Diffusion Correntropy Subband Adaptive Filtering (SAF) Algorithm over Distributed Smart Dust Networks

The diffusion subband adaptive filtering (DSAF) algorithm has attracted much attention in recent years due to its decorrelation ability for colored input signals. In this paper, a modified DSAF algorithm using the symmetry maximum correntropy criterion (MCC) with individual weighting factors is proposed and discussed to combat impulsive noise, which is denoted as the MCC-DSAF algorithm. During the iterations, the negative exponent in the Gaussian kernel of the MCC-DSAF eliminates the interference of outliers to provide a robust performance in non-Gaussian noise environments. Moreover, in order to enhance the convergence for sparse system identifications, a variant of MCC-DSAF named as improved proportionate MCC-DSAF (MCC-IPDSAF) is presented and investigated, which provides a dynamic gain assignment matrix in the MCC-DSAF to adjust the weighted values of each coefficient. Simulation results verify that the newly presented MCC-DSAF and MCC-IPDSAF algorithms are superior to the popular DSAF algorithms.


Introduction
Distributed estimation is required to get the interested estimated parameters from the data collected in distributed networks and sensors [1], and has been widely investigated and utilized in wireless sensor networks, target locations, environmental monitorings, medical applications, military applications, and other fields [2][3][4][5]. Diffusion strategy exchanges information between a current node and its neighbors in the networks, which makes it widely studied in distributed estimation [6]. A global solution in the network performs iterative operations when all node information is fused to the center, which requires a large amount of energy and communication resources [7]. Thus, many adaptive filtering (AF) algorithms in light of distributed estimation were reported like diffusion affine projection algorithms (DAPA) [8], diffusion least mean square algorithms (DLMS) [9], and diffusion subband adaptive filtering (SAF) algorithms (DSAF) [10]. DLMS was developed in line with the least mean square (LMS) for distributed estimation since the LMS is simple, but its convergence speed will sharply deteriorate for colored input signals. Although the DAPA and DSAF can improve the convergence when the input is colored, the complexity for DAPA is increased. DSAF is more popular because of its simple computational complexity that is similar to the DLMS [11][12][13][14][15]. However, the above mentioned algorithms are all developed based on the l 2 -norm optimization criterion, whose performance will be seriously degraded under non-Gaussian interference. d n (l) = u T n (l)w 0 + η n (l), n = 1, 2, 3, 4, . . . , N where u n (l) = [u n (l), u n (l − 1), . . . , u n (l − M + 1)] T denotes input vector, w 0 is the unknown vector with length M that we expect to estimate, and η n (l) is the system noise.

Review of the DSAF Algorithm
The DSAF is implemented based on a multiband structure, which is shown Figure 1. In Figure 1, ↑ I and ↓ I represent N interpolations and decimations, respectively. Considering H i (z) = ∑ M−1 m=0 h i (m)z −m , the desired signal d n (l) and inputing signal u n (l) for node n are assigned to I subband signals d n,i (l) and u n,i (l), where i = 0, 1, . . . , I − 1, I is the number of subbands. The subband desired vectors and subband input vectors for node n are expressed as: u n,i (k) = [u n,i (kI), u n,i (kI − 1), . . . , u n,i (kI − M + 1)] T , u n,i (l) for node n are handled by adaptive filterŴ n (z) whose weight vector is w n (l), which is to generate output signal y n,i (l). d n,i (l) for node n and the output signals y n,i (l) are to get d n,i,D (k) and y n,i,D (k). The subband errors e n,i,D (k) are the difference between d n,i,D (k) and y n,i,D (k). Using the synthesis filter bank G i (z) = ∑ M−1 m=0 g i (m)z −m , the fullband error e n (l) is obtained. The original and decimated sequences are denoted by variables l and k respectively, where l = kI. As we know, DSAF algorithms can be divided into two types, namely (Adapt-then-Combine) ATC and (Combine-then-Adapt) CTA [37]. In general, ATC performs better than CTA, and hence, the ATC type is employed herein.
For DSAF, its cost function for node n is in which E denotes the expected symbol. By using ATC type in Equation (4), J(k) is minimized to obtain the updated equation of the DSAF: where e n,i,D (k) is e n,i,D (k) = d n,i,D (k) − u T n,i,D (k)w n (k − 1).
N n represents the neighbor of node n. c jn denotes the combination coefficients of the N × N matrix C, where, ∑ j∈N n c jn = 1 and c jn = 0 if j ∈ N n . In other words, c jn is zero when j is not connected to n. Otherwise, the column and row of the combination coefficients c jn in C are added to one. ε is a regularization parameter which prevents the denominator from being zero, µ n is the step-size for node n. φ n (k) and w n (k) are the intermediate estimation and estimation of w 0 for node n.
Under the Gaussian noise interference, the DSAF has a decorrelation ability for the colored input signals and enjoys a faster convergence speed than DLMS. However, for the interference of non-Gaussian noise, the error sequences in the DSAF are not stabilized due to the pulse characteristics of non-Gaussian noise, which affects the convergence characteristics of DSAF. Then, a modified DSAF algorithm based on an MCC scheme with an individual weighting factor for improving the DSAF is proposed and given the name MCC-DSAF.

