Adaptive linear learning for on-line harmonicidentification:An overview with study cases

Patrice WiraLaboratoire MIPS

Universite de Haute Alsace, Mulhouse, FranceEmail: patrice.wira@ieee.org

Thien Minh NguyenLaboratoire MIPS

Universite de Haute Alsace, Mulhouse, FranceEmail: thien-minh.nguyen@uha.fr

Abstract—This work reviews Adaline-based techniques for es-timating Fourier series. The Adaline, with its linear structure andlearning, fits a Fourier series by expressing any periodic signal asa sum of harmonic terms. The learning with elementary harmonicinputs enforces the weights to converge to the amplitudes. TheAdaline therefore individually identifies the amplitudes of theharmonic terms present in the measured signal in real-time.Relevant study cases are provided. Performances are evaluatedand show that harmonic terms of the signals are efficientlyestimated.

I. I NTRODUCTION

Harmonic content is a fundamental concept in power systemanalysis, operation, and control; hence, its fast and precise es-timation is of prime importance. Consequences and problemsinduced by higher-order harmonic terms in power systemshave been well established [1]. Digital devices with highcomputational capabilities will expand the design of new andprecise harmonics identification techniques.

Fourier-based approaches are among the most fundamen-tal techniques in frequency analysis processing. However,they imply sliding window implementations and convolutionoperations which make their computational requirements aheavy burden in most applications. Furthermore, Fourier-basedapproaches only provide a response after a complete periodof the measured signal and cannot calculate the dynamiccharacteristics of measured signals over time because of theassumption that analyzed signals are stationary [2]. Since theharmonic content varies constantly in power systems, fastand real-time estimation techniques are necessary for efficientactions.

An adaptive linear element, i.e., an Adaline, perfectly learnsany linear weighted sums. Thus, if sine and cosine termswith multiple frequencies are its inputs and if a measuredsignal serves as a reference for its output, then the Adalinewill learn a Fourier series. After convergence, the weightassociated to each input, i.e., to a given harmonic term,represents its amplitude. A kind of LMS (Least Mean Square)algorithm updates its weights. This learning algorithm is notonly extremely simple but also linear, which makes learningfast and easy. The amplitude of the harmonic terms can thusbe iteratively updated, it is possible to access to them on-linein real-time applications.

The objective of this paper is to review the principles ofusing Adaline to estimate the amplitudes of Fourier series

terms. The identification of periodic signals is illustrated bytypical examples.

This paper is organized as follows. Section II briefly outlinessome key issues of the Adaline network. Section III is devotedto the principle of learning Fourier series with an Adalineand variants are reviewed. Fourier series of typical signals arelearned and their harmonic terms are individually extracted inSection IV. Section V concludes the paper.

II. BASICS OF ADAPTIVE LINEAR ELEMENTS

The architecture of the Adaline is based on a very sim-ple unit which performs a processing. This unit consists ofweights, a bias and a summation function, they are shownon Fig. 1. The processing comprise the calculation of theoutput for given inputs and the learning phase, i.e., the weightsadjustment. The Adaline is able to fit any linear relationshipsby providing a scalar output as a weighted sum of the inputsand by adapting its weights. When a multidimensional outputspace must be considered, i.e., when several outputs arerequired, several Adaline having the same inputs are used andthis is sometime referred to as a MAdaline (Multiple Adaline).

Let x and w be two vectors, respectively for the inputsand the weights. For mathematical convenience, let the firstelement ofx be equal to 1, so that the first element ofw

becomes the bias weight. At instantk, the outputy(k) is aweighted sum given by the following dot product:

y(k) = wT (k)x(k). (1)

The Adaline network is a supervised learning network thatneeds to associate a reference value for each input vector.This reference is a desired value corresponding to an inputand expressed in the Adaline’s output space. When an inputx(k) is presented to the network, the outputy(k) is calculatedand compared to the desired outputy(k) that is associated tohim. This defines the error

e(k) = y(k)− y(k) = y(k)−wT (k)x(k). (2)

The pairs of input/output valuesx(1), y(1), x(2),y(2), . . .x(Q), y(Q) represents the learning data set.Each pair can be used on-line to adapt the weights at eachiteration in order to minimizee(k). The new value of the

P. Wira and T. M. Nguyen, "Adaptive linear learning for on-line harmonic identification: An overview with study cases," International Joint Conference on Neural Networks (IJCNN 2013), Dallas, Texas, August 4-9, 2013.

