The paper deals with the estimation of amplitude and phase of an analogue multi-harmonic band-limited signal from 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 summarized expressions that enable a quick estimation at a low numerical error. The proposed algorithm for the calculation of the unknown parameters requires O((2M+1)2) flops, while the straightforward solution of the obtained equations takes O((2M+1)3) flops (M is the number of the harmonic components). It is applied in signal reconstruction, spectral estimation, system identification, as well as in other important signal processing problems. The proposed method of processing can be used for precise RMS measurements (for power and energy) of a periodic signal based on the presented signal reconstruction. The paper investigates the errors related to the signal parameter estimation, and there is a computer simulation that demonstrates the accuracy of these algorithms.
[1] J. G. Proakis and D. G. Manolakis, Digital Signal Processing:
Principles, Algorithms, Applications, 3rd ed. Englewood Cliffs, NJ:
Prentice-Hall, 1996.
[2] R. S. Prendergast, B. C. Levy, and P. J. Hurst, ÔÇ×Reconstruction of Band-
Limited Periodic Nonuniformly Sampled Signals Through Multirate
Filter Banks", IEEE Trans. Circ. Syst.-I,: vol. 51, no. 8, pp.1612-1622,
Aug. 2004.
[3] P. Marziliano, M. Vetterli, and T. Blu, ÔÇ×Sampling and Exact
Reconstruction of Bandlimited Signals With Additive Shot Noise",
IEEE Trans. Inform. Theory, vol. 52, no. 5, pp.2230-2233, May 2006.
[4] E. Margolis, Y. C. Eldar, ÔÇ×Reconstruction of nonuniformly sampled
periodic signals: algorithms and stability analysis", Electronics, Circuits
and Systems, 2004. ICECS 2004. Proceedings of the 2004 11th IEEE
International Conference, 2004, pp. 555-558.
[5] W. Sun and X. Zhou, ÔÇ×Reconstruction of Band-Limited Signals From
Local Averages", IEEE Trans. Inf. Theory, vol. 48, no. 11, pp. 2955-
2963, Nov. 2002.
[6] A.K. Muciek, ÔÇ×A Method for Precise RMS Measurements of Periodic
Signals by Reconstruction Technique With Correction", IEEE Trans.
Instrum. Meas., vol. 56, no. 2, pp.513-516, Apr. 2007.
[7] P. Petrovic, "New approach to reconstruction of nonuniformly sampled
AC signals", Proceedings of 2007 IEEE International Symposium on
Industrial Electronics (ISIE 2007), Vigo, Spain, 2007, pp. 1693-1698.
[8] A.V.D.Bos, ÔÇ×Estimation of Fourier Coefficients", IEEE Trans. Instrum.
Meas., vol. 38, no. 5, pp.1005-1007, Oct. 1989.
[9] A.V.D.Bos, ÔÇ×Estimation of complex Fourier Coefficients", IEE Proc.-
Control Theory Appl., vol. 142, no. 3, pp.253-256, May. 1995.
[10] R. Pintelon, and J. Schoukens, ÔÇ×An Improved Sine-Wave Fitting
Procedure for Characterizing Data Acquisition Channels", IEEE Trans.
Instrum. Meas., vol. 45, no. 2, pp.588-593, Apr. 1996.
[11] Y. Xiao, Y. Tadokaro, and K. Shida, ÔÇ×Adaptive Algorithm Based on
Least Mean p-Power Error Criteterion for Fourier Analysis in Additive
Noise", IEEE Trans. Signal Proc., vol. 47, no. 4, pp.1172-1181, Apr.
1999.
[12] M. Agrawal M, S. Prasad S, and S.C.D.Roy, " A simple solution for the
analytic inversion of Van der Monde and Confluent Van der Monde
matrices", IETE JOURNAL OF RESEARCH, vol. 47, no. 5, pp. 217-219,
Sep.-Oct. 2001.
