Mobile Information Systems

Volume 2018, Article ID 7328910, 9 pages

https://doi.org/10.1155/2018/7328910

## Nonparametric Blind Signal Detection Based on Logarithmic Moments in Very Impulsive Noise

^{1}School of Electronic Science and Engineering, National University of Defense Technology, Changsha 410073, China^{2}College of Electronic Engineering, Naval University of Engineering, Wuhan 430033, China

Correspondence should be addressed to Jinjun Luo; moc.kooltuo@oulnujnij

Received 2 October 2017; Accepted 5 March 2018; Published 12 April 2018

Academic Editor: Jinglan Zhang

Copyright © 2018 Jinjun Luo 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 detection problem in impulsive noise modeled by the symmetric alpha stable distribution is studied. The traditional detectors based on the second or higher order moments fail in noise, and the method based on the fractional lower order moments (FLOMs) performs poorly when the noise distribution has small values of characteristic exponent. In this paper, a detector based on the logarithmic moments is investigated. The analytical expressions of the false alarm and detection probabilities are derived in nonfading channels as well as Rayleigh fading channels. The effect of noise uncertainty on the performance is discussed. Simulation results show that the logarithmic detector performs better than the FLOM and Cauchy detectors in very impulsive noise. In addition, the logarithmic detector is a nonparametric method and avoids estimating the parameter of the noise distribution, which makes the logarithmic detector easier to implement than the FLOM detector.

#### 1. Introduction

Signal detection in noise is a classical problem in signal processing, and it is a key technology for mobile information systems. Usually, mobile information systems are designed under the assumption of Gaussian noise for mathematical convenience. However, in many applications, the background noise is non-Gaussian and shows impulsive behavior, such as man-made noise, interference from adjacent channel, and radio frequency interference (RFI) [1–3]. In impulsive noise, the conventional detection methods optimized under the Gaussian noise assumption suffer from a drastic performance degradation, which degrades the overall performance of the mobile information systems. Thus, more accurate models for impulsive noise are developed. The generalized Gaussian (GG) distribution, Gaussian mixture (GM) model, Middleton Class-A (MCA) model, and symmetric alpha stable distribution are alternative models for impulsive noise.

The distribution has a solid theoretical foundation that it satisfies the generalized central limit theorem and has an advantage of stability. In addition, empirical data prove that the distribution is a successful model for impulsive noise in mobile information systems. For example, radio frequency interference (RFI) for the embedded wireless data transceivers can be modeled using symmetric alpha stable distributions with a characteristic exponent of 1.43 [2]. The co-channel interference in a communication link in a Poison field of interferers shows very impulsive behavior and can be modeled using distributions with small characteristic exponent values [4]. In medical diagnosis or mobile health data processing, evoked potential (EP) detection is an important problem. Experimental analysis showed that the noised EP signal could be successfully modeled using distributions with characteristic exponent values between 1.06 and 1.94 [5]. In [6], the noise fitting procedure indicated that the distribution could fit the noise from GSM-R antennas much better than the Middleton Class-A model and Gaussian distributions. Due to these advantages, the distribution has received an increasing attention for modeling the impulsive noise in recent years. In this paper, we consider the signal detection problem in very impulsive distributed noise.

In the early literature, the proposed detectors for noise usually require prior knowledge of the transmitted signal. This kind of detectors mainly includes the optimal detector and the locally optimal detector under a weak signal assumption. As the probability density function (PDF) of has no closed form expression, the optimal and locally optimal detectors should be implemented through numerical methods, which results in a heavy computational burden. Thus, some locally suboptimal detectors are proposed, such as the soft limiter, hole puncher, and local Cauchy detector [3]. Some detectors based on the approximate expression for the PDF of the distribution are also investigated [7, 8].