The Proposed MCC-DSAF Algorithm
Correntropy between two random variables X and Y is defined as where κ σ denotes the Mercer kernel, and f X,Y (x, y) is the joint probability density function. A Gaussian kernel is always used and is presented as where σ is the kernel size. To use the correntropy in our algorithm, we define e n,i,D (k) = x − y, x = d n,i,D (k) and y = u T n,i (k)w n (k − 1). Then, the new cost function is defined as: where β denotes the kernel parameter related to the kernel size and β = 1 2σ 2 , ε 1 > 0 has a small value. In the global network, a large amount of communication resources and energy are required and the real-time requirements of the system are high. Thus, the above problem should be well solved in the distributed network, and the global cost function should be changed into a local cost function.
For diffusion networks, local cost function can be formulated as linear-combination of local weighted correntropy, which is expressed as: Driven by Equation (10), the increment of weight vector ∆w n (k) at instant time k is written as: Then, the updating of the MCC-DSAF based on the gradient method can be obtained, For the diffusion strategy, the local estimation w n (k) is a linear combination of intermediate-estimation φ n (k), their relationship can be expressed In such way, ∆w n (k) is written as: Comparing Equation (11) with Equation (14), the increment of intermediate estimation From Equation (12), the updating of the intermediate-estimation From Equation (13), w n (k − 1) contains more data information from neighbor nodes compared with φ n (k − 1) [7,22]. According to the diffusion in [7], by replacing φ n (k − 1) by w n (k − 1) in Equation (16), we have Using Equations (12), (15), and (17), the updating equation of the MCC-DSAF is rewritten as: When the error e n,i,D (k) is interrupted by the pulse during the weight updating process, the negative exponential action in the Gaussian kernel function term exp[−β ] makes the outlier close to zero, which ensures that the MCC-DSAF has good performance under non-Gaussian noise interference.

The Proposed MCC-IPDSAF Algorithm
The MCC-IPDSAF is proposed using the adaptive gain matrix in the MCC-DSAF to obtain faster convergence when the unknown system is sparse since the adaptive gain matrix G n (k) can get a better balance between the convergence and steady state error. The update equation for the MCC-IPDSAF is There are several methods for choosing G n (k) to render it suitable for a sparse system [33]. As we know, the main idea of adaptive gain matrix in AFs is to obtain larger step sizes for the larger filter coefficients, which guarantees their convergence. Therefore, the MCC-IPDSAF can provide a quicker convergence compared to the MCC-DSAF in the sparse system.
where α is a parameter related to system sparsity. ε 2 is a regularization to prevent the denominator from being zero.

Data Model and Assumption
Mean square analysis of the MCC-DSAF will be performed. We define the following global variables: where col {·} denotes the column vector, and diag{·} denotes a diagonal matrix . The desired signal of the entire network Herein, the P matrix is a collection of local step parameters: where ⊗ denotes the Kronecker product. Next, C is the combination coefficients matrix of N × N: and the matrix A is defined as: .
The matrix Ω(k) is given by Using the global variables, the global update equation of the MCC-DSAF can be obtained, In the analysis, we have the following assumptions: Assumption 1. All input regressions u n (l) are independent. η n (l) is independent and independent of u n (l).

Assumption 2.
Subband colored input signal u n,i (k) is close to a white signal.

Assumption 3. The Gaussian kernel function
Assumption 1 is widely used in the analysis of AF algorithms. Assumption 3 does not really apply to the proposed algorithms, since 2 ] is an error function.
The weighted error-vector for node n is expressed as: The global weighted error-vector is: Due to A = C T ⊗ I M and C 2 = 1, where C is the matrix whose coefficients are c jn , and we have A = I MN . Thus, we get the relationship of w 0 = Aw 0 . Replacing w(k) with Aφ(k) in Equation (32), then Equation (34) is modified to be: Next, the mean square analysis forw(k) will be presented.

