A NEW METHOD OF DETERMINING THE AMPLITUDEAND PHASE OF AN ALTERNATING SIGNAL
P. B. Petrovic and M. P. StevanovicUDC 621.3.018
A new method of determining the unknown amplitudes and phases of an alternating analog signal with alimited spectrum is considered. The main parameters of the signal are reconstructed from samples basedon analytical expressions. Computer modeling confirms the accuracy of the proposed algorithm. Themethod should find application when reconstructing a signal,calculating the spectrum,and the accuratemeasurement of the root mean square value (of the power and energy) of an alternating signal.Key words:signal reconstruction,Van der Monde matrix,Fourier coefficients,analytical solution,simulation.The reconstruction of signals based on measurements is the central problem in many applications,connected withsignal processing. However,the data available are often insufficient to ensure high resolution of the reconstruction [1].Many attempts have been made to find an optimal method of digitization and for re-establishing a frequency-limitedsignal,which can be represented in the form of a Fourier series (trigonometric polynomials). New methods of reconstructingperiodic signals by a nonuniform selection of readouts have been described in [2–5]. In [6],a standard matrix inversion isused as the method of reconstruction,but this requires a considerable volume of numerical calculations. It was stated in [7]that,by the correct choice of the time parameters,which enables integration to be carried out,an alternating signal can berepresented by a regular matrix form of the system of equations. This enables the reconstruction process to be carried outmuch more effectively.If the readouts are taken without errors,the algorithm proposed in the present paper can be used without any mod-ification to re-establish a periodic signal with a limited spectrum. However,in practice,the readouts are obtained with acertain error,i.e.,the signal is represented by the average value in the neighborhoods of the point at which the samplingis carried out [5]. In such a situation the proposed algorithm has to be modified,in order to obtain the best estimate of thesignal based on previously established criteria,as was done in [5,6,8–11].In this paper,we propose an algorithm for reestablishing the signal,which provides a considerable increase in com-putational efficiency over standard matrix methods. The approach is based on the use of the value obtained as a result of dig-itization of the continuous input signal at exactly defined instants of time. The process is repeated as long as it is necessaryto reconstruct the complex signal. The system of linear equations formed can be solved using the analytical and simplifiedexpressions that are derived. The method was developed for exact measurements of the root mean square (effective) valuesof periodic signals,and also their power and energy.Formulation of the Problem.We will assume that the input signal of fundamental frequency ƒ with a limited spec-trum and with the first Mharmonics can be represented by a Fourier seriess(t)=a0+∑aksin(k2πft+ψk),k=1M(1)where a0is the mean value of the input signal,and akand ψkare the amplitude and phase of the kth harmonic.Chachak Technical Faculty,Chachak,Serbia; e-mail:predragp@tfc.kg.ac.rs. Translated from Izmeritel’naya
Tekhnika,No. 8,pp. 50–55,August,2010. Original article submitted March 11,2009.
0543-1972/10/5308-0903©2010 Springer Science+Business Media,Inc.