In some practical applications, such as the communication countermeasure, and cognitive radio, the transmitted signal is not available. The aforementioned detectors cannot be used directly, and thus blind or semiblind detection techniques are required. The generalized likelihood ratio test (GLRT) is an optimal solution, but it is extremely time-consuming [9]. The blind Cauchy detector is a special case of the GLRT and only performs well for some characteristic exponent in noise. Recently, a scheme based on the fractional lower order moment (FLOM) is investigated in [10]. It performs better than the blind Cauchy detector. However, when the noise distribution has small values of the characteristic exponent, the FLOM detector performs poorly. In addition, the moment order of the FLOM should be determined, whose choice depends on the estimated characteristic exponent.

The logarithmic process was first introduced in [11]. Then the logarithmic moment (LM) was utilized for parameter estimation of the distribution [12–14]. However, its application to signal detection has not yet received much attention. In this paper, we apply the logarithmic moment to signal detection and propose a detector based on it. The performance of the logarithmic detector is investigated in detail. The main contributions include the following: (i) We derive the analytical expressions of the false alarm and detection probabilities in nonfading channels as well as Rayleigh fading channels. (ii) The effect of the noise uncertainty on the detection probability is discussed, and the detection performance with different characteristic exponents and sample sizes is also investigated. (iii) The simulation results illustrate that the logarithmic detector performs better than the Cauchy and FLOM detectors when the characteristic exponent is smaller than 1.5. (iv) The logarithmic detector is a nonparametric method and avoids estimating the parameter of the noise distribution, which make the logarithmic detector easier to implement than the FLOM detector.

The remainder of this paper is organized as follows. In Section 2, the system and noise models are presented and the conventional detectors are introduced. In Section 3, the logarithmic detector is proposed and its performance is analyzed. In Section 4, extensive simulations are carried out. Finally, conclusions are drawn in Section 5.

#### 2. System Model and Conventional Detectors

##### 2.1. System Model

In this paper, the word “signal detection” refers to deciding whether a signal is present or not in the received data. It can be formulated as a binary hypothesis testing problem, described by : signal absent and : signal present. It can be written aswhere is the received data sample at time , represents the impulsive noise, is the transmitted signal, and is the channel gain between the transmitter and receiver.

The impulsive noise is modeled by an independent and identically distributed (i.i.d.) distribution. Since the received signal is usually a sum of multiple nonline of sight (NLOS) signals, can be modeled as a Gaussian random variable with zero mean according to the central limit theorem and is assumed to be independent of the noise .

The distribution is described by its characteristic function:where . is the characteristic exponent and determines the thickness of the distribution tail. When is smaller, the tail is more thick and the pulse characteristic is more noticeable. is the scale parameter, also known as the dispersion parameter. It indicates the degree of the dispersion spread which is similar to the variance in a Gaussian distribution.

The PDF of the distribution is the inverse Fourier transform of the characteristic function, namely,

The PDF does not have a closed form expression except for and . In the two special cases, the distribution reduces to Gaussian distribution and Cauchy distribution, respectively.

##### 2.2. Conventional Detectors

In blind signal detection, the transmitted signal and channel gain are usually not available, so the likelihood ratio test (LRT) cannot be used directly. We can first estimate the unknown parameters and then apply the likelihood ratio test, which is known as the generalized likelihood ratio test, defined as [9]where and are the maximum likelihood estimates of and . Due to the numerical evaluation of and the parameter estimation, the GLRT detector is too complex to be applied for real-time applications. In the special case of , the GLRT detector becomes a Cauchy detector [9], namely,

In the case of , the second and higher order moments do not exist, so the traditional detector based on these moments cannot be used. However, the fractional lower moments (FLOMs) of any order less than do exist. Then, the FLOM detector is proposed [10], which is defined aswhere is the order of the fractional moment. In order to ensure the existence of the FLOM statistic variance, the allowable value of should be limited between 0 and . The choice of depends on which could be estimated by some methods [12, 13].

#### 3. Nonparametric Logarithmic Detector and Performance Analysis

##### 3.1. Logarithmic Moment-Based Detector

The logarithmic moment has been successfully applied to parameter estimation of the distribution, but few studies investigate its application to signal detection. In this paper, we proposed a detector based on the logarithmic moment (LM). The LM detector is defined aswhere is the detection threshold which could be determined by a given . Compared with the FLOM detector, this detector does not need to decide the moment order , which is convenient in practical applications.

