statistical phase-shifting step estimation algorithm based on the continuous wavelet transform for...

Statistical phase-shifting step estimation algorithmbased on the continuous wavelet transform for

high-resolution interferometry metrology

Bicheng Chen and Cemal Basaran*Electronic Packaging Laboratory, State University of New York at Buffalo, Buffalo, New York 14260, USA

*Corresponding author: [email protected]

Received 1 November 2010; accepted 13 December 2010;posted 21 December 2010 (Doc. ID 137421); published 31 January 2011

We propose a statistical phase-shifting estimation algorithm for temporal phase-shifting interferometry(PSI) based on the continuous wavelet transform (CWT). The proposed algorithm explores spatial infor-mation redundancy in the intraframe interferogram dataset using the phase recovery property on thepower ridge of the CWT. Despite the errors introduced by the noise of the interferogram, the statisticalpart of the algorithm is utilized to give a sound estimation of the phase-shifting step. It also introducesthe usage of directional statistics as the statistical model, which was validated, so as to offer a betterestimation compared with other statistical models. The algorithm is implemented in computer codes, andthe validations of the algorithmwere performed on numerical simulated signals and actual phase-shiftedmoiré interferograms. The major advantage of the proposed algorithm is that it imposes weaker condi-tions on the presumptions in the temporal PSI, which, under most circumstances, requires uniform andprecalibrated phase-shifting steps. Compared with other existing deterministic estimation algorithms,the proposed algorithm estimates the phase-shifting step statistically. The proposed algorithm allows thetemporal PSI to operate under dynamic loading conditions and arbitrary phase steps and also withoutprecalibration of the phase shifter. The proposedmethod can serve as a benchmarkmethod for comparingthe accuracy of the different phase-step estimation methods. © 2011 Optical Society of AmericaOCIS codes: 050.5080, 000.5490, 100.5070, 100.7410, 120.2880, 120.3180.

1. Introduction

Phase-shifting interferometry (PSI) is an importanttechnique in high-precision optical metrology. Tem-poral PSI was first introduced by Bruning et al. [1]to overcome the λ=2 sensitivity limits of the fringecounting methods in the interferometry for the appli-cation in the surface quality measurement of the op-tical components. In the Bruning et al. paper, errorsources are enumerated, including electrical noisein the detectors, mechanical drifts, atmospheric tur-bulence and the noise in the laser source. It is alsoemphasized that the error can be significant whenthe sensitivity reaches to λ=100. The arctangent func-tion is used in Bruning’s paper to reconstruct the

phase information. The least-squares estimation(LSE) method was first introduced by Morgan [2]to minimize the intensity errors nonlinearly betweenthe imaged intensity map and the estimated one onthe parameter of the measured phase. The commonlyused version of LSE was established by Greiven-kamp [3]. Greivenkamp uses the same interferenceequation but minimizes the intensity errors accord-ing to three parameters (background, cosine, andsine function of the phase shifter). Greivenkamp’sLSE is linear and removes several restrictions whendesigning the phase steps.

Lai and Yatagai apply additional interference set-ups to detect the shifted phase, while using Fouriertransform methods [4]. Larkin added preprocessingand postprocessing to improve the robustness ofLai and Yatagai’s Fourier-transform-based method[5]. Farrell and Player once again pointed out that

0003-6935/11/040586-08$15.00/0© 2011 Optical Society of America

586 APPLIED OPTICS / Vol. 50, No. 4 / 1 February 2011

the error in the phase steps can cause a distortion ofthe results, and they suggested an interpixel ellipsefitting estimation method [6]. The phase step can beunknown and unequal. Wei et al. gave a least-squares algorithm based on the interpixel relation-ship by means of a simple intensity transformbetween the sampled interferograms [7].

Iterative LSE was proposed by Kong and Kim, andconsiders both the phase and phase shift as param-eters [8]. The iterative method and its applicationwas also documented by Rivera et al. [9]. Based onthe initially induced phase step (e.g., from the line-arity table of piezoelectric actuator), Greivenkamp’sLSE is first used to estimate the wavefront phases,and a modified Greivenkamp linear LSE is used toestimate the phase step afterward. The iterative pro-cess is therefore built on these two LSE models to ob-tain the final wavefront phases upon the convergencecondition. The idea is generalized and extended byWang and Han [10]. Another phase-shift extractionmethod, based on the assumed object wave, is pro-posed by Cai et al. [11,12]. The Cai et al.method usesthe mean of the differences between two interfero-grams to calculate the phase difference. An iterativeprocess is available using the estimated phase stepsto reconstruct the phase distribution iteratively. Xuet al. proposed a noniterative version based on thesame principle [13].