903
When signal (1) is digitized and a system of equations of the same form is set up to determine the 2M+ 1 unknowns(the amplitudes and phases of the Mharmonics,and also the mean value of the signal),we obtains(tl)=a0+∑aksin(2kπftl+ψk),k=1M(2)where l=⎯⎯⎯1,2⎯⎯⎯M+1,and tlis the instant when the input signal is sampled.Equation (2) can be represented in the abbreviated forms(tl)=a0+∑ak(sinαk,lcosψk+cosαk,lsinψk);k=1M(3)αk,l=2kπftl=kαl;k=1,M,l=1,2M+1.(4)The quantity αk,lis a variable,which is determined by the instant when sampling occurs and by the frequency of theprocessed signal. The determinant of the system of 2M+ 1 equations with the number of unknowns obtained in this way hasthe forma0a1sinα1,1...aMsinαM,1a1cosα1,1...aMcosαM,1aa1sinα1,2...aMsinαM,2a1cosα1,2...aMcosαM,2X=0=.......a0a1sinα1,2M+1...aMsinαM,2M+1a1cosα1,2M+1...aMcosαM,2M+1=a0(a1a2⋅...⋅aM−1aM)2X2M,where1sinα1,1...sinαM,1cosα1,1...cosαM,11sinα1,2...sinαM,2cosα1,2...cosαM,2X2M+1==.......1sinα1,2M+1...sinαM,2M+1cosα1,2M+1...cosαM,2M+11sinα1...sinMα1cosα1...cosMα11sinα2...sinMα2cosα2...cosMα2=........1sinα2M+1...sinMα2M+1cosα2M+1...cosMα2M+1(5)(6)Formula (5) recalls the form of the well-known Van der Monde determinant [12–15]. This form has been investi-gated in [6],which was essentially the first step to determining the Fourier coefficients. All these papers only give an approachto the solution of the original and inverse Van der Monde determinants. Below we present new analytic and simplified expres-sions,which provide an exact solution of system of equations (3) and use these determinants as the starting point. Hence,theexpressions obtained enable one to eliminate the use of standard procedures for solving systems of equations,as consideredin [6],and hence a powerful processor and a long calculation time are not required.The Determinants of the Van der Monde Matrix. Consider the Mth order Van der Monde determinant [14,15]:1DM=ΔMx12x12x2...M−1...x1M−1...x2=......M−1...xMMM−11x2=......1xMx2Mk=j+1j=1∏∏(xk−xj).904
We will introduce the following symbols for the co-determinant (adjunct):Di,j=cof(DM);Di,j=(−1)i+jΔi,j,where Δi,jis the co-factor (minor),formed from the determinant ΔMby eliminating the ith row and jth column. If we expandΔMin the last row,we obtainΔM=1DM,1+xMDM,2+...+xrM−1DM,r+...+xMM−1DM,M.On the other hand,ΔM=(xM−x1)(xM−x2)...(xM−xM−1)k∏MM−2=j+1∏(xk−xj);j=1MM∏−2DM,M=(xk−xj);k∏=j+1j=1ΔM=(xM−x1)(xM−x2)...(xM−xM−1)DM,M;DM,M−1=−(x1+x2+...+xM−1)DM,M;. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .DM,r=(−1)M−r∑(x1+x2+...+xM−1)M−rDM,M;. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .DM,1=(−1)M−1(x1x2...xM−1)DM,M.As a result,DM,r=(−1)M−r(x1x2...xM−1)∑DM,M/(x1x2...xM−1)r−1;ΔM,r=(x1x2...xM−1)∑DM,r/(x1x2...xM−1)r−1.It further follows that1xq11...x1−...x1M−11xq1...x1−1...x1M−1....................................1xqp−1...xp−1−1...xMp−−11ΔM=1xp...xqp−1...xMp−1=1xqp1+1...xp−+1...xMp+−11;....................................1xM...xqM−1...xMM−11xqM...xM−1...xMM−11xp...xqp−1...xMp−1Δp,q=(−1)M−p(x1...xp−1xp+1...xM)∑[(x1...xp−1xp+1...xM)q−1]−1××ΔM[(xM−xp)(xM−1−xp)...(xp+1−xp)(xp−xp−1)...(xp−x1)]−1,905
whenceΔp,q=ΔM(x1...xp−1xp+1...xM)(xp−x1)(xp−x2)...(xp−xp−1)(xp−xp+1)...(xp−xM)Dp,q=(−1)p+qΔp,q,∑[(x1...xp−1xp+1...xM)q−1]−1;i.e.,all the inverted products,ordered with respect to q– 1 in inverse order,are summed.Derivation of the Analytic Expressions.For the system of equations obtained using the proposed method of pro-cessing the input signal (3),instead of the variables xwe must substitute trigonometric functions of the variables α1,α2,α3,...,α2M+1,presented in (4). (It should be noted that determinant (5) can also be written using Euler’s formulas.)The co-determinants,which are necessary to solve (3),have the forms(t1)sinα1...sinMα1cosα1...cosMα1s(t2)sinα2...sinMα2cosα2...cosMα2X2M+1,1=;.......s(t2M+1)sinα2M+1...sinMα2M+1cosα2M+1...cosMα2M+11s(t1)sin2α1...sinMα1cosα1...cosMα11s(t2)sin2α2...sinMα2cosα2...cosMα2=;etc.........1s(t2M+1)sin2α2M+1...sinMα2M+1cosα2M+1...cosMα2M+1X2M+1,2Another form of writing this is:11X2M+1,1=s(t1)X11+s(t2)X2+...s(t2M+1)X2M+1,11where X11,X2,...,X2M+1are the co-determinants,formed from the co-determinant X2M+1,1after eliminating the corre-qsponding row and the first column. The other co-determinants can be obtained using Xp– the co-factor of X2M+1. After inten-sive mathematical calculations (as in [7]),we obtain2M+12MX1p=M(M+1)j=k+1k=1p+q(−1)(−1)222M(M−1)2M+1∏∏∏k=1k≠psinαj−αk2×sinαp−αk2×∑⎧⎫⎪α1+...+αp−1+αp+1+...+α2M+1⎪cos⎨−(α1+...+αp−1+αp+1+...+α2M+1)M⎬,2⎪⎪⎭⎩where the summation is carried out for all Mon the factor {α2,α3,...,α2M+1}.For 2 ≤q≤M+ 1:2M+12MXqp=M(M+1)j=k+1k=1p+q(−1)(−1)222M(M−1)+12M+11≤k≠p∏∏∏sinαj−αk2×sinαp−αk2906
