Algorithm for Fourier coefficient estimation

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Published in IET Signal ProcessingReceived on 26th June 2009Revised on 5th October 2009doi: 10.1049/iet-spr.2009.0161

ISSN 1751-9675

Algorithm for Fourier coefficient estimationP.B. Petrovic M.R. StevanovicTechnical Faculty Cacak, Svetog Save 65, 32000 Cacak, SerbiaE-mail: [email protected]

Abstract: This study deals with the estimation of amplitude and phase of an analogue multi-harmonic band-limited signalfrom irregularly spaced sampling values. To this end, assuming the signal fundamental frequency is known in advance (i.e.estimated at an independent stage), a complexity-reduced algorithm for signal reconstruction in time domain is proposed.The reduction in complexity is achieved owing to completely new analytical and summarised expressions that enable aquick estimation at a low numerical error. The proposed algorithm can be applied in signal reconstruction, spectralestimation, system identification, as well as in other important signal processing problems. The proposed method ofprocessing can be used for precise root mean square measurements (for power and energy) of a periodic signal basedon the presented signal reconstruction. The authors investigate the errors related to the signal parameter estimation, andthere is a computer simulation that demonstrates the accuracy of these algorithms.

1 Introduction

Estimating the amplitude and phase of a signal accuratelyeven when the frequencies contained in the signal arealready known is very important in many areas [1]. Inpower systems, estimation of the harmonic components isnecessary to ensure the quality of power supply. When thefrequencies contained in the target signal are in arelationship of an integer multiple of a fundamentalfrequency or in a harmonic relationship, the magnitude andphase of the signal can be obtained by the well-knowndiscrete Fourier transform (DFT) and short-time DFT(simple and cost-efficient tools for these applications) [1].By means of the DFT, the Fourier coefficients can beobtained at every sampling time with a sliding fashionusing a frequency sampling structure.

Many attempts have been made with respect to applicationof sampling techniques, supported by optimal methods ofreconstruction of band-limited signals in the form of aFourier series (trigonometric polynomials). In [2] a newtechnique for reconstruction of a band-limited signal fromits periodic non-uniform samples is described. Using amodel for the non-uniform sampling process, thereconstruction was viewed in terms of a perfectreconstruction multirate filter bank problem. Marzilianoet al. [3] considered sampling of continuous time-periodicband-limited signals, which contain additive noise. Bymodelling the shot noise as a stream of Dirac pulses,Marziliano et al. [3] showed that the sum of a band-limitedsignal with a stream of Dirac pulses falls into the class ofsignals that contain a finite rate of innovation, that is, afinite number of degrees of freedom. In [4] two newalgorithms were introduced for perfect reconstruction of aperiodic band-limited signal from its non-uniform samples.The reconstruction can be obtained by using the Lagrange

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interpolation formula for trigonometric polynomials, andwhen the number of sampling points N is odd and itsatisfies N ≥ 2M + 1, Lagrange interpolation forexponential polynomials results in a complex-valuedinterpolation function. In [5], there is a description of aniterative frame algorithm for the reconstruction of band-limited signals from local averages with symmetricaveraging function, which gives bounds estimates on thealiasing error when a direct frame reconstruction is used forreconstruction of non-band-limited signals.

This paper analyses the sampling of an analoguemulti-harmonic input signal, as opposed to the method basedon the integration of the input analogue signal [6], along withthe problem of subsequent reconstruction of the processedsignals. In [6] the standard matrix inversion is used as themethod for reconstruction, requiring very intensive numericalcalculation. It was noticed in [7] that the ac signal integrationmethod produces a regular form of system matrix in thederived system of equations, provided an adequate choice ismade regarding the time parameters within which theintegration is done; this makes it possible to have much moreefficient reconstruction procedure subsequently.

If the samples were assumed to be measured without errors,the presented algorithm, without any further modifications,can be used for signal reconstruction of periodic band-limited signals, which is the situation that occurs insimulation. In a real environment and when the measuringis performed in practice, the samples are measured witherror. This happens as a result of the error that is caused bythe imprecision, introduced by the voltage and currenttransducers, used to adapt the measured signals to themeasuring system – the error of conversion, which occursbecause of errors that appear during the processing of theanalogue signal through various digital circuits, as well asduring the determining of the digital equivalent, and so

IET Signal Process., 2011, Vol. 5, Iss. 2, pp. 138–149doi: 10.1049/iet-spr.2009.0161

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forth. The sampled values that are obtained in practice maynot be the exact values of the signal at sampling points, butonly averages of the signal near these points [5]. In thiscase, the suggested algorithm must be modified, in order tobe able to determine the best signal estimate, according tothe criterion assumed, just like in [5, 6, 8–11].

The proposed algorithm for estimation is based on the useof the values that were obtained as a result of the sampling ofthe continual input signal. This kind of processing must bedone as many times as needed to enable the determinationof the unknown signal parameters. The obtained system oflinear equations can be simply solved by using the derivedanalytical and summarised expressions. For this reason, theproposed method offers a significant improvement incomputational efficiency over the standard reconstructionalgorithms, at a lower numerical error. The method isdesigned for very accurate root mean square (RMS)measurements of periodic signals, and can be applied forprecise measurements of other important quantities such aspower and energy.

2 Problem formulation

Let us assume that the input signal of the fundamentalfrequency f is band-limited to the first M harmoniccomponents. This form of continuous signal with a complexharmonic content can be represented as a sum of theFourier components as follows (Table 1)

s(t) = a0 +∑M

k=1

ak sin(k2pft + ck) (1)

By sampling the signal (1) and by forming a system ofequations of the same form, in order to determine the2M + 1 unknowns (amplitudes and phases of the Mharmonic, as well as the average value of the signal), weobtain

s(tl) = a0 +∑M

k=1

ak sin(k2pftl + ck ) (2)


where l ¼ 1, 2, . . . , 2M + 1 and tl is the time moment in whichthe sampling of the input analogue signal is done. The s(tl)value represents the value of the processed signal at themoment when the sampling is performed. The obtainedrelation can be represented in the short form as

a0 +∑M

k=1

ak (sinak,l cosck + cosak,l sinck ) = s(tl) (3)

where

ak,l = 2kpftl = kal; (k = 1, 2, . . . , M );

(l = 1, 2, . . . , 2M + 1) (4)

The ak,l are the variables determined by the moment atwhich the sampling is done, as well as by the frequencyof the corresponding harmonic of the input periodicsignal. The system determinant for the system of2M + 1 unknown parameters (3) can be represented as (see (5))

where (see (6))A similar form of the obtained determinant (6) was the

object of study in [6]. In its form, determinant (6)resembles the well-known Van der Monde determinant.This determinant was the subject of a very intensiveanalysis [12–15]. However, all of these papers wereproposing only the procedures that relied on programprocedures (mostly iterative ones) in determining the value

Table 1 Nomenclature used in the following parts of the paper

Symbol Meaning

F fundamental frequency of processing signal

s(t) band-limited input analogue signal

M number of signal spectrum components

ADC analogue-to-digital converter

a0 average value of the input signal

ak amplitude of the kth harmonic

k number of the harmonic

ck phase angle of the kth harmonic

X system =

a0 a1 sina1,1 a2 sina2,1 . . . aM sinaM ,1 a1 cosa1,1 a2 cosa2,1 . . . aM cosaM ,1

a0 a1 sina1,2 a2 sina2,2 . . . aM sinaM ,2 a1 cosa1,2 a2 cosa2,2 . . . aM cosaM ,2

..