1

+-

å...

{1

sin(ωk)

cos(ωk)

sin(Nωk)

cos(Nωk)y(k)

y(k)

harmonic inputsAdaline network

e(k)

Fig. 1. Adaline architecture with harmonic terms as inputs

weight vector is updated from its previous value according tothe µ-LMS rule or to theα-LMS rule, i.e., respectively

w(k + 1) = w(k) + µ e(k) x(k), (3)

w(k + 1) = w(k) + αe(k)x(k)

‖x(k)‖2, (4)

whereµ and α are learning rates. Theα-LMS algorithm isonly a normalized version of theµ-LMS algorithm. Normal-izing the inputx before applying it to the network leads tothe same result using theµ-LMS algorithm. These learningrules come from the LMS algorithm and is called the Widrow-Hoff learning rule [3]. Approximately, the Adaline convergesto least squares error whenk → ∞ [4], [5]. The maincharacteristic of LMS algorithm is that it safes the error andit reduces the average quadratic error. Variants have beenrecapitulated in [6].

In (3) and (4),w(k+1) is the new value that will take theweight vector from its previous valuew(k) which representsthe memory of the network. In the weight update process, thelearning rate gives more or less importance to the innovationterm based on the error compared to the memory termw(k).Therefore, the values of the ratesµ or α are chosen between0 and 1.

III. R EVIEW OF ADALINE -BASED APPROACHES TO

HARMONIC CONTENT ESTIMATION

The last decades have seen many studies about harmonicdistortion identification techniques to improve power quality.In this literature, a harmonic term is defined as a component ofa periodic wave having a frequency that is an integral multipleof the fundamental power line frequency. In the following, wefocus on on-line and iterative algorithms to estimate harmonicterms in real-time applications.

A. General case and principle of operation

Any periodic, distorted waveform can be expressed as asum of pure sinusoids. The sum of sinusoids is referred toas a Fourier series. The Fourier analysis permits a periodicdistorted waveform to be decomposed into an infinite seriescontaining a DC component, a fundamental component (50/60Hz for power systems) and its integer multiples called the har-monic components. The harmonic numbern usually specifiesa harmonic component, which is the ratio of its frequency tothe fundamental frequency.

An ideal power signal, i.e., a voltage or a current, is asinusoidal signal of periodT (scalar)

y(k) = a sin(ωk + φ) (5)

where a is the amplitude,ω = 2π/T represents the actualangular frequency, andφ is the initial phase angle. This signalis measured and digitalized with a sampling frequency offs,the time interval between two successive samples is thusTs =1/fs.

A non-ideal power signal contains harmonic terms and noiseand can generally be approximated by

y′(k) = a′0+a′

1sin(ωk+φ1)+

∑N

n=2a′n sin(nωk + φn)+η(k)

(6)wherea′

0is the DC component andη(k) represents a noise.

Each harmonic component is defined by its amplitudea′nand its phase angleφn. Practically, the sum of the harmoniccomponentsa′n sin(nωk + φn) is limited (to n = N ).

According to Fourier, every periodic signal can be estimatedby a functionf :

f(k) = a0 +∑∞

n=1an cos(nωk) +

∑∞

n=1bn sin(nωk) (7)

wherea0 is the DC part andn is called then-th harmonic.The sum of the termsan cos(nωk) is the even part and thesum of the termsbn sin(nωk) is the odd part of the signal.Rearranging the even and the odd part gives (8) which is awell known result:

f(k) = c0 +

∞∑

n=1

cn cos(nωk − θn), (8)

with cn the harmonic amplitudes andθn the phase angles:

c0 = a0, cn =√

a2n + b2n, θn = tan−1

(

bnan

)

. (9)

f(k) is a weighted sum of sinusoidal terms. It is therefore alinear relationship that can be fitted by an Adaline but with alimited number ofn = N terms. Sine and cosine terms withunit amplitude are provided as the Adaline’s input vector:

x =[

1 sin(ωk) cos(ωk) . . . sin(Nωk) cos(Nωk)]T

.(10)

The outputy(k) of the Adaline is compared to the measuredsignaly(k) in order to adapt the weights. After learning andconvergence, the signal is estimated by the functionf(k) =w

∗Tx and we obtain:

w∗ =

[

a0

2b1 a1 b2 a2 . . . bN aN

]T. (11)

This approach allows to handle the most general case, i.e.,to estimate the amplitudes of the complete harmonic contentof a signal with an Adaline. The Adaline can be designed tofit some precise harmonic content. Indeed, specific cosine andsine terms must be expressed, even with delays (i.e.,φn) andused as inputs. After learning and convergence, the Adalinefits the weighted sum and the weights represent the amplitudeoh each of them.