[13] I. Gohberg and V. Olshevsky, ÔÇ×The fast generalized Parker-Traub
algorithm for inversion of Van der Monde and related matrices", J.
Complexity, vol. 13, no.2, pp. 208-234, June 1997.
[14] C. S. Jog, ÔÇ×On the explicit determination of the polar decomposition in
n-dimensional vector spaces", JOURNAL OF ELASTICITY , vol.66, no.
2, pp.159-169, Dec. 2002.
[15] V.E. Neagoe, ÔÇ×Inversion of the Van der Monde matrix", IEEE Signal
Processing Letters, vol. 3, no. 4, pp.119-120, Apr. 1996.
[16] H. C. So, K. W. Chan, Y. T. Chan, and K. C. Ho, ÔÇ×Linear Prediction
Approach for Efficient Frequency Estimation of Multiple Real
Sinusoids: Algorithms and Analyses", IEEE Trans. Signal Proc., vol.
53, no. 7, pp.2290-2305, July 2005.
[17] B. Wu and M. Bodson, "Frequency estimation using multiple source and
multiple harmonic components", American Control Conference, 2002.
Proceedings of the 2002, vol.1, 2002, pp. 21-22.
[18] G. Seber, Linear Regression Analysis, New York; Wiley, 1977.
[19] S.J.Reeves and L. P. Heck, ÔÇ×Selection of Observations in Signal
Reconstruction", IEEE Trans. Signal Proc., vol. 43, no. 3, pp.788-791,
Mart 1995.
[20] Y. S. Poberezhskiy and G. Y. Poberezhskiy, ÔÇ×Sampling and Signal
Reconstruction Circuits Performing Internal Antialiasing Filtering and
Their Influence on the Design of Digital Receivers and Transmitters",
IEEE Trans. Circ. Sys.-I, vol. 51, no. 1, pp. 118-129, Jan. 2004.
[21] Y. Xiao, R. K. Ward, L. Ma, and A. Ikuta, "A New LMS-Based Fourier
Analyzer in the Presence of Frequency Mismatch and Applications",
IEEE Trans. Circ. Syst.-I, vol. 52, no. 1, 2005, pp.230-245, Jan. 2005.
[22] Y. Xiao, R.K. Ward, and L. Xu, "A new LMS-based Fourier analyzer in
the presence of frequency mismatch", ISCAS-03, Proceedings of the
2003 International Symposium on Circuits and Systems, vol. 4, 2003,
pp.369-372.
[23] P. Petrovic, ÔÇ×New Digital Multimeter for Accurace Measurement of
Synchronously Sampled AC Signals", IEEE Trans. Instrum. Meas., vol.
53, no.3, 2004, pp.716-725, June 2004.
[24] T. Daboczi, "Uncertainty of Signal Reconstruction in the Case of Jitter
and Noisy Measurements", IEEE Trans.on Instrum.Meas., vol. 47, no. 5,
pp.1062-1066, Dec. 1999.
[25] G. Wang, W. Han, "Minimum Error Bound of Signal Reconstruction",
IEEE Signal Proc. Lett., vol. 6, no. 12, pp. 309-311, Dec. 1999.
[26] N. J. Nigham, Accuracy and Stability of Numerical Algorithms, 2nd ed,
SIAM, 2002.
[27] H. G. Feichtinger, ÔÇ×Reconstruction of band-limited signals from
irregular samples, a short summary", 2nd International Workshop on
Digital Image Processing and Computer Graphics with Applications,
1991, pp. 52-60.
[28] T. Cooklev, ÔÇ×An Efficient Architecture for Orthogonal Wavlet
Transforms", IEEE Signal Proc. Lett., vol. 13, no. 2, pp. 77-79, Feb.
2006.
[29] J. Schoukens, Y. Rolain, G. Simon, and R. Pintelon, ÔÇ×Fully Automated
Spectral Analysis of Periodic Signals, IEEE Trans. Instrum. Meas., vol.
52, no. 4, pp.1021-1024, Aug. 2003.