. ... ..

.. . . ..

. ... ..

.. . . ..

.

a0 a1 sina1,2M+1 a2 sina2,2M+1 . . . aM sinaM ,2M+1 a1 cosa1,2M+1 a2 cosa2,2M+1 . . . aM cosaM ,2M+1

∣∣∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣∣= a0(a1a2 . . . aM−1aM )2X 2M=1 (5)

X 2M+1 =

1 sin a1,1 sin a2,1 . . . sin aM ,1 cos a1,1 cos a2,1 . . . cos aM ,1

1 sin a1,2 sin a2,2 . . . sin aM ,2 cos a1,2 cos a2,2 . . . cos aM ,2

..

. ... ..

.. . . ..

. ... ..

.. . . ..

.

1 sin a1,2M+1 sin a2,2M+1 . . . sin aM ,2M+1 cos a1,2M+1 cos a2,2M+1 . . . cos aM ,2M+1

∣∣∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣∣

=

1 sin a1 sin 2a1 . . . sin Ma1 cos a1 cos 2a1 . . . cos Ma1

1 sin a2 sin 2a2 . . . sin Ma2 cos a2 cos 2a2 . . . cos Ma2

..

. ... ..

.. . . ..

. ... ..

.. . . ..

.

1 sin a2M+1 sin 2a2M+1 . . . sin Ma2M+1 cos a2M+1 cos 2a2M+1 . . . cos Ma2M+1

∣∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣(6)

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of the original and inverted Van der Monde’s matrix with animproved efficiency. In this paper, we derive completely newanalytical and summarised expressions for exactly solving thesystem of equations (3) by using this determinant as thestarting point of the analysis. Owing to relations derived inthis way, it is not necessary to use the standard procedurefor solving the system of equations, as suggested in [6].This procedure, in the case of an extremely complexspectral content of a signal, would require a powerfulprocessor and enough time for processing.

2.1 Determinants of the Van der Monde matrix

Let us consider the Van der Monde determinant of M order[14, 15]

DM =DM =

1 x1 x21 . . . xM−1

1

1 x2 x22 . . . xM−1

2

..

. ... ..

.. . . ..

.

1 xM x2M . . . xM−1

M

∣∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣=

∏Mk=j+1

∏M−1

j=1

(xk − xj)

(7)

If we introduce the following notation

Di, j = cof (DM ); Di, j = (−1)i+jDi, j (8)

where Di,j is the cofactor (minor) which is obtained fromdeterminant DM after the i row and j column have beeneliminated. The expansion of DM in the last row yields

DM = 1DM ,1 + xM DM ,2 +·· ·+ xr−1M DM ,r +·· ·+ xM−1

M DM ,M

(9)

On the other hand, we have

DM = (xM − x1)(xM − x2)...(xM − xM−1)∏M−1

k=j+1

∏M−2

j=1

(xk − xj)

(10)

DM ,M =∏M−1

k=j+1

∏M−2

j=1

(xk − xj) (11)

DM = (xM − x1)(xM − x2) . . . (xM − xM−1)DM ,M (12)

DM,M is the coefficient along with xM−1M in developing DM

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from the degrees of xM. This yield

DM ,M−1 = −(x1 + x2 + · · · + xM−1)DM ,M (13)

DM ,M−2 = (−1)2∑

(x1 + x2 + · · · + xM−1)2DM ,M (14)

DM ,r = (−1)M−r∑

(x1 + x2 + · · · + xM−1)M−rDM ,M (15)

DM ,1 = (−1)M−1(x1x2 . . . xM−1)DM , M (16)

This infers that

DM ,r = (−1)M−r(x1x2 . . . xM−1)∑ 1

(x1x2 . . . xM−1)r−1

DM ,M

(17)

DM ,r = (x1x2 . . .xM−1)∑ 1

(x1x2 . . . xM−1)r−1

DM ,r (18)

Now, we can determine Dp,q and Dp,q

DM =

1 x1 . . . xq−11 . . . xM−1

1

..

. ...

. . . ...

. . . ...

1 xp . . . xq−1p . . . xM−1

p

..

. ...

. . . ...

. . . ...

1 xM . . . xq−1M . . . xM−1

M

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

= (−1)n−p

1 x1 . . . xq−11 . . . xM−1

1

..

. ...

. . . ...

. . . ...

1 xp−1 . . . xq−1p−1 . . . xM−1

p−1

1 xp+1 . . . xq−1p+1 . . . xM−1

p+1

..

. ...

. . . ...

. . . ...

1 xM . . . xq−1M . . . xM−1

M

1 xp . . . xq−1p . . . xM−1

p

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣⇒ Dp,q = (−1)M−p(x1 . . . xp−1xp+1 . . . xM )

×∑ 1

(x1 . . . xp−1xp+1 . . . xM )q−1

× DM

(xM − xp)(xM−1 − xp) . . . (xp+1 − xp)

(xp − xp−1) . . . (xp − x1)

(19)

It follows that (see (20))

Dp,q = DM

(xp − x1)(xp − x2) . . . (xp − xp−1)(xp − xp+1) . . . (xp − xM )

× (x1 . . . xp−1xp+1 . . . xM )∑ 1

(x1 . . . xp−1xp+1 . . . xM )q−1

Dp,q = (−1)p+qDp,q (20)


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(the sum of all inverted (q 2 1) products of different indices).

2.2 Derivation of the new relationsfor solving the observed system of equations

In the case of system of equations that has been formedfollowing the suggested concept of processing of the input


signals (3), what we obtain is that, instead of the x variablesin the above expressions, it is necessary to take in thetrigonometric values for a1, a2, a3, . . . , a2M+1, as it wasdefined in (4). The given determinant (6) can be transformedin the following way (by using Euler’s formulas) (see (21))

The cofactors that correspond to the observed system ofequations (3) are (see (22) and (23))

X 2M+1 = 1

22Me−M (p/2)i

1 ea1i − e−a1i e2a1i − e−2a1i . . . eMa1i − e−Ma1i

1 ea2i − e−a2i e2a2i − e−2a2i . . . eMa2i − e−Ma2i

..

. ... ..

.. . . ..

.

1 ea2M+1i − e−a2M+1i e2a2M+1i − e−2a2M+1i . . . eMa2M+1i − e−Ma2M+1i

∣∣∣∣∣∣∣∣∣∣ea1i + e−a1i e2a1i + e−2a1i . . . eMa1i + e−Ma1i

ea2i + e−a2i e2a2i + e−2a2i . . . eMa2i + e−Ma2i

..

. ...

