Median-Difference Correntropy for DOA under the Impulsive Noise Environment

e source localization using direction of arrival (DOA) of target is an important research in the eld of Internet ofings (IoTs). However, correntropy suers the performance degradation for direction of arrival when the two signals contain the similar impulsive noise, which cannot be detected by the dierence between two signals.is paper proposes a new correntropy, called the median-dierence correntropy, which combines the generalized correntropy and the median dierence. e median dierence is dened as the deviation between the sampling value and the median of the signal, and it intuitively reects the abnormality of impulsive noise. en, the median dierence is combined with the generalized correntropy to form a new weighting factor that can eectively suppress the amplitude level of impulsive noise. To improve the robustness of the algorithm, an adaptive kernel size is also integrated into the weighting factor to obtain the optimal local feature. e inuence of adaptive kernel sizes on the proposed algorithm is simulated, and the comparison between three typical direction-of-arrival estimation algorithms is conducted. e results show that the accuracy of the median-dierence correntropy is signicantly superior to the correntropybased correlation and the phased fractional lower-order moment for a wide range of alpha-stable distribution noise environments.


Introduction
e source localization using direction of arrival (DOA) of target is an important research in the eld of Internet of ings (IoTs).e direction-of-arrival (DOA) approaches based on acoustics have many applications including radar, sonar, seismic exploration, navigation, and sound source tracking [1][2][3].DOA estimation is usually regarded as a problem of signal matching, and the performance is signi cantly in uenced by noise.A majority of existing DOA estimations are based on the concept that noise follows Gaussian distribution [4,5].Since the Gaussian process has second-order and higher-order statistics, the traditional DOA algorithms can easily evaluate the signal characteristics according to second-order statistics [6].e multiple signal classi cation (MUSIC) algorithm [7,8] and estimation method of signal parameters via rotational invariance techniques (ESPRIT) are the basic subspace algorithms which have good performance [9,10].MUSIC is the representative of the noise subspace algorithm, and ESPRIT is the representative of the signal subspace algorithm.
Because of atmospheric noise, electromagnetic interference, sea clutter, car ignitions, and o ce equipment, the signal is corrupted by the extremely impulsive noise that exhibits irregularity in time domain.In addition, the probability density functions of impulsive noise decay with heavy tails and do not follow a common Gaussian distribution [11].erefore, alpha-stable distribution is usually used to de ne impulsive noise [12].e conventional covariance matrix is calculated from the second-order statistics of the signal, which may be in nite when the data are corrupted by the extremely impulsive noise [13,14].In addition, the conventional DOA algorithms cannot be decomposed into the signal subspace and the noise subspace with covariance matrix.us, it has become increasingly important to study DOA under the impulsive noise environment.
e relevant statistical algorithms mainly focus on the existence of statistics.e fractional lower-order statistics (FLOS) algorithms [15] exhibit a more desirable performance than the second-order statistics for alpha-stable distribution [16].When the signal contains impulsive noise, the signal bears the fractional lower-order statistics.e FLOS algorithms employ the minimum dispersion (MD) criterion to suppress impulsive noise, such as the robust covariation in ROC-MUSIC [17], the fractional lower-order statistics-(FLOM-) based MUSIC, and the phased fractional lower-order moment (PFLOM).
e FLOM algorithmbased MUSIC obtains a finite covariance by suppressing one of two cross-correlation signals when the characteristic exponent of alpha-stable distribution ranges from 1 to 2 [18].
e PFLOM algorithm gets the accurate DOA estimation with circular symmetrical signals, embedded in the additive impulsive noise [19].However, investigators demonstrate that the performance of FLOM and PFLOM algorithms depends on the relationship between the parameter of fractional lower-order moment and the characteristic exponent of alpha-stable distribution.If the characteristic exponent is unknown, the performance of FLOM and PFLOM algorithms seriously decreases.e correntropy criterion [20] is a relatively simple method that can measure the local similarity between two signals [21,22].Because it has the properties of M-estimation, the correntropy has been widely used in the impulsive noise environment [23].Zhang et al. [24] investigated a narrowband model based on the generalized correntropy which is called the correntropy-based correlation (CRCO) in impulsive noise environment.e CRCO algorithm imposes a correntropy operator on the covariance matrix to depress impulsive noise.
e generalized correntropy is suitably for dealing with the template matching between the received signal and the template signal [25].Since DOA estimation is a matching problem between two signals, the generalized correntropy is incapable of measuring the difference between the outliers and fails to suppress impulsive noise.us, when data contain the similar impulsive noise, the cross-correlation of the generalized correntropy is infinite.
In order to solve the problem that correntropy cannot distinguish the similar impulsive noise, we propose a median-difference correntropy (MDCO) algorithm.e MDCO depending on the inner product of vectors measures the similarity of multidimensional properties from input space.A weighting factor of a median difference is defined and evaluates the similarity between the sample value and the median of the signal.e median difference intuitively reflects the abnormality of impulsive noise to guarantee that the autocorrelation is finite.
en, MDCO derives the weighting factor of the generalized correntropy from the correntropy criterion which suppresses the larger impulsive noise.Hence, the weighted covariance function of signal employs the generalized correntropy instead of second-order statistics.ese two criteria map the signal from the low-dimensional space to an infinite-dimensional reproducing kernel Hilbert space (RKHS) with impulsive noise, thus including higher-order signal statistics.
e adaptive kernel size is applied to the weighting factor for MDCO which can obtain the optimal local feature.At convergence, MDCO is unbiased and it is applicable to achieve CRLB under various parameters.e main work is summarized as follows: (1) We propose a median-difference correntropy (MDCO) algorithm, which can effectively combine the generalized correntropy and the median difference to suppress impulsive noise (2) To improve the robustness of MDCO, we also introduce a novel adaptive kernel size into the weighting factor of the generalized correntropy and the median difference e rest of this paper is organized as follows: e problem model is defined with DOA in Section 2. In Section 3, we present the median-difference correntropy (MDCO) for DOA estimation under impulsive noise environment.
e performance evaluation is presented in Section 4. Finally, conclusions are drawn in Section 5.

