1、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. Them
2、ethod 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, simula
3、tion.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 digiti
4、zation 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 25. In 6, a standard matrix inversion isused as t
5、he method of reconstruction, but this requires a considerable volume of numerical calculations. It was stated in 7that, 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 equatio
6、ns. 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
7、 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 establ
8、ished criteria, as was done in 5, 6, 811.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
9、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 me
10、asurements 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 with a limited spec-trum and with the first M harmonics can be represented by a Fourier series(1)where a
11、0is the mean value of the input signal, and akand kare the amplitude and phase of the kth harmonic.st a a k ftkkkM( ) sin ( ),=+ +=012 Measurement Techniques, Vol. 53, No. 8, 2010A NEW METHOD OF DETERMINING THE AMPLITUDEAND PHASE OF AN ALTERNATING SIGNALP. B. Petrovic and M. P. Stevanovic UDC 621.3.
12、018Chachak Technical Faculty, Chachak, Serbia; e-mail: predragptfc.kg.ac.rs. Translated from IzmeritelnayaTekhnika, No. 8, pp. 5055, August, 2010. Original article submitted March 11, 2009.0543-1972/10/5308-09032010 Springer Science+Business Media, Inc. 903When signal (1) is digitized and a system o
13、f equations of the same form is set up to determine the 2M + 1 unknowns(the amplitudes and phases of the M harmonics, and also the mean value of the signal), we obtain(2)where l =1,2M +1, and tlis the instant when the input signal is sampled.Equation (2) can be represented in the abbreviated form(3)
14、(4)The quantity k,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 form(5)where(6)Formula (5) recalls the form of the well-kno
15、wn Van der Monde determinant 1215. 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 simp
16、lified 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
17、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:Dxx xxx xxx xxxMMMMMM MMkjMkjjM= = = =+ =111112112222121111. . . . .().=+ +111111 1222 221 21 21 21sin . sin cos . cossin . sin cos . cos. . . . .sin . s
18、in cos . cos. MMMMMMM MX2111 1 11 112 2 12 2121 21 121 21111MMMMMMM M+ +=sin . sin cos . cossin . sin cos . cos. . . . .sin . sin cos . cos, , , =aaa a aMM M012 122( . ) ,XX =+aa a a aaa a a aaa a a aMM M MMM M MMMM MMM0 1 11 1 1 11 10 1 12 2 1 12 20 1 12 1 2 1 1 12 1 2sin . sin cos . cossin . sin c
19、os . cos. . .sin . sin cos . cos, , , +=1kl l lkft k k M l M,;,.= = = +21121st a alkklkklkkM( ) (sin cos cos sin );,=+ +=01 st a a k ftlklkkM( ) sin ( ),=+ +=012 904We will introduce the following symbols for the co-determinant (adjunct):where i,jis the co-factor (minor), formed from the determinant
20、 Mby eliminating the ith row and jth column. If we expandMin the last row, we obtainOn the other hand,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .As a result,It further follows that + MM pM p p pp p px xx x x xxx xx( )(
21、 ).( )( ).( ) ,11 111=+ + pqMppp M pp Mqxx x x xx x x,( ) ( . . ) ( . . )1111 111 11= = +Mq MppqpMMMqMMq MppqpMppqxx xxx xxx xxx xxxxxx11111111111111 1111111111111. . . . . . . . . . . . . . . . . . . . .+ 1111 11111. . . . . . . .;xxx xxx xpMMMqMMppqpM=Mr M Mr M rxx x D xx x,( . ) /( . ) .12 1 12 1
22、 1D xxx D xxxMrMrMMM Mr( ) ( . ) /( . ) ;=112 1 12 1 1DxxDMMMMM,( ) ( . ) .1112 11=DxxxDMrMrMMrMM( ) ( . ) ;= + + +112 1DxxxDMM M MM,( . ) ;= + + +112 1= MM M MMMMxxxxxx D( )( ).( ) ;,12 1DxxMMkjMkjjM,();=+ =112= =+ =MM M MMkjMkjjMxxxxxx xx( )( ).( ) ( );12 1112= + + +MMMM MrMr MMMMDxD xD xD11211, ,
23、 ,. . .D cof D Dij M ijijij,(); () ,=+1905whencei.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 variabl
24、es x we must substitute trigonometric functions of the variables 1, 2, 3,., 2M+1, presented in (4). (It should be noted that determinant (5) can also be written using Eulers formulas.)The co-determinants, which are necessary to solve (3), have the formAnother form of writing this is:where X11, X12,
25、., X12M+1are the co-determinants, formed from the co-determinant X2M+1,1after eliminating the corre-sponding row and the first column. The other co-determinants can be obtained using Xpq the co-factor of X2M+1. After inten-sive mathematical calculations (as in 7), we obtainwhere the summation is car