[30] S. J. Reeves, ÔÇ×An Efficient Implementation of the Backward Greedy
Algorithm for Sparse Signal Reconstruction", IEEE Signal Proc. Lett.,
vol. 6, no. 10, pp. 266-268, Oct. 1999.
[31] P. Petrovic, ÔÇ×New procedure for estimation of amplitude and phase of
compelx ac signals", in Proc. I2MTC, Austin, TX, USA, 2010, pp.464-
469.
[32] K.J.Coakley, C.M.Wang, PD. Hale and T.S. Clement, ÔÇ×Adaptive
characterization of jitter noise in sampled high-speed signals", IEEE
Trans. Instrum. Meas., vol. 52, no. 5, pp.1537-1547, Oct. 2003.
[33] H. Z. Hoseini, I. Kale and O. Shoaei, ÔÇ×Modeling of Switched-Capacitor
Delta-Sigma Modulators in SIMULINK", IEEE Trans. Instrum. Meas.,
vol. 54, no. 4, pp.1646-1654, Aug. 2005.
[34] G. Vendersteen and R. Pintelon, ÔÇ×Maximum likelihood estimator for
jitter noise models", IEEE Trans. Instrum. Meas., vol. 49, no. 6,
pp.1282-1284, Dec. 2000.
[35] R.M. Hidalgo, J.G. Fernandez, R.R. Rivera, and H.A. Larrondo, ÔÇ×A
Simple Adjustable Window Algorithm to Improve FFT Measurements",
IEEE Trans. Instrum. Meas., vol. 51, no. 1, pp.31-36, Jan. 1996.
[36] N.C.F. Tse and L.L. Lai, "Wavelet-Based Algorithm for Signal
Analysis", EURASIP Journal on Advances in Signal Processing, Vol.
2007, Article ID 38916, 10 pages, 2007.
[1] J. G. Proakis and D. G. Manolakis, Digital Signal Processing:
Principles, Algorithms, Applications, 3rd ed. Englewood Cliffs, NJ:
Prentice-Hall, 1996.
[2] R. S. Prendergast, B. C. Levy, and P. J. Hurst, ÔÇ×Reconstruction of Band-
Limited Periodic Nonuniformly Sampled Signals Through Multirate
Filter Banks", IEEE Trans. Circ. Syst.-I,: vol. 51, no. 8, pp.1612-1622,
Aug. 2004.
[3] P. Marziliano, M. Vetterli, and T. Blu, ÔÇ×Sampling and Exact
Reconstruction of Bandlimited Signals With Additive Shot Noise",
IEEE Trans. Inform. Theory, vol. 52, no. 5, pp.2230-2233, May 2006.
[4] E. Margolis, Y. C. Eldar, ÔÇ×Reconstruction of nonuniformly sampled
periodic signals: algorithms and stability analysis", Electronics, Circuits
and Systems, 2004. ICECS 2004. Proceedings of the 2004 11th IEEE
International Conference, 2004, pp. 555-558.
[5] W. Sun and X. Zhou, ÔÇ×Reconstruction of Band-Limited Signals From
Local Averages", IEEE Trans. Inf. Theory, vol. 48, no. 11, pp. 2955-
2963, Nov. 2002.
[6] A.K. Muciek, ÔÇ×A Method for Precise RMS Measurements of Periodic
Signals by Reconstruction Technique With Correction", IEEE Trans.
Instrum. Meas., vol. 56, no. 2, pp.513-516, Apr. 2007.
[7] P. Petrovic, "New approach to reconstruction of nonuniformly sampled
AC signals", Proceedings of 2007 IEEE International Symposium on
Industrial Electronics (ISIE 2007), Vigo, Spain, 2007, pp. 1693-1698.
[8] A.V.D.Bos, ÔÇ×Estimation of Fourier Coefficients", IEEE Trans. Instrum.
Meas., vol. 38, no. 5, pp.1005-1007, Oct. 1989.
