the inﬂuence of different past-based subspace trackers on ......160 of complete decompositions....

Adv. Radio Sci., 10, 153–158, 2012www.adv-radio-sci.net/10/153/2012/doi:10.5194/ars-10-153-2012© Author(s) 2012. CC Attribution 3.0 License.

Advances inRadio Science

The influence of different PAST-based subspace trackerson DaPT parameter estimation

M. Lechtenberg and J. Gotze

Information Processing Lab, TU Dortmund University, Otto-Hahn-Str. 4, 44227 Dortmund, Germany

Correspondence to:M. Lechtenberg ([email protected])

Abstract. In the context of parameter estimation, subspace-based methods like ESPRIT have become common. They re-quire a subspace separation e.g. based on eigenvalue/-vectordecomposition. In time-varying environments, this can bedone by subspace trackers. One class of these is based on thePAST algorithm. Our non-linear parameter estimation algo-rithm DaPT builds on-top of the ESPRIT algorithm. Evalu-ation of the different variants of the PAST algorithm showswhich variant of the PAST algorithm is worthwhile in thecontext of frequency estimation.

1 Introduction

Parameter estimation is a common task in signal processing.It is used in DoA-applications (direction of arrival) like radarand sonar or in frequency estimation applications like radiotransmission or mass spectrometry – just to name a few. Weare researching parameter estimation in the context of powerquality applications, i.e. finding the signal components nextto the fundamental frequency to extrapolate the transmissionsystem’s state and fault-conditions.

Next to transformation-based parameter estimation viaFourier methods like the DFT (discrete Fourier transfor-mation) and STFT (short-time Fourier transformation) orwavelets, there are parameter estimators based on subspaceanalysis. Estimators based on the separation of subspaceshave advantages that justify the higher computational com-plexity: They have a significantly higher resolution notbounded by the sampling frequency and the window lengthand do not suffer fromspectral leakage.

Common subspace-based parameter estimators are MU-SIC (multiple signal classification, Schmidt, 1986) and ES-PRIT (estimation parameters via rotational invariance tech-niques, Paulraj et al., 1985). MUSIC identifies the fre-quency components excluded from the noise subspace, ES-PRIT identifies the components from the signal subspace.

For both algorithms, noise and signal subspace have to beseparated. A wide spread way to do this is to perform a de-composition into eigenvectors and separate those in noise andsignal. Apart from the expensive eigenvalue decomposition,in a time-varying environment, this can also be done by effi-cient subspace trackers.

A subspace tracker can be circumscribed EV (eigenvector)updating, but this is not exactly true. The vectors generatedby subspace trackers (in general) form one possible basis ofthe signal’s space. This basis does not have to be the EV ofthe correlation matrix, nor does it have to be orthogonal. Theproperties of the basis depend on the subspace tracker used.YAST (yet another subspace tracker, Badeau et al., 2008)and PROTEUS (plane rotation-based EVD-updating scheme,Champagne and Liu, 1998) generate an orthonormal basis,whereas PAST (projection approximation subspace tracking,Yang, 1995b) in its original form provides a basis that onlyapproximates orthonormality over time.

Depending on the application, an orthonormal basis is notneeded. To verify that our application does not require anorthonormal basis, we compared the influence of differentsubspace trackers all based on the mentioned PAST algo-rithm. Variants of the original PAST algorithm introduceforced orthonormality (OPAST,orthonormal PAST, Abed-Meraim et al., 2000) and additional steps for numericalstability (OPAST-PGS,PAST with pairwise Gram-Schmidt-orthogonalization, Bartelmaos et al., 2005). We also in-cluded an unrolled version of PAST reducing computationalcomplexity (PASTd,PAST with deflation, Yang, 1995a) and– as reference – pure eigenvalue decomposition (EVD).

Section two describes the environment in whose contextthe subspace trackers are compared. Section three presentsthe different expansion stages of the subspace tracker. Thesimulation environment is depicted section four and the sim-ulation results are presented in five.

Published by Copernicus Publications on behalf of the URSI Landesausschuss in der Bundesrepublik Deutschland e.V.

