Frequency domain maximum correntropy criterion spline adaptive filtering

A filtering algorithm based on frequency domain spline type, frequency domain spline adaptive filters (FDSAF), effectively reducing the computational complexity of the filter. However, the FDSAF algorithm is unable to suppress non-Gaussian impulsive noises. To suppression non-Gaussian impulsive noises along with having comparable operation time, a maximum correntropy criterion (MCC) based frequency domain spline adaptive filter called frequency domain maximum correntropy criterion spline adaptive filter (FDSAF-MCC) is developed in this paper. Further, the bound on learning rate for convergence of the proposed algorithm is also studied. And through experimental simulations verify the effectiveness of the proposed algorithm in suppressing non-Gaussian impulsive noises. Compared with the existing frequency domain spline adaptive filter, the proposed algorithm has better performance.

www.nature.com/scientificreports/ Non-Gaussian noise models are usually classified into heavy-tailed non-Gaussian noise (e.g., Alpha-stable, Laplace, Cauchy, etc.) and light-tailed non-Gaussian noise (i.e., binary, uniform, etc.). This paper mainly focuses on heavy-tail noise. The comparison of the probability density function of heavy tail noise as shown in Fig. 1. We can know that compared with other noise models, alpha-stable distribution noise has heavier tails and sharp peaks. Therefore, an alpha-stable distribution is uitilized for the experiment. An alpha-stable distribution, with α ∈ (0, 2] is a characteristic exponent representing the stability index which determines the strength of impulse, β ∈ [−1, 1] is a symmetry parameter, ι > 0 is a dispersion parameter, and ̺ is a location parameter 22 . Which can be expressed as Where In this paper, the parameters of alpha-stable distribution are set as follows, α = 1.6, β = 0, ι = 0.05, ̺ = 0 . We adding to the output of the unknown system, an independent white Gaussian noise v(n) with the signal to noise ratio (SNR=30dB). In adaptive system, the frequency domain weight is initialized as   Figure 2 shown the MSE curves in different δ parameters. This work is carried out in without non-Gaussian noise environment. With the parameter δ increase, the convergence performance of MSE curve becomes better. However, when the parameters δ increase to a certain value, the MSE curve will not change any more. In subsequent experiments, setting the parameter δ = 6.
(1)  www.nature.com/scientificreports/ Figure 3 shown the MSE curves comparison between FDSAF, the proposed FDSAF-MCC, and SAF-MCC under without non-Gaussian noise environment. The convergence performance of the three algorithms is consistent, but the running time of FDSAF-MCC is 0.13309 ms, which is shorter, than the FDSAF, which is 0.13661 ms, than the SAF-MCC, which is 0.5064 ms.
The FDSAF-MCC algorithm to track the weight of FIR filtering with non-Gaussian noise environment, as shown in Fig. 4. The proposed algorithm has good tracking performance. The FDSAF-MCC algorithm to track the spline knots with non-Gaussian noise environment, as shown in Fig. 5. The proposed algorithm has good tracking performance. Figure 6 shown the MSE curves comparison between FDSAF, the proposed FDSAF-MCC, and SAF-MCC under non-Gaussian noise environment. The convergence performance of FDSAF algorithm is bad, the MSE curve oscillates randomly. The proposed FDSAF-MCC and SAF-MCC algorithms have better convergence performance. Figure 7 shown the MSE curves comparison between FDSAF, the proposed FDSAF-MCC, and SAF-MCC under non-Gaussian noise and a sudden change. The proposed FDSAF-MCC and SAF-MCC algorithms have better convergence performance. After a sudden change, the MSE curve converges of the FDSAF-MCC and the SAF-MCC algorithms, the MSE curve divergent of the FDSAF algorithm.    www.nature.com/scientificreports/ The saturated nonlinearity is described by p described the nonlinear input, f(p) described the nonlinear output. Figure 8 shown the MSE curves comparison between FDSAF, the proposed FDSAF-MCC, and SAF-MCC under impulsive noise environment. The convergence performance of FDSAF algorithm is poor. FDSAF algorithm cannot suppress impulsive noise. The proposed FDSAF-MCC algorithm has the better convergence performance than SAF-MCC algorithm. Figure 9 shown the MSE curves comparison between FDSAF, the proposed FDSAF-MCC, and SAF-MCC under impulsive noise and a sudden change. The convergence performance of the FDSAF and the SAF-MCC algorithm is poor. FDSAF algorithm cannot suppress a sudden change with non-Gaussian noise. The proposed FDSAF-MCC algorithm shown a convergence trend, but the convergence effect is not very good after sudden change.