Problem Formulation
It is considered that a uniform linear array of M isotropic acoustic sensors receives the far-field signal generated by P(M > P) narrowband sources.en, Figure 1 illustrates the linear array model.e first array sensor is a reference sensor, and the received signal of the mth sensor can be expressed as where s p (t) is the pth signal at time t, n m (t) is the noise of the mth sensor, τ mp denotes the time delay of the mth acoustics sensor relative to the reference sensor for the pth signal, and f is the center frequency of signal.e received signal of acoustics can be expressed as where Impulsive noise N(t) in ( 2) is usually defined as symmetric alpha-stable distribution.However, the probability density function of symmetric alpha-stable distribution cannot be expressed by a general expression [24].erefore, it is generally introduced by its characteristic function which can be expressed as where α is the characteristic exponent of symmetric alphastable distribution whose range is 0 < α ≤ 2.Moreover, the smaller the α becomes, the more impulsive the non-Gaussian noise will be.c is the dispersion.Usually, the received data contain very little impulsive noise and can also be represented as where Many DOA estimation algorithms usually use the covariance to represent the second-order statistics of the signal.
e conventional covariance matrix can be given as where the superscript [•] H represents transpose, the notation E[•] represents the expectation operation, and R s and R n are the covariance matrix of signal and noise, respectively.If the noise contains impulsive noise, the signal and noise cannot be completely orthogonal.e conventional DOA algorithms that describe signal with impulsive noise cannot be decomposed into the signal subspace and the noise subspace by the covariance matrix.erefore, the covariance matrix ( 7) can also be represented as where R w denotes the covariance term generated by outliers and the covariance R is dominated by impulsive noise when some elements of covariance R w is much larger than the corresponding elements of A(θ)R s A H (θ) + R e .Because natural noise and man-made noise often include outliers, some elements of matrix R w may have a large value which gives rise to false DOA estimation.From Figure 1, we can see that the received data of the acoustic sensors suffer impulsive noise with large values For example, because data b 1 and c 2 which are located on the different sensors are both impulsive noise at the same time, the cross-correlation with a relatively large value is infinite: where Y and Z are two sampling points of total snapshots and R co mM (t 2 ) represents the cross-correlation of data b 1 and c 2 at time t 2 .Assuming that only one acoustic source impinges on the array, the signal X m (t 2 ) is a delayed signal of X M (t 1 ) with time Δt when the signal X M (t 1 ) is received by the array sensor M at time t 1 .Furthermore, array sensors m and M both contain impulsive noise at time t 2 .At this point, the noise covariance R w (t 2 ) is dominant so that the noise subspace would spread to the signal subspace, causing the characteristics of the signal to be covered.In addition, the autocorrelation of data a 1 , b 1 , and c 1 is also nonexistent.
e goal is to search for an efficient strategy of suppressing impulsive noise that makes it possible to lim where ψ(R w ) indicates the operator of suppressing impulsive noise.In principle, as long as R w provides a small