154 M. Lechtenberg and J. Gotze: Influence of PAST variants on DaPT

2 Application environment

Modern energy transmission systems can be characterized bytwo main features. On the one hand, they were designed andconstructed for a more static system state with constant 75centralized generation – on the other hand, nowadays, theyare stressed by decentralizedgreen generation (e.g. solar,wind, biomass), economic interests (e.g. stock exchange) andcross-border transmission. Decades ago, transmission man-agement was a question of several minutes and could easily80 be done by humans tweaking a few parameters (like theexcitation of a power plant’s machine). Today’s transmis-sion grid management depends on a lot more factors and issometimes a question of seconds. To overcome these newchallenges, it is desirable to have a system of measurementunits being 85 able to characterize the system state withoutthe need of human interpretation.

The signal model describing a power transmission is com-posed ofsuperposed sinusoids:

x(n) =

p∑i=1

cie2πj ·fi

nfs +wawgn(n) (1)

x(n) =

p∑i=1

cie2πj ·fi

nfs +wawgn(n) (2)

x(n) = A(n,f )c(n)+wawgn(n) (3)

(scalar model (1) with rankp, complex amplitudeci = aiejϕi

and noisewawgn(n); vector model (2) with window lengthNand vectorn = [n...n+N −1]; matrix-vector model (3) withN × p matrix A = [a(f1)...a(fp)], where vectorsa(f ) =[e

2πj ·f nfs ...e

2πj ·f n+N−1fs

]T

). The dominating sinusoid will

naturally be the fundamental frequency (50 Hz/60 Hz, de-pending on the region). Next to this sinusoid, there may beinterim sinusoids. These originate in oscillations from ca-pacitances and inductances characterizing the transmissionlines and routing devices triggered by changes (intended orby fault) of the net configuration. They may also be the resultof not correctly synchronized/well-controlled areas of the net(interarea modes).

2.1 Database assisted parameter estimation

We have designed a database-assisted parameter estimationmethod (DaPT,Lechtenberg and Gotze, 2011). This algo-rithm rates incoming parameter estimations over time andprovides a noise-reduced parameter estimation with reliablerank estimation.

This database has entries for each signal component con-taining frequencyfi and (complex) amplitudeci . Additionalfields hold

1. Frequency drift: this field tracks the change of thefrequency estimations against the database field forthreshold-based frequency field updates.

2. Phase difference: it tracks the change of the phase gra-dient as second indicator for a misestimated frequency.The phase is assumed to be constant, so a linearlychanging phase estimation (based on the database’s fre-quency) indicates a mismatch.

3. Severity of frequency drift/phase difference: indicatorsfor the prior mentioned threshold-based updating rules.

4. Quality of entry: this is the actual rating. It rates howoften a frequency was part of the last estimation recur-sions. Based on that, components can be marked re-liable (i.e.present), accounted for rank estimation andoutput.

In every recursion, the incoming frequency estimation ismapped to the most suitable database entry. Ideally, bothvalues match neglecting the measurement noise. For entriesthat have been successfully mapped, the quality is increased,for the other decreased. For the drift/difference properties,the estimation is compared to the entry’s features and expo-nentially weighted added to the field. Severity is updated likethe quality mentioned before. After the drift fields have beenupdated, the drift’s/difference’s severity is checked whetherthe frequency component needs to be updated. Dependingon the quality, the relevant (or reliable/present) entries areselected for output and counted for rank estimation.

2.2 Estimating parameters via rotational invariancetechniques

Remembering the signal model in Eq. (3), the signal can alsobe described by subspaces like in

x(n) = A(n,f )c(n)+wawgn(n) = S(n)+σ 2awgnI . (4)

Within that equation,S(n) describes the signal subspace,which can be written as a product of the eigenvectorsW(n)

and the eigenvalues (on a diagonal matrix)D(n):

S(n) = W(n)D(n)W(n)H . (5)

Estimating frequencies (more general:parameters) via ro-tational invariance relates to the eigenvectors of the autocor-relation matrix of the input samples. In terms of subspacesdescribed by eigenvectors, one time-step can be described bya multiplication with a diagonal matrix8 with diagonal en-

triese2πj ·

fifs :

S(n+1) = W(n)D(n)8W(n)H . (6)

In other words, the multiplication with8 is a rotation inthe complex plane or a phase-shift in the time domain. Thisis visualized by Fig. 1.