. . . ...

ea2M+1i + e−a2M+1i e2a2M+1i + e−2a2M+1i . . . eMa2M+1i + e−Ma2M+1i

∣∣∣∣∣∣∣∣∣∣= 1

22Me−M (p/2)i

1 ea1i − e−a1i e2a1i − e−2a1i . . . eMa1i − e−Ma1i∣∣

ea1i + e−a1i e2a1i + e−2a1i . . . eMa1i + e−Ma1i∣∣

= 1

22Me−M (p/2)i

1 ea1i − e−a1i e2a1i − e−2a1i . . . eMa1i − e−Ma1i ea1i e2a1i . . . eMa1i∣∣ ∣∣

= (−1)M

22Me−M (p/2)i

1 e−a1i e−2a1i . . . e−Ma1i ea1i e2a1i . . . eMa1i∣∣ ∣∣

= (−1)(M (M−1)/2)

22Me−M (p/2)i

e−Ma1i e−(M−1)a1i . . . e−a1i 1 ea1i e2a1i . . . eMa1i∣∣ ∣∣

= (−1)(M (M−1)/2)

22Me−M (p/2)ie−M (a1+···+a2M+1)i

1 ea1i e2a1i . . . e(M−1)a1i eMa1i e(M+1)a1i . . . e2Ma1i∣∣ ∣∣

= (−1)(M (M − 1)/2)

22Me−M (p/2)ie−M (a1 + · · · + a2M+1)i ∏2M+1

j=k+1

∏2M

k=1

(eaj i − eak i)

= (−1)(M (M + 1)/2)iM

2Me−M (a1 + · · · + a2M+1)i ∏2M+1

j=k+1

∏2M

k=1

e(aj+ak/2)i2i sinaj − ak

2

( )

= (−1)(M (M + 1)/2)iM

2Me−M (a1 + · · · + a2M+1)ieM (a1 + · · · + a2M+1)i2(2M (2M+1)/2)i(2M (2M+1)/2) ∏2M+1

j=k+1

∏2M

k=1

sinaj − ak

2

⇒ X2M+1 = (−1)(M (M+1)/2)22M2 ∏2M+1

j=k+1

∏2M

k=1

sinaj − ak

2(21)

X 2M+1,1 =

s(t1) sin a1 sin 2a1 . . . sin Ma1 cos a1 cos 2a1 . . . cos Ma1

s(t2) sin a2 sin 2a2 . . . sin Ma2 cos a2 cos 2a2 . . . cos Ma2

..

. ... ..

.. . . ..

. ... ..

.. . . ..

.

s(t2M+1) sin a2M+1 sin 2a2M+1 . . . sin Ma2M+1 cos a2M+1 cos 2a2M+1 . . . cos Ma2M+1

∣∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣(22)

X 2M+1,2 =

1 s(t1) sin 2a1 . . . sin Ma1 cos a1 cos 2a1 . . . cos Ma1

1 s(t2) sin 2a2 . . . sin Ma2 cos a2 cos 2a2 . . . cos Ma2

..

. ... ..

.. . . ..

. ... ..

.. . . ..

.

1 s(t2M+1) sin 2a2M+1 . . . sin Ma2M+1 cos a2M+1 cos 2a2M+1 . . . cos Ma2M+1

∣∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣(23)

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and so on. The cofactors given above based on the followingdevelopment can be written as

X2M+1,1 = s(t1)X 11 + s(t2)X 1

2 + · · · + s(t2M+1)X 12M+1 (24)

X 11, X1

2, . . . , X 12M+1 are the cofactors, obtained from cofactor

X2M+1,1 after the corresponding row as well as the firstcolumn has been eliminated. The second cofactor is derivedfrom the expansion of X2M+1 along such a column. For thispurpose, we must determine X q

p as cofactors of X2M+1.After the intensive mathematical calculation (Appendix) weobtain (see (25))

(the summing is done by all of the Ms of the set a1, a2, . . . ,a2M+1).

For 2 ≤ q ≤ M + 1 (see (26))

q = M + r + 1 ^ 1 ≤ r ≤ M − 1

Xqp = (−1)(M (M−1)/2)22M (M−1)+1

×∏2M+1

j=k+1

∏2Mk=1 sin(aj − ak/2)∏2M+1

1≤k=p sin(ap − ak/2)

×∑

cosa1 + · · · + ap−1 + ap+1 + · · · + a2M+1

2

{

− (a1 + · · · + ap−1 + ap+1 + · · · + a2M+1)M−r

}q = 2M + 1

Xqp = (−1)(M (M − 1)/2)22M (M−1)+1

×∏2M+1

j=k+1

∏2Mk=1 sin(aj − ak/2)∏2M+1


× cosa1 + · · · + ap−1 + ap+1 + · · · + a2M+1

2(27)

Based on (analytical) relation derived in this way theunknown parameters of the signal (amplitude, phase), canbe determined through a simple division of the expressionthat represents solution of the adequate co-determinantswith the expression that represents an analytical solution tothe system determinant. The analysed determinants and co-determinants of the system (3) were used only as thestarting function (of the polynomial form), to the purpose ofobtaining explicit and summarised analytical expressionswhich can be used further to perform the calculation of theunknown signal parameters (practically online for the

142


known frequency spectrum of input signals). It is a fact thatthe obtained system of equations (3) can be described, afterthe processing, by a special form of the determinant (whichis summarised as the Van der Monde’s determinant). Thisfact enables factoring and application of transformationsthat can be applied only on determinants.

2.3 Verification of derived relations

Table 2 presents a comparison of the results obtained throughthe application of the derived relations for solving the observedsystem of equations (3) and Gaussian elimination with partialpivoting (GEPP) algorithm, offered in the MATLAB programpackage itself (all of the calculations are done in IEEE standarddouble floating point arithmetic with unit round-offu ≃ 1.1 × 10−16). This represents a practical verification ofthe proposed algorithm for a case of ideal sampling (withoutan error in taking the value of the sample and determiningthe frequency of the processed signal).

The values in columns separated by commas correspondto the solution for derived relations with different orders(taking that M ¼ 7, f ¼ 50 Hz, tl ¼ 0.001 s). This meansthat in the column for X 1

l values 77.3751539, 65.8192372,296.0730198, 82.9370773, and so on correspond to X1

1 ¼

77.3751539, X 12 ¼65.8192372, X1

3 ¼ 296.0730198, X14 ¼

82.9370773, respectively.As it can be seen from Table 2, the derived relations

produce solutions that are practically identical to theprocedure that is most commonly used in solving system oflinear equations. The difference in the obtained values wasequal to 1 × 10−14.

3 Proposed reconstruction algorithm anduncertainty analysis

The proposed algorithm for signal reconstruction is presentedin the form of a flowchart in Fig. 1.

For the proposed algorithm, the order of the highest Mharmonic component in the processed signal spectrum mustbe known in advance or adopted in advance, accepting thatan M determined in this way is bigger than the expected(real) value. One of the well-known methods can be used toestimate the frequency spectrum. Two accurate frequencyestimation algorithms for multiple real sinusoids in whitenoise based on the linear prediction approach have beendeveloped in [16]. The first algorithm minimises theweighted least squares (WLS) cost function, subject to ageneralised unit-norm constraint. At the same time, thesecond method is a WLS estimator with a monic constraint.Both algorithms give very close frequency estimates whose

X 1p = (−1) p+q(−1)(M (M+1)/2)22M (M−1)

∏2M+1j=k+1

∏2Mk=1 sin(aj − ak/2)∏2M+1

k=1k=p

sin(ap − ak/2)

×∑

cosa1 + · · · + ap−1 + ap+1 + · · · + a2M+1

2− (a1 + · · · + ap−1 + ap+1 + · · · + a2M+1)M

{ }(25)

X qp = (−1) p+q(−1)(M (M+1)/2)22M (M−1)+1

∏2M+1j=k+1

∏2Mk=1 sin(aj − ak/2)∏2M+1


×∑

sina1 + · · · + ap−1 + ap+1 + · · · + a2M+1

2− (a1 + · · · + ap−1 + ap+1 + · · · + a2M+1)M+q−1

{ }(26)


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accuracies attain Cramer–Rao lower bound for whiteGaussian noise. A modified parameter estimator based on amagnitude phase-locked loop principle was proposed in[17]. It showed that the modified algorithm providedtracking improvements for situations in which thefundamental component of the signal became small ordisappeared for certain periods of time.