Convergence Analysis
The expectation is simultaneously exerted on both sides of (35), then we get It can be seen from Assumption 3 that Ω(k) is independent of U(k). Thereby, we obtain where ].
From Assumption 1, we can find that the expectation of the last term of (36) is close to zero. Therefore, (36) is rewritten as In Equation (38) ] R U should be stable for all n, which means that µ n should satisfy the following equation then, where λ max denotes the maximum eigenvalue of R U . If the l 1 norm of the weight w n (k) is smaller than τ, we have |e n,i,D (k)| = |d n,i,D − u T n,i (k)w n (k − 1)| ≤ u n,i (k) 1 w n (k − 1) 1 + |d n,i,D (k)| ≤ τ u n,i (k) 1 + |d n,i,D (k)|. (41) Next, let u T n,i (k)u n,i (k) = u n,i (k) 2 2 . Therefore, the following condition for the stability of the MCC-DSAF should satisfy

Steady-State Performance
Steady-state behavior of MCC-DSAF is studied herein. Σ denotes any symmetric positive definite weighting matrix, t 2 Σ denotes the weighting squared Euclidean norm t 2 Σ = t T Σt. Then, considering the Σ-Euclidean-norm on both sides of Equation (35) where and and

Z(k) = Ω(k)H(k)U(k)U T (k).
We take the expectation on both sides of Equation (43) and find: Herein, let E[Σ ] = Σ . The bvec{} operator is to convert a block matrix into a single-column vector. The operator denotes the block Kronecker product. From bvec[QΣP T ] = (P Q)ξ in [40], ξ = bvec[Σ] and applying the bvec{} operator to each item on the right side of (48) yields bvec bvec By applying the bvec{} operator to right side of Equation (48) without ξ, the final expression is represented by matrix Q: where bvec[Σ] = ξ, and applying the bvec{} operator to left side of Equation (48), we get bvec[Σ ] = ξ . Finally, we have Also, applying bvec{} operator to the second term on the right side of Equation (47) yields Thus, R is According to the above analysis, Equation (47) is obtained, The mean square deviation (MSD) for node n is given by Here, m n is defined as where b n,N denotes the n-th column vector of I N , vec stacks the columns of its matrix into a column vector. Let ξ = m n in Equation (57). When k approaches infinity, the MSD for node n is The MSD of the entire network is defined by the average of all node MSDs:

Simulation
The effectiveness of the proposed algorithms is verified through experimental simulation. Figure 2 shows the network topology with 20 nodes. The location coordinates for the nodes in a squared area The combination coefficients c jn are obtained by Metropolis criterion [11,41], where c jn = 1 max (N n , N j ) , if n = j and c jn = 1 − ∑ j =n c jn , if n = j. The unknown system has a length of M = 128 and the calculation of sparsity is ζ( [42,43] with ζ(w 0 ) = 0.751. Figure 3 shows the non-Gaussian noise distribution of P r = 0.01 and P r = 0.1 for node n = 1.