Experiment 3.
In order to further verify the convergence performance under non-Gaussian noise and/or a sudden change environment, this example compares the FDSAF and FDSAF-MCC under different input signals. The input signal x(n) is generated by (4) L(s) = −2.8e12s 3 + 4.6e18s 2 + 6.4e21s + 3.2e27 s 7 + 1e4s 6 + 2.6e9s 5 + 1.2e13s 4 + 1.2e18s 3 + 2.1e21s 2 + 9.4e23s + 9.7e26 www.nature.com/scientificreports/ a ∈ [0, 1) is a correlation coefficient determining the correlation relation between x(n) and x(n − 1) , and ξ(n) is a white Gaussian stochastic process with zero mean and unitary variance. When a = 0 , the input signal x(n) is the white noise. When a is close to 1, the input signal x(n) is colored noise. In this experiment, the cases of a = 0, 0.9 are considered. The MSE curves of FDSAF-MCC and FDSAF are compared in the experiment. Figure 10 shown the MSE curves comparison between FDSAF and the proposed FDSAF-MCC under white noise input and colored noise input. The convergence performance of the FDSAF algorithm and the proposed FDSAF-MCC algorithm is compared under the same parameter a. When a = 0 , input with the white noise, the proposed FDSAF-MCC algorithm has the better convergence performance than the FDSAF algorithm. When a = 0.9 , input with the colored noise, the proposed FDSAF-MCC algorithm has the better convergence performance than the FDSAF algorithm. Figure 11 shown the MSE curves comparison between FDSAF and the proposed FDSAF-MCC with different a under non-Gaussian noise and a sudden change. The convergence performance of the proposed FDSAF-MCC algorithm better than the FDSAF algorithm when in the same a.