The methods reviewed above can be divided intotwo categories: temporal methods and temporal-spatial methods. The temporal methods rely on anestablished phase step to give pixelwise estimationson the phase distribution. The temporal-spatialmethods use the spatial data redundancy to first es-timate the phase step and then to reconstruct thephase distribution from the temporal samples [5].The proposed algorithm in this paper belongs to thetemporal-spatial method. The flow chart of the pro-posed algorithm is shown in Fig. 1. Compared withthe existing algorithms, the proposed algorithm uti-lizes the continuous wavelet transform (CWT) to es-timate the wavefront phases first, which is operatedusing spatial data correlation within an interfero-gram frame. Later, the phase steps are estimatedstatistically upon the phase difference distributionsbetween temporal sampled interferogram frames.The parameter estimation part of the algorithm wasdemonstrated here using the maximum likelihoodestimation (MLE). However, any statistical param-eter estimation method or nonparametric estimationcan be used instead. The proposed method allows anarbitrary phase step but does not involve any itera-tive process, which differs from most of the existingtemporal-spatial methods. Furthermore, the pro-posed algorithm is a framework for building a robustphase-shifting algorithm, which can allow the PSI tooperate under a dynamic loading condition. Accord-ing to our literature survey, this is the first time thatthe CWT is reported to be used to help in the estima-tion of the phase-shifting steps.

2. Interferometry Phase Least-Squares Estimation

The structure and notation of the interference equa-tion are introduced in this section. They are criticalelements to get an understanding of the proposedalgorithm. This section also lays out some essential

Fig. 1. Flow chart of the proposed phase-shifting step estimationalgorithm.

1 February 2011 / Vol. 50, No. 4 / APPLIED OPTICS 587

concepts for the proposed algorithm, especially onthe formulation and parameterization of the LSEused in the field of phase-shift estimations. In com-mon circumstances, the temporal PSI samples multi-ple intensity values on the same spatial point atdifferent times and with shifted phase steps. Forthe detector using a charge coupled device (CCD),each sampling is an accumulation process of the elec-trical charges that are proportional to the intensity ofthe incident photon flux integrated over the exposuretime. The reconstruction of the temporal phase-shifted interferograms using the least-squares for-mula is widely used [3,14]. This serves in the laststep in the proposed algorithm. The interferenceequation for the temporal PSI can be written as [14]

Ikði; jÞ ¼ aði; jÞ þ bði; jÞ cos½ϕði; jÞ þΔϕk�; ð1Þ

where k is the number of phase shifting k ¼ 0, 1, …,K − 1 (K is the total shifting number) and ði; jÞ are thepixel coordinates of the image collected by the CCD.Ikði; jÞ, aði; jÞ, bði; jÞ denote the intensity value, back-ground intensity, and the fringe visibility.

Equation (1) can be written into a linear system byusing a trigonometric identity:

Ikði; jÞ ¼ aði; jÞ þ bði; jÞ cosðΔϕkÞ cos½ϕði; jÞ�− bði; jÞ sinðΔϕkÞ sin½ϕði; jÞ�: ð2Þ

The LSEs of the wavefront phases are performed oneach individual spatial sampling point ði; jÞ. For aphase step larger than 3, the linear equation systemis overdetermined and the mean square error (MSE)is defined on the difference between the sampled in-tensity values and estimated intensity values, asshown in Eq. (3). The least-squares principle is thenapplied, as shown in Eq. (4):

MSE ¼ Σi;j½Ieði; jÞ − Ikði; jÞ�2; ð3Þ

−2Σi;j

∂Ikði; kÞ∂cn

Ikði; jÞ ¼ 0; n ¼ 1; 2; 3;…; ð4Þ

where

cði; jÞ ¼24 aði; jÞbði; jÞ cosϕði; jÞbði; jÞ sinϕði; jÞ

35; ð5Þ

where Ieði; jÞ is the estimated intensity. The minimi-zation is carried out on the parameters cði; jÞ inEq. (5).

