Journal of Sensors

Volume 2018, Article ID 6804197, 6 pages

https://doi.org/10.1155/2018/6804197

## Fast Extraction for Skewed Source Signals Using Conditional Expectation

College of Communication Engineering, Army Engineering University of PLA, Nanjing 210007, China

Correspondence should be addressed to Qiao Su; moc.liamxof@018usoaiq

Received 14 December 2017; Revised 12 June 2018; Accepted 2 July 2018; Published 22 July 2018

Academic Editor: Fanli Meng

Copyright © 2018 Qiao Su et al. 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

The extraction of the stochastic source signals whose probability density functions (PDFs) are skewed is very important in many applications such as biomedical signal processing and mechanical fault diagnosis. This paper shows that the skewed source signal with the maximal absolute value of skewness can be fast extracted by a proposed algorithm using conditional expectation. Compared with the existing conditional expectation-based algorithms, the proposed one possesses two main advantages. One is that it does not require the prior knowledge of the positive support of the desired source, namely the time indices where the source of interest is positive. The other is that it can be employed both in the determined and underdetermined cases. Furthermore, the proposed algorithm is mainly based on the first- and second-order statistics and does not need the preprocessing so that the computational cost is significantly low. Simulation results show the superiority of the proposed algorithm over the existing methods and indicate that the proposed algorithm also performs well in the underdetermined case when the number of sensors is slightly less than that of sources.

#### 1. Introduction

The target of independent source extraction is to estimate a specific source from the observations mixed by the source signals, where the source signals are mutually independent. It can be applied in various areas such as speech and image processing, biomedical signal processing, mechanical fault diagnosis, and wireless communication. Due to these wide application fields, independent source extraction has gained much attention in the past few decades [1, 2]. In some applications especially biomedical signal processing and mechanical fault diagnosis, it is of significance to extract the stochastic source signals with the skewed probability density functions (PDFs). For instance, the mechanical vibrations derived from defective bearings, which are desired to be extracted in vibration analysis, may have asymmetric PDF [3], and the fetal electrocardiogram (FECG) signal has quite different skewness compared to the maternal electrocardiogram (MECG) signals [4], where the FECG requires to be estimated from the mixtures of the FECG and the MECG. The existing extraction algorithms mainly employ the second- and higher-order statistics by exploiting the statistical independence of the sources, and some of them need to preprocess the mixture data such as FastICA [5]. Recently, Zarzoso et al. [6] and Xu et al. [7] and Xu and Shen [8] proposed a class of more computationally efficient algorithms for independent source extraction based on the first-order statistics as well as the conditional expectation. But before performing this class of algorithm, it requires to know the time indices where the source of interest is positive, that is, the positive support of the desired source, which reduces the practicability of this class of algorithm.

In this paper, we propose a new extraction algorithm using the conditional expectation for the skewed source signal with the maximal absolute value of skewness. It does not require the prior knowledge of the positive support of the desired source and can be applied both in the determined and underdetermined cases. It should be noted that after obtaining the source signal with the maximal absolute value of skewness, the source signal with the second maximal absolute value of skewness can also be extracted by the proposed algorithm from the mixtures which subtract the component of the estimated source (e.g., through linear regression as in [6]). Likewise, if required, the other skewed source signals with different skewness can be gained sequentially. For simplicity, this paper assumes that the skewed source signal with the maximal absolute value of skewness is the desired source. The proposed algorithm obtains the desired column vector of the mixing matrix corresponding to the source of interest by the conditional expectation and retrieves the desired source by the minimum mean-squared error-based (MMSE) beamforming approach [9]. Through several iterations with the initial value of the desired column vector of the mixing matrix gotten by the estimated approximately purely positive or negative interval, the estimation of the desired source can be derived accurately. The proposed algorithm is rather cost-effective, since it is mainly based on the first- and second-order statistics and does not require the preprocessing. Simulation results validate the superiority of the proposed algorithm over the existing methods and show that the proposed algorithm even performs well in the underdetermined case when the number of sensors is close to that of sources.

