Compdsp Kootsookos Frequency-Estimation

8/4/2019 Compdsp Kootsookos Frequency-Estimation

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July 30th, 2004 comp.dsp conference 1

Frequency Estimation Techniques

Peter J. Kootsookos

[email protected]


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Talk Summary

Some acknowledgements

What is frequency estimation?o What other problems are there?

Some algorithms

o Maximum likelihoodo Subspace techniques

o Quinn-Fernandes

Associated problemso Analytic signal generation

Kay / Lank-Reed-Pollon estimators

o Performance bounds: Cramr-Rao Lower Bound


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Some Acknowledgements

Eric Jacobson for his presence on comp.dsp and

for his work on the topic. Andrew Reilly for his presence on comp.dsp and

for analytic signal advice.

Steven M. Kay for his books on estimation and

detection generally, and published research work onthe topic.

Barry G. Quinn as a colleague and for his work thetopic.

I. Vaughan L. Clarkson as a colleague and for hiswork on the topic.

CRASys Now defunct Cooperative ResearchCentre for Robust & Adaptive Systems.


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Talk Summary



Some algorithms


o Quinn-Fernandes





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What is frequency estimation?

Find the parameters A, w, f, and s2 in

y(t) = A cos [w(t-n) +f)] +e(t)

where t= 0..T-1, n =T-1/2and e(t) is a noisewith zero mean and variance s2.

qis used to denote the vector [Awfs2]T.


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What other problems are there?

y(t) = A cos [w(t-n) +f)] +e(t) What about A(t) ?

o Estimating A(t) is envelope estimation (AMdemodulation).

o If the variation of A(t) is slow enough, the problemof estimating wand estimating A(t) decouples.

What about w(t)?o This is the frequency tracking problem.

Whats e(t) ?o Usually assumed additive, white, & Gaussian.

o Maximum likelihood technique depends onGaussian assumption.


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What other problems are there? [continued]

Amplitude-varying example: conditionmonitoring in rotating machinery.


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Frequency tracking example: SONAR

Thanks to BarryQuinn & TedHannan for theplot from their

book TheEstimation &Tracking ofFrequency.


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Multi-harmonic frequency estimation

y(t) = SAmcos [mw(t-n) +fm)] +e(t)

For periodic, but not sinusoidal, signals.

Each component is harmonically relatedto the fundamental frequency.

p

m=1


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Multi-tone frequency estimation

y(t) = SAmcos [wm(t-n) +fm)] +e(t)

Here, there are multiple frequencycomponents with no relationshipbetween the frequencies.

p

m=1


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Talk Summary



Some algorithms


o Quinn-Fernandes





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The Maximum Likelihood Approach

The likelihood function for this problem, assuming that

e(t) is Gaussian is

L(q) = 1/((2p)T/2|Ree|)exp((Y(q))TR-1ee(Y(q))/ 2)

whereRee=The covariance matrix of the noise e

Y= [y(0) y(1) y(T-1)]T

= [A cos(f) A cos(w+f) A cos(w(T-1) +f)]T

Yis a vector of the date samples, and is a vector ofthe modeled samples.


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The Maximum Likelihood Approach [continued]

Two points to note:

The functional form of the equation

L(q) = 1/((2p)T/2

|Ree|)

exp((Y(q))T

R-1

ee(Y(q))/ 2)

is determined by the Gaussian distribution ofthe noise.

If the noise is white, then the covariancematrix Ris just s2I a scaled identity matrix.


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Often, it is easier to deal with the log-likelihood function:

(q) =(Y(q))TR-1ee(Y(q))

where the additive constant, and multiplying constanthave been ignored as they do not affect the position

of the peak (unless s is zero or infinite).

If the noise is also assumed to be white, the maximumlikelihood problem looks like a least squares problemas maximizing the expression above is the same as

minimizing

(Y(q))T(Y(q))


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If the complex-valued signal model is used,

then estimating wis equivalent to maximizingthe periodogram:

P(w) =|Sy(t) exp(-iwt) |2

For the real-valued signal used here, thisequivalence is only true as Ttends to infinity.

t=0

T-1


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Talk Summary



Some algorithms


o Quinn-Fernandes

Associated problems

o Analytic signal generation Kay / Lank-Reed-Pollon estimators



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Subspace Techniques

The peak of the spectrum produced by spectralestimators other than the periodogram can be used forfrequency estimation.

Signal subspace estimators use either

PBar(w) = v*(w) RBarv(w)or

PMV(w) = 1/( v*(w) RMV-1 v(w) )

where v(w) = [ 1 exp(iw)exp(i2w) .. exp(I(T-1)w)]and an estimateof the covariance matrix is used.

^

^

Note:If Ryy is full rank, the PBar is thesame as the periodogram.


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Subspace Techniques - Signal

Bartlett:

RBar =Slkeke*k

Minimum Variance:

RMV-1 =S1/lkeke*k

Assuming there are pfrequency components.

^

^

k=1

p

k=1

p


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Subspace Techniques - Noise

Pisarenko:

RPis-1 = ep+1 e*p+1

Multiple Signal Classification (MUSIC):

RMUSIC-1 =Seke*k

Assuming there are pfrequency components.