Median-Difference Correntropy (MDCO)
In order to solve impulsive noise, the structure of this section is as follows: firstly, the median-difference correntropy (MDCO) is proposed.Next, we summarize some properties for MDCO.Finally, the application of MDCO for DOA estimation is designed.
Normally, if the signal follows Gaussian distribution, there is a 99.73% probability that results in X ′ ∈ (μ ′ − 3σ ′ , μ ′ + 3σ ′ ), where X ′ is the collection of Gaussian distribution signal, μ ′ is the mean value, and σ ′ is the standard deviation (μ ′ and σ ′ have no relationship to the similar parameters behind At this point, we can simply set a threshold ε to measure the similarity between the signals and set ε � 6σ ′ .Definition 1.For two variables Y and Z of total snapshots X, where Y, Z ∈ X, if |Y − Z| < ε, we say that the variable Y is similar to Z. As shown in Table 1, the similarity of the variables Y and Z is divided into four cases.If Y, Z ∈ X ′ , (X ′ ∈ X), the variables Y and Z are considered normal values.But, if Y, Z ∈ X ′ + W, where X ′ + W ∈ W, we consider the variables Y and Z to be the outliers containing impulsive noise.One of our goals is to suppress the three abnormal cases that contain impulsive noise.

Median-Difference Correntropy (MDCO).
In this section, we present a median-difference correntropy (MDCO) algorithm for DOA with alpha-stable distribution.e correntropy is a new method for nonlinear and local optimal measurement of two random variables Y and Z.It is defined as follows: where κ(., .) is a translation function of the shift-invariant Mercer kernel [26] and p Y,Z (., .) is the joint probability density function of Y and Z.In this paper, the Gaussian density function is used as kernel function.e Gaussian kernel creates a RKHS [26] with universal approximating capability.
e Gaussian kernel is numerically stable and usually gets reasonable results.In this paper, we use a Gaussian kernel to suppress impulsive noise.
Assuming that the random variables Y and Z follow symmetric alpha-stable distribution whose characteristic exponent is 1 < α < 2, the median-difference correntropy (MDCO) can be considered as where σ is the kernel size of Gaussian kernel.e variable B is the average of l 1 − norm of the preprocessing data which can approximately represent the median of the received data.
Because impulsive noise contained in the signal is sparse, the variable B can effectively weigh the amplitude of the signal.e variables |Y| − B and |Z| − B reflect the median difference to which the received signal deviates from the median B. erefore, the variables |Y| − B and |Z| − B are called the weighting factors of the median difference.e correntropy Y − Z measures the similarity between the signals and is called the weighting factor of the generalized correntropy.e median difference evaluates the deviation between the sampling value and the median of the signal, and it intuitively reflects the abnormality of impulsive noise.
en, the median difference is combined with the generalized correntropy to form a new weighting factor, which can effectively suppress the amplitude level of impulsive noise.
en, the median difference and the generalized correntropy are combined with the traditional covariance, and a novel generalized weighted covariance is obtained.
To simplify (12), the median differences |Y| − B and |Z| − B can be combined.e variable |Y| + |Z| − 2B reflects the deviation degree of |Y| + |Z|.e relaxation of (12) can be expressed as where |Y| and |Z| are the absolute value of the signal and σ 1 and σ 2 are the kernel sizes that control the scale of the metric.e short-time energy method is a common method to detect impulsive noise in time domain [27].We use a shorttime average energy method to preprocess the received data.However, the data processed by the short-time average energy method does not directly be used as the input of MDCO.e short-time average energy indicates that the energy of the signal is average in a short segment.e shorttime average energy in the kth segment can be expressed as Wireless Communications and Mobile Computing where X(i) is the sampling sequence of the original signal.By the formula above, we can obtain the average energy with I data points.
If the signal energy is much larger than the short-time average energy, the preprocessing signal X pre can be represented as erefore, in this study, B can be expressed as where N is the total sampling size of snapshots.e signal with impulsive noise does not participate in calculating the median of the signal.erefore, for a majority of data, the median B is less than ε/2 due to preprocessing.e key to the median-difference correntropy is that the kernel size can work in the confidence interval to eliminate impulsive noise.
e kernel size is usually related to the dispersion coefficient c by considering the local feature of the signal.We take a modified Sigmoid function to make the kernel size adaptive [24,[28][29][30].en, kernel sizes σ 1 and σ 2 can be expressed as where g 1 and g 2 indicate the scale of the modified Sigmoid function and h 1 and h 2 control the monotonicity of the modified Sigmoid function [28].e shrinkage direction and rate are determined by h 1 and h 2 , respectively.
e adaptive median-difference correntropy provides a new metric criterion for impulsive noise.And the M-estimators of the autocorrelation and cross-correlation of random variables Y and Z are in existence associated with the generalized correntropy and median difference.
In order to prove the effectiveness of MDCO in theory, five crucial properties of MDCO are listed as follows: Property 1 (C1).When the noise follows temporally stationary zero-mean white Gaussian processes, MDCO is reduced to traditional second-order statistics, and the performance of MDCO is equal to traditional signal subspace algorithms (see Appendix A for this proof ).