The key idea of Esprit is to estimate the rotation8. Dueto noise,8 will have off-diagonal elements. So an EVD isperformed on8. The frequencies can be extracted from these

Adv. Radio Sci., 10, 153–158, 2012 www.adv-radio-sci.net/10/153/2012/

M. Lechtenberg and J. Gotze: Influence of PAST variants on DaPT 155Matthias Lechtenberg: Evaluation of PAST variants for DaPT 3

n

n+1

4 8 12 16

−0.3

−0.28

−0.26

−0.24

−0.22

−0.2

−0.18

−0.16Wi([1 ... N-1])

Wi([2 ... N])

Fig. 1. Rotation of exemplary eigenvector from time-step n to n+1

The key idea of Esprit is to estimate the rotation Φ. Due tonoise, Φ will have off-diagonal elements. So an EVD is per-formed on Φ. The frequencies can be extracted from theseeigenvalues by calculating their angle. These frequencies arethe input of DaPT.145

Phase and amplitude estimation is obtained by a simpleLS approach referring the signal model Eq. (3). Since timeindex and frequency are known, the matrix A can be built.Together with the samples the complex amplitude can be es-timated.150

3 Subspace Tracking

The autocorrelation matrix of the input samples is definedas Rxx =E

[xxH

]. The expectation value is a statistical

expression that cannot be calculated for a stream of samples,but it can be estimated e.g. via exponential filtering:

Rxx≈ Rxx(n) =βRxx(n−1)+xxH (7)

(β is called forgetting factor). An ordinary EVD can be ap-plied to this approximated autocorrelation matrix to generatethe eigenvectors for ESPRIT. Depending on the algorithmused, this costs O(N2) to O(N3), which is expensive con-155

sidering that in most cases N � p. Subspace trackers real-ize the extraction of eigenvectors with computational costsin the range of O(p2) by an recursive concept of calculatingprojections and deviations to a previous calculation insteadof complete decompositions.160

This paper will focus on one branch of subspace trackers,i.e. PAST and derivations of PAST. This is in order to be ableto benchmark the influence of features like orthonormalityof the subspace basis on the parameter estimation withouthaving deviations due to different calculation approaches.165

3.1 PAST

The projection approximation subspace tracking concept(Yang, 1995b) is based on a non-constraining cost function:

J (W) =E‖x−WWHx‖2 = tr(Rxx)−tr(WHRxxW

)+ tr(WHRxxW ·WHW

)(8)

(tr(...) is the trace operator). This cost function has some im-portant properties: it is minimized if and only if W describesa basis of the signal subspace; there are no more constraintsregarding W, especially not orthonormality. In consequence170

J(W) is minimized for every basis of the signal subspace,not only the basis consisting of the eigenvectors of the auto-correlation matrix.

To find the minimum of the cost function J , a straightfor-ward recursive gradient-descend method is applied (step sizeµ). The Gradient of J is written

5J =[−2Rxx+RxxWWH+WWHRxx

]W (9)

(here, W is short for W(n− 1)). The resulting updatingequation for a W approaching the minimum is given as fol-lows:

W(n) = W(n−1)−µ ·5J(n−1) (10)

By rearranging this equation with the projection y = WHxand the approximation WHW≈ I, this becomes

W(n) = W(n−1)−µ[x(n)−W(n−1)y(n)]yH(n)(11)

With this knowledge (Eq. (11)) and the approximationconcept of the autocorrelation matrix (Eq. (7)) the cost func-tion can be reformulated as an exponentially weighted LS(least squares) problem

J ′(W(n)) =

n∑i=1

βn−i‖x(i)−W(n)y(i)‖2 (12)

This is also known from adaptive filtering. The minimizationof this cost function is expressed by the following equations

W(n) = Rxy(n)R−1yy (n) (13)

Rxy(n) =βRxy(n−1)+xyH (14)

Ryy(n) =βRyy(n−1)+yyH (15)

The PAST algorithm by Yang is based on the RLS (recur-sive LS) algorithm described by the Eq. (13) and the inver-175

sion of Rxy . Whereas the W in the first definition of J (Eq.(8)) was not constrained to hold orthogonal columns but pro-duced an orthogonal solution, the reformulated J ′ (Eq. (12))only approaches an orthogonal solution while convergencedepends on SNR and β.180

This section shall only give the idea of PAST. An im-plementation can be found in other publications like Yang(1995b) or Abed-Meraim et al. (2000). The computationalcomplexity is 3Np+O

(p2), see Table 1.

