Majority of spectrum sensing schemes are focused on primary user detection in Gaussian noise. The Gaussian distribution fails to satisfactorily describe some noise in practice. For instance, the Gaussian distribution cannot be used to model Radio Frequency (RF) noise and low frequency atmospheric noise. This is primarily due to the fact that the noise in practice is likely to generate observations of high magnitude than what can be produced from the Gaussian distribution. High magnitude observations are also referred to as impulsive noise. This conveys that the probability distribution function (pdf) of such noise has heavy tails. It can be stated that the pdf of the
For accurate detection of primary user (PU) signal under noise uncertainty many detection schemes are designed for additive Gaussian noise. The effect of threshold selection over the performance of spectrum sensing in cognitive radio network (CRN) using energy detector (ED) is studied in
Recently, under the category of nonparametric detection, some detection schemes based on the Goodnessoffit test (GoFT) have been proposed. They include the AndersonDarling (AD) detection, the KolmogorovSmirnov (KS) test, the detection based on ordered statistics and so on. These schemes perform better than energy detection in a Gaussian environment
This work considers the problem of spectrum sensing with the assumption that the noise follows a nonGaussian distribution with heavier tails. The main contributions in this paper are summarized as follows:
The effect of nonGaussian noise on the energy detector’s performance in noise uncertain environment is investigated.
To overcome the effect of noise uncertainty, the spectrum sensing algorithm using adaptive fuzzy threshold proposed in
The effect on the probability of detection due to the characterization of the statistical moments in the noise distribution is investigated.
Performance evaluation of the conventional and fuzzy threshold methods are carried out through simulations under different noise uncertainty conditions.
The best method out of these techniques under varying noise uncertainty conditions is identified.
The rest of this paper is organized as follows. Section 2 gives the outline on the nonGaussian noises. Section 3 and 4 detail the system model for the spectrum sensing and the proposed method. The results are presented and discussed in Section 5 and concluding remarks are provided in Section 6.
Pearson introduced kurtosis to understand whether a given distribution is Gaussian or not. Three conditions proposed by Pearson for a distribution to be called as Gaussian are (i) kurtosis, (ii) skewness, (iii) the distance between mean and mode (modal divergence). Skewness and kurtosis are the higher order moments that are used to determine the difference between other distributions and the Gaussian distribution
Since the Laplace distribution is the most popular distribution and is frequently used in engineering studies this work focuses on primary user signal detection in Laplacian noise. The Laplace distribution is within the class of generalized Gaussian distributions.
The Gaussian pdf is given as
The Laplacian pdf is given by
Where,
Normally, a heavier tailed noise has a larger degree of nonGaussianity, also known as kurtosis, defined as
Equivalently the kurtosis coefficient minus 3 is referred to as excess kurtosis. As the kurtosis increases the tails are heavier and k as 6 (excess kurtosis is 3) corresponds to Laplacian noise, while when the pdf is Gaussian the excess kurtosis is 0.
The detection of the PU activity in the presence of noise by the secondary user (SU) is modeled as a binary hypothesis testing problem as given below:
where v[n]’s are independent and identically distributed (i.i.d.) noises, with nonGaussian distribution of zero mean and variance
The energy detector (ED) is the most commonly used method for PU detection as it has very low computational complexity. The energy detector under Laplacian noise (EDL) is discussed in
The test statistic of energy detector under Laplacian noise is given as
The energy measurement
Where, P gives the power of the PU signal, N gives the number of samples, variance
In practical scenarios, it is not possible to accurately know the statistics of the noise v[n]. This creates the uncertainty in noise in the detection that may vary with time. The noise uncertainty is modelled such that the noise power is distributed in a single interval as given below.
Where,
The SNR wall for an energy detector under Laplacian noise is given in
Considering the worstcase noise uncertainty Eqn (6) and (7) which gives false alarm and the probability of detection respectively are modified.
The decision on PU activity by the CR device is usually made by a conventional ED using a single threshold. But for accurate detection it is necessary to fix an optimum threshold level, as the detection performance is dependent on the threshold level. The optimum threshold cannot be easily defined due to the problem of noise uncertainty. The situation worsens with low SNR, since the noise uncertainty leads to SNR Wall which is defined as the SNR below which the detector cannot provide reliable detection. In addition, the detection performance is dissatisfactory under the nonGaussian noise in comparison to the Gaussian noise
To overcome the effect of noise uncertainty in CR systems several studies have been proposed as discussed in Section 1 of the paper. In this study a fuzzy thresholding scheme is investigated under nonGaussian scenarios. This study is the first of its kind which uses a fuzzy thresholding approach for PU detection in a nonGaussian environment. The two thresholds are therefore formulated from Eqn (6) as
Where,
The decision process using the two thresholding scheme is given below:
The fuzzy decision F is defined as
Where, Z is the output of the fuzzy logic system, λ_{F }is the threshold set for taking fuzzy logic decision.
The final decision X=D+F
The decision in the confused region is taken using two antecedents called 1) Credibility 2) SNR available at SU. The output of the fuzzy logic system is the consequent which gives the possibility of the primary user activity.
The credibility C is defined as
If the value of C is high, then H_{1} is favoured and if the value of C is low H_{0} is favoured.
The fuzzy rule base and the membership for the antecedents and consequents are detailed in
Antecedent 1 (Credibility) 
Antecedent 2 (SNR) 
Consequent (Likelihood of the presence of PU) 
Low 
Low 
Very Low 
Low 
Medium 
Low 
Low 
High 
Medium 
Medium 
Low 
Low 
Medium 
Medium 
Medium 
Medium 
High 
High 
High 
Low 
Medium 
High 
Medium 
High 
High 
High 
Very High 
This section details the results obtained through MATLAB simulations to analyze the performance of fuzzy two threshold scheme under nonGaussian noise.
The spectrum sensing is carried out in a CR environment using energy detection. The detection probability is used as a standard of measurement to determine the sensing accuracy. The primary transmitter signal is considered as a sinusoidal pilot signal of known frequency. The noise is modeled as nonGaussian. In the simulations the Laplacian noise is considered, as it is a special case of nonGaussian noise. It has pdf with excess kurtosis of 3 while the pdf of the Gaussian has the excess kurtosis as 0
Parameters 
Values 