Property 2 (C2).
If the variable Y is an outlier rather than Z, the MDCO can eliminate impulsive noise.R w (t) of MDCO approaches zero; that is to say, R MDCO is close to zero at this point (see Appendix B for this proof ).

Property 3 (C3, C4
).When the variables Y and Z both contain impulsive noise, R w (t) and R MDCO both approach zero (see Appendix C for this proof).Property 4. When the variables Y and Z both contain impulsive noise, the MDCO is more effective than the CRCO for noise suppression (see Appendix D for this proof ).
Property 5. Assuming that the random variables Y and Z follow the symmetric alpha-stable distribution, R MDCO is bounded and the MDCO has the generalized correntropy statistics (see Appendix E for this proof ).
Because of the abovementioned properties, the generalized weighted covariance R MDCO has finite autocorrelation and cross-correlation at all moments.When the sampling data contain impulsive noise, R w (t) of MDCO approaches zero; that is to say, R MDCO is close to zero at this point.If two signals contain the different degrees of impulsive noise at the same time, MDCO can induce a very small weighted covariance by means of the median difference operator and then obtain a convergent covariance matrix.As shown in Table 2, the performances of the MDCO is listed for different noise cases.In summary, the MDCO has reliably good performance in all cases.

Performance Simulations
In this experiment, we compare the MDCO-MUSIC to FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC and test the performance of the proposed MDCO-MUSIC under different parameter conditions.In our experiment, we evaluate the performance of the algorithm from five aspects, namely, various kernel sizes of MDCO-MUSIC, GSNR, snapshots of sampling, different characteristic exponents of alpha-stable distribution, and angular separation of two direction angles.
e resolution probability can well define the performance of the four algorithms.We use a popular resolution criterion to measure the spatial spectrum which can be given as where θ 1 and θ 2 are the two independent direction angles, θ m � (θ 1 + θ 2 )/2 is the midangle between them, and P(θ) is Wireless Communications and Mobile Computing the spatial spectrum.e two direction angles are resolvable if the result on the left of the equation is smaller than that on the right; otherwise, the two direction angles are not resolvable.RMSE is the deviation criterion between the observation and the true value and can be given by where L represents the total number of MC run and  θ 1 (l) and  θ 2 (l) are estimations of θ 1 and θ 2 in the lth MC run, respectively.
Kozick and Sadler has come to a closed-form expression of CRLB for the impulsive noise [31], which can be expressed as    6 Wireless Communications and Mobile Computing where S d (t) � diag s 1 (t), s 2 (t), . . ., s P (t)   is a diagonal matrix of the received signal, the subscript θ of A is omitted from A(θ), D � [za(θ 1 )/zθ 1 , . . ., za(θ P )/zθ P ] is the differential of the array manifolds A, Re • { } is the real part, and the coefficient I c can be expressed as x dx, in which x � |e| is the modulus of the impulsive noise and f(x) is probability density function of x.Note that I 1 � 1/2c with α � 1 and I 2 � 3/5c 2 with α � 2. For simplicity, the coefficient I c for 1 < α < 2 can be approximated by first-order linear interpolation with I 1 and I 2 [32].