Fig. 1. Rotation of exemplary eigenvector from time-stepn to n+1.

eigenvalues by calculating their angle. These frequencies arethe input of DaPT.

Phase and amplitude estimation is obtained by a simple LSapproach referring the signal model Eq. (3). Since time indexand frequency are known, the matrixA can be built. Togetherwith the samples the complex amplitude can be estimated.

3 Subspace tracking

The autocorrelation matrix of the input samples is defined asRxx = E

[xxH

]. The expectation value is a statistical expres-

sion that cannot be calculated for a stream of samples, but itcan be estimated e.g. via exponential filtering:

Rxx ≈ Rxx(n) = βRxx(n−1)+xxH (7)

(β is calledforgetting factor). An ordinary EVD can be ap-plied to this approximated autocorrelation matrix to generatethe eigenvectors for ESPRIT. Depending on the algorithmused, this costsO(N2) to O(N3), which is expensive con-sidering that in most casesN � p. Subspace trackers realizethe extraction of eigenvectors with computational costs in therange ofO(p2) by an recursive concept of calculating pro-jections and deviations to a previous calculation instead ofcomplete decompositions.

This paper will focus on one branch of subspace trackers,i.e. PAST and derivations of PAST. This is in order to be ableto benchmark the influence of features like orthonormalityof the subspace basis on the parameter estimation withouthaving deviations due to different calculation approaches.

3.1 PAST

The projection approximation subspace trackingconcept(Yang, 1995b) is based on a non-constraining cost function:

J (W) = E‖x −WWH x‖2= tr(Rxx)−

tr(WH RxxW

)+ tr

(WH RxxW ·WH W

)(8)

(tr(...) is the trace operator). This cost function has some im-portant properties: it is minimizedif and only if W describesa basis of the signal subspace; there are no more constraintsregardingW, especially not orthonormality. In consequenceJ (W) is minimized for every basis of the signal subspace,not only the basis consisting of the eigenvectors of the auto-correlation matrix.

To find the minimum of the cost functionJ , astraightfor-ward recursive gradient-descend method is applied (step sizeµ). The Gradient ofJ is written

5J =

[−2Rxx +RxxWWH

+WWH Rxx

]W (9)

(here,W is short forW(n−1)). The resulting updating equa-tion for aW approaching the minimum is given as follows:

W(n) = W(n−1)−µ ·5J (n−1) (10)

By rearranging this equation with the projectiony = WH x

and the approximationWH W ≈ I , this becomes

W(n) = W(n−1)−µ[x(n)−W(n−1)y(n)]yH (n) . (11)

With this knowledge (Eq. 11) and the approximation con-cept of the autocorrelation matrix (Eq. 7) the cost functioncan be reformulated as an exponentially weighted LS (leastsquares) problem

J ′(W(n)) =

n∑i=1

βn−i‖x(i)−W(n)y(i)‖2. (12)

This is also known from adaptive filtering. The minimiza-tion of this cost function is expressed by the following equa-tions

W(n) = Rxy(n)R−1yy (n) (13)

Rxy(n) = βRxy(n−1)+xyH (14)

Ryy(n) = βRyy(n−1)+yyH . (15)

The PAST algorithm by Yang is based on the RLS (recur-sive LS) algorithm described by the Eq. (13) and the inversionof Rxy . Whereas theW in the first definition ofJ (Eq. 8) wasnot constrained to hold orthogonal columns but produced anorthogonal solution, the reformulatedJ ′ (Eq. 12) only ap-proaches an orthogonal solution while convergence dependson SNR andβ.

This section shall only give the idea of PAST. An im-plementation can be found in other publications likeYang(1995b) or Abed-Meraim et al.(2000). The computationalcomplexity is 3Np+O(p2), see Table 1.

3.2 OPAST

As mentioned before, the original PAST algorithm does notproduce orthogonal results in every run. The OPAST (or-thogonal PAST) by Abed-Meraim et al.(2000) is based on

www.adv-radio-sci.net/10/153/2012/ Adv. Radio Sci., 10, 153–158, 2012

156 M. Lechtenberg and J. Gotze: Influence of PAST variants on DaPT

Table 1. Comparison of the algorithms computational complexities(N window length andp rank, noteN � p).