Target false alarm probability 
0.05 

Total numbers of samples N 
100 

SNR range of interest 
20 to 5dB 

Excess Kurtosis 
Gaussian 
0 
NonGaussian 
3 
The method proposed in
SNR 
PD using Single Thresholding under Gaussian Noise 
PD using Single Thresholding under Non Gaussian Noise 
PD using Fuzzy Thresholding under Gaussian Noise 
PD using Fuzzy Thresholding under NonGaussian Noise 
15 
0.04 
0.05 
0.15 
0.12 
10 
0.06 
0.06 
0.18 
0.15 
Realizing Eqn (1017) in MATLAB and by appropriately choosing the membership functions and the fuzzy rule base in the fuzzy inference system the proposed adaptive thresholding using fuzzy logic method for ED under Laplacian noise, provides reliable detection compared to the other methods when the noise uncertainty increases. Another interesting observation drawn is on the improvement in signal detection in the low SNR regimes.
From
Kurtosis refers to the size of the tails on a distribution. The tails of a distribution quantify the number of events which have appeared that are outside the standard range. If the distributions of event outcomes have lots of occurrences of outlier results, this causes heavy tails on the bellshaped distribution curve. This is referred to as excess kurtosis. A kurtosis of 3 is termed as mesokurtic. Standard normal distributions are mesokurtic. If the kurtosis is >3 it can be visualized as a thin bellshaped curve with a high peak termed as leptokurtic. One of the most wellknown leptokurtic distributions is Student's t distribution, Laplace distribution. When kurtosis <3 is recognized as platykurtic with broad peak and thick tails. All uniform distributions are platykurtic.
The detection probability achieved for the mesokurtic, platykurtic and leptokurtic distributions are plotted in
In the leptokurtic distribution, like the Laplace distribution, the tails approach asymptotically to zero more slowly than a Gaussian. Hence the number of outliers are more than the normal distribution. Thus, the uniform distribution with fewer numbers of outlier’s shows higher PU signal detection compared to normal and Laplace distributions.
In this paper, some typical impairments in PU detection, such as the effect of a nonGaussian noise and noise uncertainty are investigated. Majority of spectrum sensing schemes are focused on primary user detection in Gaussian noise. But in practical wireless communication scenarios the noise usually has a heavy tailed nature. The Laplacian noise is an important nonGaussian noise distributions and hence it is frequently used in engineering studies. This work focuses on primary user signal detection in Laplacian noise. Through Monte Carlo simulations it is observed that a nonGaussian noise noticeably affects the performance of ED. Also a fractional change in noise uncertainty degrades the performance of energy detector. Hence to combat the above degradation in energy detector’s performance the spectrum sensing algorithm using adaptive fuzzy thresholding which was primarily studied under Gaussian noise is extended under a nonGaussian noise environment. Through simulations it is shown that by appropriately choosing the membership functions and the fuzzy rule base in the fuzzy inference system the proposed adaptive thresholding using fuzzy logic method for Laplacian energy detector provides reliable detection. The effect of the probability of detection due to the characterization of the skewness and kurtosis are also studied. Results show that as the kurtosis increases, the tails are heavier and it degrades the PU detection. The study limits to the use of single antenna in the sensing process. Hence the effect of using multiple antennas in the sensing process and in fading scenarios can be considered for future investigation in a nonGaussian noise environment.