The LSE formulation here follows Greivenkamp’schoice of parameters. As shown in Eq. (5), this kind ofLSE is effective for combining the temporal informa-tion to isolate the phase information ϕði; jÞ from thebackground noises aði; jÞ and bði; jÞ. Therefore, it isused in the proposed algorithm for the interferogramreconstruction (see Fig. 1). However, as shown in

Eq. (4), we assume that the MSE is only affectedby the background variation aði; jÞ, the fringe visibi-lity bði; jÞ, and the phase term ϕði; jÞ. Under this for-mulation, the phase-shift step Δϕk is assumed to beaccurate and should not contribute to the error. Inreal experimental conditions, the piezoelectric actua-tor calibration error [15–17] and the fast driftinginterference patterns under dynamic loading condi-tions can make the static phase-shift step assump-tion invalid.

Because the phase-shift step is assumed accuratein traditional PSI, the inference of the phase-shiftstep is critical to the accuracy of the measurement,which depends on the way of introducing the phaseshifting itself. Phase shifting in the interferometrycan be introduced by the translation and the tiltingof the optical components (i.e., mirror, wedge, andgrating) [18]. For example, in moiré interferometrywith phase shifting [19], a nanoscale resolutionpiezoelectric motion actuator and controller are usedhere to motorize the movement of the beam splittinggrating to introduce phase shifting. Under this con-figuration, the estimation of the phase steps can becalculated from the relation between the shiftedphase and the translation step of the beam splittinggrating [19–21], and it is given as

Δϕ ¼ 4π δΛ ; ð6Þ

where Δϕ is the shifted phase; Λ is the beam split-ting period in nanometer; δ is the translation step ofthe beam splitter grating. For the estimation of thephase step to be unbiased, the phase shifting shouldonly be introduced by the piezoelectric device andthere should be no errors from the miscalibrationof the actuator and its feedback controller, which isdifficult (and sometimes impossible) to satisfy in areal-world testing situation, in which environmentalvibration and thermal noise can be large enough tointroduce a large bias into Eq. (6). The proposed al-gorithm is invented to solve this problem effectivelyby considering the shifted phase as a variable, whichcan be exacted from the measurement datasets.

3. Wavefront Phase Estimation Using the ContinuousWavelet Transform

In the proposed algorithm, the CWT is integratedinto the algorithm for wavefront phase estimation(see Fig. 1). The CWT is implemented in a scanningmanner (i.e., 1D CWT) to demonstrate the workingprinciple of the proposed phase-shifting estimationframework. Nonetheless, 2D CWT can be integratedin the framework to replace the 1D CWT used here.Watkins et al. have shown that there are severalproperties with CWT suitable for extracting phasesfrom the sinusoidal modulated signal, and the li-terature has reported several successful applicationsusing it for processing the interference fringepatterns [20,22–26]. Compared with the Fourier-transform-based phase extraction methods, thewavelet-based methods give higher resolution whenextracting localized phase information [20]. The


definition of CWT is to convolute the signal with ascaling wavelet family:

Wf ða; bÞ ¼Z þ∞

−∞

f ðtÞψ�a;bðtÞdt; ð7Þ

where f ðtÞ is the input signal, ψa;bðtÞ is the waveletfunction, and a and b are the wavelet scaling andtranslational parameters, respectively. Wf ða; bÞ isthe wavelet transformation as a function of a; b. f �is the complex conjugate of function f . The functionψa;b is scaled and translated from a mother functionψ , and the relationship between them is

ψa;bðxÞ ¼1ffiffiffia

p ψ�x − ba

�: ð8Þ

The complex Morlet wavelet is chosen here becauseof its sinusoidal feature weighted by the Gaussiandistribution function. The mother wavelet functioncan be written as

ψðtÞ ¼ 1ffiffiffiffiffiffi2π

p expð−jω0tÞ exp�−t2

2

�; ð9Þ

where ω0 ¼ 5:336 following [27] in order to get thenormalized result. Although the complex Morletwavelet is not admissible in the strict sense, it can beconsidered admissible in a loosely satisfying sense.