EVD O(N2)

PAST 3Np+O(p2)

OPAST 4Np+O(p2)

OPAST-PGS 5Np+O(p2)

PASTd 4Np+O(p2)

the same cost function, but introduces an additional orthog-onalizing step. This step is informally written likeW =

W(WH W)−0.5, with its brackets including the exponentcalledinverse square root.

This orthogonalizing step ensures an orthogonal basis(represented byW) in every time-step. This additional cal-culation is incorporated in the updating algorithm of PAST.The implementation can be found in the corresponding pub-lication. The computational complexity is 4Np+O(p2), seeTable 1.

3.3 OPAST-PGS

Due toBartelmaos et al.(2005), a problem of OPAST is itssensitivity to numerical rounding errors. These are originatedin the inverse square root concept. The authors propose apairwise Gram-Schmidt orthogonalization(OPAST-PGS) tooppose this problem.

Remembering that the recursion is indexed byn and therank is p, two indices are defined:kn = mod(n,p) andkn+1 = mod(n+1,p). With these indices, thekn+1-vectorof W is made orthogonal to thekn-vector, before it is nor-malized:

wkn+1(n) = wkn+1(n)−wkn(n)wHkn

(n)wkn+1(n) (16)

wkn+1(n) =wkn+1(n)

‖wkn+1(n)‖. (17)

The computational complexity is 5Np+O(p2) (Table 1).

3.4 PASTd

Another derivation of the PAST algorithm was proposed byYang (1995a). This approach focuses on a further reduc-tion of computational costs by unrolling the PAST algorithm.Thisdeflationdescribes the sequential estimation of the prin-cipal components (or: basis-vectors) beginning with the mostdominant. The most dominant is defined by the maximal(pseudo-)eigenvalue corresponding to the principal compo-nent.

The process is the same as for the original PAST, exceptthat it is done for each vector ofW separately. So, PASTis performed for the most dominant vector, then a projection

of the current data onto this vector is removed from the datavector. The current vector is now excluded and the processwill be repeated for the new dominant vector. This methodresults in less matrix computations resulting in a computa-tional complexity of 4Np+O(p). Note that it isO(p) andnotO(p2).

4 Simulation environment

The signal will always embody a 50 Hz-component (sloppyfor fundamental system frequency ideally being 50 Hz or60 Hz, depending on the region), slightly varying over time,and spontaneous, temporally present components with ran-dom parameters.

The simulation is Monte-Carlo based and performs 400iterations, each having 20 sections. The following results areconsistent and bias-free.

4.1 Signal generator

The signal generator is based on a concept of random sec-tions. A default section has a length of 5 s. The actual sec-tions will have a length that is a small, random integer multi-ple of the default length. Each section varies the parameters(frequency, amplitude) of the fundamental component a lit-tle. For each section, a small random number of additionalcomponents is added and/or removed. In consequence, theadditional components will endure a small random number ofsections. If an additional component endures more than onesection, its parameters are also slightly changed randomly.

The noise applied to the generated signal is based on theAWGN (additional white Gaussian noise) model and is lev-eled 20 dB.

4.2 Exemplary estimation

Figure 2a–b shows a typical estimation of such a signal.When viewing the total simulation time, no details of theestimation are observable. To get an insight into these algo-rithms, two cut-outs of typical estimations will be discussed.

Since the database concept is non-linear and relies on itsown statistics, especially the first estimations of a compo-nent have a high influence. Outliers from pure incoming fre-quency estimations can have high impact at first. Figure 3shows this for – in this case – faulty initial estimations ofPAST(d). When the database fields become statistically reli-able, the output values become consistent again.

To visualize the principle of the phase difference based fre-quency updating, Fig. 4 gives a cut-out of a phase estimation.The solid line is not constant but linearly in-/decreasing (withsteps). The gradient of this line is a hint on the deviation be-tween the true frequency and the current estimation. At thepositions where steps occur, this gradient has been used toupdate the estimation.


M. Lechtenberg and J. Gotze: Influence of PAST variants on DaPT 157Matthias Lechtenberg: Evaluation of PAST variants for DaPT 5

50 100 150 time [s]0

1000

2000

frequency [Hz]

50 100 150 time [s]0

150

300

amplitude [V]

Fig. 2. A typical estimation picked from one of the simulations.