[9] A.V.D.Bos, ÔÇ×Estimation of complex Fourier Coefficients", IEE Proc.-
Control Theory Appl., vol. 142, no. 3, pp.253-256, May. 1995.
[10] R. Pintelon, and J. Schoukens, ÔÇ×An Improved Sine-Wave Fitting
Procedure for Characterizing Data Acquisition Channels", IEEE Trans.
Instrum. Meas., vol. 45, no. 2, pp.588-593, Apr. 1996.
[11] Y. Xiao, Y. Tadokaro, and K. Shida, ÔÇ×Adaptive Algorithm Based on
Least Mean p-Power Error Criteterion for Fourier Analysis in Additive
Noise", IEEE Trans. Signal Proc., vol. 47, no. 4, pp.1172-1181, Apr.
1999.
[12] M. Agrawal M, S. Prasad S, and S.C.D.Roy, " A simple solution for the
analytic inversion of Van der Monde and Confluent Van der Monde
matrices", IETE JOURNAL OF RESEARCH, vol. 47, no. 5, pp. 217-219,
Sep.-Oct. 2001.
[13] I. Gohberg and V. Olshevsky, ÔÇ×The fast generalized Parker-Traub
algorithm for inversion of Van der Monde and related matrices", J.
Complexity, vol. 13, no.2, pp. 208-234, June 1997.
[14] C. S. Jog, ÔÇ×On the explicit determination of the polar decomposition in
n-dimensional vector spaces", JOURNAL OF ELASTICITY , vol.66, no.
2, pp.159-169, Dec. 2002.
[15] V.E. Neagoe, ÔÇ×Inversion of the Van der Monde matrix", IEEE Signal
Processing Letters, vol. 3, no. 4, pp.119-120, Apr. 1996.
[16] H. C. So, K. W. Chan, Y. T. Chan, and K. C. Ho, ÔÇ×Linear Prediction
Approach for Efficient Frequency Estimation of Multiple Real
Sinusoids: Algorithms and Analyses", IEEE Trans. Signal Proc., vol.
53, no. 7, pp.2290-2305, July 2005.
[17] B. Wu and M. Bodson, "Frequency estimation using multiple source and
multiple harmonic components", American Control Conference, 2002.
Proceedings of the 2002, vol.1, 2002, pp. 21-22.
[18] G. Seber, Linear Regression Analysis, New York; Wiley, 1977.
[19] S.J.Reeves and L. P. Heck, ÔÇ×Selection of Observations in Signal
Reconstruction", IEEE Trans. Signal Proc., vol. 43, no. 3, pp.788-791,
Mart 1995.
[20] Y. S. Poberezhskiy and G. Y. Poberezhskiy, ÔÇ×Sampling and Signal
Reconstruction Circuits Performing Internal Antialiasing Filtering and
Their Influence on the Design of Digital Receivers and Transmitters",
IEEE Trans. Circ. Sys.-I, vol. 51, no. 1, pp. 118-129, Jan. 2004.
[21] Y. Xiao, R. K. Ward, L. Ma, and A. Ikuta, "A New LMS-Based Fourier
Analyzer in the Presence of Frequency Mismatch and Applications",
IEEE Trans. Circ. Syst.-I, vol. 52, no. 1, 2005, pp.230-245, Jan. 2005.
[22] Y. Xiao, R.K. Ward, and L. Xu, "A new LMS-based Fourier analyzer in
the presence of frequency mismatch", ISCAS-03, Proceedings of the
2003 International Symposium on Circuits and Systems, vol. 4, 2003,
pp.369-372.
[23] P. Petrovic, ÔÇ×New Digital Multimeter for Accurace Measurement of
Synchronously Sampled AC Signals", IEEE Trans. Instrum. Meas., vol.
53, no.3, 2004, pp.716-725, June 2004.
[24] T. Daboczi, "Uncertainty of Signal Reconstruction in the Case of Jitter
and Noisy Measurements", IEEE Trans.on Instrum.Meas., vol. 47, no. 5,
pp.1062-1066, Dec. 1999.