In order to recalculate unknown parameters (amplitude andphase) of the processed periodic signals, it is necessary tohave the results of the sampling of the input analoguesignals s(tl) (2). The samples of the input signal areobtained by means of sampling in a precisely defined timemoments, which are referred in relation to the detectedmoment of zero-crossing. The values of the derivedexpressions depend on the measured frequency f, becausethe values of the determinant elements are calculated basedon coefficient al, according to (5). Apart from this, otherparameters of the derived system of equations will not bedependent on the frequency of the carrier signal and thestarting moment of integration of the input signal, as wasthe case in [7].

Owing to the presence of the error in determining thesamples s(tl) and variables al, which is caused by theirdependence on the carrier frequency f of the processedsignal, in the practical applications of the proposedalgorithm, we need to have the best estimate of the givenvalues, according to the criterion assumed. This can bedone by means of recalculation of the values s(tl) and al,through N passages (N is arbitrary). In this process, weform series s(tl)n and anl (1, . . . , N ), as it is given in theproposed algorithm. The random errors Dn of measurementsare unbiased, E(Dn) ¼ 0 have the same variancevar(Dn) ¼ s2, and are not mutually correlated. Under theseassumptions, the least squares (LS) estimator minimises theresidual sum of squares [18]

S(s(tl)) =∑N

n=1

pn(s(tl) − s(tl)n)2 (28)

S(al) =∑N

n=1

pn(al − anl)2 (29)

Table 2 Verification of the derived expression for solving a

system of equations with which the reconstruction of the observed

signal is done

X2M+1 X 1l ; l = 1, . . . , 15

proposed

algorithm

96.2746562 77.3751539; 240.7369977;

585.7387744; 1048.6023899;

1601.6571515; 2122.6636989;

2515.8984526; 2655.5007221;

557.8121810; 2190.4815359;

1686.1064501; 1129.5098525;

644.9058063; 279.7307319;

94.0066154

GEPP

algorithm

96.2746562 77.3751539; 240.7369977;

585.7387744; 1048.6023899;

1601.6571515; 2122.6636989;

2515.8984526; 2655.5007221;

557.8121810; 2190.4815359;

1686.1064501; 1129.5098525;

644.9058063; 279.7307319;

94.0066154


where pn ¼ m/s2, m . 0 (m is arbitrary). By minimising thefunction S, we obtain the LS estimators s(tl), al of the valuess(tl) and al as

s(tl) =∑N

n=1 pns(tl)n∑Nn=1 pn

(30)

al =∑N

n=1 pnanl∑Nn=1 pn

(31)

The value of N will depend on the required speed ofprocessing – the higher the N, the more precise theestimation of the value is. In particular, the LS method hasbeen applied to a variety of problems in the realengineering field due to its low computational complexity.In the concrete case, the estimation procedure does notrequire the matrix inversion and is considerably lessdemanding from the processor aspect, than the methods

Fig. 1 Flowchart of the proposed reconstruction algorithm

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described in [19]. In addition to this, when the proposedalgorithm is used in simulations, the estimation of the givenvariables (30), (31) will not be necessary, which willsignificantly reduce the processing time needed for itsrealisation. The proposed solution can be modified, in orderto reduce the error in determining the samples of the inputsignal. In [20] it was shown that the implementation ofsampling and reconstruction with internal antialiasingfiltering radically improves performances of digitalreceivers, enabling reconstruction with a much lower error.

It is necessary to note that the frequencies of the proposedsignal may show some differences in relation to the givenones – that is to say that a large or a small frequencymismatch (FM) may exist in real applications. In [21, 22], anew LMS-based Fourier analyser is proposed. This analyserworks simultaneously – on one side, it estimates thediscrete Fourier coefficients and on the other side, itaccommodates the FM. This analyser can very wellcompensate for the performance degeneration because ofFM. However, in the LS procedure proposed here, theestimation of the samples s(tl) and variables al is done byminimising the S function (28) and (29). Furthermore, whenthis is done, the derived analytical expressions are used todetermine the unknown Fourier coefficients. In addition tothis, with every passage of the described algorithm (Fig. 1),the moment of sampling is referred to the detected zero-crossing of the processed signal, and its basic frequency isalso calculated at the same time. In this way, thedetermination of the unknown parameters of the processedperiodic signal is less dependent on the possible FM, whencompared to the algorithm analysed in [21, 22] (theparameters analysed in our paper are less inter-dependent).However, if the signal-to-noise ratio was to be very lowand accompanied by a marked FM, it would still bepossible to modify the described estimation procedure in away presented in [21, 22], without adding any requirementsto the realisation.

Fig. 2 shows the influence of the error in determining thefrequency of the carrier signal on the relative error whiledetermining the value of system determinant, for variousharmonic contents of the input periodic signal. Theimmunity of algorithm could be improved by applying amore complex algorithm for the detection of signal zero-

Fig. 2 Relative error in calculation of the system determinant asfunction of error in synchronisation with frequency of fundamentalharmonic of the input signal (f ¼ 50 Hz, tl ¼ 0.001 s)

144


crossing moments [23]. In [24] special attention is givento uncertainty analysis for the calibration of high-speedcalibration systems. The effect of the uncertainty created bythe time base generator ( jitter) can be modelled as non-stationary additive noise. Daboczi [24] also developed amethod to calculate an uncertainty bound around thereconstructed waveform, based on the required confidencelevel. The error that appears because of the supposed non-idealities that exist in the suggested reconstruction model iswithin the boundaries specified by [24, 25]. A sensitivityfunction is commonly formulated assuming noise-free data.This function provides point-wise information about thereliability of the reconstructed signal before the actualsamples of the signal are taken. In [25], the minimumerror bound of signal reconstruction is derived assumingnoise data.

Unlike the procedure described in [7], the algorithmproposed here is much less sensitive to the variation in thefrequency of the carrier signal. The moments tl in which thesampling of the input signal is done can be completelyrandom (asynchronous) and independent of the frequencyof the processed signal, because of the way in which theyare defined in (4). The interval between two consecutivesamples is actually dependent primarily on the speed of theS/H (sample and hold) circuit and the analogue-to-digital(AD) conversion circuit, with which a numeric equivalentto the sample of the input signal is formed. However, becauseof practical realisation and the way in which real sigma–deltaADC functions, the tl moments can be defined as tl ¼ ltsample,where tsample ¼ 1/fS ( fS is the sampling frequency). The startof the sampling must be referred in relation to the detectedzero-crossing of the input signal.