Starting with results from the phase shifting [referto the interference equation Eq. (1)], we set the inputsignal as f ðtÞ ¼ cosðϕðtÞÞ. It has been shown that thepoint of maximum magnitude (called the “ridge”) ofthe CWT

��Wf ða; bÞ��max

has the following properties,

that is, the phase at position b can be computed bychoosing the largest response of wavelet banks[25,28]:

argWf ðamax; bÞ ¼ ϕðbÞ; ð10Þ

where amax is the scaling at��Wf ða; bÞ

��max

. Equa-

tion (10) shows that the ridge of the maximum powerof the CWT can retrieve the phase information.

Phase estimation from the CWT suffers the errorsfrom the inappropriate choices of the scaling factor a,the finite length of the signal input, and the spatialdiscontinuities of the input signal. In practice, theimplementation of different ridge detection schema-ta [29,30] can effectively reduce the errors introducedby inappropriate choices of the scaling factor a. It isassumed that the errors happening here follow thestatistical distribution. The statistical analysis inthe next section validates this.

4. Statistical Analysis of the Phase Step Distribution

The phase information itself offers no statisticalproperties, but the phase difference at the same sam-pling location over the entire interferogram shouldfollow the same statistical distribution because of

the deterministic nature of the phase step. This isdescribed in the interference intensity equation[Eq. (1)]. This is also the basic assumption for PSI[11,12]. The phase step is assumed to be constantover the space at the specific sampling moment,but the measured phase step is prone to includeerrors. Hence, the phase step can be treated as astatistical variable, which is expressed in Eq. (11):

Δϕkði; jÞ ¼ δk þ εkði; jÞ; ð11Þ

where Δϕkði; jÞ is the phase difference between twointerferograms calculated by the CWT, δk is the truevalue of the phase-shifting step, and εkði; jÞ is theerror for each estimation calculated from the phasedifferences using CWT.

After aggregating the phase difference results, theprobability distribution of the measured phase-shiftstep can be established. The error term in Eq. (11) isassumed to follow certain statistical distribution. Asshown in the flow chart (see Fig. 1), both parametricestimation and nonparametric estimation can beused to estimate δk. In parametric estimation, aparameterized population distribution is requiredto fit the sample probability distribution. By con-trast, nonparametric estimation makes no assump-tions on the population distribution. In this paper,the parametric estimation, combined with the MLE,is chosen as the statistical estimation method todemonstrate the feasibility and effectiveness of theproposed algorithm. However, any other statisticalestimation can be used in the proposed algorithm,as shown in Fig. 1. The MLE is generally consistent,which means that a larger sample size will generallyhelp increase the accuracy of the estimation [31]. Theprinciple of MLE is to find the estimator vector θ tomaximize the likelihood function Lðθ;ωÞ, which is re-lated to the probability function pðω; θÞ around theobservation vector ω [32–34]. Considering that theobservations are independent to one another (i.e.,pðω; θÞ ¼ Π

npnðωn; θÞ, where n is the number of ob-

servations), the MLE is as shown in Eq. (12) (thelog-likelihood function is used):

Xn

∇θ lnðpnðωn; θÞÞ ¼ 0: ð12Þ

The variance of the MLE estimator can be judged bythe asymptotic covariance matrix, which can bederived from the Fisher information. The Fisherinformation is the variance of the Fisher scoresθ ¼ ∇θ lnðpnðωn; θÞÞ. Under regular conditions, theFisher information matrix can be simplified asshown in Eq. (13) and the asymptotic covariance ma-trix is equal to the inverse of the Fisher informationmatrix [31]:

FðθÞ ¼ Eð½∇θ lnðpðω; θÞÞ�½∇θ lnðpðω; θÞÞ�T jθÞ; ð13Þ

where Eð…jθÞ denotes the expectation over a vari-able with respect to the probability function pðω; θÞ.


For parametric parameter estimation, it is neces-sary to make an assumption on the population distri-bution. There are several distributions that are goodcandidates for the phase step, and they are listed inTable 1. Although all three candidates belong to thefamily of exponential distributions, the von Misesdistribution (or circular normal distribution) consid-ers the data as circular data, which is more closelyrelated to the data structure of the phase information[35,36].

