Identification of LTI Autonomous All Pole System Using Eigenvector Algorithm

This paper presents a method for identification
of a linear time invariant (LTI) autonomous all pole system
using singular value decomposition. The novelty of this paper
is two fold: First, MUSIC algorithm for estimating complex
frequencies from real measurements is proposed. Secondly,
using the proposed algorithm, we can identify the coefficients
of differential equation that determines the LTI system by
switching off our input signal. For this purpose, we need only
to switch off the input, apply our complex MUSIC algorithm
and determine the coefficients as symmetric polynomials in the
complex frequencies. This method can be applied to unstable
system and has higher resolution as compared to time series
solution when, noisy data are used. The classical performance
bound, Cramer Rao bound (CRB), has been used as a basis for
performance comparison of the proposed method for multiple
poles estimation in noisy exponential signal.

Authors:

• Sudipta Majumdar

Keywords:

• MUSIC algorithm
• Cramer Rao bound
• frequency estimation.

References:

[1] G. Ravi Shankar Reddy and Rameshwar Rao, ”Instantaneous
frequency estimation based on time-varying auto regressive model
andWAX-Kailath algorithm”, Signal Processing: An International
Journal, vol. 8, Issue 4, pp. 43-66, 2014.
[2] Z. K. Peng, G. Meng, F. L. Chu, Z. Q. Lang, W. M. Zhang
and Y. Yang, ”Polynomial Chirplet Transform with application
to instantaneous frequency estimation”, IEEE Transactions on
Instrumentation and Measurements, vol. 60, n0. 9, pp. 3222-3229,
September 2011.
[3] Biao Huang, Steven X. Ding, and S. Joe Qin, ” Closed-loop
subspace identification: an orthogonal projection approach”,
Journal of Process Control, 15 ,pp. 53-66,(2005).
[4] Pedro M. Ramos and A. Cruz Serra, ”Comparision of frequency
estimation algorithms for power quality assesment”,Measurement,
42 (2009), pp. 1312-1317.
[5] Paavo Alku and Jouni Pohjalainan, ”Formant frequency
estimation of high pitched vowels using weighted linear
prediction”, J. Acoust. Soc. Am., 134(2), pp. 1295-1313, August
2013.
[6] J. R. Jensen, M.G. Christensen and S. H. Jensen, ”Nonlinear least
squares methods for joint DOA and pitch estimation: IEEE Trans
on Audio, Speech, Lang. Process, vol. 21, no. 5, pp. 923-933, May
2013.
[7] Petre Stoica, Randolph L. Moses, Benjamin Friedlander, Torsten,
Soderstrom, ” Maximum likelihood estimation of the parameter
of multiple sinusoids from noisy measurements”, IEEE Trans.
on Acoustics, Speech and Signal Processing, vol. 37, no. 3, pp.
378-392, March, 1989.
[8] Peter Stoica and Arye Nehorai, ”Music, maximum likelihood and
Cramer Rao bound”, IEEE Transactions on Acoustics, Speed and
Siganl Processing, vol. 37, no. 5, pp.720-741, May 1989,
[9] Arye Nehorai and David Starer, ”Adaptive pole estimation”,
IEEE Transactions on Acoustics, Speech and Signal Processing,
vol. 38, no. 5, pp. 825-838, May 1990.
[10] S. M. Kay,”Modern Spectrum Estimation: Theory and
Applications”, Prentice Hall 1988.
[11] Monson H. Hayes, Statistical Digital Signal Processing and
Modeling, John Wiley and Sons, 1996.
[12] Daeyong Kim, Madihally J. Narashima and Donal C. Cox, ”An
improved single frequency estimator”, IEEE Signal Processing
Letters, vol. 3, no. 7, July 1996, pp. 212-214.
[13] Bin Liao and Shing Chow Chan, ”An improved
eigendecomposition based algorithms for frequencies estimation
of two sinusoids”, IEEE Communication Letters, vol.17, no.3,
March 2013, pp. 557-560.
[14] H. L. Vantrees, Detection, Estimation and Modulaion Theory,
New York, Wiley, 1968.
[15] Yonina C. Edlar, ”Uniformly Improving the Cramer-Rao Bound
and Maximum-Likelihood Estimation”, IEEE Transactions on
Signal Processing”, vol. 54, no. 8, pp. 2943-2956, August, 2006.