3.1 Numerical complexity of proposed algorithm

The derived relations require a total of (2M+ 1) (4M3 + 2M+ 1)

multiplications and (2M + 1) 22M − 1

22MM

( )( )additions,

to be realised. However, because of the method used todetermine the unknown parameters of the processed signals(as specified by algorithm in Fig. 1), the necessary numberof numeric operations is significantly reduced. Namely, thederived relations have many common factors in the formedproducts, so that they are abbreviated during the divisionoperations (Appendix). For this reason, the total number ofoperations is reduced to 18M2 + 12M + 2, meaning that theproposed algorithm required 9/2(2M + 1)2 flops.

The analytical expressions (16)–(27) are correct, whereasthe error that occurs in their implementation appearsbecause of the numeric procedure used in their calculation(in realising addition and multiplication). The values of thepossible error are defined and analysed in [26], although itwas also proved that the algorithms that solve theVandermond-like systems (3) are much more accurate (butno more backward stable) than GEPP (which requires2/3(2M + 1)3 flops) or algorithm with QR factorisation(which requires 4/3(2M + 1)3 flops).

It is well known that the computational load of the iterativestep involving fast Fourier transforms (FFTs) does not changewith the number of sampling values for some of the non-matrix implementation, but the speed of convergence isimproved if more points are available [27]. A larger numberof sampling points will result in the appearance of largematrices; the same will happen in the case of an inputsignal with large spectrum. Therefore the computational


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load for standard matrix methods (either iterative or thoseusing pseudo-inverse matrices) increases quickly. Thus,they may be extremely efficient for the situation with a fewsampling points, but slow if there are many samplingpoints. Quite the contrary happens for the methodsproposed here. The suggested algorithm is non-iterative andtherefore much faster. Only the number of unknownsdefines the number of samples required by the proposedalgorithm in order to perform the reconstruction in theobserved system (3) (2M + 1 unknowns are parameters).This is the main reason why we think that our derivedanalytical solution is more computationally attractive formoderately sized problems [26]. Moreover, this featuremakes it feasible for large reconstruction problems. Inaddition, the method is free from the effects of spectralleakage, which is a common problem in the reconstructionalgorithm based on the use of FFT [27].

3.2 Computing time

The suggested algorithm can be applied in operation withsigma–delta ADC, thus enabling high resolution and speedin processing of input signals. This is an importantdifference to be taken into consideration when comparingits implementation in this approach (to processing) to theresults presented in [7]. The time needed to perform thenecessary number of samplings of the input signal thatis the object of reconstruction can be defined as(2M + 1)tsample, which represents the value approximateto the time needed for reconstruction (in simulation). Inpractical applications of the proposed algorithm, thedetermined time for the reconstruction of the processingsignal ought to be increased by the time necessary toestimate the variables s(tl) and al (this time is directlydependent on the value N ), and the time interval Dt whichis necessary to perform all the other recalculationsaccording to the proposed algorithm (Fig. 1). Owing to allthat has been said, the reconstruction time can be defined asN (2M + 1)tsample + Dt ≃ N (1/f ) + Dt, because of thenecessary synchronisation with the zero-crossing of theinput signal. The speed of the proposed algorithm makes itas fast as the algorithms analysed in [28, 29].

Reeves [30] gives a measurement of the required processortime, in the realisation of the matrix method in thereconstruction of signals, in the form in which it isimplemented in many program packages. The methodsuggested by this paper does not require any specialmemorisation of the transformation matrix, nor does itrequire recalculation of the inversion matrix. In this way, itis much more efficient in implementation and it is notlimited only to sparse matrices.


This procedure can be used for the spectral analysis as well,where it is possible to find out the amplitude and the phasevalues of the signal harmonic, based on the set (predicted)system of equations. By taking a systematic approach inconducting the described procedure, it is possible toestablish the exact spectral content and, after this has beendone, to perform the optimisation of the proposedalgorithm. With this, the algorithm will be adapted to thereal form of the signal. The accuracy of signalreconstruction can be guaranteed in practical applications innoisy environments with the use of a powerful processorwith adequate filtering.

4 Simulation results

Additional testing of the proposed algorithm was carried outby simulation in the program packages MATLAB andSIMULINK (version 7.0). Since measurement is corruptedby noise, the reconstruction is an estimation task, that is,the reconstructed signal may vary, depending on the actualnoise record. We investigate the issue of noise and jitter onthe estimation method. The presence of the noise and jittercauses false detection of signal zero-crossing moments,giving an incorrect calculation of derived relations. Apartfrom this, an error occurs in determining the value of thesamples of the processed signal. This error appears becauseof the imprecision in determining the time moment atwhich sampling is done, as well because of the errorintroduced by the real ADC.

Since the suggested reconstruction algorithm is practicallybased on the standard hardware components with which thesampling of the input analogue signal is done, the modelthat was used in simulation was the system shown in Fig. 3.A sigma–delta ADC with corresponding circuit for S/Hwas used to perform the conversion; at the same time, themeasuring of the frequency of the carrier signal was donein the way suggested in [23], that is, by using comparator(Schmitt-triggers), we can detect the passing of the multi-harmonic input signal through zero and in this way detectthis frequency.

The input signal is formed as a superposition ofharmonic components within the ‘multi-harmonic inputsignal’ block. Since the presence of the noise and jittercauses false detection of signal zero-crossing moments,there is a separate model that simulates their impact onthe signal that is processed according to the proposedalgorithm. All Taylor expansions of jitter are biased [31];therefore the correct way to include jitter is to include avariable delay in the signal path. The model by whichthe jitter presence is simulated is made of the pulsegenerator and random number, whose outputs are taken

Fig. 3 SIMULINK model of the circuit for the realisation of the proposed reconstruction algorithm

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to the circuit with the variable transport delay. Fig. 3shows the addition of white Gaussian noise to thecomplex periodic signal. In the original toolbox, all thepossible noise sources (mainly the contributions ofthe operational amplifiers and of the voltage references)were supposed to be white. The noise power in theblock ‘band-limited white noise’ is the height of thepower spectral density (PSD), expressed in V2/Hz. Byusing the simple relay block, shown in Fig. 3, the signalformed in this way is transformed into a series of right-angle impulses, by means of which the frequency ofthe fundamental harmonic was measured within the‘S-function’ block [23].

A sigma–delta ADC model described in [32] was used(the effective resolution of the ADC during the simulationwas 24 bit and sampling rate fS ¼ 1 kHz) to performthe sampling and to determine the numeric equivalent forthe value of the sample of the input power signal. In thecourse of the simulation conducted in this way, the outputPSD of the ideal, thermal noise affected and clock jitteraffected was in the range of 2100 to 2170 dB forthe signal-to-noise distortion ratio ranged between 85and 96 dB.

A quantiser circuit is introduced in the ADC model. Thequantisation error is a very important problem, because thereconstruction algorithm proposed here is of a quitesophisticated form and some operations, like determinantcalculation for example, are badly conditioned task andmay considerably amplify the quantisation errors.

After the values of the input signals samples weredetermined through the above-described ADC model, thesewere introduced in the ‘subsystem3’ block, together withthe information about the measured frequency of the inputsignal, so that the values of the unknown amplitudes andphases of the input periodic signal can be established, basedon the derived relations. During the simulation, theparameters of the input signal correspond to the valuesgiven in Table 3.

