part 3part 3. spectrum . spectrum estimationestimation 3.1...
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ENEE630 ENEE630 PartPart--33
Part 3Part 3. Spectrum . Spectrum EstimationEstimation3.1 3.1 Classic Methods for Spectrum EstimationClassic Methods for Spectrum Estimation
Electrical & Computer EngineeringElectrical & Computer Engineering
University of Maryland, College Park
Acknowledgment: ENEE630 slides were based on class notes developed by Profs. K.J. Ray Liu and Min Wu. The slides were made by Prof. Min Wu,
ith d t f M W i H Ch C t t i @ d d
UMD ENEE630 Advanced Signal Processing (ver.1111)
with updates from Mr. Wei-Hong Chuang. Contact: [email protected]
LogisticsLogistics
Last Lecture: lattice predictorcorrelation properties of error processes©
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– correlation properties of error processes– joint process estimator in lattice– inverse lattice filter structureat
ed b
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inverse lattice filter structure
Today: S t ti ti b k d d l i l th d63
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– Spectrum estimation: background and classical methods
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Summary of Related Readings on PartSummary of Related Readings on Part--IIII
2.1 Stochastic Processes and modelingHaykin (4th Ed) 1.1-1.8, 1.12-1.14 Hayes 3.3 – 3.7 (3.5); 4.7
2.2 Wiener filteringHaykin (4th Ed) Chapter 2Hayes 7.1, 7.2, 7.3.1
2 3 2 4 Li di ti d L i D bi i2.3-2.4 Linear prediction and Levinson-Durbin recursionHaykin (4th Ed) 3.1 – 3.3Hayes 7.2.2; 5.1; 5.2.1 – 5.2.2, 5.2.4– 5.2.5
2.5 Lattice predictorHaykin (4th Ed) 3.8 – 3.10
UMD ENEE630 Advanced Signal Processing (F'10) Parametric spectral estimation [3]
Hayes 6.2; 7.2.4; 6.4.1
Summary of Related Readings on PartSummary of Related Readings on Part--IIIIII
Overview Haykins 1.16, 1.10
3 1 Non-parametric method3.1 Non-parametric methodHayes 8.1; 8.2 (8.2.3, 8.2.5); 8.3
3 2 P t i th d3.2 Parametric methodHayes 8.5, 4.7; 8.4
3.3 Frequency estimationHayes 8.6
Review – On DSP and Linear algebra: Hayes 2.2, 2.3
UMD ENEE630 Advanced Signal Processing (F'10) Parametric spectral estimation [4]
– On probability and parameter estimation: Hayes 3.1 – 3.2
Spectrum Estimation: BackgroundSpectrum Estimation: Background
Spectral estimation: determine the power distribution in frequency of a random process©
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frequency of a random process– E.g “Does most of the power of a signal reside at low or high
frequencies?” “Are there resonances in the spectrum?”
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Applications:– Needs of spectral knowledge in spectrum domain non-causal
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Wiener filtering, signal detection and tracking, beamforming, etc.– Wide use in diverse fields: radar, sonar, speech, biomedicine,
geophysics, economics, …
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g p y , ,
Estimating p.s.d. of a w.s.s. process is equivalent to estimate autocorrelation at all lags
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Spectral Estimation: ChallengesSpectral Estimation: Challenges
When a limited amount of observation data are available– Can’t get r(k) for all k and/or may have inaccurate estimate of r(k)©
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Can t get r(k) for all k and/or may have inaccurate estimate of r(k)– Scenario-1: transient measurement (earthquake, volcano, …) – Scenario-2: constrained to short period to ensure (approx.)
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stationarity in speech processing
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Observed data may have been corrupted by noise
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Spectral Estimation: Major ApproachesSpectral Estimation: Major Approaches
Nonparametric methods– No assumptions on the underlying model for the data
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p y g– Periodogram and its variations (averaging, smoothing, …)– Minimum variance method
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Parametric methods– ARMA, AR, MA models
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– Maximum entropy method
Frequency estimation (noise subspace methods)For harmonic processes that consist of a sum of sinusoids orC
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– For harmonic processes that consist of a sum of sinusoids or complex-exponentials in noise
High-order statistics
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Example of Speech SpectrogramExample of Speech Spectrogramu
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Figure 3 of SPM May’98 Speech Survey
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0.1 0.3 0.5
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Section 3.1 Classical Nonparametric MethodsSection 3.1 Classical Nonparametric Methods©
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3) Recall: given a w.s.s. process {x[n]} with
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The power spectral density (p.s.d.) is defined as
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As we can take DTFT on a specific realization of a random process,What is the relation between the DTFT of a specific signal and the
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What is the relation between the DTFT of a specific signal and the p.s.d. of the random process?
