Skip to main content
SpringerLink
Log in

Cramér-Rao bound for damped/undamped exponential process with prior-knowledge

  • Published:
Multidimensional Systems and Signal Processing Aims and scope Submit manuscript

Abstract

Introducing prior-knowledge of some damped/undamped poles in the estimation of the parameters of a multi-poles sinusoidal model is an important problem in various applications. The principle is to orthogonally project the data onto the noise space associated with the known poles. As the Cramér-Rao Lower Bound (CRB) gives a benchmark against which algorithms performance can be compared, it is useful to derive the CRB associated with this model, named Prior-CRB (P-CRB). In particular, we analyze this bound in the close subspaces context, i.e., when the known poles are close to the unknown ones and we show that the P-CRB is lower-bounded by the CRB without (interfering) known poles and higher-bounded by the CRB with no a priori but which takes into account the known poles as parameters of interest. In addition, we show that for damped/undamped exponential process the sensibility of the P-CRB to the closeness of the subspaces is simply relied to the diagonal terms of the Fisher Information Matrix (FIM) associated only with the complex amplitudes of the model. So, by discarding these terms, we regularize the P-CRB and we can derive and analyze closed-form expressions (asymptotics and exacts) of this bound in the case of known complex-amplitude parameters.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Bell, K. L. (May 1995). Performance Bounds in Parameter Estimation with Application to Bearing Estimation, PhD Thesis. Fairfax: George Mason University.

  • Bouleux, G., & Boyer, R. (2006). Analysis of prior-subspace estimation schemes. Proceedings of the fourth IEEE workshop on sensor array and multi-channel processing (SAM-2006).

  • Boyer R., Abed-Meraim K. (2004) Audio modeling based on delayed sinusoids. IEEE Transactions on Speech and Audio Processing 12(2): 110–120

    Article  Google Scholar 

  • de Carvalho, E., Cioffi, J., & Slock, D. T. M. (2000). Cramér-Rao bounds for blind multichannel estimation. Proceedings of IEEE global telecommunications conference (GLOBECOM 2000) (Vol. 2, pp. 1036–1040), 27 November–1 December, San Francisco, USA.

  • Cavassila S., Deval S., Huegen C., Ormondt D., van Graveron-Demilly G. (1999) The beneficial influence of prior knowledge on the quantitation of the in vivo magnetic resonance spectroscopy signals. Investigative Radiology 34(3): 242–264

    Article  Google Scholar 

  • Chen H., Van Huffel S., Vandewalle J. (1997) Improved methods for exponential parameter estimation in the presence of known poles and noise. IEEE Transactions on Signal Processing 45(5): 1390–1393

    Article  Google Scholar 

  • Chen H., Van Huffel S., van Ormondt D., de Beer R. (1996) Parameter estimation with prior knowledge of known signal poles for the quantification of NMR spectroscopy data in the time domain. Journal of Magnetic Resonance, Series A 119(2): 225–234

    Article  Google Scholar 

  • Cramér H. (1946) Mathematical methods of statistics. Princeston University Press, Princeston

    MATH  Google Scholar 

  • DeGroat R.D., Dowling E.M., Linebarger D.A. (1993) The constrained MUSIC problem. IEEE Transactions on Signal Processing 41(03): 1445–1449

    Article  MATH  Google Scholar 

  • Dowling E.M., DeGroat R.D., Linebarger D.A. (1994) Exponential parameter estimation in the presence of known components and noise. IEEE Transactions on Antennas and Propagation 42(5): 590–599

    Article  Google Scholar 

  • Golub G.H., Van Loan C.F. (1996) Matrix computations (3rd ed). Johns Hopkins University Press, Baltimore

    MATH  Google Scholar 

  • Hansen P.C. (1987) The truncated SVD as a method for regularization. BIT 27: 534–553

    Article  MATH  MathSciNet  Google Scholar 

  • Hua, Y., & Sarkar, T. K. (May 1990). Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise. IEEE Transactions on Acoustic, Speech and Signal Pocessing, 38(5), 814–824.