#### 2. Data Model

Consider the instantaneous linear mixture shown by where is composed of source signals which are mutually independent, consists of mixtures received by sensors, is the unknown mixing matrix, and the superscript represents the transpose operator. In this paper, we consider that the source signals are stochastic with the unimodal continuous PDFs, since many signals such as some vibrational signals [10] and ECG signals [11] possess these characteristics. For convenience, we further assume that the source signals are stationary with zero mean and unit variance. Note that the assumption of stationary source signals is reasonable, since some nonstationary sources can also be divided to be several stationary blocks and we can tackle these stationary blocks separately. For example, some ECG signals can be regarded to be stationary in the duration of one heartbeat, despite they are nonstationary signals [12]. Our goal is to extract the skewed source signal which has the maximal absolute value of skewness from the accessible mixtures.

#### 3. The Proposed Algorithm

Since the first-order statistics-based algorithms [6–8] generally extract the source of interest under the condition that the mixing matrix is a unitary matrix, they can be only used in the determined case. In this paper, we remove this condition and obtain the column vector of the mixing matrix corresponding to the desired source based on the conditional expectation shown by where denotes the expectation operator, is the desired source, is the th column vector of the mixing matrix, , and . Obviously, and are the constants according to the stationary assumption of the sources. The proof of (2) can be easily deduced from the assumptions of the sources, that is, where is the unit vector in which the th entry is 1 and the other entries are 0. The proof of the other equation when in (2) is similar with (3). Then, we estimate the desired source by the MMSE beamforming approach, which is where is the estimation of , is the covariance matrix of the mixtures, and the superscript −1 stands for the inversion operator. Thus, when and and the positive or negative support of the desired source are provided, the desired source can be estimated by (2) and (4). Actually, and can be easily figured out when the PDF of the desired source is known. The values of for some normalized distributions are shown in [6]. However, when the PDF of the desired source is unknown, and are not obtainable. In this case, we can only get the direction of the vector by (2) which is the same with the direction of or . Since the estimated has the correct direction and unknown size, it will lead to the ambiguous amplitude of the estimated by (4). Fortunately, this indeterminacy of amplitude for estimating the desired source is allowable in many applications. For simplicity, we set , and then (2) denotes the direction of . Unless stated otherwise, in the rest of this section, estimating refers to estimate the direction of the vector .

In practical applications, the complete information about the positive or negative support of the desired source is extremely hard to be acquired. However, it is more possible to get a subset of the samples of the desired source in which the positive samples are more than the negative samples obviously, or it is the opposite. We can see that this subset is close to a purely positive or negative set. We define the correct index classification ratio as in [6], where is the number of the positive samples in the subset and is the total number of the samples in the subset. It was suggested in [6] that when is close to 0 or 1, the subset can be also used to estimate the desired source and the estimation performance is only slightly worse than that of employing the complete information about the positive or negative support of the desired source. Similarly, if we get the subset like this, we can use the information about this subset to roughly estimate . Fortunately, for a signal with unimodal continuous skewed distribution, we can utilize the asymmetry of its PDF to get the subset which is close to be purely positive or negative.

Figure 1 shows the PDF of unimodal continuous skewed distribution, where denotes the skewness defined by in which and are the mean and standard deviation of the random variable , respectively. , , and in Figure 1 represent the positively skewed distribution, the negatively skewed distribution, and the symmetric distribution, respectively. Since we consider the case when is zero mean and unit variance, the definition of skewness is reduced to be . When is subject to a positively skewed distribution (), we take a finite set of samples, , generated by the distribution of into account. As the skewness reflects the asymmetry of a PDF, a larger absolute value of skewness means stronger asymmetry of a PDF. Thus, it is easily deduced that most samples in are smaller than zero, and the proportion of the negative samples increases when rises. When randomly extracting some samples from to form a subset, we can find that the subset may get close to a purely negative set. And with the increase of , it will be more likely that the samples in this subset are completely negative. Likewise, in the case with , we can obtain the similar results. These provide the possibility to extract the source with the maximal absolute value of skewness based on the conditional expectation. Nevertheless, the random extraction is unstable, because this method is probabilistic. Instead, we separate the whole samples into several equivalent intervals and test all the intervals to find out the one closest to be purely positive or negative. Note that the size of each interval should be appropriate. If the size of each interval is too large, there will be high probability to contain the positive and negative values simultaneously. On the contrary, too small size of each interval cannot present the skewness characteristic of the skewed signal, since exhibiting the skewness requires a certain quantity of samples. The proper size should be adjusted depending on the nature of the signal, mainly its PDF. In the following section, we show that this proper size and the number of the intervals can be gotten by the simulations.