Experimental Setup.
e linear array is set to M � 6 omnidirectional acoustic sensors which are placed in a half of the wavelength at the center frequency of the signal.We consider the case that the number of narrowband sources is  Wireless Communications and Mobile Computing two which follow temporally stationary zero-mean white Gaussian processes.e central frequency of signal is set to f � 100 Hz.In order to distinguish the signal, the power of each signal is different.e symmetric alpha-stable distribution is used to model impulsive noise with the characteristic exponent α � 1.4 and dispersion c � 1. e GSNR is set to GSNR � 5 dB.e sampling frequency of each array element is F s � 240 Hz.All experiments are carried out through two directions of arrival associated with θ 1 � − 10 ∘ and θ 2 � 15 ∘ .e snapshots are snaps � 512.Assume that the parameter p of the fractional lower-order statistics is equal to 1.1 in FLOM-MUSIC and PFLOM-MUSIC algorithms.In every experiment, 500 Monte Carlo (MC) runs are performed.Furthermore, we compare the resolution probability and the root-mean-square error (RMSE) between the different algorithms.

Kernel Parameters of MDCO.
In this experiment, we test the performance of MDCO-MUSIC with respect to the different kernel parameters of the weighting factors for the separate characteristic exponents α � 1.4 and α � 1.8.In  Wireless Communications and Mobile Computing order to eliminate impulsive noise, it is necessary to satisfy the requirement that the large local weighting factor of the median difference and generalized correntropy associate with the small kernel sizes σ 2 1 and σ 2 2 ; that is, h 1 and h 2 must be less than zero.Because the weighting factor of the median difference and generalized correntropy are independent parameters, the performance can be analyzed separately.Figures 2 and 3 show that the appropriate g 1 and h 1 for the median difference can effectively suppress impulsive noise and evidently decrease RMSE.From Figure 2, we can see that g 1 ∈ [8,11] would get the best result with three different values of h 1 .e convergence of g 1 is not affected by h 1 .Empirically, it is revealed that impulsive noise can be effectively alleviated in Figure 3, when Figures 4 and 5 illustrate the robustness of MDCO-MUSIC in terms of the appropriate g 2 and h 2 for the generalized correntropy.MDCO-MUSIC gains the optimal performance when the kernel parameter g 2 varies from 9 to 12. Figure 5 shows that RMSE gains a more significant decrease if h 2 ∈ [− 1.4, − 1].Usually, it is a common method that the optimal kernel parameters are obtained by experiments.erefore, we set g 1 � 10, h 1 � − 0.4, g 2 � 10, and h 2 � − 1.2 in other experiments.e MDCO-MUSIC algorithm yields particularly robust estimation to impulsive noise and can accurately estimate the DOA when the GSNR is 2 dB.e RMSE of MDCO-MUSIC is much smaller than that of FLOM-MUSIC and PFLOM-MUSIC.It is clear from Figure 6 that that the performance of MDCO coincides with the CRLB at high SNR.

Performance Comparison versus Characteristic Exponent of Alpha-Stable Distribution.
In this simulation, we aim at evaluating the performance of the characteristic exponent on the basis of probability of resolution and RMSE.e characteristic exponent of the symmetric alpha-stable distribution ranges from 1 to 2. Figure 7 shows that the performance of MDCO-MUSIC outperforms the FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC both in resolution probability and in RMSE.Moreover, the smaller the α becomes, the worse the performance of FLOM-MUSIC and PFLOM-MUSIC will be.Another observation is that it is beneficial to employ MDCO-MUSIC rather than PFLOM-MUSIC if the source signal is not circularly symmetrical.Furthermore, when α approaches 1, FLOM-MUSIC and PFLOM-MUSIC gain a large RMSE if the parameters of the fractional lower-order statistics cannot automatically adapt.
e result shows that the accuracy of the PFLOM-MUSIC algorithm is influenced by the relationship of the FLOM parameter p and the characteristic exponent α.Because of the contribution of two weighting factors, MDCO-MUSIC is not affected by the variety of characteristic exponent and can effectively estimate the DOA in any case.

