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July 30th, 2004 comp.dsp conference 1
Frequency Estimation Techniques
Peter J. Kootsookos
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July 30th, 2004 comp.dsp conference 2
Frequency Estimation Techniques
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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July 30th, 2004 comp.dsp conference 3
Frequency Estimation Techniques
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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July 30th, 2004 comp.dsp conference 4
Frequency Estimation Techniques
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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July 30th, 2004 comp.dsp conference 5
Frequency Estimation Techniques
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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July 30th, 2004 comp.dsp conference 6
Frequency Estimation Techniques
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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July 30th, 2004 comp.dsp conference 7
Frequency Estimation Techniques
What other problems are there? [continued]
Amplitude-varying example: conditionmonitoring in rotating machinery.
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July 30th, 2004 comp.dsp conference 8
Frequency Estimation Techniques
What other problems are there? [continued]
Frequency tracking example: SONAR
Thanks to BarryQuinn & TedHannan for theplot from their
book TheEstimation &Tracking ofFrequency.
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July 30th, 2004 comp.dsp conference 9
Frequency Estimation Techniques
What other problems are there? [continued]
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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July 30th, 2004 comp.dsp conference 10
Frequency Estimation Techniques
What other problems are there? [continued]
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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July 30th, 2004 comp.dsp conference 11
Frequency Estimation Techniques
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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July 30th, 2004 comp.dsp conference 12
Frequency Estimation Techniques
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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July 30th, 2004 comp.dsp conference 13
Frequency Estimation Techniques
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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July 30th, 2004 comp.dsp conference 14
Frequency Estimation Techniques
The Maximum Likelihood Approach [continued]
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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July 30th, 2004 comp.dsp conference 15
Frequency Estimation Techniques
The Maximum Likelihood Approach [continued]
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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July 30th, 2004 comp.dsp conference 16
Frequency Estimation Techniques
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 problems
o Analytic signal generation Kay / Lank-Reed-Pollon estimators
o Performance bounds: Cramr-Rao Lower Bound
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July 30th, 2004 comp.dsp conference 17
Frequency Estimation Techniques
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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July 30th, 2004 comp.dsp conference 18
Frequency Estimation Techniques
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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July 30th, 2004 comp.dsp conference 19
Frequency Estimation Techniques
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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July 30th, 2004 comp.dsp conference 20
Frequency Estimation Techniques
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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July 30th, 2004 comp.dsp conference 21
Frequency Estimation Techniques
Quinn-Fernandes [continued]
The algorithm can be interpreted as finding the
maximum of a smoothed periodogram.
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July 30th, 2004 comp.dsp conference 22
Frequency Estimation Techniques
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 problems
o Analytic signal generation Kay / Lank-Reed-Pollon estimators
o Performance bounds: Cramr-Rao Lower Bound
F E i i T h i
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July 30th, 2004 comp.dsp conference 23
Frequency Estimation Techniques
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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July 30th, 2004 comp.dsp conference 24
Frequency Estimation Techniques
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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July 30th, 2004 comp.dsp conference 25
Frequency Estimation Techniques
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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July 30th, 2004 comp.dsp conference 26
Frequency Estimation Techniques
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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July 30th, 2004 comp.dsp conference 27
Frequency Estimation Techniques
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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July 30th, 2004 comp.dsp conference 28
Frequency Estimation Techniques
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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July 30th, 2004 comp.dsp conference 29
Frequency Estimation Techniques
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.
Frequency Estimation Techniques
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July 30th, 2004 comp.dsp conference 30
Frequency Estimation Techniques
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)^
Frequency Estimation Techniques
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July 30th, 2004 comp.dsp conference 31
Frequency Estimation Techniques
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^
Frequency Estimation Techniques
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July 30th, 2004 comp.dsp conference 32
Frequency Estimation Techniques
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.
Frequency Estimation Techniques
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July 30th, 2004 comp.dsp conference 33
Frequency Estimation Techniques
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.
Frequency Estimation Techniques
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July 30th, 2004 comp.dsp conference 34
Frequency Estimation Techniques
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 problems
o Analytic signal generation Kay / Lank-Reed-Pollon estimators
o Performance bounds: Cramr-Rao Lower Bound
Frequency Estimation Techniques
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July 30th, 2004 comp.dsp conference 35
Frequency Estimation Techniques
Thanks!
Thanks to Lori Ann, Al and Rick forhosting and/or organizing this get-
together.
Frequency Estimation Techniques
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July 30th 2004 comp dsp conference 36
Frequency Estimation Techniques
Good-bye!