Ensemble Average of Squared Fourier MagnitudeEnsemble Average of Squared Fourier Magnitude
p.s.d. can be related to the ensemble average of the squared Fourier magnitude |X()|2
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Mn Mm
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Note: for each frequency f, PM(f) is a random variable
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Ensemble Average of PEnsemble Average of PMM(f)(f)©
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Now, what if M goes to infinity?
P.S.D. and Ensemble Fourier MagnitudeP.S.D. and Ensemble Fourier Magnitude©
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Thus
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3.1.1 Periodogram Spectral Estimator3.1.1 Periodogram Spectral Estimator©
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3) (1) This estimator is based on (**)
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Given an observed data set {x[0], x[1], …, x[N-1]},the periodogram is defined as
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An Equivalent Expression of PeriodogramAn Equivalent Expression of Periodogram20
04) The periodogram estimator can be given in terms of )(kr
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poorer since they involve fewer terms of lag products in the averaging operation
Exercise: to show this from the periodogram definition in last page
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UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [21]
Exercise: to show this from the periodogram definition in last page
(2) Filter Bank Interpretation of Periodogram(2) Filter Bank Interpretation of Periodogram©
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For a particular frequency of f0:21
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complex exponential
Frequency Response of h[n]Frequency Response of h[n]©
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3) )()1(exp)(i)(sin)( 0
0 ffNjffNffNfH
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– Center frequency is f0
– 3dB bandwidth 1/N
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Periodogram: Filter Bank PerspectivePeriodogram: Filter Bank Perspective
Can view the periodogram as an estimator of power spetrum that has a built-in filterbank©
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p– The filter bank ~ a set of bandpass filters– The estimated p.s.d. for each frequency f0 is the power of one
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p q y 0 poutput sample of the bandpass filter centering at f0
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UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [27]
E.g. White Gaussian ProcessE.g. White Gaussian Process[Lim/Oppenheim Fig.2.4] Periodogram of zero-mean white Gaussian noise using N-point data record: N=128, 256, 512, 1024
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The random fluctuation (measured by variance) of the i d d t d ith i i N
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periodogram does not decrease with increasing N periodogram is not a consistent estimator
(3) How Good is Periodogram for Spectral Estimation?(3) How Good is Periodogram for Spectral Estimation?00
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?)( p.s.d. will, If PER fPPN
Estimation: Tradeoff between bias and varianced by
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Performance of Periodogram: SummaryPerformance of Periodogram: Summary
The periodogram for white Gaussian process is an unbiased estimator but not consistent
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– The variance does not decrease with increasing data length– Its standard deviation is as large as the mean (equal to the quantity
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to be estimated)
Reasons for the poor estimation performance– Given N real data points, the # of unknown parameters {P(f0), … 30
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Similar conclusions can be drawn for processes with arbitrary p.s.d. and arbitrary frequencies– Asymptotically unbiased (as N goes to infinity) but inconsistent
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3.1.2 Averaged Periodogram3.1.2 Averaged Periodogram
As one solution to the variance problem of periodogram– Average K periodograms computed from K sets of data records
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And the K sets of data records are
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UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [32]
}],[{ 1K
Performance of Averaged PeriodogramPerformance of Averaged Periodogram
– If K sets of data records are uncorrelated with each other,we have: ( fi = i/L )
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KK for 0Varand,Var i.e., KK for 0Var and ,Var i.e.,
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UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [33]
,,i.e., consistent estimate
Practical Averaged PeriodogramPractical Averaged Periodogram
Usually we partition an available data sequence of length Ninto K non overlapping blocks each block has length L (i e N=KL)©
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– into K non-overlapping blocks, each block has length L (i.e. N=KL)
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Periodogram averaging is also known as the Bartlett’s method
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Averaged Periodogram for Fixed Data SizeAveraged Periodogram for Fixed Data Size
Given a data record of fixed size N, will the result be better if we segment the data into more and more subrecords?