5. Validation Using Simulated and Experimental Data

In this section, the proposed algorithm is validatedusing both simulated and experimental data. Theproposed algorithm is implemented in MATLABcode. Unlike the experimental data, whose truephase-shift step is unknown, the advantage of usingthe numerically simulated data is that they providetrue values to be compared with. In the validationprocess of the numerical data, the simulated inputsignal was a single- frequency sinusoidal signal witha signal length of L ¼ 1008 (which was the samelength as the detector of the lab setup in the experi-mental data below). The data are synthetically signalgenerated with Eq. (14):

s½n� ¼ 12

�1þ cos

�2π · 10 ·

nLþ δk

��; ð14Þ

where s½n� is the synthetic signal; n is the spatial co-ordinate; δk ¼ 0, 2=5π, 4=5π, 6=5π are the phase-

shifting steps; and L is the length of the signal.The plot of the signals is as shown in Fig. 2.

The CWT was performed on the input signals hereat δ1 ¼ 0 and δ3 ¼ 4=5π. The differences of the phasewere taken both in angular scalar values and in vec-tor values. For the normal distribution and Laplacedistribution, the scalar angular values were used.For the von Mises distribution, which is based on cir-cular statistics, the vector values were used. Thewavefront phases from CWT were plotted in Fig. 3.The distribution fittings, using a different distribu-tion on the phase-shifting step, were plotted in Fig. 4.The parameters were estimated by the MLE usingEq. (12), and the asymptotic covariance matrixwas calculated using Eq. (13). The estimated param-eters and variances are listed in Table 2.

The results from the numerical testing showedthat the choice of the population distribution was im-portant in order to obtain a good result. The normaldistribution assumption on the phase difference dis-tribution was not a fair assumption on the simulatedsignal. The von Mises distribution gave a better es-timation. The Laplace distribution gave the exactphase-shift steps with very small estimation var-iance (refer to Table 2). The reason for this is thatCWT operation introduces only a very small amountof the error during operation. The error from theCWTwas explained earlier. In contrast, the real datain the next section showed that the error term εkði; jÞin Eq. (11) followed different error patterns because

Fig. 2. (Color online) Input signal at δ1 ¼ 0 and δ3 ¼ 4=5π.Fig. 3. (Color online) CWT estimation of the phase difference forδ3 ¼ 4=5π compared with the true value.

Table 1. Probability Distribution Candidates for the Parametric Parameter Estimation

Probability Density Function MLE Parameter 1 MLE Parameter 2

Normal distributionf ðϕÞ ¼ 1ffiffiffiffiffiffiffiffi

2πσ2p exp

�−

ðϕ−μÞ22σ2

� μ̂ ¼ Pni¼1 ϕi σ̂2 ¼ 1

n

Pni¼1ðϕi − μ̂Þ2

Laplace distributionf ðϕÞ ¼ 1

2b exp�−

jϕ−μjb

� μ̂ ¼ medianðϕiÞ b̂ ¼ 1n

Pni¼1 jϕi − μ̂j

Von Mises distribution f ðϕÞ ¼ 12πI0ðκÞ exp½κðϕ − μÞ� μ̂ ¼ arctan

�Pni¼1

sinϕiPni¼1

cos αϕ

�I1ðκÞI0ðκÞ ¼

Pni¼1

cosðϕi−μ̂Þn

a

aWhere Iα is the modified Bessel function of the first kind.


of the optical defects and the variation of the spatialfrequency in the interference pattern, which was notpresented in the numerical simulation shown above.

The experimental data used here were taken froma real moiré interferometry experiment. For detailsof the moiré interferometry and the experiment con-figuration used to obtain the data, please refer to ourprevious paper [19]. On the one hand, the phase-shifting step was introduced by the translation ofthe piezoelectric device by 100nm at each temporalsampling, which could be inferred fromEq. (6); on theother hand, the phase drift was also introducedby the loading condition of the experiment (the ther-mal expansion of the experiment stage, and airflow), which was undeterminable before the actualexperiments. The specimen was a copper sheet(50mm × 25mm with 1mm thickness) under currentstressing (26 A) in the longitudinal direction. Be-cause this paper addresses the phase-shifting stepestimation, the details of the experiment will notbe included. The phase step estimated from Eq. (6)was 1:5080 rad. The CCD sensor was a Pulnix TM-1040 which has one million valid pixel points. Thearea of interest was the image row 300 to 400, which

included 101,808 individual sampling points (seeFig. 5). From our perspective, the size of the sam-ples is large enough for a statistically significantinference.