×∑⎧⎫⎪α1+...+αp−1+αp+1+...+α2M+1⎪sin⎨−(α1+...+αp−1+αp+1+...+α2M+1)M+q−1⎬.2⎪⎪⎭⎩. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . For q= 2M+ 12M+12MXqp=M(M−1)j=k+1k=1(−1)222M(M−1)+12M+11≤k≠p∏∏∏sinαj−αk2×sinαp−αk2×cosα1+...+αp−1+αp+1+...+α2M+12.The result obtained shows that there is no need to use the standard procedure for solving the system of equations,asproposed in [7,8]. If the spectral composition of the signal is exceptionally complex,it is necessary to use a highly efficientprocessor for this procedure and sufficient time to carry out the process. The expressions derived practically on-line enableone to determine the unknown parameters – the amplitudes and phases using the following formulas:a0=X2M+1,1X2M+1;ak=1X2M+12X22M+1,k+1+X2M+1,M+k+1;ψk=arctanX2M+1,M+k+1X2M+1,k+1.A Check of the Results.Calculations showed agreement (to 10–14) of the results of calculations of X2M+1and X1l(l= 1,...,15),carried out using the above formulas and the GEPP (Gaussian elimination with partial pivoting) algorithm,which is usually used for these purposes (all the calculations were carried out in the standard IEEE arithmetic with doubleaccuracy). Hence,a practical check of this algorithm was carried out for the case of an ideal sample,ignoring quantizationerrors and errors in measuring the frequency of the fundamental harmonic.The proposed algorithm requires a knowledge of the number of higher harmonic components Mof the signal beingprocessed or it must be taken to be greater than the expected (real) value. For this purpose,one must use any well-knownmethod of estimating the frequency spectrum. Two such algorithms for estimating the spectrum of a sinusoidal signal in thepresence of white noise are described in [16]. A modified parametric estimate,based on phase adjustment of the frequency,was proposed in [17].In view of the presence of an error when averaging the samples s(tl) and the variables αl,it is necessary to obtaintheir dependence on the carrier frequency ƒ of the signal fairly accurately. In this case the estimation procedure does notrequire matrix inversion or a highly efficient processor,unlike the procedure considered in [18]. In Fig. 1,we show how theerror in measuring the frequency ƒ affects the relative error in calculating the system determinant for a different number ofharmonics of the input alternating signal. The stability of the algorithm to this error can be increased using a complex algo-rithm for distinguishing the instant when the signal passes through zero (zero-crossing moments) [19]. The times tlat whichthe input signal is sampled may be quite random (asynchronous) and independent of the frequency ƒ when using the methodby means of which they are determined in (4). The beginning of the digitization process should correspond to the passage ofthe input signal through zero.A Numerical Estimate of the Complexity of the Proposed Algorithm. The expressions derived require (2M+ 1)×⎛2M1⎛2M⎞⎞×(4M+ 2M+ 1) multiplications and (2M+1)⎜2−⎜⎟summations to calculate them. Nevertheless,the method2⎝M⎟⎠⎠⎝by which the unknown parameters of the processed signal are determined requires much fewer calculations,since theexpressions contain many common coefficients in the products formed and they can be shortened. As a result,the total num-3907
HzFig. 1. Graphs of the relative error in calculating the determinant of the system as afunction of the error in synchronization with the frequency of the fundamental inputsignal ƒ = 50 Hz; the time quantization step is 1 msec (Mis the number of harmonics).
