# Representing Smoothed Spectrum Estimate with the Cauchy Integral

Authors: Ming Li
Journal: Mathematical Problems in Engineering
 (2012)

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## 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.


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## Body

## 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

Letx(t) be a signal for -∞<t<∞. Let Sxx(ω) be its PSD, where ω=2πf is radian frequency and f is frequency. Then, by using the Fourier transform, Sxx(ω) is computed by
(2.1)Sxx(ω)=|∫-∞∞x(t)e-jωtdt|2,j=-1.
In practical terms, if x(t) is a random signal, Sxx(ω) may never be achieved exactly because a PSD is digitally computed only in a finite interval, say, (T1, T2) for T1≠T2. Therefore, one can only attain an estimate of Sxx(ω).Denote byS^xx(ω) an estimate of Sxx(ω). Then,
(2.2)S^xx(ω)=|∫T1T2x(t)e-jωtdt|2.
Without generality losing, we assume T1=0 and T2=T. Thus, the above becomes
(2.3)S^xx(ω)=|∫0Tx(t)e-jωtdt|2.
In the discrete case, one has the following for a discrete signal x(n) [21–23]:
(2.4)S^xx(ω)=|∑n=0N-1x(n)e-jωn|2.Because(2.5)|∑n=LN+L-1x(n)e-jωn|2≠|∑n=MN+M-1x(n)e-jωn|2forL≠M,S^xx(ω) is usually a random variable. One way of reducing the variance of S^xx(ω) is to smooth S^xx(ω) by a smoother function denoted by G(ω). Denote by S~xx(ω) the smoothed PSD estimate. Let * imply the operation of convolution. Then, S~xx(ω) is given by
(2.6)S~xx(ω)=S^xx(ω)*G(ω).Assume thatS~xx(ω) is differentiable any time for -∞<ω<∞. Then, by using the Taylor series at ω=ω0, S^xx(ω) is expressed by
(2.7)S^xx(ω)=∑l=0∞S^xx(l)(ω0)l!(ω-ω0)n.
Therefore,
(2.8)S~xx(ω)=∑l=0∞S^xx(l)(ω0)l!(ω-ω0)n*G(ω).
Let ω-ω0=ω1. Then,
(2.9)(ω-ω0)n*G(ω)=ω1n*G(ω1+ω0).
Thus, we have a theorem to represent S~xx(ω) based on the Cauchy integral.Theorem 2.1.
SupposeS^xx(ω) is differentiable any time at ω0. Then, the smoothed PSD, that is, S~xx(ω), may be expressed by
(2.10)S~xx(ω)=∑l=0∞S^xx(l)(ω0)l!ω1l*G(ω1+ω0)=∑l=0∞S^xx(l)(ω0)∫0ω1(ω1-ωτ)ll!G(ωτ+ω0)dωτ.Proof.
The Cauchy integral in terms ofG(ωτ+ω0) is in the form
(2.11)∫0ω1(ω1-ωτ)ll!G(ωτ+ω0)dωτ=∫0ωτdωτ⋯∫0ωτG(ωτ+ω0)dωτ︸l+1.
That may be taken as the convolution between ω1l/l! and G(ω1+ω0). Thus,
(2.12)ω1ll!*G(ω1+ω0)=∫0ω1(ω1-ωτ)ll!G(ωτ+ω0)dωτ.
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 approximateS~xx(ω) by a finite series given by
(2.13)S~xx(ω)≈∑l=0LS^xx(l)(ω0)∫0ω1(ω1-ωτ)ll!G(ωτ+ω0)dωτ.
From the above theorem, we have the following corollary.Corollary 2.2.
SupposeS^xx(ω) is differentiable any time at ω=0. Then, S~xx(ω) may be expressed by
(2.14)S~xx(ω)=∑l=0∞S^xx(l)(0)l!ωl*G(ω)=∑l=0∞S^xx(l)(0)∫0ω(ω-ωτ)ll!G(ωτ)dωτ.
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 functionG(ω) may consider the approximation described by the Cauchy integral with the finite Taylor series (2.13). In addition, the smoother function G(ω) can also be taken as a solution to the integral equation (2.14), which is worth being investigated in the future.



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*Source: 673049-2012-11-29.xml*