1

2

3

Fig. 3. A cut-out of a typical frequency estimation. (1) and (3)show aberrations due to erroneous parameter estimation and howthey converge when estimations become reliable (here from PASTdand PAST); (2) points to the ideal dotted line and the EVD/OPAST-(PGS) estimation.

15 17 19 time [s]1.2

1.6

phase [rad]

Fig. 4. A cut-out of a typical phase estimation. In the solid line,linearly increasing trends can be seen and the steps where DaPTreaches the threshold for phase difference severity and uses the gra-dient to correct the frequency estimation.

The solid line is not constant but linearly in-/decreasing (withsteps). The gradient of this line is a hint on the deviation be-tween the true frequency and the current estimation. At thepositions where steps occur, this gradient has been used toupdate the estimation.270

5 Simulation Results

To summarize the results of the Monte-Carlo simulation,three cases will be presented. For the first, we measured thedeviation of the frequency estimation and the theoretical val-

Fig. 5. Deviation of frequency estimation against true values,summed and averaged.

Fig. 6. Number of frequency components not found by the estima-tion per algorithm, summed and averaged.

ues from the signal generator. We summed them for all con-275

temporaneous components and averaged the sums over thesimulation time and all simulations. Figure 5 presents theresults.

First of all, it emerges that the differences between the dif-ferent algorithms are small. If there are findings at all, then280

EVD has slight disadvantages and (O)PAST has slight advan-tages. EVD might suffer from the position of the exponentialfiltering. In EVD the autocorrelation matrix itself is expo-nentially filtered, which is indirect since not Rxx but the EVare desired. The other algorithms perform that step directly285

on the basis of the subspace.Also, we counted the number of frequency components

that were not found by the estimator. Reasons may be (from


Matthias Lechtenberg: Evaluation of PAST variants for DaPT 5

50 100 150 time [s]0

1000

2000

frequency [Hz]

50 100 150 time [s]0

150

300

amplitude [V]


1

2

3


15 17 19 time [s]1.2

1.6

phase [rad]













Fig. 3. A cut-out of a typical frequency estimation. (1) and (3) showaberrations due to erroneous parameter estimation and how theyconverge when estimations become reliable (here from PASTd andPAST); (2) points to the ideal dotted line and the EVD/OPAST-(PGS) estimation.


50 100 150 time [s]0

1000

2000

frequency [Hz]

50 100 150 time [s]0

150

300

amplitude [V]


1

2

3


15 17 19 time [s]1.2

1.6

phase [rad]














5 Simulation results

To summarize the results of the Monte-Carlo simulation,three cases will be presented. For the first, we measured thedeviation of the frequency estimation and the theoretical val-ues from the signal generator. We summed them for all con-temporaneous components and averaged the sums over thesimulation time and all simulations. Figure 5 presents theresults.

First of all, it emerges that the differences between the dif-ferent algorithms are small. If there are findings at all, thenEVD has slight disadvantages and (O)PAST has slight advan-tages. EVD might suffer from the position of the exponentialfiltering. In EVD the autocorrelation matrix itself is expo-nentially filtered, which is indirect since notRxx but the EV


50 100 150 time [s]0

1000

2000

frequency [Hz]

50 100 150 time [s]0

150

300

amplitude [V]


1

2

3


15 17 19 time [s]1.2

1.6

phase [rad]















50 100 150 time [s]0

1000

2000

frequency [Hz]

50 100 150 time [s]0

150

300

amplitude [V]


1

2

3


15 17 19 time [s]1.2

1.6

phase [rad]














are desired. The other algorithms perform that step directlyon the basis of the subspace.

Also, we counted the number of frequency componentsthat were not found by the estimator. Reasons may be (fromthe signal’s perspecitve) a two weak amplitude or (from thestatistical perspective) too high variance in the estimations.The numbers are per second and include the mandatory non-hits before DaPT regards them as present (resulting in an off-set in the figure). In Fig. 6, the most significant fact is, thatPASTd cannot keep up with the others. Also, EVD leavesa better mark. Both facts might be justified by the handlingof estimations with high variance. A high variance in the es-timations may confuse the sequential processing of PASTd.The joint cost of the serial cost functions might be higherthan the cost of the (matrix-) PAST.