[25] G. Wang, W. Han, "Minimum Error Bound of Signal Reconstruction",
IEEE Signal Proc. Lett., vol. 6, no. 12, pp. 309-311, Dec. 1999.
[26] N. J. Nigham, Accuracy and Stability of Numerical Algorithms, 2nd ed,
SIAM, 2002.
[27] H. G. Feichtinger, ÔÇ×Reconstruction of band-limited signals from
irregular samples, a short summary", 2nd International Workshop on
Digital Image Processing and Computer Graphics with Applications,
1991, pp. 52-60.
[28] T. Cooklev, ÔÇ×An Efficient Architecture for Orthogonal Wavlet
Transforms", IEEE Signal Proc. Lett., vol. 13, no. 2, pp. 77-79, Feb.
2006.
[29] J. Schoukens, Y. Rolain, G. Simon, and R. Pintelon, ÔÇ×Fully Automated
Spectral Analysis of Periodic Signals, IEEE Trans. Instrum. Meas., vol.
52, no. 4, pp.1021-1024, Aug. 2003.
[30] S. J. Reeves, ÔÇ×An Efficient Implementation of the Backward Greedy
Algorithm for Sparse Signal Reconstruction", IEEE Signal Proc. Lett.,
vol. 6, no. 10, pp. 266-268, Oct. 1999.
[31] P. Petrovic, ÔÇ×New procedure for estimation of amplitude and phase of
compelx ac signals", in Proc. I2MTC, Austin, TX, USA, 2010, pp.464-
469.
[32] K.J.Coakley, C.M.Wang, PD. Hale and T.S. Clement, ÔÇ×Adaptive
characterization of jitter noise in sampled high-speed signals", IEEE
Trans. Instrum. Meas., vol. 52, no. 5, pp.1537-1547, Oct. 2003.
[33] H. Z. Hoseini, I. Kale and O. Shoaei, ÔÇ×Modeling of Switched-Capacitor
Delta-Sigma Modulators in SIMULINK", IEEE Trans. Instrum. Meas.,
vol. 54, no. 4, pp.1646-1654, Aug. 2005.
[34] G. Vendersteen and R. Pintelon, ÔÇ×Maximum likelihood estimator for
jitter noise models", IEEE Trans. Instrum. Meas., vol. 49, no. 6,
pp.1282-1284, Dec. 2000.
[35] R.M. Hidalgo, J.G. Fernandez, R.R. Rivera, and H.A. Larrondo, ÔÇ×A
Simple Adjustable Window Algorithm to Improve FFT Measurements",
IEEE Trans. Instrum. Meas., vol. 51, no. 1, pp.31-36, Jan. 1996.
[36] N.C.F. Tse and L.L. Lai, "Wavelet-Based Algorithm for Signal
Analysis", EURASIP Journal on Advances in Signal Processing, Vol.
2007, Article ID 38916, 10 pages, 2007.
@article{"International Journal of Information, Control and Computer Sciences:51730", author = "Predrag B. Petrović", title = "AC Signals Estimation from Irregular Samples", abstract = "The paper deals with the estimation of amplitude and phase of an analogue multi-harmonic band-limited signal from 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 summarized expressions that enable a quick estimation at a low numerical error. The proposed algorithm for the calculation of the unknown parameters requires O((2M+1)2) flops, while the straightforward solution of the obtained equations takes O((2M+1)3) flops (M is the number of the harmonic components). It is applied in signal reconstruction, spectral estimation, system identification, as well as in other important signal processing problems. The proposed method of processing can be used for precise RMS measurements (for power and energy) of a periodic signal based on the presented signal reconstruction. The paper investigates the errors related to the signal parameter estimation, and there is a computer simulation that demonstrates the accuracy of these algorithms.
", keywords = "Band-limited signals, Fourier coefficient estimation, analytical solutions, signal reconstruction, time.", volume = "5", number = "8", pages = "841-9", }