A signal containing the first seven harmonics was used,with the fundamental frequency f ¼ 50 Hz. Table 3 definesthe amplitude and phase values of the signal. Thesuperposed noise and jitter will, in simulation performed inthis way, cause a relative error in detection on fundamentalfrequency of 0.0001%. It can be seen that the accuracy ofthe proposed algorithm is within the limits that are attainedin processing a signal of this form in [29, 33]. The errors inthe amplitude and phase detection are mainly because ofthe error while measuring the input signal samples and theerror during determining the value of the derived equations.

Table 3 Simulation results of signal reconstruction by the

proposed algorithm

Harmonic

number

Amplitude,

Vpp

Phase,

rad

Proposed reconstruction

algorithm

Amplitude

error, %

Phase

error, %

1 1 p 0.0016 0.0021

2 0.73 p/3 0.0023 0.0018

3 0.64 0 0.0021 0.0022

4 0.55 p/6 0.0017 0.0019

5 0.32 p/4 0.0018 0.0024

6 0.27 p/12 0.0023 0.0017

7 0.14 0 0.0024 0.0023

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5 Conclusion

The algorithm proposed in this paper is a new complexity-reduced algorithm for estimation of the Fourier coefficient.The derived analytical expression opens a possibility toperform fast calculations of the basic parameters of signals(the phase and the amplitude), with a low numeric error.All the necessary hardware resources can be satisfied by aDSP of standard features and real sigma–delta ADC. Thesuggested concept can be used as a separate algorithm aswell, for the spectral analysis of the processed signals.Based on the identified parameters of the AC signals, wecan establish all the relevant values in the electric utilities(energy, power, RMS values). The measurement uncertaintyis a function of the error in synchronisation withfundamental frequency of processing signal (because of thenon-stationary nature of the jitter-related noise and whiteGaussian noise) and the error that occurs duringdetermining the values of the samples of the processedsignal. The simulation results show that the proposedalgorithm can offer satisfactory precision in thereconstruction of periodic signals in a real environment.

6 References

1 Proakis, J.G., Manolakis, D.G.: ‘Digital signal processing: principles,algorithms, applications’ (Prentice-Hall, Englewood Cliffs, NJ, 1996, 3rdedn.)

2 Prendergast, R.S., Levy, B.C., Hurst, P.J.: ‘Reconstruction of band-limited periodic nonuniformly sampled signals through multirate filterbanks’, IEEE Trans. Circ. Syst.-I, 2004, 51, (8), pp. 1612–1622

3 Marziliano, P., Vetterli, M., Blu, T.: ‘Sampling and exact reconstructionof bandlimited signals with additive shot noise’, IEEE Trans. Inf.Theory, 2006, 52, (5), pp. 2230–2233

4 Margolis, E., Eldar, Y.C.: ‘Reconstruction of nonuniformly sampledperiodic signals: algorithms and stability analysis’. Proc. 2004 11thIEEE Int. Conf. Electronics, Circuits and Systems, 2004, (ICECS2004), 13–15 December 2004, pp. 555–558

5 Sun, W., Zhou, X.: ‘Reconstruction of band-limited signals from localaverages’, IEEE Trans. Inf. Theory, 2002, 48, (11), pp. 2955–2963

6 Muciek, A.K.: ‘A method for precise RMS measurements of periodicsignals by reconstruction technique with correction’, IEEE Trans.Instrum. Meas., 2007, 56, (2), pp. 513–516

7 Petrovic, P.: ‘New approach to reconstruction of nonuniformly sampledAC signals’. Proc. 2007 IEEE Int. Symp. Industrial Electronics (ISIE2007), Vigo, Spain, 2007, pp. 1693–1698

8 Bos, A.V.D.: ‘Estimation of Fourier coefficients’, IEEE Trans. Instrum.Meas., 1989, 38, (5), pp. 1005–1007

9 Bos, A.V.D.: ‘Estimation of complex Fourier coefficients’, IEE Proc.Control Theory Appl., 1995, 142, (3), pp. 253–256

10 Pintelon, R., Schoukens, J.: ‘An improved sine-wave fitting procedurefor characterizing data acquisition channels’, IEEE Trans. Instrum.Meas., 1996, 45, (2), pp. 588–593

11 Xiao, Y., Tadokaro, Y., Shida, K.: ‘Adaptive algorithm based on leastmean p-power error criterion for Fourier analysis in additive noise’,IEEE Trans. Signal Proc., 1999, 47, (4), pp. 1172–1181

12 Agrawal, M., Prasad, S., Roy, S.C.D.: ‘A simple solution for the analyticinversion of Van der Monde and confluent Van der Monde matrices’,IETE J. Res, 2001, 47, (5), pp. 217–219

13 Gohberg, I., Olshevsky, V.: ‘The fast generalized Parker–Traubalgorithm for inversion of Van der Monde and related matrices’,J. Complexity, 1997, 13, (2), pp. 208–234

14 Jog, C.S.: ‘On the explicit determination of the polar decomposition inn-dimensional vector spaces’, J. Elast., 2002, 66, (2), pp. 159–169

15 Neagoe, V.E.: ‘Inversion of the Van der Monde matrix’, IEEE SignalProcess. Lett., 1996, 3, (4), pp. 119–120

16 So, H.C., Chan, K.W., Chan, Y.T., Ho, K.C.: ‘Linear predictionapproach for efficient frequency estimation of multiple real sinusoids:algorithms and analyses’, IEEE Trans. Signal Process., 2005, 53, (7),pp. 2290–2305

17 Wu, B., Bodson, M.: ‘Frequency estimation using multiple source andmultiple harmonic components’. Proc. 2002 American Control Conf.,2002, vol. 1, pp. 21–22

18 Seber, G.: ‘Linear regression analysis’ (Wiley, New York, 1977)


www.ietdl.org

19 Reeves, S.J., Heck, L.P.: ‘Selection of observations in signalreconstruction’, IEEE Trans. Signal Process., 1995, 43, (3),pp. 788–791

20 Poberezhskiy, Y.S., Poberezhskiy, G.Y.: ‘Sampling and signalreconstruction circuits performing internal antialiasing filtering andtheir influence on the design of digital receivers and transmitters’,IEEE Trans. Circ. Syst.-I, 2004, 51, (1), pp. 118–129

21 Xiao, Y., Ward, R.K., Ma, L., Ikuta, A.: ‘A new LMS-based Fourieranalyzer in the presence of frequency mismatch and applications’,IEEE Trans. Circ. Syst.-I, 2005, 52, (1), pp. 230–245

22 Xiao, Y., Ward, R.K., Xu, L.: ‘A new LMS-based Fourier analyzer inthe presence of frequency mismatch’. Proc. 2003 Int. Symp. Circuitsand Systems 2003, (ISCAS’03), vol. 4, pp. 369–372

23 Petrovic, P.: ‘New digital multimeter for accurate measurement ofsynchronously sampled AC signals’, IEEE Trans. Instrum. Meas.,2004, 53, (3), pp. 716–725

24 Daboczi, T.: ‘Uncertainty of signal reconstruction in the case of jitter andnoisy measurements’, IEEE Trans. Instrum. Meas., 1998, 47, (5),pp. 1062–1066