System Identification
The algorithms use four-subband cosine modulation filter banks. White, colored and speech signals are used as input in this section. The colored signal u n (l) is realized by Gaussian white noise via a first-order system with its transform function of H(z) = 1 1 − 0.95z −1 . The variance of the input signal and the variance of the Gaussian noise are given in Figure 4. Zero-mean Gaussian noise v n (l) and the impulsive noise z n (l) are used to construct the measurement noise η n (l). The impulsive noise z n (l) is obtained by using the Bernoulli process φ n (l) and the Gaussian process q n (l), which is defined as z n (l) = q n (l)φ n (l). The probability density function of the Bernoulli process is P{φ n = 0} = 1 − P r , P{φ n = 1} = P r . The signal-to-noise ratio (SNR) and signal-to-interference ratio (SIR) used in [18] Figure 5 gives the performance of the DSAF, MCC-DSAF, and MCC-IPDSAF for colored input signals, where P r = 0.01. Their step-size parameters are 0.1, 0.0158 and 0.0075 to get nearly the same initial convergence. The DSAF is severely degraded under non-Gaussian noise interference. The proposed MCC-DSAF can better suppress non-Gaussian interferences. The steady-state error of MCC-IPDSAF is smaller than MCC-DSAF, which is attributed to the adaptive gain matrix that reassigns the gains to each coefficient. Figure 6 illustrates the performance of diffusion sign error-LMS (DSE-LMS) [44], DMCC [21], diffusion affine projection sign algorithm (DAPSA) [45], MCC-DSAF, and MCC-IPDSAF for colored input signals for P r = 0.01. Their step-size parameters are 0.0017, 0.065, 0.11, 0.029, and 0.015. The Gaussian kernel σ DMCC = 2 and the DAPSA projection order is 4. DAPSA converges faster than the DSE-LMS and DMCC, but it converges slower than the proposed MCC-DSAF. MCC-DSAF's convergence is significantly faster than DSE-LMS,DMCC, and DAPSA. It can be verified that the subband algorithms can speed-up the convergence. When the proposed MCC-IPDSAF maintains the same convergence speed with MCC-DSAF, its steady-state error is smaller than that of the MCC-DSAF.  DSAF shows the worst behavior. The DSSAF and IWF-DSSAF algorithms have almost the same performance as each other, and the IPDSSAF and IWF-IPDSSAF algorithms are similar, this is because IWF-DSSAF and IWF-IPDSSAF are subband variants of DSSAF and IPDSSAF, respectively, and subband splitting of white signals has almost no effect when the impulsiveness of non-Gaussian noise is not particularly large. The steady-state error of the proposed MCC-DSAF is smaller than the DSSAF and IPDSSAF because it is realized based on MCC, which can resist non-Gaussian noise. As a result, the behavior of MCC-IPDSAF is better than the other algorithms.   Figure 7. Figure 9 discusses the behavior of DSAF, DSSAF, IWF-DSSAF, MCC-DSAF, IPDSSAF, IWF-IPDSSAF, and MCC-IPDSAF for a colored input signal, where P r = 0.01. The step-size parameters of these algorithms are 0.1, 0.42, 0.055, 0.0158, 0.3,0.06, and 0.0075, respectively. The DSAF has been severely degraded, while the proposed MCC-DSAF has a smaller steady-state-error than those of DSSAF, IWF-DSSAF, and IPDSSAF. Due to the advantages of adaptive gain matrix and MCC schemes, the proposed MCC-IPDSAF outperforms all other algorithms. Figure 10 shows the performance of DSAF, DSSAF, IWF-DSSAF, MCC-DSAF, IPDSSAF, IWF-IPDSSAF, and MCC-IPDSAF for a colored input signal, where P r = 0.1. The step size parameters of these algorithms are 0.1, 0.25, 0.06,0.021, 0.25, 0.0765, and 0.01, respectively. Compared with Figure 9, when the impulsiveness of non-Gaussian noise is increased, the steady-state error and convergence for all algorithms become worse. However, the behavior of the proposed MCC-IPDSAF algorithm is the best. Figure 11 shows the tracking behavior of DSAF, DSSAF, IWF-DSSAF, MCC-DSAF, IPDSSAF, MCC-IPDSAF, IWF-IPDSSAF for a colored input signal, where P r = 0.01. The unknown system changes when the iterations reach to 20,000, the proposed MCC-DSAF and MCC-IPDSAF still have good tracking performance.     Figure 12 gives a highly correlated real speech signal and the sparse channel. The real speech signal is the input, and the sampling frequency is 8 KHz, and the sample length is 4.8 × 10 4 . From Figure 13, DSAF fails to converge, while the steady state error of the proposed MCC-DSAF is better than those of the DSSAF and IWF-DSSAF for the non-Gaussian interference and speech input. The behavior of MCC-IPDSAF is superior to the other algorithms. Although the non-stationarity of speech input affects the behavior of the mentioned algorithms, the experiment result verifies the feasibility and effectiveness of the MCC-DSAF and MCC-IPDSAF algorithms.

Conclusions
In this paper, the maximum correntropy criterion and an individual weighting factor have been taken to construct a new cost function within the distributed subband adaptive filtering framework, which is named MCC-DSAF. The developed MCC-DSAF has been well derived and analyzed. The proposed MCC-DSAF algorithm can not only effectively suppress non-Gaussian noise interference, but also outperforms DSSAF and IWF-DSSAF with respect to convergence and MSD. Moreover, the proportionate adaption scheme is also introduced into MCC-DSAF to get MCC-IPDSAF, which further enhances the behavior of MCC-DSAF for identifying sparse systems. The convergence analysis and the steady-state behavior of MCC-DSAF are presented. The estimation behaviors of the algorithms are verified and the simulation results demonstrate that the proposed MCC-DSAF and MCC-IPDSAF are superior to the mentioned, popular DSAF algorithms. The algorithms in this paper will provide a better effect in the fields of radar, medical, wireless sensor networks, smart dust networks, distributed channel estimations, and hydroacoustics, etc.