P. Wira and T. M. Nguyen, "Adaptive linear learning for on-line harmonic identification: An overview with study cases," International Joint Conference on Neural Networks (IJCNN 2013), Dallas, Texas, August 4-9, 2013.

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Generally, power system signals present a limited numberof specific harmonics. Adalines with specific harmonic inputsis therefore very convenient for estimating the amplitude ofcurrent distortions.

B. Adaline-based approaches for harmonic estimation

In power systems, identifying the harmonics allows to sepa-rate the disturbing higher-order harmonics introduced by non-linear loads from the fundamental term carrying the electricenergy [1]. These operations are necessary for monitoring andensuring electric power quality. Efficient methodologies forthe analysis and measurement of the basic electric magnitudesin are required. Methods with short computation time forreal-time calculation must be employed for the generationof compensating currents in order to instantaneously re-injectthem, most often with shunt active power filtering schemes [7].

The following shows how Adaline-based approaches can bejudiciously used for estimating the frequency/harmonic contentof power system signals. Frequency estimation means estimat-ing the fundamental frequency and tracking its fluctuationsand deviations. Harmonics identification means estimating theamplitude and the phase of the harmonic terms contained inthe signal.

The use of an Adaline to learn the Fourier series of thesignal model given by (6) has been introduced in [8]. Thiswork corresponds to the general approach detailed abovewhere a decaying DC quantity is added to the signal model.An additional element (-kTs) is therefore introduced in theAdaline input vector and allows to efficiently track the ampli-tude and the phase of 6 harmonic terms. A similar approachis proposed in [9], where a signal model with a differentexpression of the decaying quantity is used. This leads to themodification of one element of the input vector. The estimationerror is also fed back recurrently in order to enhance the inputvector by 3 elements (e(k), e(k− 1) ande(k− 2)). The verysimplest approach, based on the Fourier series, is also usedin [10] and in [11]. In this last work, only the two weightselements of the fundamental component are updated, henceit is independent of the number of harmonic orders present.In [12], the same approach and the same signal model is used,one Adaline is used for harmonic estimation, another Adalineis used for predicting the line voltage.

The S-Adaline proposed in [13] is able to synchronize itselfwith time-varying signals to the frequency deviation for on-line tracking of single phase reactive power. It contains a fun-damental angular frequency deviation measurement algorithmthat is used to generate sine and cosine terms of the inputvector of the Adaline. These terms are thus in phase with thefundamental term of the measured signal. In [2], two Adalinesin a cascaded two-stage approach is used. In the first stage,an Adaline implements the Prony’s method for tracking thefundamental frequency of the measured signal. In the secondstage, an Adaline learns the Fourier series decomposition ofthe signal with the very simplest approach for estimating theamplitudes of the harmonics.

The previous approaches identify the harmonics in the mea-sured signal reference frame. This means that the measuredsignal, i.e., the current, is directly expanded into a Fourierseries which is learned by an Adaline. However, the measuredsignal can be converted into another reference frame beforebeing expanded, learned and approximated by an Adaline. Ifthe principle remains the same, the conversion of the signalin a different reference frame allows to highlight more or lesssome parts of the signal. The current is thus converted into avirtual power space by multiplying the measured current bya sine term in [14]. In another method of [14], 2 Adalinesserves to estimate the Fourier series of the instantaneous PQ-powers [7] which requires the measure of the currents andof the voltages for the 3 phases. In [6], measured current ofthe 3 phases is converted into a current expressed in the DQ-space with the Park transform. A Complex Adaline is proposedin [15]. This approach estimates the fundamental frequencyof a power system with an input vector composed of sineand cosine terms. To produce the input vectors and deal withthe decaying DC term, the Park transformation is used. Thetwo weights associated to the fundamental frequency are usedthrough a Hamming filter to calculate the amplitude of thefundamental term.