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We examine for a real-valued stationary process:
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UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [36]
n
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Mean of Averaged PeriodogramMean of Averaged Periodogram©
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Mean of Averaged Periodogram (cont’d)Mean of Averaged Periodogram (cont’d)©
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3) fkrkwfPE )}(][{DTFT)](ˆ[ PER AV 1
multiplication in time
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dPfW )()(21
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convolution in frequency
Biased estimate (both averaged and regular periodogram)– The convolution with the window function w[k] lead to the mean of30
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– The convolution with the window function w[k] lead to the mean of the averaged periodogram being smeared from the true p.s.d
Asymptotic unbiased as L
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– To avoid the smearing, the window length L must be large enough so that the narrowest peak in P(f) can be resolved
This gives a tradeoff between bias and variance
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UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [39]
gSmall K => better resolution (smaller smearing/bias) but larger variance
NonNon--parametric Spectrum Estimation: Recapparametric Spectrum Estimation: Recap
Periodogram – Motivated by relation between p.s.d. and squared magnitude of DTFT
of a finite-size data record– Variance: won’t vanish as data length N goes infinity ~ “inconsistent”– Mean: asymptotically unbiased w r t data length N in generalMean: asymptotically unbiased w.r.t. data length N in general
equivalent to apply triangular window to autocorrelation function(windowing in time gives smearing/smoothing in freq domain)
unbiased for white Gaussian unbiased for white Gaussian
Averaged periodogram– Reduce variance by averaging K sets of data record of length L eachReduce variance by averaging K sets of data record of length L each– Small L increases smearing/smoothing in p.s.d. estimate thus higher
bias equiv. to triangular windowing
UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [41]
Windowed periodogram: generalize to other symmetric windows
Case Study on NonCase Study on Non--parametric Methodsparametric Methods Test case: a process consists of narrowband components
(sinusoids) and a broadband component (AR)– x[n] = 2 cos(1 n) + 2 cos(2 n) + 2 cos(3 n) + z[n]
where z[n] = a1 z[n-1] + v[n], a1 = 0.85, 2 = 0.1 /2 = 0 05 /2 = 0 40 /2 = 0 421/2 = 0.05, 2/2 = 0.40, 3/2 = 0.42
– N=32 data points are available periodogram resolution f = 1/32
Examine typical characteristics of various non-parametric pspectral estimators
(Fig.2.17 from Lim/Oppenheim book)
UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [42]
UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [43]
3.1.3 Periodogram with Windowing3.1.3 Periodogram with Windowing
Review and Motivation
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The periodogram estimator can be given in terms of )(k
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krkrknxnxN
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)()(][][)(0nN 0for k
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Solution: weigh the higher lags less
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UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [44]
– Trade variance with bias
WindowingWindowing Use a window function to weigh the higher lags less
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Effect: periodogram smoothing – Windowing in time Convolution/filtering the periodogram
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g g p g– Also known as the Blackman-Tukey method
Common Lag WindowsCommon Lag Windows Much of the art in non-parametric spectral estimation is in
choosing an appropriate window (both in type and length)
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Table 2.1 common lag window (from Lim-Oppenheim book)
Discussion: Estimate r(k) via Time AverageDiscussion: Estimate r(k) via Time Average Normalizing the sum of (N-k) pairs
by a factor of 1/N ? v.s. by a factor of 1/(N-k) ?Biased (low variance) Unbiased (may not non-neg. definite)
• Hints on showing the non-negative definiteness: using )(kr
definiteness: using to construct correlation matrix
)(1 kr
UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [47]
HW#8)(For 2 kr
3.1.43.1.4 Minimum Variance Spectral Estimation Minimum Variance Spectral Estimation (MVSE)(MVSE)