##### 3.2. Performance Analysis of the Logarithmic Detector in Nonfading Channels

The statistic is the sum of independent random variables. When is large enough, the statistic follows an asymptotic Gaussian distribution under either hypothesis due to the central limit theorem. To calculate the false alarm and detection probabilities, the mean and variance of the statistic should be calculated first.

Under hypothesis , the mean and variance of can be calculated as (see Appendix for detail)

Under hypothesis , the mean of can be expressed aswhere

The variance of can be computed aswhere

Then the false alarm and detection probabilities can be calculated as

The expression of has a closed form, so the threshold can be easily calculated from this expression, where the parameters and can be estimated by many methods [12, 13]. Although the expression of has no closed form, it can be evaluated by numerical methods. Through evaluating the expression of , we can conveniently obtain the relationship between the detection probability and the sample size under different GSNRs, which gives us useful guideline for the choice of the sample size in practical applications.

In the following, we consider the effect of noise uncertainty on the logarithmic detector. Noise uncertainty is mainly caused by temperature variation, nonlinearity of components, initial calibration error, and interference [15, 16]. In general, a detector with a fixed threshold is designed based on the nominal noise power which is estimated in a finite time, so the performance will degrade when the actual noise power does not equal the nominal noise power.

From (8), (9), and (16), the threshold for logarithmic detector depends on the knowledge of the parameters and which should be estimated by taking a large number of samples but some residual uncertainty still exists. Next, we only analyze the effect of uncertainty of on the detection performance; a similar procedure can be carried out for the parameter .

We denote the nominal noise dispersion or the estimated noise dispersion by and the actual noise dispersion by . In the presence of noise uncertainty, , where is the noise uncertainty coefficient, and its value in dB is denoted by . Usually, is uniformly distributed in the interval [10, 17]. Then, the PDF of can be expressed as

According to the relationship between and , we can easily obtain the PDF of as follows:

In nonpresence of noise uncertainty, the false alarm and detection probabilities can be calculated using (16) and (17). When the noise uncertainty does exist, the noise dispersion is no longer a constant value. In this case, we view (16) and (17) as conditional probabilities, where the condition is that is fixed. Then and are functions of the actual noise dispersion , and they can be rewritten as and , respectively. By averaging (16) and (17) over (19), the probabilities of false alarm and detection under noise uncertainty can be obtained aswhere and . and do not have a closed form expression, but they can be evaluated by numerical methods.

##### 3.3. Performance Analysis of the Logarithmic Detector in Rayleigh Fading Channels

In Rayleigh fading channels, remains the same as that of (16) in nonfading channels. The average detection probability can be evaluated by averaging the conditional in the nonfading case as given in (17) over the channel gain. In this case, in (17) can be regarded as a function of , rewritten as . The average detection probability in Rayleigh fading channels can be obtained as follows:

Since the channel gain is Rayleigh distributed, its PDF is shown aswhere . So, can be calculated as

It is difficult to find a closed form expression for , but we can evaluate it by numerical methods. Specially, when the is low enough where and , can be simplified as

Because is not a function of , the integral will be greatly simplified.

#### 4. Simulation Results and Analysis

In this section, simulation results are presented to show the performance of the logarithmic detector. The channel is assumed fading free, unless otherwise specified. The transmitted signal is a Gaussian process with zero-mean and variance . The results are based on 10000 Monte–Carlo simulations with a generalized signal-to-noise ratio (GSNR) defined as [10].

Figure 1 shows the detection probability versus the GSNR for the logarithmic detector in nonfading channels, where and . The curves are obtained from the simulation using (7) and by evaluating the analytical expressions in (16) and (17). The theoretical results are consistent with the Monte–Carlo simulation results very well for different and , which proves the accuracy of the analytical expressions of the false alarm and detection probabilities. In addition, for a fixed false alarm probability, the detection probability increases with the increment of when , whereas it decreases with the increment of when . Therefore, the relationship between the detection probability and depends on the values of .