Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 673049, 5 pages

doi:10.1155/2012/673049

Research Article

Representing Smoothed Spectrum Estimate with the Cauchy Integral

Ming Li1,2

¹School of Information Science & Technology, East China Normal University, No. 500, Dong-Chuan Road, Shanghai 200241, China

²Department of Computer and Information Science, University of Macau, Padre Tomas Pereira Avenue, Taipa, Macau, China

Received 1 October 2012; Accepted 20 November 2012

Academic Editor: Sheng-yong Chen

Copyright © 2012 Ming Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Estimating power spectrum density (PSD) is essential in signal processing. This short paper gives a theorem to represent a smoothed PSD estimate with the Cauchy integral. It may be used for the approximation of the smoothed PSD estimate.

1. Introduction

Estimating power spectrum density (PSD) of signals plays a role in signal processing. It has applications to many issues in engineering [1–21]. Examples include those in biomedical signal processing, see, for example, [1–3, 6, 12, 13]. Smoothing an estimate of PSD is commonly utilized for the purpose of reducing the estimate variance, see, for example, [22–29]. By smoothing a PSD estimate, one means that a smoothed estimate of PSD of a signal is the PSD estimate convoluted by a smoother function [30, 31]. This short paper aims at providing a representation of a smoothed PSD estimate based on the Cauchy’s integral.

2. Cauchy Representation of Smoothed PSD Estimate

Let be a signal for . Let be its PSD, where is radian frequency and is frequency. Then, by using the Fourier transform, is computed by

In practical terms, if is a random signal, may never be achieved exactly because a PSD is digitally computed only in a finite interval, say, (, ) for . Therefore, one can only attain an estimate of .

Denote by an estimate of . Then,

Without generality losing, we assume and . Thus, the above becomes

In the discrete case, one has the following for a discrete signal [21–23]:

Because

is usually a random variable. One way of reducing the variance of is to smooth by a smoother function denoted by . Denote by the smoothed PSD estimate. Let imply the operation of convolution. Then, is given by

Assume that is differentiable any time for . Then, by using the Taylor series at , is expressed by

Therefore,

Let . Then,

Thus, we have a theorem to represent based on the Cauchy integral.

Theorem 2.1. Suppose is differentiable any time at . Then, the smoothed PSD, that is, , may be expressed by

Proof. The Cauchy integral in terms of is in the form

That may be taken as the convolution between and . Thus,

Therefore, (2.10) holds. This completes the proof.

The present theorem is a theoretic representation of a smoothed PSD estimate. It may yet be a method to be used in the approximation of a smoothed PSD estimate. As a matter of fact, we may approximate by a finite series given by

From the above theorem, we have the following corollary.

Corollary 2.2. Suppose is differentiable any time at . Then, may be expressed by

The proof is omitted since it is straightforward when one takes into account the proof of theorem.

3. Conclusions

We have presented a theorem with respect to a representation of a smoothed PSD estimate of signals based on the Cauchy integral. The theorem constructively implies that the design of a smoother function may consider the approximation described by the Cauchy integral with the finite Taylor series (2.13). In addition, the smoother function can also be taken as a solution to the integral equation (2.14), which is worth being investigated in the future.

Acknowledgments

This work was supported in part by the 973 plan under the Project Grant no. 2011CB302800 and by the National Natural Science Foundation of China under the Project Grant nos. 61272402, 61070214, and 60873264.

References

R. Kramme, K. P. Hoffmann, and R. S. Pozos, Springer Handbook of Medical Technology, Springer, New York, NY, USA, 2012.

Wang and S. P. Patrick, Eds., Pattern Recognition, Machine Intelligence and Biometrics, Springer, Berlin, Germany; Higher Education Press, Beijing, China, 2011.

V. Capasso and D. Bakstein, An Introduction to Continuous-Time Stochastic Processes: Theory, Models, and Applications to Finance, Biology, and Medicine, Birkhauser, Berlin, Germany, 2005.

N. Rosario, H. Mantegna, and H. E. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge University Press, New York, NY, USA, 2000.

L. Anselin and S. J. Rey, Eds., Perspectives on Spatial Data Analysis, Springer, New York, NY, USA, 2010.

C. Cattani and J. Rushchitsky, Wavelet and Wave Analysis as Applied to Materials with Micro or Nanostructure, World Scientific, Singapore, 2007.

C. Cattani, “Harmonic wavelet approximation of random, fractal and high frequency signals,” Telecommunication Systems, vol. 43, no. 3-4, pp. 207–217, 2010.

C. Cattani, “On the existence of wavelet symmetries in Archaea DNA,” Computational and Mathematical Methods in Medicine, vol. 2012, Article ID 673934, 21 pages, 2012.