25 Wang, G., Han, W.: ‘Minimum error bound of signal reconstruction’,IEEE Signal Process. Lett., 1999, 6, (12), pp. 309–311

26 Nigham, N.J.: ‘Accuracy and stability of numerical algorithms’ (SIAM,2002, 2nd edn.)

27 Feichtinger, H.G.: ‘Reconstruction of band-limited signals fromirregular samples, a short summary’. Second Int. Workshop on DigitalImage Processing and Computer Graphics with Applications, 1991,pp. 52–60

28 Cooklev, T.: ‘An efficient architecture for orthogonal wavelettransforms’, IEEE Signal Process. Lett., 2006, 13, (2), pp. 77–79

29 Schoukens, J., Rolain, Y., Simon, G., Pintelon, R.: ‘Fully automatedspectral analysis of periodic signals’, IEEE Trans. Instrum. Meas.,2003, 52, (4), pp. 1021–1024

30 Reeves, S.J.: ‘An efficient implementation of the backward greedyalgorithm for sparse signal reconstruction’, IEEE Signal Process.Lett., 1999, 6, (10), pp. 266–268

31 Coakley, K.J., Wang, C.M., Hale, P.D., Clement, T.S.: ‘Adaptivecharacterization of jitter noise in sampled high-speed signals’, IEEETrans. Instrum. Meas., 2003, 52, (5), pp. 1537–1547

32 Hoseini, H.Z., Kale, I., Shoaei, O.: ‘Modeling of switched-capacitordelta–sigma modulators in SIMULINK’, IEEE Trans. Instrum. Meas.,2005, 54, (4), pp. 1646–1654

33 Hidalgo, R.M., Fernandez, J.G., Rivera, R.R., Larrondo, H.A.: ‘Asimple adjustable window algorithm to improve FFT measurements’,IEEE Trans. Instrum. Meas., 1996, 51, (1), pp. 31–36

7 Appendix

We must determine Xqp for 1 ≤ q ≤ M + 1. Therefore

Xqp = (−1)p+qEq

p (32)

where Eqp is obtained from X2M+1 overturning p row and q

column. We know that X2M+1 ¼ E2M+1.

E1p =

e−M (p/2)i

22Mea1i − e−a1i · · · eMa1i − e−Ma1i∣∣ ea1i

+ e−a1i e2a1i + e−2a1i · · · eMa1i + e−Ma1i∣∣p

(33)

The index p shows that p row is eliminated from thisdeterminant.

E1p =

e−M (p/2)i

22Mea1i − e−a1i · · · eMa1i − e−Ma1i ea1i∣∣

+ e−a1i e2a1i + e−2a1i · · · eMa1i + e−Ma1i∣∣p

= (−1)M

2Me−M (p/2)i

× e−a1i · · · e−Ma1i ea1i e2a1i · · · eMa1i∣∣ ∣∣

p


= (−1)M (M+1)/2

2Me−M (p/2)i

× e−Ma1i · · · e−a1i ea1i · · · eMa1i∣∣ ∣∣

p

= (−1)M (M+1)/2

2Me−M (p/2)ie−M (a1+···+ap−1+ap+1+···+a2M+1)i

× 1 ea1i · · · e(M−1)a1i e(M+1)a1i · · · e2Ma1i∣∣ ∣∣

p

= (−1)M (M+1)/2

2Me−M (p/2)ie−M (a1+···+ap−1+ap+1+···+a2M+1)i

×Dp,M+1(xj = eaj i)

(34)

Dp,M+1(xj = eaj i) = D2M+1

(xp − x1)(xp − x2) . . . (xp − xp−1)

(xp − xp+1) . . . (xp − x2M+1)

× (x1 . . . xp−1xp+1 . . . x2M+1)

×∑ 1

(x1 . . . xp−1xp+1 . . . x2M+1)M

∣∣∣∣xj=eaj i (35)

D2M+1(xj = eaj i) = 2M (2M+1)eM (2M+1)(p/2)i

× eM (a1+a2+···+a2M+1)i∏2M+1

j=k+1

∏2M

k=1

sinaj − ak

2(36)

(xp − x1)(xp − x2) . . . (xp − xp−1)(xp − xp+1) . . .

(xp − x2M+1)|xj=eaj i = (−1)M 22M eMapi

× e1/2(a1+···+ap−1+ap+1+···+a2M+1)i∏2M+1

k=1k=p

sinap − ak

2(37)

D2M+1(xj = eaj i)

= (−1)M 2M (2M+1)eM (p/2)ieM (a1+a2+···+a2M+1)i

×∏2M+1

j=k+1

∏2M

k=1

sinaj − ak

2

× (−1)M 2−2M e−Mapie−1/2(a1+···+ap−1+ap+1+···+a2M+1)i

× 1∏2M+1k=1k=p

sin(ap − ak/2)e(a1+···+ap−1+ap+1+···+a2M+1)i

×∑

e−(a1+···+ap−1+ap+1+···+a2M+1)i|M

⇒ E1p = (−1)M (M+1)/222M (M−1)

×∏2M+1

j=k+1

∏2Mk=1 sin(aj − ak/2)∏2M+1

k=1k=p

sin(ap − ak/2)

× e1/2(a1+···+ap−1+ap+1+···+a2M+1)i

×∑

e−(a1+···+ap−1+ap+1+···+a2M+1)i|M (38)

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or

E1p = (−1)M (M+1)/222M (M−1)

∏2M+1j=k+1

∏2Mk=1 sin(aj − ak/2)∏2M+1

k=1k=p

sin(ap − ak/2)

×∑

cosa1 + · · · + ap−1 + ap+1 + · · · + a2M+1

2

{

−(a1 + · · · + ap−1 + ap+1 + · · · + a2M+1)M

}(39)

It follows that

X 1p

X 2M+1

= (−1)p+1

22M

1∏2M+11≤k=p sin(ap − ak/2)

×∑

cosa1 + · · · + ap−1 + ap+1 + · · · + a2M+1

2

{

−(a1 + · · · + ap−1 + ap+1 + · · · + a2M+1)M

}(40)

Now, we can determine Eqp for 2 ≤ q ≤ M + 1 (see (41))

It follows that

(Dp,M−q+2−Dp,M+q)(xj =eaj i)

= D2M+1∏2M+11≤k=p (xp−xk )

(x1x2 ...xp−1xp+1 ...x2M+1)

×

∑ 1

(x1x2 ...xp−1xp+1 ...x2M+1)M−q+1

−∑ 1

(x1x2 ...xp−1xp+1 ...x2M+1)M+q−1

⎧⎪⎪⎨⎪⎪⎩

⎫⎪⎪⎬⎪⎪⎭|xj=eaj i

⇒Eqp= (−1)(M (M+1)/2)22M (M−1)+1

∏2M+1j=k+1

∏2Mk=1 sin(aj−ak/2)∏2M+1

1≤k=p sin(ap−ak/2)

×∑

sin

a1+···+ap−1+ap+1+···+a2M+1

2−(a1+···+ap−1+ap+1

+···+a2M+1)M+q−1

⎧⎪⎪⎨⎪⎪⎩

⎫⎪⎪⎬⎪⎪⎭ (42)