Key Idea: The noise subspace is orthogonal to the signalsubspace, so zeros of the noise subspace will indicatesignal frequencies.

^

^ M

k=p+1

While Pisarenko is not statistically efficient, itis very fast to calculate.


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Quinn-Fernandes

The technique of Quinn & Fernandes assumes that the

data fits the ARMA(2,2) model:y(t)by(t-1) + y(t-2) =e(t)ae(t-1) +e(t-2)

1. Set a1 = 2cos(w).2. Filter the data to form

zj(t) = y(t) +ajzj(t-1) zj(t-2)3. Form bj by regressing ( zj(t) +zj(t-2) ) on zj(t-1)

bj = St( zj(t) +zj(t-2) ) zj(t-1) /Stzj2(t-1)

4. If |aj -bj| is small enough, set w = cos-1(bj/ 2),otherwise set aj+1 =bj and iterate from 2.


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Quinn-Fernandes [continued]

The algorithm can be interpreted as finding the

maximum of a smoothed periodogram.


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Talk Summary



Some algorithms


o Quinn-Fernandes

Associated problems



F E i i T h i


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Associated Problems

Other questions that need answering are:

What happens when the signal is real-valued, andmy frequency estimation technique requires acomplex-valued signal?

o Analytic Signal generation

How well can I estimate frequency?

o Cramer-Rao Lower Bound

o Threshold performance

F E i i T h i


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Associated Problems: Analytic Signal Generation

Many signal processing problems already use analytic signals:

communications systems with in-phase and quadraturecomponents, for example.

An analytic signal, exp(i-blah), can be generated from a real-valuedsignal, cos(blah) , by use of the Hilbert transform:

z(t) = y(t) + i H[ y(t) ]

where H[.] is the Hilbert transform operation.

Problems occur if the implementation of the Hilbert transform ispoor. This can occur if, for example, too short an FIR filter isused.

F E ti ti T h i


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Associated Problems: Analytic Signal Generation [continued]

Another approach is to FFT y(t) to obtain Y(k). From

Y(k), form

Z(k) = 2Y(k) for k = 1 to T/2 - 1

Y(k) for k = 0

0 for k = T/2to T

and then inverse FFT Z(k) to find z(t).

Unless Y(k) is interpolated, this can cause problems.

Makes sure the DC term iscorrect.

F E ti ti T h i


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Associated Problems: Analytic Signal Generation [continued]

If you know something about the signal (e.g. frequency

range of interest), then use of a band-pass Hilberttransforming filter is a good option.

See the paper by Andrew Reilly, Gordon Fraser &Boualem Boashash, Analytic Signal Generation :Tips & Traps IEEE Trans. on ASSP, vol 42(11),pp3241-3245

They suggest designing a real-coefficient low-pass filter

with appropriate bandwidth using a good FIR filteralgorithm (e.g. Remez). The designed filter is thenmodulated with a complex exponential of frequencyfs/4.

F E ti ti T h i


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Kays Estimator and Related Estimators

If an analytic signal, z(t), is obtained, then thesimple relation:

arg( z(t+1)z*(t) )

can be used to find an estimate of thefrequency at time t.

See this by writing:

z(t+1)z*(t) = exp(i (w(t+1) +f) ) exp(-i (wt +f) )= exp(iw)

F E ti ti T h i


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Kays Estimator and Related Estimators [continued]

What Kay did was to form an estimator

w= arg( w(t) z(t+1)z*(t) )

where the weights, w(t), are chosen tominimize the mean square error.

Kay found that, for very small noise

w(t) = 6t(T-t) / (T(T2-1))

which is a parabolic window.

ST-2

t=0

^

Freq enc Estimation Techniq es


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Kays Estimator and Related Estimators [continued]

If the SNR is known, thenits possible to choose

an optimal set ofweights.

For infinite noise, therectangular window isbest this is the Lank-Reed-Pollon estimator.

The figure shows how theweights vary with SNR.



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Associated Problems: Cramer-Rao Lower Bound

The lower bound on the variance ofunbiased estimators of the frequency a

single tone in noise is

var(w) >= 12s2 / (T(T2-1)A2)^



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Associated Problems: Cramer-Rao Lower Bound [continued]

The CRLB for the multi-harmonic case is:

var(w) >= 12s2 / (T(T2-1) m2Am2)

So the effective signal energy in this caseis influenced by the square of theharmonic order.

Sp

m=1^



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Associated Problems: Threshold Performance

Key idea: Theperformance degradeswhen peaks in thenoise spectrum exceedthe peak of the

frequency component.

Dotted lines in the

figure show theprobability of thisoccurring.



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Associated Problems: Threshold Performance [continued]

For the multi-harmoniccase, two thresholdmechanisms occur: thenoise outlier case andrational harmonic

locking.

This means that,sometimes, , 1/3,2/3, 2 or 3 times thetrue frequency isestimated.



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Talk Summary



Some algorithms


o Quinn-Fernandes

Associated problems





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Thanks!

Thanks to Lori Ann, Al and Rick forhosting and/or organizing this get-

together.



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July 30th 2004 comp dsp conference 36


Good-bye!

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Compdsp Kootsookos Frequency-Estimation