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Ming Li, "Representing Smoothed Spectrum Estimate with the Cauchy Integral", Mathematical Problems in Engineering, vol. 2012, Article ID 673049, 5 pages, 2012. https://doi.org/10.1155/2012/673049
Representing Smoothed Spectrum Estimate with the Cauchy Integral
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.
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 .
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.
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.
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.
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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.