Methods
FDSAF-MCC filtering. The basic structure of SAF in Fig. 12 is the cascade of a linear adaptive filter and a nonlinear cubic CR-spline interpolation function 23 . The structure of the frequency domain maximum correntropy criterion spline adaptive filter (FDSAF-MCC) is shown in Fig. 13. Frequency domain adaptive filtering (FDAF), used to linear module of the SAF 24 . During this process, the filtering and parameter updating are performed every M instants. Let the length of data buffer equal to the length of FIR filter weight, M. The input    www.nature.com/scientificreports/ As shown in Fig. 13, the nonlinear spline interpolation contains table look-up and interpolation two procedures. The look-up table (LUT) is made up of N + 1 control points (knots) defined as G j = [g x,j , g y,j ] T , (j = 0, 1, . . . , N) . The subscripts x and y denote abscissa and ordinate, respectively. The abscissas are uniformly distributed with an interval x . The LUT process will calculate the spline interval index i j and the local abscissa u j according to s(kM + j) at instant n = kM + j, ⌊·⌋ denotes the floor operator. Then vectorizing all normalized abscissas u j and the corresponding local spline knots g i,j of all s(kM + j) in data buffer. u j = [u 3 j , u 2 j , u j , 1] T is the local abscissa vector derived from s(kM + j) . The normalized abscissas u j are vectorized as U(k) ∈ R 4×M which is In a similar way, u j = [3u 2 j , 2u j , 1, 0] T is the differential vector of a local abscissa. We denote U (k) ∈ R 4×M the differential matrix of normalized abscissas, which can be expressed as And we denote G(k) ∈ R 4×M the spline knot matrix, which can be described by, , g i,j+2 , g i,j+3 ] T . After FIR filtering in frequency domain, the intermediate variables s(k) will enter the spline interpolation, obtaining the output signal y(k), from instant n = kM + 1 to n = kM + M , which can be written as The spline interpolation function is represented by ϕ(·) , y(kM + j) denotes the output of spline filter at instant n = kM + j , (·) ii represents the column vector made up from the diagonal elements of the matrix, sum r (·) represents the column vector derived from the summation of the matrix by rows, and sum c (·) represents the row vector derived from the summation of the matrix by columns.
Cubic spline curves, which mainly include B-spline 25 and Catmul-Rom (CR) spline 26 . Because of the feature that CR-spline passes through all of the control points, CR-spline may has much better performance in local approximation with respect to B-spline 27 . Therefore, CR-spline is the only one considered in the paper. The basis matrix C FDSAF-MCC adaptive. As shown in Fig.14 the structure of FDSAF-MCC nonlinear system identification, we will derive the parameter updating rules. The error e(k) can be written as  www.nature.com/scientificreports/ e(kM + j) = d(kM + j) − y(kM + j) , at instant n = kM + j . d(k) = [d(kM + 1), d(kM + 2), . . . , d(kM + M)] T is the desired output. An maximum correntropy cost function [19][20][21] which is insensitive to impulsive noises, given by The kernel size δ > 0 , and √ 2πδ is the normalization. We take the derivative of Eq. (17) with respect to e(k) , and the result described by e d (k) e d (k) = [e d (kM + 1), e d (kM + 2), . . . , e d (kM + M)] T . As there is no FFT transformation in the process of spline interpolation, so the parameter updating of spline knot is the same of time domain SAF. In order to match the data block description in FDSAF-MCC, the parameter updating can be expressed in the vectorization form. We take the derivative of J(k) with respect to g i,j 15 , described Then the derivative of J(k) with respect to G(k) can be expressed as a vectorization form Therefore, the parameter updating of the spline knots matrix can be expressed as The learning rate µ g on the updating of G(k) . We take the derivative of J(k) with respect to w(k) In which, we denote the back propagation error e s (kM + j) = e d (kM + j)φ(s(kM + j)) , then it can be vectorized as = e d (kM + j) ∂e(kM + j) ∂y(kM + j) ∂y(kM + j) ∂s(kM + j) ∂s(kM + j) ∂w(k) www.nature.com/scientificreports/ The learning rate µ w on the updating of w F , and 0 ∈ R M×1 = [0, 0, . . . , 0] T is a zero vector with length of M.

FDSAF-MCC brief.
In order to explain the FDSAF-MCC algorithm more clearly, a brief summary of FDSAF-MCC is given in Algorithmic 1.
Convergence performance. The convergence performance of the linear module and the nonlinear module are considered separately. According to the cost function in the parameter updating, �e(k + 1)� 2 < �e(k)� 2 must be satisfied during filtering. We take the first order Taylor series expansion of �e(k + 1)� 2 We ignore h.o.t. which represents high order terms, satisfied �e(k + 1)� 2 < �e(k)� 2 The bound on learning rate µ g for spline knots complies with In a similar way, we take the Taylor series expansion of �e(k + 1)� 2 about w(k) , at instant k We ignore h.o.t. which represents high order terms, condition of �e(k + 1)� 2 < �e(k)� 2 , obtained = �e(k)� 2 (1 − µ g �exp(− e 2 (k) 2δ 2 )C T U(k)� 2 ) + h.o.t.

Conclusion
This paper proposed a new frequency domain maximum correntropy criterion spline adaptive filtering. To suppression non-Gaussian impulsive noise along with having comparable operation time. Instead of using the lest mean square (LMS), the new algorithm employed the maximum correntropy criterion (MCC) as a cost function. Three experimental methods were used to verify the effectiveness of the proposed algorithm in suppressing impulsive noise. Compared with the existing frequency domain spline adaptive filtering algorithm, the proposed algorithms provided better robustness against alpha stable noise. The proposed algorithm has a good effect in suppressing the alpha stable noise in the above-mentioned environments, but it may have a problem of poor convergence in suppressing light-tailed noise. It is necessary to further explore a more appropriate cost function and construct a filtering algorithm to suppress light-tailed models noise. In the future, for the light-tailed noise model, we will be carried out in the frequency domain spline adaptive filtering algorithm.