The algorithm was performed on the experimentaldata in the same manner as it was for the simulateddata. The probability distributions of the phase-shiftstep are plotted in Fig. 6, and the estimated param-eters and variances calculated from the Fisher infor-mation are listed in Table 3.

6. Discussion

The results show that the phase-shift step is a statis-tical variable instead of a deterministic one. In thenumerical case, the proposed algorithm gave theexact value of the phase-shift value. In the experi-mental case, all three estimations from different un-derlying population distribution assumptions gavedifferent numbers, and they deviated from the origi-nal estimation, which was inferred from a controlfeedback circuit of the piezoelectric actuator usingEq. (6). The variance was due to the error term inEq. (11), which can be introduced either by any opti-cal defects within the instruments or by a digitalsampling process. Based upon the comparison be-tween three preferred distribution candidates, boththe Laplace and von Mises distributions have signif-icant values in estimation and accuracy. The valida-tion only implemented MLE and Fisher informationcovariance calculation. However, most of the para-metric and nonparametric parameter estimationmethods can be used to replace the statistical estima-tion block. The decoupling of different components isone of the major advantages of the proposed algo-rithm. The MLE calculated the results based onthe likelihood function built on the sample frequency.A larger sample size, usually from a digital detector,decreased the variance of the estimated parameter.

Fig. 4. (Color online) Probability distribution of the CWT estima-tion of the phase-shift step in the numerical simulation forδ3 ¼ 4=5π.

Table 2. Parameter Estimate for the Numerical Simulation

Distribution Parameter EstimateStandardError Real

Normal μ̂ 2.524 0.011 2.5133σ̂ 0.340 0.008 N/A

Laplace μ̂ 2.5133 0.0002 2.5133b̂ 0.1523 0.0048 N/A

Von Mises μ̂ 2.5204 0.0104 2.5133κ̂ 9.7344 0.4208 N/A

Fig. 5. Moiré interferometry data at (a) δ1 and (b) δ3.


In contrast, the deterministic method cannot explorethis feature from the PSI data source. The signifi-cance of the study includes building the phase stepestimation on the statistical ground and, also, beingable to establish the phase-shift step variabledirectly from the interferograms. From the sameperspective, the existing algorithm can also be inter-preted as building a statistical model on the intensityvalue. For example, the LSE on the MSEs of the in-tensity values, using phase steps as the minimiza-tion parameters [8,10], can be treated equally asan LSE on the phase steps, considering that the noiseis included in the intensity term and the noise followsa normal distribution [34]. Because the intensityvalues were determined by multiple variables [referto Eq. (1)], it will be difficult to establish a large num-ber of samples with the same statistical structures ofthe errors. The estimation from the proposed algo-rithm gives a simple structure of the estimatedparameter [refer to Eq. (11)]. This allows the re-searchers and engineers to design or utilize a morespecific and precise statistical estimation procedure.

7. Conclusions

In this paper, a statistical phase-shifting step estima-tion algorithm, based on the CWT for high-resolutionPSI was proposed, and a statistical analysis on thephase-shifting step in temporal PSI, based on CWT,was included. The proposed algorithm is an open fra-mework, treating the phase-shifting step as a regularstatistical variable. Inside this framework, thestatistical estimation component can be easily re-placed by other advanced parametric or nonpara-metric methods. Every point in the interferogramcan be included in the samples to build up the statis-tical population. In the algorithm, the wavefrontphases are exacted based on the maximum powerridge property of the CWT. The error could by intro-duced largely by the disruption in the data in the in-tensity map caused by optical defects on the lightpath. This was validated by utilizing the algorithmboth in the numerical simulated data and experi-mental data. In the numerically simulated data,the exact phase-shifting step was obtained up to foursignificant figures. In the experimental study, the ex-perimental data from the phase-shifting moiré inter-ferometry were put though the algorithm. This gavea larger variance compared with the numerical one.However, a large sampling size (around 100,000 sam-ples) for each interferogram enables a statisticallysignificant inference based on MLE and the Fisherinformation based on asymptotic variance calcula-tion. The validation showed the proposed methodwas enabled to estimate the phase steps accurately.The proposed methods can not only be used as astand-alone method to estimate the phase, but alsoto provide a new statistical tool for temporal PSI.

This project has been sponsored by the UnitedStates Navy Office of Naval Research (ONR) Ad-vanced Electrical Power System under the directionof Terry Ericsen.

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