Noise variance estimation based on measured maximums of sampled subsets

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Abstract

In this paper, an estimation of the Gaussian noise variance based on observed (measured) maximums of subsets of samples is given. Circumstances of the measurement environment being limited, only maximums of subsets of samples are available and the non-constant variance of the Gaussian noise can be estimated. In the case of power line noise, the variance of the zero mean Gaussian noise is a periodic function of the a-priory known parameterization.

Variance function parameters estimation is computed in two steps, first the estimation formula of the constant variance Gaussian noise is applied to a certain subset of samples and second, the least mean square (LMS) criterion is applied to fit the parametrized variance function to estimated variances.

The maximum likelihood estimation (MLE) criterion is applied to derive estimators of the variance function parameters. Beside that, the quotient of the variance of the zero mean Gaussian noise and its maximums is evolved explicitly.

Experimental results on real and simulated data are given to demonstrate their accuracy.

Introduction

The noise variance estimation presented in this paper was the problem that had to be copped with in power line modelling, see [1]. The measurement environment was affected by specific constraints impeding the procedure of data sampling and leading to the below described parameter estimation task. Details are given in subsequent sections.

An explicit formula for noise variance estimation based on maximums of samples is not known in the literature. Such formula is derived in this paper. As a consequence, an explicit formula for the ratio between the noise variance and the variance of noise maximums is given.

As already mentioned, the problem of variance estimation arose in the statistical modelling of the power line for the purpose of digital communication, see [1]. Characteristics of the corona noise are measured with a digital oscilloscope and there is not enough memory available to store the measured data in a longer time period. To overcome this problem, only the maximums of samples are stored. Therefore, certain noise characteristics must be estimated from maximums of samples, see Section 5.

Notations used in his paper are the following. Capital letters X, Xk, etc. denote random variables. The probability density function (PDF) of a random variable X is denoted by f, Xf, and the cumulative density function (CDF) of a random variable X is denoted by F, XF. When PDF exists, (d/dξ)F(ξ)=f(ξ) by definition (for details, see [2]).

The outline of the paper is as follows: an exact problem statement along with rather complicated notations is given in the next Section. In Section 3, we prove the estimation formula required to solve the mentioned variance estimation task, while in Section 4 we apply the theoretical result to define the noise estimation procedure. Finally, in Section 6, the proposed procedure is verified on (a) simulated and (b) real (power line noise measurement) data. We end the paper by short concluding Section and remarks on further work.

Section snippets

Notations and problem statement

The time dependent periodic variance function σ(t)2 of the zero mean Gaussian noise X={Xt:t∈[0,∞)} is to be estimated. The period T of the function σ is a-priory known. The noise X is sampled at nm equidistant times in each time interval of length T; N intervals of length T are sampled, samples are independent. The distance between two adjacent samples is △t=T/nm. The random variable Xt at t=tk,i,j is denoted by Xk,i,j and the sample taken is denoted by xk,i,j. Samples of each sampling interval

Variance estimation formula

The noise variance estimation formula based on the sample maximums is given in this Section. To derive the estimation formula, the following lemma is needed.

Lemma 1

Let Xk, 1≤km, be mutually independent random variables with CDFs Fk(ξ). The CDF of a random variable Ym=maxk{Xk} is F(η)=∏k=1mFk(η).

Proof

Recall that the maximum of random variables is again a random variable, see [2]. Since max{X1,…,Xm}=max{max{X1,…,Xm−1},Xm}, it is enough to proof the lemma for m=2 and the general result follows by induction.

Variance function parameters estimation

Assume the notation of Section 2. The goal of this Section is to apply the result of the Theorem 1 to estimate parameters p∈P of the parametric variance function σp such that the estimated function fits given data best is some sense, see Section 2.

According to the Lemma 1, the CDF of a random variable Xk,i sampled to obtain a maximum of samples {xk,i,1,…,xk,i,m} (see Eq. (1)) is therefore a product of CDFs samples xk,i. Since samples xk,i,j are independent, their maximums xk,i are also

Application of the proposed method to impulse noise measurement

Communication channels are quite often characterized as environments with a fluctuation noise. Such noise is represented with the white Gaussian noise with a variable variance. The measured power spectrum density (PSD) of the noise gives an average power of the noise in the estimated interval. The time characteristic of such noise in a selected interval cannot be determined from the measured PSD. In that circumstances, the time-variable variance of the noise must be determined from the measured

Experimental results

To evaluate proposed estimation formulae , , variance parameters p were estimated for 1. simulated data and 2. for real data.

The software package Mathematica (Wolfram research) was used to generate samples and to perform the computations, see web page [5].

Conclusion

A procedure for the Gaussian noise variance estimation is given in the paper. An unbiased variance estimation formula based on maximums of samples is derived. The MLE criterion is used to derive estimation formulae and the LMS procedure is applied to complete the estimation results. To evaluate the accuracy of the proposed estimation formulae, estimation results computed on simulated samples are given.

References (5)

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