Performance Comparison versus Snapshots of
Sampling.In this simulation, the performance of MDCO-MUSIC, FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC about snapshots is tested under symmetric alpha-stable distribution.Considering that the snapshots start at 50 and end at about 1000. Figure 8 illustrates that MDCO-MUSIC gains a more significant enhancement in resolution probability with the increasing number of snapshots and a more evident decrease in RMSE compared to the other three algorithms.Consequently, MDCO-MUSIC can work effectively for a small number of snapshots.

Performance Comparison versus Angular Separation.
Figure 9 depicts the performance of four algorithms when the direction angle of the second signal is gradually away from that of the first angle.e first direction of arrival is θ 1 � − 10 ∘ .As expected, the MDCO-MUSIC can effectively estimate the two directions of arrival with 7 ∘ angle separation.Furthermore, MDCO-MUSIC requires a smaller angle separation threshold than FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC associated with a fixed probability of resolution, and then MDCO-MUSIC gains a lower RMSE than other algorithms associated with a fixed angular separation threshold.rough the RMSE curve in the above experiment, MDCO can effectively converge to a stable trend under different parameters, which ensure the boundedness and improve the robustness of the algorithm.

Conclusion
e issue of DOA in the presence of the abnormal similarity is solved through the adaptive median-difference correntropy.e MDCO combines the weighting factor of the median difference and generalized correntropy to suppress impulsive noise.e median difference evaluates the deviation between the sampling value and the median of the signal, and it intuitively reflects the abnormality of impulsive noise.
e generalized correntropy measures the similarity between the two signals.
ese two weighting factors map the signal to an infinite-dimensional space with impulsive noise, thus including much statistical information.By controlling the adaptive kernel size, MDCO can effectively deal with DOA.We also prove that the MDCO satisfies some robust properties and applies the MDCO to DOA estimation combined with MUSIC.Experimental results illustrate that the MDCO-MUSIC is more robust than the FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC at a much lower GSNR and in the extremely impulsive noise environment.

B. Proof of Property 2
If the variable Y is an outlier rather than erefore, the weighting factor of generalized correntropy Y − Z and median difference |Y| − B is very small which can effectively suppress outlier Y. Let R MDCO be expressed as where R MDCO results in a small value that is close to zero.In summary, the MDCO algorithm performs better than the CRCO algorithm under impulsive noise environments, and results in R MDCO ⟶ 0.

E. Proof of Property 5
Under the premise of Property 4, there exists 0 < η < 1 which makes Meanwhile, R CRCO is bounded with [24] and can be expressed as Combining (E.3) with (E.2), the authors can easily get which means that R MDCO is bounded.

Figure 1 :
Figure 1: Model of the uniform linear array.

Figure 4 : 4 ,
Figure 4: Performance comparison of the proposed algorithm versus kernel parameter g 2 .

Figure 5 :
Figure 5: Performance comparison of the proposed algorithm versus kernel parameter h 2 .

Figure 6 :
Figure 6: Performance comparison of four algorithms versus GSNR: (a) probability resolution of four algorithms; (b) RMSE of four algorithms.

Figure 7 :
Figure 7: Performance comparison of four algorithms versus α values: (a) probability resolution of four algorithms; (b) RMSE of four algorithms.

Figure 8 :Figure 9 :
Figure 8: Performance comparison of four algorithms versus snapshot values: (a) probability resolution of four algorithms; (b) RMSE of four algorithms.

1 )
Although the variables Y and Z involve impulsive noise, the median difference of |Y| − B and |Z| − B works well on it.

Table 1 :
Comparison of similarity between two variables.

Table 2 :
Comparison of performance between CRCO and MDCO.
To prove Property 4, it is only necessary to prove that |R mdco w | < |R crco w | when the sampling data contains impulsive noise.According to Property 3, e authors only need to prove |R MDCO | < |R CRCO |.For the MDCO and CRCO algorithms, the authors have For the generalized covariance of the two variables Y and Z at a given moment, (D.1) can be rewritten as When the variables Y and Z contain impulsive noise, the authors can get exp(− |W Y | 2 /2σ 2 ) < 1 and exp(− |W Z | 2 /2σ 2 ) < 1. en,