26、ried out for all M on the factor 2, 3, ., 2M+1.For 2 q M + 1:XpqpqMMMMjkkMjkMpkkpM= +=+() ()sinsin()()11 2221221112121121+ + +cos. .( . . ) , 11121111212pp Mpp MMXppqMMMMjkkMjkMpkkkpM112211212112111 222= +=+=+() ()sinsin()()XXX X211 11122121211MMst st st+=+ +,( ) ( ) . ( ) ,X21211 11 122 22 221 21 2
27、1 21 211212MMM MM Mst M Mst M Mst M M+ + +=,( ) sin . sin cos . cos( ) sin . sin cos . cos. . . . . .( ) sin . sin cos . cos; etc.X21111 11 122 22 221 21 21 21 21MMM MM Mst M Mst M Mst M M+ + +=,( ) sin . sin cos . cos( ) sin . sin cos . cos.( ) sin . sin cos . cos; Dpqpqpq,() ,= +1= + + +pqMppM ppM
28、qp p pp pp pMxx x x xx x xxxxx xx xx xx,( . . ) ( . . )( )( ).( )( ).( );111 111 1112 1 1906. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . For q = 2M + 1The result obtained shows that there is no need to use the standard procedure for solv
29、ing 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
30、the unknown parameters the amplitudes and phases using the following formulas:A Check of the Results. Calculations showed agreement (to 1014) 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) a
31、lgorithm,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
32、 of the fundamental harmonic.The proposed algorithm requires a knowledge of the number of higher harmonic components M of 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 spectr
33、um. 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 var
34、iables l, it is necessary to obtaintheir dependence on the carrier frequency 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 th
35、e frequency 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
36、 (zero-crossing moments) 19. The times tlat whichthe input signal is sampled may be quite random (asynchronous) and independent of the frequency 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 s
37、ignal through zero.A Numerical Estimate of the Complexity of the Proposed Algorithm. The expressions derived require (2M + 1) (4M3+ 2M + 1) multiplications and summations to calculate them. Nevertheless, the methodby which the unknown parameters of the processed signal are determined requires much f
38、ewer calculations, since theexpressions contain many common coefficients in the products formed and they can be shortened. As a result, the total num-()2121222MMMM+aaMMkMMk MMk kMMkMk021121 21211221 12 21 12111= + =+ + +XXXXXXX,; ; arctan .+ +cos. . 111212pp MXpqMMMMjkkMjkMpkkpM=+=+()sinsin()()12221
39、221112121121+ + +sin. .( . . ) . 111211112112pp Mpp MMq907ber 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, a
40、nd itis 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 l, and by the interval t required to carryout all the calculations using this
41、 algorithm. Hence, to recover the signal the time required is N(2M + 1)ts+ t N/ + t,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 r
42、ealized using a dSPACE DS1104 chip, which contains an MPC8240 processor, operating witha clock frequency of 250 MHz. When processing a signal having M = 7 harmonics with a sampling frequency s= 1 kHz,the calculation of the unknown amplitudes and phases took approximately 19.5 msec.The Results of Mod
43、eling. An additional check of the algorithm was carried out using modeling in the Matlab andSimulink (version 7.0) software. In practice, the algorithm is based on the use of standard apparatus components, by means908HzFig. 1. Graphs of the relative error in calculating the determinant of the system
44、 as afunction of the error in synchronization with the frequency of the fundamental inputsignal = 50 Hz; the time quantization step is 1 msec (M is 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, po
45、lyharmonic 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.of which the input analog signal is sampled (Fig. 2). The carrier signal frequency is measured using the method de
46、scribedin 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 effectiv
47、eresolving power of the ADC in the modeling is 24 bits and the sampling frequency s= 1 kHz). In the modeling, the spectralpower density of ideal thermal noise and clock-signal jitter were in the range of (100170) dB, whereas the signal-to-noiseratio (signal-to-noise distortion ratio SNDR) was in the
48、 range 8596 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 = 50 Hz. The table shows the amplitudesand phases of each harmon