  • Laudadio T., Selen Y., Vanhamme L., Stoica P., Van Hecke P., Van Huffel S. (2004) Subspace-based MRS data quantitation of multiplets using prior knowledge. Journal of Magnetic Resonance 168: 53–65

    Article  Google Scholar 

  • Linebarger D.A., DeGroat R.D., Dowling E.M., Stoica P., Fudge G.L. (1995) Incorporating a priori information into MUSIC-algorithms and analysis. Signal Processing 46(1): 85–104

    Article  MATH  Google Scholar 

  • Liu R.C., Brown L.D. (1993) Nonexistence of informative unbiased estimators in singular problems. Annals of Statistics 21(1): 1–13

    Article  MATH  MathSciNet  Google Scholar 

  • Majda G., Wei M. (1989) A simple procedure to eliminate known poles from a time series. IEEE Transactions on Antennas and Propagation 37(10): 1343–1344

    Article  MathSciNet  Google Scholar 

  • McCloud M.L., Scharf L.L. (2002) A new subspace identification algorithm for high-resolution DOA estimation. IEEE Transactions on Antennas and Propagation 50(10): 1382–1390

    Article  MathSciNet  Google Scholar 

  • Oh S.K., Un C.K. (1993) A sequential estimation approach for performance improvement of eigenstructure-based methods in array processing. IEEE Transactions on Signal Processing 41: 457–463

    Article  Google Scholar 

  • Rao C.R. (1946) Information and accuracy attainable in the estimation of statistical parameters. Bulletin of the Calcutta Mathematical Society 37: 81–91

    Google Scholar 

  • Roy R., Kailath T. (1989) Esprit—estimation of signal parameters via rotational invariance techniques. IEEE Transactions on Acoustics, Speech, and Signal Processing 37(7): 984–995

    Article  Google Scholar 

  • Stoica P., Händel P., Nehorai A. (1995) Improved sequential MUSIC. IEEE Transactions on Aerospace and Electronic Systems 31(4): 1230–1239

    Article  Google Scholar 

  • Stoica P., Marzetta T.L. (2001) Parameter estimation problems with singular information matrices. IEEE Transactions on Signal Processing 49(1): 87–90

    Article  MathSciNet  Google Scholar 

  • Stoica, P., & Moses, R. L. (2005). Spectral analysis of signals. Prentice Hall.

  • Stoica P., Nehorai A. (1989) MUSIC, maximum likelihood, and Cramér-Rao bound. IEEE Transactions on Signal Processing 37(5): 720–741

    Article  MATH  MathSciNet  Google Scholar 

  • Uber, J. A. (December 2003). Estimation of the dimensionality of the signal subspace, PhD Thesis. Fairfax: George Mason University.

  • van Tongeren, B. P. O., de Beer, R., Mehlkopf, A. F., & van Ormondt, D. (1993). Prior knowledge modelling for MR signal processing. Proceedings of the IEEE benelux, ProRISC workshop on circuits, systems and signal processing (pp. 257–262).

  • Yeredor, A. (August 21–23, 2006). On the role of constraints in system identification. Proceedings of the 4th international workshop on total least squares and errors-in-variables modeling. Leuven: Arenberg castle.

  • Ying-Xian, Y., & Pandit, S. M. (1995). Cramér-Rao lower bounds for a damped sinusoidal process. IEEE Transactions on Signal Processing, 43(4).

  • Van Huffel S., Decanniere C., Chen H., Van Hecke P. (1994) Algorithm for time-domain NMR data fitting based on total least squares. Journal of Magnetic Resonance A 110: 228–237

    Article  Google Scholar 

  • Van Der Veen, A. -J, Deprettere, E. F., & Swindlehurst, A. L. (1993). Subspace-based signal analysis using singular value decomposition. Proceedings of the IEEE, 81(9).

Download references

Author information

Authors and Affiliations

  1. Laboratoire des Signaux et Systemes, CNRS - Université Paris-Sud, Supélec, 3, rue Joliot-Curie, 91190, Gif-sur-Yvette, France

    Rémy Boyer

Authors
  1. Rémy Boyer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rémy Boyer.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Boyer, R. Cramér-Rao bound for damped/undamped exponential process with prior-knowledge. Multidim Syst Sign Process 20, 351–374 (2009). https://doi.org/10.1007/s11045-008-0078-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11045-008-0078-8

Keywords

Search

Navigation