Recall: filter bank perspective of periodogram– The periodogram can be viewed as estimating the p s d byThe periodogram can be viewed as estimating the p.s.d. by
forming a bank of narrowband filters with sinc-like response– The high sidelobe can lead to “leakage” problem:
large output power due to p.s.d outside the band of interest
MVSE designs filters to minimize the leakage from out-of-MVSE designs filters to minimize the leakage from out ofband spectral components– Thus the shape of filter is dependent on the frequency of interest
d d t d tiand data adaptive(unlike the identical filter shape for periodogram)
UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [49]
– MVSE is also referred to as the Capon spectral estimator
Main Steps of MVSE MethodMain Steps of MVSE Method
Design a bank of bandpass filters Hi(f) with center frequency fi so thati
– Each filter rejects the maximum amount of out-of-band power– And passes the component at frequency fi without distortion
Filter the input process { x[n] } with each filter in the filter bank and estimate the power of each output processbank and estimate the power of each output process
Set the power spectrum estimate at frequency fi to be the power estimated above divided by the filter bandwidth
UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [50]
Formulation of MVSEFormulation of MVSE
The MVSE designs a filter H(f) for each f f i t t ffrequency of interest f0
minimize the output power
dffPfH )()( 221
1
minimize the output power
dffPfH )()(21
subject to 1)( 0 fH
(i.e., to pass the components at f0 w/o distortion)
UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [51]
Deriving MVSE SolutionsDeriving MVSE Solutions
UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [53]
Solution of MVSE (cont’d)Solution of MVSE (cont’d)
The optimal filter: eRhT 1
The optimal filter:
f
eReh
TH 1
TTHTH 1It follows that eRRhhRh TTHTH 1
ehH 1 eRe
ehTH 1
UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [59]
MVSE: SummaryMVSE: Summary
If choosing the bandpass filters to be FIR of length p, its 3dB-b.w. is approximately 1/pg p, pp y p
Thus the MVSE ismatrixncorrelatio
is ˆ ppR
eRe
pfPTH 1MV
ˆ)(
matrix ncorrelatio
)2exp(
1fj
e
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eRe
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e
MVSE is a data adaptive estimator and provides improved resolution over periodogram– Also referred to as “High-Resolution Spectral Estimator”
UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [60]
Also referred to as High Resolution Spectral Estimator– Does not assume a particular underlying model for the data
Recall: Case Recall: Case Study on NonStudy on Non--parametric parametric Methods Methods Test case: a process consists of narrowband components
(sinusoids) and a broadband component (AR)– x[n] = 2 cos(1 n) + 2 cos(2 n) + 2 cos(3 n) + z[n]
where z[n] = a1 z[n-1] + v[n], a1 = 0.85, 2 = 0.1 /2 = 0 05 /2 = 0 40 /2 = 0 421/2 = 0.05, 2/2 = 0.40, 3/2 = 0.42
– N=32 data points are available periodogram resolution f = 1/32
Examine typical characteristics of various non-parametric pspectral estimators
(Fig.2.17 from Lim/Oppenheim book)
UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [61]
UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [62]
Deriving MVSE SolutionsDeriving MVSE Solutions
UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [53]
Output Power From H(f) filterOutput Power From H(f) filter
From the filter bank perspective of periodogram:0
)1(
2][)(Nn
fnjenhfH
Thus
dffPelhekh fljfkj )(][][0
221
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1)1(
)1( )1( 2
1 )(][][Nk Nl
dfefPlhkh
0 0
)(][][ klrlhkhEquiv. to filter r(k) with { h(k) h*(-k) } and evaluate at
)1( )1(
)(][][Nk Nl
klrlhkh
UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [54]
and evaluate at output time k=0
MatrixMatrix--Vector Form of MVSE FormulationVector Form of MVSE Formulation
Define
The constraint can be written in vector form as 1ehH
)( 0fH
Thus the problem becomes
hRh THmin subject to 1ehH
hj
UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [56]
Solution of MVSESolution of MVSE )1(2Re ehhRhJ HTHdef
Use Lagrange multiplier approach for solving the constrained optimization problem
– Define real-valued objective function s.t. the stationary condition can be derived in a simple and elegant way based on the theorem for complex derivative/gradient operatorsp g p
)1()1(min*
,ehehhRhJ HHTH
h
eheRh HT 1
11 and
)1()1( * heehhRh HHTH
00either * ehRJ T
eReT
TH
1
11
00or
00either
**
*
eRhJ
ehRJTTH
h
h
eRe
eRhTH
T
1
1
UMD ENEE630 Advanced Signal Processing (ver.1111) Nonparametric spectral estimation [58]
00 ehRehR THT