The Adaline for frequency estimation and harmonic identi-fication can be used in a different way by replacing the Fourierseries expression by a recursive linear expression of the signal.Considering a measured signal of the type given by (5), threeconsecutive samplesy(k), y(k − 1) and y(k − 2) meet therelationship

y(k) = (2 cosω0Ts)y(k − 1)− y(k − 2) (12)

The inputs of the Adaline therefore becomex =[ y(k − 1) y(k − 2) ]T and its output is compared to thereference signaly(k) with is the measured signal at in-stantk. After minimizing the error, the weights converge tow

∗ = [ w∗

1w∗

2 ]T = [ 2 cosω0Ts −1 ]T . The frequencycan thus be obtained from the first element ofw

∗. Indeed,f = (2πTs)

−1cos−1(w∗

1). As can be seen, it is simple and

therefore well suited for the frequency estimation problem.However, it is sensitive to noise because based on an idealexpression of a current waveform.

This simple principle is used in [16] where the Adalineinputs are enhanced by additional harmonic terms to takeaccount of a decaying DC component and of harmonic dis-tortions present in the power system signal. Fundamentalfrequency estimation is thus achieved. In [17], a tapped delayline of the measured current is used to generate the inputsfor the Adaline. Power quality event detection is thus possiblewith an Adaline with only 4 inputs.

In [18], a tapped delay line of a larger size is used togenerate the Adaline inputs for disturbance detection. Theidentification of the power system frequency is achieved byanother Adaline that combines delayed signal measures andsine and cosine inputs. More recently, [19] proposes anapproach for frequency estimation but not with an Adaline. Itis a least-squares approach that uses 3 consecutive measures

P. Wira and T. M. Nguyen, "Adaptive linear learning for on-line harmonic identification: An overview with study cases," International Joint Conference on Neural Networks (IJCNN 2013), Dallas, Texas, August 4-9, 2013.

3

-1

-0.5

0

0.5

1

-2

-1

0

1

2

0 500 1000 1500 2000 2500 3000 3500 4000-0.5

0

0.5

1

1.5

iteration (k)

Fig. 2. Performance of an Adaline with 41 weights in approximatingasinusoidal waveform disturbed by 3 harmonics

of the signal and that calculates once per iteration the solution(i.e., the coefficients equivalent to the weights of the Adaline)by using a pseudo-inverse computing.

Finally, we can say that:

• There are two main approaches, one based on learningand approximating the amplitudes of the Fourier seriesused to express the signal, and one based on learningand approximating an iterative expression of the signal.

• Learning the Fourier series is more appropriate to har-monic terms identification while learning the iterativeexpression is more appropriate to fundamental frequencyestimation.

• Approximating Fourier series can be tackled in the mea-sured signal’s but also in other specific reference frames.

• All the reviewed approaches use a sophisticated learningrule with an adaptive learning rate.

IV. STUDY CASES

All these approaches are well suited for power systemapplications where typical nonlinear loads generate very loweven harmonics and where triple harmonics can be ignored inthree-phase circuits [1]. As a consequence, appropriate inputterms can be specified for the Adaline. Once identified, the har-monic terms can be compensated individually according to thestrategy objective (full or selective harmonic compensation,power factor and unbalance compensation...). It is obviousthat the dimension of the weight vector to be updated on eachiteration depends on the number of harmonics to be estimated.The complexity of the implementation must be compliant toreal-time constraints.

In order to show the effectiveness of the approach, differentcases are investigated thereafter. At the beginning, the Adalineis initially untrained withw(0) = 0. For convenient reasons,the learning rate is chosen as constant:α = 0.85 for all thestudied cases. Finally, the MSE (Mean Square Error) of theestimation is used as a measure of overall performance.

A. A main sinusoid with harmonics of ranks 3, 5 and 7

As a first example, we propose to identify the harmoniccontent of a signal composed of a fundamental sinusoidal

-1

0

1

-2

0

2

0 500 1000 1500 2000 2500 3000 3500 4000-0.5

0

0.5

1

1.5

iteration (k)

Fig. 3. Performance of an Adaline with 5 specific inputs in approximatinga sinusoidal waveform disturbed by 3 harmonics

term where harmonics of rank 3, 5 and 7 have been addedas follows:

s(k) = sinωk +1

3sin 3ωk +

1

5sin 5ωk +

1

7sin 7ωk (13)

with a fundamental frequencyf = 50 Hz. The signal issampled withTs = 0.0002 s. and 4000 samples are used.

If only the frequency of the fundamental term is knownand if we suppose that the signal does only contain harmonicterms of rank less than 20, then its harmonic content can beestimate by an Adaline that fits the classical Fourier seriesgiven by (7) with 20 terms. This means that the Adaline willtake 41 inputs, 1 DC term, 20 cosine and 20 sine terms. Thevalues of the coefficientsa0, an and bn will be estimated bylearning.