Fig. 2. Simulink-model of the circuit for realizing the proposed signal recovery algorithm:1,2,3,5) pulse,random number,polyharmonic input signal and noise generators; 4) variabledelay; 6) adder; 7) comparator (a Schmidt trigger); 8) sigma-delta analog-digital converter;9) null element; 10) processing unit.
ber of operations is reduced by 18M2+ 12M+ 2. This means that the algorithm requires 9(2M+ 1)2/2 operations with afloating point.
The time taken to carry out the necessary number of samples of the input signal can be defined as (2M+ 1)ts,and it
is approximately equal to the time required to reestablish the signal when modeling. In practical applications of this algorithm,this interval must be increased by the time required to estimate the variables s(tl) and αl,and by the interval Δtrequired to carryout all the calculations using this algorithm. Hence,to recover the signal the time required is N(2M+ 1)ts+ Δt≈N/ƒ + Δt,taking into account the need for synchronization with the instant when the signal passes through zero. The speeds of the pro-posed algorithm and of the algorithms analyzed in [20,21] are comparable.
The algorithm was realized using a dSPACE DS1104 chip,which contains an MPC8240 processor,operating with
a clock frequency of 250 MHz. When processing a signal having M= 7 harmonics with a sampling frequency ƒs= 1 kHz,the calculation of the unknown amplitudes and phases took approximately 19.5 msec.
The Results of Modeling.An additional check of the algorithm was carried out using modeling in the Matlab and
Simulink (version 7.0) software. In practice,the algorithm is based on the use of standard apparatus components,by means908
TABLE 1. Results of Modeling of Signal Recovery Using the Proposed Algorithm
Error,%Number of harmonicsAmplitude,arb. unitsPhase,radamplitudephase123456710.730.640.550.320.270.14ππ/30π/6π/4π/1200.00160.00230.00210.00170.00180.00230.00240.00210.00180.00220.00190.00240.00170.0023of which the input analog signal is sampled (Fig. 2). The carrier signal frequency is measured using the method describedin[19],i.e.,using a comparator (a Schmitt trigger). The circuit enables the transit of a polyharmonic signal through zero tobe recorded and hence enables its frequency to be determined.
The sigma-delta analog-to-digital converter (ADC) model employed is described in detail in [22] (the effective
resolving power of the ADC in the modeling is 24 bits and the sampling frequency ƒs= 1 kHz). In the modeling,the spectralpower density of ideal thermal noise and clock-signal jitter were in the range of –(100–170) dB,whereas the signal-to-noiseratio (signal-to-noise distortion ratio – SNDR) was in the range 85–96 dB. During the modeling the input signal parametershad the values shown in the table. These data were adapted to the values of the signals used in [21,23],for comparison withthe results of an analysis and check of the proposed algorithm.
The signal had the first seven harmonics and a fundamental frequency ƒ = 50 Hz. The table shows the amplitudes
and phases of each harmonic. Additional noise and jitter during the modeling led to a relative error of 0.0001% in determin-ing the frequency ƒ. It follows from the data that the accuracy achieved in reconstructing the signal is in the limits indicatedin [21,23] for this type of signal. The error which arises when calculating the amplitudes and phases is mainly due to theerrors in taking the samples of the input signal and inaccuracy in the expressions derived.
Conclusions.The proposed algorithm is promising for solving problems of reconstructing signals and in precision
measurements of periodic signals,particularly complex signals. Using this algorithm,one can determine the energy,powerand root mean square value of the alternating voltage of an electric network. Possible errors are due to deficiencies in syn-chronization with the fundamental frequency of the signal due to the effect of Gaussian white noise and jitter. The results ofmodeling confirmed that the proposed algorithm can provide satisfactory accuracy in reconstructing a periodic signal underpractical conditions.
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