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158 M. Lechtenberg and J. Gotze: Influence of PAST variants on DaPT6 Matthias Lechtenberg: Evaluation of PAST variants for DaPT

0 200 400 amplitude [V]0

2

number

Fig. 7. Summed and averaged not-found components plottedagainst the amplitude: EVD, PAST, OPAST, OPAST-PGS,

PASTd.

the signal’s perspecitve) a two weak amplitude or (from thestatistical perspective) too high variance in the estimations.290

The numbers are per second and include the mandatory non-hits before DaPT regards them as present (resulting in an off-set in the figure). In Fig. 6, the most significant fact is, thatPASTd cannot keep up with the others. Also, EVD leavesa better mark. Both facts might be justified by the handling295

of estimations with high variance. A high variance in the es-timations may confuse the sequential processing of PASTd.The joint cost of the serial cost functions might be higherthan the cost of the (matrix-) PAST.

High variances (indicating less reliable estimations) are300

most likely for low amplitudes (in relation to noise and domi-nant components). The last case surveys the unfound compo-nents against the amplitude, as (relative) amplitude is a goodindicator for less reliable estimations. Figure 7 approves theprevious results. PASTd is especially unsuitable for low am-305

plitudes which may hint high variances in estimations. Allthe other perform equally well with slight advantages forEVD and (O)PAST. OPAST-PGS does not seem to bring ad-vantages.

6 Conclusions310

We have simulated the performance of different subspacetrackers, based on the PAST theory, to evaluate the impact ofthe properties of the subspace basis presented to the ESPRITalgorithm. The estimated components supply the input ofour DaPT non-linear parameter estimation post-processing.315

The subspace’s basis of the signal generated from the (es-timated) autocorrelation matrix of the input samples is notunique. The subspace can be described by various basis’,some of them orthogonal, some even orthonormal and oneof them being the signal eigenvectors of the autocorrelation320

matrix.The simulations have shown, that it is not necessary for the

subsequent steps (ESPRIT, DaPT) to be provided with exact

eigenvectors (EVD). The basis presented to ESPRIT doesnot even have to be orthogonal (OPAST(-PGS)). The clas-325

sic PAST algorithm, which does not guarantee an orthonor-mal basis, performs as well as the variations providing an or-thonormal basis and as EVD providing actual eigenvectors.

A closer look reveals, that the sequential PASTd performsworse due to the lack of a joint cost function. The or-330

thonormal OPAST is marginally better than PAST, becausethe approximation WHW ≈ I is more accurate for reallyorthonormal columns of W. The algorithm with additionalpairwise Gram-Schmidt orthogonalization does not improvethe results.335

In future work, the phase- and amplitude estimation shallbe investigated further, i.e. other algorithms as LS relatingthe signal model. Also, cooperative estimations of regionallydisplaced measurements shall be investigated.

Acknowledgements. This work was supported by DFG (German340

Research Foundation).

References

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Badeau, R., Richard, G., and David, B.: Fast and Stable YASTAlgorithm for Principal and Minor Subspace Tracking, IEEETransactions on Signal Processing, 56, 3437 – 3446, doi:10.1109/TSP.2008.925924, 2008.

Bartelmaos, S., Abed-Meraim, K., and Attallah, S.: Mobile local-350

ization using subspace tracking, in: 2005 Asia-Pacific Confer-ence on Communications, pp. 1009 –1013, doi:10.1109/APCC.2005.1554216, 2005.

Champagne, B. and Liu, Q.-G.: Plane rotation-based EVD updat-ing schemes for efficient subspace tracking, IEEE Transactions355

on Signal Processing, 46, 1886 – 1900, doi:10.1109/78.700961,1998.

Lechtenberg, M. and Gotze, J.: Database Assisted Frequency Es-timation, in: 4th IEEE International Conference on ComputerScience and Information Technology, 2011.360

Paulraj, A., Roy, R., and Kailath, T.: Estimation Of Signal Parame-ters Via Rotational Invariance Techniques, in: IEEE Conferenceon Circuits, Systems and Computers, pp. 83 – 89, 1985.