After convergence, the amplitudes are obtained fromw

∗: a0 = 0.0, an=1,2...20 = {0.0} and bn=1,2...20 ={1.0, 0.0, 0.3333, 0.0, 0.2000, 0.0, 0.1429, 0.0, · · · 0.0}.These coefficients perfectly represent the harmonic contentof s(k) in (13). Results obtained with an Adaline with 41inputs are represented by Fig. 2. The error between theoriginal signals(k) and the one estimated by the Adaline isrepresented. Additionally, the signal and its estimation, butalso the evolution of the Adaline’s weights are shown. Theweights that converge to zero correspond to the harmoniccomponents that are not present in the signal. It can be seenthat after 3 periods of signal, the MSE is less than 0.1 (s(k)varies between -1 and 1). Furthermore, the MSE is less than8e−11 over the two last periods.

On the other hand, if we know which harmonics are presentin the signal for estimating their amplitude, then we preciselyspecify them as the inputs of the Adaline. In the case ofs(k) given by (13), knowing that it is composed of a DCcomponent, of a the fundamental term withf = 50 Hz and ofharmonics of rank 3, 5 and 7 allows to defined the followinginputs:

x(k) =[

1 sinωk sin 3ωk sin 5ωk sin 7ωk]T

. (14)

After learning, the resulting weights are:

w∗ =

[

0 1.0 0.3333 0.2000 0.1429]T

. (15)

P. Wira and T. M. Nguyen, "Adaptive linear learning for on-line harmonic identification: An overview with study cases," International Joint Conference on Neural Networks (IJCNN 2013), Dallas, Texas, August 4-9, 2013.

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One can see that the Adaline affects a null weight to theDC component and correctly estimates the amplitude of theharmonic terms (fundamental and higher-order harmonics).The estimated signal, the error and the weights evolution arerepresented by Fig. 3. The MSE is less than 0.04 after 3periods and is less than 3e-011 over the two last periods. Withonly 5 inputs, the degrees of freedom of the Adaline are lessthan with 41 inputs and the convergence is therefore faster.

The computational complexity and computing time are notthe same in both cases. Intensive simulations tests show thatthere is a ratio of approximately 1/12 between the time tolearn with 5 and with 41 inputs under the same computingconditions.

Similar tests have been lead to signals with a fundamentalterm and up to 30 randomly chosen higher-oder harmonics.An Adaline with 30 sine and cosine terms is able to estimatethese signals with an final MSE of less than5e−5 (for signalsvarying between -1.5 and 1.5). Knowing the present harmonicsallows to precisely specify the inputs in order to obtain theiramplitude and allows to reduce the computational complexityand a faster convergence. Obviously, delayed harmonic termscan be taken into account. As an example,1

7sin(7ωk) has

been replaced by17sin(7ωk− π/4) in (13). The results show

that with the appropriate inputs, the Adaline is perfectly ableto estimate the signal with the same precision.

B. Case of a pure square wave

The example of a periodic square wave that varies between0 and 1 ant that rises for the first time atk = 0 is interestingbecause in theory it is estimated by the following series:

f(k) =1

2+

2

π(sinωk +

1

3sin 3ωk +

1

5sin 5ωk + · · ·) (16)

which gives in other words, a0 = 1/2,an = 0, ∀n, and bn=1,2,3,4,5,6,7,... ={0.6366, 0.0, 0.2122, 0.0, 0.1273, 0.0, 0.0909, 0.0...}.

A square wave signal withf = 40 Hz and composedof 8000 samples has been artificially synthesized withTs = 0.0002 s. Assuming that such a signal is madeof up to N = 20 harmonic terms, an Adaline with 41weights can estimate it. After learning with an Adalineconfigured like this, we obtain fromw∗: a0 = 0.4996,and an=1,2,3,...20 = {−0.0089, 0.0021, −0.0664, 0.0204,−0.0052, −0.0033, 0.0012, · · · } and bn=1,2,3,...20 ={0.6253, 0.0091, 0.2111, 0.0062, 0.1271, 0.0040, 0.0888,· · · }. Furthermore, all amplitudes of harmonic terms of rankover 7 are less than 0.0533. It can be noticed that the values ofan andbn whenn takes an even value are close to zero. Thus,the estimated values of the amplitudes perfectly correspondsto the theoretical Fourier series of (16). Results are illustratedby Fig. 4 and the MSE between the perfect square signal andthe output of the Adaline, i.e., the recovered signal, is 0.0641for the 2 last periods.