It follows that

X qp

X2M+1

= (−1)p+q

22M−1

1∏2M+11≤k=p sin(ap−ak/2)

×∑

sin

a1+···+ap−1+ap+1+···+a2M+1

2−(a1+···+ap−1+ap+1

+···+a2M+1)M+q−1

⎧⎪⎪⎨⎪⎪⎩

⎫⎪⎪⎬⎪⎪⎭; 2≤q≤M

(43)

Eqp = e−(M−1)(p/2)i

22M−1 1 ea1i − e−a1i . . . e(q−2)a1i − e−(q−2)a1i eqa1i − e−qa1i . . .∣∣

eMa1i − eMa1i ea1i + e−a1i . . . eMa1i + e−Ma1i∣∣p

= e−(M−1)(p/2)i

2M 1 ea1i − e−a1i . . . e(q−2)a1i − e−(q−2)a1i eqa1i − e−qa1i . . . eMa1i − e−Ma1i ea1i∣∣. . . e(q−2)a1i e(q−1)a1i + e−(q−1)a1i eqa1i . . . eMa1i

∣∣p

= (−1)M−1

2Me−(M−1)(p/2)i

1 e−a1i . . . e−(q−2)a1i e−qa1i . . . e−Ma1i ea1i . . .∣∣

e(q−2)a1i e(q−1)a1i + e−(q−1)a1i eqa1i . . . eMa1i∣∣p

= (−1)M−1

2Me−(M−1)(p/2)i

1 e−a1i . . . e−(q−2)a1i e−qa1i . . . e−Ma1i ea1i . . . eMa1i∣∣ ∣∣

p

+(−1)M−11 e−a1i . . . e−Ma1i ea1i . . . e(q−2)a1i eqa1i . . . eMa1i∣∣ ∣∣

p

{ }

= (−1)M−1

2Me−(M−1)(p/2)i

(−1)(M (M−1)/2)e−Ma1i . . . e−qa1i e−(q−2)a1i . . . e−a1i 1 ea1i . . . eMa1i∣∣ ∣∣

p

+(−1)M−1(−1)M (M−1)/2e−Ma1i . . . e−a1i 1 ea1i . . . e(q−2)a1i eqa1i . . . eMa1i∣∣ ∣∣

p

{ }

= (−1)(M (M+1)/2)

2Me−(M−1)(p/2)ie−M (a1+···+ap−1+ap+1+···+a2M+1)i

×1 . . . e(M−q)a1i e(M−q+2)a1i . . . e(M−1)a1i . . . e2Ma1i∣∣ ∣∣

p

− 1 . . . e(M−1)a1i eMa1i . . . e(M+q−2)a1i e(M+q)a1i . . . e2Ma1i∣∣ ∣∣

p

{ }

= (−1)(M (M+1)/2)

2Me−(M−1)(p/2)ie−M (a1+···+ap−1+ap+1+···+a2M+1)i{(Dp,M−q+2 − Dp,M+q)(xj = eaj i)} (41)

148 IET Signal Process., 2011, Vol. 5, Iss. 2, pp. 138–149

& The Institution of Engineering and Technology 2011 doi: 10.1049/iet-spr.2009.0161

www.ietdl.org

while

Xqp

X 2M+1

= (−1)p+q

22M−1

1∏2M+11≤k=p sin(ap−ak/2)

sina1+···+ap−1+ap+1+···+a2M+1

2; for q=M +1 (44)

For q ¼ M + r + 1

Eqp = 1 sina1 . . . sin Ma1 cosa1 . . . cos(r − 1)a1 cos(r + 1)a1 . . . cos Ma1

∣∣ ∣∣p

= e−M (p/2)i

22M−1 1 ea1i − e−a1i . . . eMa1i − e−Ma1i ea1i + e−a1i . . . e(r−1)a1i + e−(r−1)a1i∣∣

e(r+1)a1i + e−(r+1)a1i . . . eMa1i + e−Ma1i∣∣p

= e−M (p/2)i

2M 1 ea1i − e−a1i . . . era1i − e−ra1i . . . eMa1i − e−Ma1i ea1i . . . e(r−1)a1i e(r+1)a1i . . . eMa1i∣∣ ∣∣

p

= (−1)M−1

2Me−M (p/2)i

1 e−a1i . . . e−(r−1)a1i era1i − e−ra1i e−(r+1)a1i . . . e−Ma1i∣∣

ea1i . . . e(r−1)a1i e(r+1)a1i . . . eMa1i |p

= (−1)M−1

2Me−M (p/2)i

(−1)M−11 e−a1i . . . e−(r−1)a1i e−(r+1)a1i . . . e−Ma1i ea1i . . . eMa1i∣∣ ∣∣

p

− 1 e−a1i . . . e−Ma1i ea1i . . . e−(r−1)a1i e(r+1)a1i . . . eMa1i∣∣ ∣∣

p

{ }

= (−1)M−1

2Me−M (p/2)i

(−1)M−1(−1)(M (M−1)/2)e−Ma1i . . . e−(r+1)a1i e−(r−1)a1i . . . e−a1i 1 ea1i . . . eMa1i∣∣ ∣∣

p

−(−1)(M (M−1)/2)e−Ma1i . . . e−a1i 1 ea1i . . . e−(r−1)a1i e(r+1)a1i . . . eMa1i∣∣ ∣∣

p

{ }

= − (−1)M−1

2Me−M (p/2)i(−1)(M (M−1)/2)eM (a1+···+ap−1+ap+1+···+a2M+1)i(Dp,M−r+1 + Dp,M+r+1)

⇒ q = M + r + 1 ^ 1 ≤ r ≤ M − 1

Eqp = (−1)(M (M−1)/2)22M (M−1)+1

∏2M+1j=k+1

∏2Mk=1 sin(aj − ak/2)∏2M+1


×∑

cosa1 + · · · + ap−1 + ap+1 + · · · + a2M+1

2− (a1 + · · · + ap−1 + ap+1 + · · · + a2M+1)M−r

{ }

q = 2M + 1

Eqp = ( − 1)(M (M−1)/2)22M (M−1)+1

∏2M+1j=k+1

∏2Mk=1 sin(aj − ak/2)∏2M+1

1≤k=p sin(ap − ak/2)cos

a1 + · · · + ap−1 + ap+1 + · · · + a2M+1

2(45)

It follows that

for q = M + r + 1 ^ 1 ≤ r ≤ M − 1

×Xq

p

X 2M+1

= (−1)p+r+1

22M−1

1∏2M+11≤k=p sin(ap − ak/2)

×∑

cosa1 + · · · + ap−1 + ap+1 + · · · + a2M+1

2− (a1 + · · · + ap−1 + ap+1 + · · · + a2M+1)M−r

{ }(46)

for q = 2M + 1

X qp

X 2M+1

= (−1)p+M+1

22M−1

1∏2M+11≤k=p ap − ak/2

cosa1 + · · · + ap−1 + ap+1 + · · · + a2M+1

2(47)

IET Signal Process., 2011, Vol. 5, Iss. 2, pp. 138–149 149doi: 10.1049/iet-spr.2009.0161 & The Institution of Engineering and Technology 2011

Algorithm for Fourier coefficient estimation

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