Schmidt, R.: Multiple emitter location and signal parameter estima-tion, IEEE Transactions on Antennas and Propagation, 34, 276 –365

280, 1986.Yang, B.: An extension of the PASTd algorithm to both rank and

subspace tracking, IEEE Signal Processing Letters, 2, 179 – 182,doi:10.1109/97.410547, 1995a.

Yang, B.: Projection approximation subspace tracking, IEEE Trans-370

actions on Signal Processing, 43, 95 – 107, doi:10.1109/78.365290, 1995b.

Fig. 7. Summed and averaged not-found components plottedagainst the amplitude: EVD, PAST, OPAST, OPAST-PGS,

PASTd.

High variances (indicating less reliable estimations) aremost likely for low amplitudes (in relation to noise and domi-nant components). The last case surveys the unfound compo-nents against the amplitude, as (relative) amplitude is a goodindicator for less reliable estimations. Figure 7 approves theprevious results. PASTd is especially unsuitable for low am-plitudes which may hint high variances in estimations. Allthe other perform equally well with slight advantages forEVD and (O)PAST. OPAST-PGS does not seem to bring ad-vantages.

6 Conclusions

We have simulated the performance of different subspacetrackers, based on the PAST theory, to evaluate the impact ofthe properties of the subspace basis presented to the ESPRITalgorithm. The estimated components supply the input ofour DaPT non-linear parameter estimation post-processing.The subspace’s basis of the signal generated from the (es-timated) autocorrelation matrix of the input samples is notunique. The subspace can be described by various basis’,some of them orthogonal, some even orthonormal and oneof them being the signal eigenvectors of the autocorrelationmatrix.

The simulations have shown, that it is not necessary for thesubsequent steps (ESPRIT, DaPT) to be provided with exacteigenvectors (EVD). The basis presented to ESPRIT doesnot even have to be orthogonal (OPAST(-PGS)). The clas-sic PAST algorithm, which does not guarantee an orthonor-mal basis, performs as well as the variations providing an or-thonormal basis and as EVD providing actual eigenvectors.

A closer look reveals, that the sequential PASTd performsworse due to the lack of a joint cost function. The orthonor-mal OPAST is marginally better than PAST, because the ap-proximationWH W ≈ I is more accurate for really orthonor-mal columns ofW. The algorithm with additional pairwiseGram-Schmidt orthogonalization does not improve the re-sults.

In future work, the phase- and amplitude estimation shallbe investigated further, i.e. other algorithms as LS relatingthe signal model. Also, cooperative estimations of regionallydisplaced measurements shall be investigated.

Acknowledgements.This work was supported by DFG (GermanResearch Foundation).

References

Abed-Meraim, K., Chkeif, A., and Hua, Y.: Fast orthonormal PASTalgorithm, IEEE Signal Proc. Let., 7, 60–62, doi:10.1109/97.823526, 2000.

Badeau, R., Richard, G., and David, B.: Fast and Stable YAST Al-gorithm for Principal and Minor Subspace Tracking, IEEE T.Signal Proces., 56, 3437–3446, doi:10.1109/TSP.2008.925924,2008.

Bartelmaos, S., Abed-Meraim, K., and Attallah, S.: Mobile local-ization using subspace tracking, in: 2005 Asia-Pacific Confer-ence on Communications, 1009–1013, doi:10.1109/APCC.2005.1554216, 2005.

Champagne, B. and Liu, Q.-G.: Plane rotation-based EVD updatingschemes for efficient subspace tracking, IEEE T. Signal Proces.,46, 1886–1900, doi:10.1109/78.700961, 1998.

Lechtenberg, M. and Gotze, J.: Database Assisted Frequency Es-timation, in: 4th IEEE International Conference on ComputerScience and Information Technology, 2011.

Paulraj, A., Roy, R., and Kailath, T.: Estimation Of Signal Parame-ters Via Rotational Invariance Techniques, in: IEEE Conferenceon Circuits, Systems and Computers, 83–89, 1985.

Schmidt, R.: Multiple emitter location and signal parameter estima-tion, IEEE T. Antenn. Propag., 34, 276–280, 1986.

Yang, B.: An extension of the PASTd algorithm to both rank andsubspace tracking, 9, 179–182, doi:10.1109/97.410547, 1995a.

Yang, B.: Projection approximation subspace tracking, IEEE T. Sig-nal Proces., 43, 95–107, doi:10.1109/78.365290, 1995b.


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