1

)}({

jjyy

N

Nm

jmxx

jN

eWeN

emEN

eIE

mN

nyywwxx mmmE

1

0][][]}[{

Page 14: 2-FFT-Based Power Spectrum Estimation

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Constantinides© 14

AGC

DSP

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DSP

Analysis on Bias

The asterisk denotes convolution. The bias is then given as the difference

between the expected mean and the true mean PSDs at some frequency.

)()(*)(21

)()({

2kj

yyjj

yy

kjyy

kjN

eeWeN

eeIEB

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Constantinides© 15

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DSP

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DSP

Example For example take a rectangular window

then ,

which, when convolved with the true PSD, gives the mean periodogram, ie a smoothed version of the true PSD.

22

)2/sin()2/sin(

)(

N

eW j

Page 16: 2-FFT-Based Power Spectrum Estimation

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DSP

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DSP

Example

Note that the main lobe of the window has a width of

and hence as

we have

at every point of continuity of the PSD.

N/2N

)()}({lim jyy

jN

NeeIE

Page 17: 2-FFT-Based Power Spectrum Estimation

Professor A G

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DSP

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DSP

Asymptotically unbiased

Thus is an asymptotically unbiased estimator of the true PSD.

The result can be generalised as follows.

)( jN eI

Page 18: 2-FFT-Based Power Spectrum Estimation

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DSP

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DSP

Windows & Estimators

For the window to yield an unbiased estimator it must satisfy the following:

1) Normalisation condition

2) The main lobe width must decrease as 1/N

1

0

2 ][N

nNnw

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DSP

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DSP

The Variance

The Variance refers to the question on the “goodness” of the estimate:

Does its variance of the estimate decrease with N? ie does the expression below tend to zero as N tends to infinity?

22 )})({(}))({()}(var{ jN

jN

jN eIEeIEeI

Page 20: 2-FFT-Based Power Spectrum Estimation

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DSP

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DSP

Analysis on Variance

If the process is Gaussian then (after very long and tedious algebra) it can be shown that

where

AeIEeI jN

jN 2)})({()}(var{

2)(*)( )()()(

21

deWeWeN

A jjjyy

Page 21: 2-FFT-Based Power Spectrum Estimation

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Constantinides© 21

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Analysis

Hence it is evident that as the length of data tends to infinity the first term remains unaffected, and thus the periodogram is an inconsistent estimator of the PSD.

Page 22: 2-FFT-Based Power Spectrum Estimation

Professor A G

Constantinides© 22

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DSP

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DSP

Example

For example for the rectangular window taken earlier we have

where

CeIEeI jN

jN 2)})({()}(var{

21

)1(][

sin

)(sin

N

Nmyy mN

mNC

Page 23: 2-FFT-Based Power Spectrum Estimation

Professor A G

Constantinides© 23

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DSP

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DSP

Decaying Correlations

If has for then for we can write above

From which it is apparent that

ny 0yy 0mm

0mN

22

sinsin

1)()}(var{

NN

eeI jyy

jN

elsewheree

eeI j

yy

jyyj

N

,0

)(2

)()}(var{ 2

2

Page 24: 2-FFT-Based Power Spectrum Estimation

Professor A G

Constantinides© 24

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DSP

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DSP

Variance is large

Thus even for very large windows the variance of the estimate is as large as the quantity to be estimated!

Page 25: 2-FFT-Based Power Spectrum Estimation

Professor A G

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DSP

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DSP

Smoothed Periodograms Periodograms are therefore inadequate for

precise estimation of a PSD. To reduce variance while keeping estimation

simplicity and efficiency, several modifications can be implemented

a)     Averaging over a set of periodograms of (nearly) independent segments

b)     Windowing applied to segments c)     Overlapping the windowed segments for

additional averaging

Page 26: 2-FFT-Based Power Spectrum Estimation

Professor A G

Constantinides© 26

AGC

DSP

AGC

DSP

Welch-Bartlett Procedure

Typical is the Welch-Bartlett procedure as follows.

Let be an ergodic process from which we are given data points for the signal .

1)     Divide the given signal into blocks each of length .

2)     Estimate the PSD of each block 3)     Take the average of these estimates

nyM ][ny

NMK /N

Page 27: 2-FFT-Based Power Spectrum Estimation

Professor A G

Constantinides© 27

AGC

DSP

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DSP

Welch-Bartlett Procedure

Step 2 can take different forms for different authors.

For the Welch-Bartlett case the periodogram is suggested as

21

0][

1)(

N

n

jnr

jrN

emxN

eI

Page 28: 2-FFT-Based Power Spectrum Estimation

Professor A G

Constantinides© 28

AGC

DSP

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DSP

Welch-Bartlett Procedure

where the segment is a windowed portion of

 

  And is the overlap. (Strictly the Bartlett case has a

rectangular window and no overlap).

][nxrny

)]([][][ 0NNrnynwnxr

0N

Page 29: 2-FFT-Based Power Spectrum Estimation

Professor A G

Constantinides© 29

AGC

DSP

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DSP

Comments FFT-based Spectral estimation is limited by

a) the correlation assumed to be zero beyond the measurement length and

b) the resolution attributes of the DFT. Thus if two frequencies are separated by

then a data record of length

is required.(Uncertainty Principle)

/2N

Page 30: 2-FFT-Based Power Spectrum Estimation

Professor A G

Constantinides© 30

AGC

DSP

AGC

DSP

Narrowband Signals The spectrum to be estimated is some cases

may contain narrow peaks (high Q resonances) as in speech formants or passive sonar.

The limit on resolution imposed by window length is problematic in that it causes bias.

The derived variance formulae are not accurate