C. Cattani, E. Laserra, and I. Bochicchio, “Simplicial approach to fractal structures,” Mathematical Problems in Engineering, vol. 2012, Article ID 958101, 21 pages, 2012.

C. Cattani, “Fractional calculus and Shannon wavelet,” Mathematical Problems in Engineering, vol. 2012, Article ID 502812, 26 pages, 2012.

C. Toma, “Advanced signal processing and command synthesis for memory-limited complex systems,” Mathematical Problems in Engineering, vol. 2012, Article ID 927821, 13 pages, 2012.

Z. Liao, S. Hu, D. Sun, and W. Chen, “Enclosed Laplacian operator of nonlinear anisotropic diffusion to preserve singularities and delete isolated points in image smoothing,” Mathematical Problems in Engineering, vol. 2011, Article ID 749456, 15 pages, 2011.

S. Hu, Z. Liao, D. Sun, and W. Chen, “A numerical method for preserving curve edges in nonlinear anisotropic smoothing,” Mathematical Problems in Engineering, vol. 2011, Article ID 186507, 14 pages, 2011.

E. G. Bakhoum and C. Toma, “Specific mathematical aspects of dynamics generated by coherence functions,” Mathematical Problems in Engineering, vol. 2011, Article ID 436198, 10 pages, 2011.

P. Flandrin, Time-Frequency/Time-Scale Analysis, Academic Press, San Diego, Calif, USA, 1999.

P. Flandrin and P. Borgnat, “Time-frequency energy distributions meet compressed sensing,” IEEE Transactions on Signal Processing, vol. 58, no. 6, pp. 2974–2982, 2010.

P. Borgnat, P. Flandrin, P. Honeine, C. Richard, and J. Xiao, “Testing stationarity with surrogates: a time-frequency approach,” IEEE Transactions on Signal Processing, vol. 58, no. 7, pp. 3459–3470, 2010.

M. Orini, R. Bailon, L. Mainardi, P. Laguna, and P. Flandrin, “Characterization of dynamic interactions between cardiovascular signals by time-frequency coherence,” IEEE Transactions on Biomedical Engineering, vol. 59, no. 3, pp. 663–673, 2011.

J. Xiao and P. Flandrin, “Multitaper time-frequency reassignment for nonstationary spectrum estimation and chirp enhancement,” IEEE Transactions on Signal Processing, vol. 55, no. 6, pp. 2851–2860, 2007.

G. Rilling and P. Flandrin, “Sampling effects on the empirical mode decomposition,” Advances in Adaptive Data Analysis, vol. 1, no. 1, pp. 43–59, 2009.

E. A. Robinson, “A historical perspective of spectrum estimation,” Proceedings of the IEEE, vol. 70, no. 9, pp. 885–907, 1982.

S. K. Mitra and J. F. Kaiser, Handbook For Digital Signal Processing, Wiley, Chichester, UK, 1993.

A. Papoulis, Signal Analysis, McGraw-Hill, New York, NY, USA, 1977.

M. S. Bartlett, “Periodogram analysis and continuous spectra,” Biometrika, vol. 37, no. 1-2, pp. 1–16, 1950.

J. Niu, K. Li, W. D. Jiang, X. Li, G. Y. Kuang, et al., “A new method of micro-motion parameters estimation based on cyclic autocorrelation function,” Science China Information Sciences. In press

P. C. Mu, D. Li, Q. Y. Yin, and W. Guo, “Robust MVDR beamforming based on covariance matrix reconstruction,” Science China Information Sciences. In press

D. Wang and Y. Wu, “Effects of finite samples on the resolution performance of the rank reduction estimator,” Science China Information Sciences. In press

X. Zhang, Y. Zhang, J. Zhang, S. Y. Chen, D. Chen, and X. Li, “Unsupervised clustering for logo images using singular values region covariance matrices on Lie groups,” Optical Engineering, vol. 51, no. 4, article 047005, 2012.

N. M. Kwok, X. Jia, D. Wang, S. Y. Chen, Q. P. Ha, and G. Fang, “Visual impact enhancement via image histogram smoothing and continuous intensity relocation,” Computers & Electrical Engineering, vol. 37, no. 5, pp. 681–694, 2011.

A. Papoulis, “Minimum-bias windows for high-resolution spectral estimates,” IEEE Transactions on Information Theory, vol. 19, no. 1, pp. 9–12, 1973.

D. J. Thomson, “Spectrum estimation and harmonic analysis,” Proceedings of the IEEE, vol. 70, no. 9, pp. 1055–1096, 1982.