C. Case of real noisy signals

1) A real noisy square wave signal: The example of a nonperfect square wave that varies between -0.3 and 0.3 which

-2

0

2

-2

0

2

0 1000 2000 3000 4000 5000 6000 7000 8000-2

0

2

iteration (k)

Fig. 4. Performance of an Adaline with 41 weights for learning theFourierseries of a pure square signal

-0.5

0

0.5

-0.5

0

0.5

0 1 2 3 4 5 6 7 8

4

-0.5

0

0.5

iteration (k.10 )

Fig. 5. Performance of an Adaline with 41 weights for learning theFourierseries of a real square signal

f = 40 Hz is chosen. It is a real signal, disturbed by noise,and composed of 80 000 samples measured with a samplingperiod ofTs = 0.0002 s.

As in the previous cases, an Adaline with the same pa-rameters and under the same conditions is used to estimate itsconventional Fourier series. Results with 41 weights (N= 20)are given by Fig. 5 and the MSE is 0.0396 for the 2 last periodsof the signal. It can be noticed that the noise still remains onthe recovered signal. It has not been removed because it hasnot been modeled in one of the weights as a harmonic term.

2) A current measured on a nonlinear low power load:The harmonic content of a typical current of a nonlinear loadpower device is estimated thereafter. This current is measuredon one of the 3 phases of a nonlinear load characterized byf=50 Hz, 380 V, 20 A and 4000 samples are used (Ts =6.0000e-005 s.).

This signal is analyzed with an Adaline with the sameparameters and under the same conditions. For a Fourier serieswith N = 20, an Adaline with 41 weights leads to a MSE of0.1931 over the last two periods. In a second test, a Fourierseries withN = 60 is estimated by an Adaline with 121weights. Results are represented by Fig. 6 where the harmonicterms are also as a histogram. They are estimated in less than3 periods. The MSE over the last two periods is 0.1093 whichrepresents an error of 0.45% for the current (on a range of 24A).

3) An Electrocardiogram: The electrocardiogram (ECG)is a graphical representation of the electrical activity of the

P. Wira and T. M. Nguyen, "Adaptive linear learning for on-line harmonic identification: An overview with study cases," International Joint Conference on Neural Networks (IJCNN 2013), Dallas, Texas, August 4-9, 2013.

5

-10

-5

0

5

10

-20

-10

0

10

20

-10

0

10

20

iteration (k)

0 500 1000 1500 2000 2500 3000 3500 4000

5 15 25 350

1

Fig. 6. Adaline performance for learning the Fourier series of aload current

-1

0

1

2

0

1

2

3

0 500 1000 1500 2000 2500 3000-0.5

0

0.5

1

1.5

iteration (k)

5 15 25 350

1

Fig. 7. Adaline performance for learning the Fourier series of anECG

heart that offers information about the state of the cardiacmuscle. This representation consists of a base line and severaldeflections and waves. Our example is characterized by afrequencyf=1.20 Hz and a mean of 1.0509 and 3000 sampleshave been acquired and sampled withTs =0.0100 s. Once, thesignal has been filtered in order to remove the residual noise,an Adaline with 41 input terms is employed to estimate itsFourier series.

After learning, the signal is estimated byw∗. The firstweight, a0 = 1.0515, represents the true value of the DCcomponent of the signal. Other results and the normalizedcn coefficients are shown by Fig. 7. Less than 3 periods ofthe signal are necessary to converge and a MSE of 0.0306 isobtained at the end.

V. CONCLUSION

This paper investigates the performance of Adaline learningsystems in estimating Fourier series. The Adaline is a linearadaptive neuron whose structure perfectly fits the Fourierseries formalism in expressing any periodic signal as a sum ofharmonic terms. The Adaline parameters have the ability to bephysically interpretable. Its weights represent the amplitude.Thus, with a LMS learning algorithm which requires low com-putational costs, the Adaline on-line identifies the amplitudesof the harmonic terms. This is illustrated by relevant studycases where Fourier series of typical sinusoidal waveforms,

real power system currents and square waves have beenefficiently estimated. Tests with other types of signals, suchas ECG, demonstrate the good performance of the Adalinein estimating harmonics. Whatever the type of the signal, theharmonics are identified with a MSE of less than 0.45% andin less than 3 periods of the signal.

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