 Research
 Open Access
 Published:
Two variants of the IIR spline adaptive filter for combating impulsive noise
EURASIP Journal on Advances in Signal Processingvolume 2019, Article number: 8 (2019)
Abstract
It has been pointed out that the nonlinear spline adaptive filter (SAF) is appealing for modeling nonlinear systems with good performance and low computational burden. This paper proposes a normalized least Mestimate adaptive filtering algorithm based on infinite impulse respomse (IIR) spline adaptive filter (IIRSAFNLMM). By using a robust Mestimator as the cost function, the IIRSAFNLMM algorithm obtains robustness against nonGaussian impulsive noise. In order to further improve the convergence rate, the setmembership framework is incorporated into the IIRSAFNLMM, leading to a new setmembership IIRSAFNLMM algorithm (IIRSAFSMNLMM). The proposed IIRSAFSMNLMM inherits the benefits of the setmembership framework and leastM estimate scheme and acquires the faster convergence rate and effective suppression of impulsive noise on the filter weight and control point adaptation. In addition, the computational burdens and convergence properties of the proposed algorithms are analyzed. Simulation results in the identification of the IIRSAF nonlinear model show that the proposed algorithms offer the effectiveness in the absence of nonGaussian impulsive noise and robustness in nonGaussian impulsive noise environments.
Introduction
Due to their concise design and low complexity, the adaptive linear filters have gained wide attention in system modeling and identification [1, 2]. The adaptive linear filter is conventionally modeled as a finite impulse response (FIR) filter or an infinite impulse response (IIR) filter. Its tap weights are updated iteratively by using adaptive algorithms such as the least mean square (LMS) algorithm, normalized least mean square (NLMS) algorithm, and affine projection algorithm (APA). However, in the case of nonlinear system, linear models are inadequate and suffer from the performance losses due to the failure to model the nonlinearity. Hence, in order to model the nonlinearity, several adaptive nonlinear structures have been presented such as truncated Volterra adaptive filters (VAF) [3], neural networks (NNs) [4], blockoriented architecture [5], and spline adaptive filters (SAF) [6–9]. Truncated VAF, originated from the Taylor series expansion, is one of the most used model for the nonlinearity. However, its implementation is limited because of a huge complexity requirement, in particular, for highorder Volterra models. To overcome the drawbacks of the truncated VAF, one of the most wellknown structure is the blockoriented nonlinear architecture, which can be represented by the connections of linear timeinvariant (LTI) models and memoryless nonlinear functions. There are several basic classes of the blockoriented nonlinear structure including the Wiener model [10], the Hammerstein model [11], and the variants originated from these two classes in accordance with different topologies (i.e., parallel, feedback, and cascade). Specifically, the Wiener model consists of a cascade of a linear LTI filter followed by a static nonlinear function which sometimes is deemed as linearnonlinear (LN) model, and the Hammerstein model comprises a static nonlinear function connected behind a linear LTI filter which usually is considered as nonlinearlinear (NL) model. The cascade model, such as linearnonlinearlinear (LNL) model or nonlinearlinearnonlinear (NLN) model, has been proved to be more suitable for the generality of the model to be identified [12]. NNs are a flexible application for modeling nonlinearity, but it suffers from a large computational cost and difficulties in online adaptation.
Recently, combining the blockoriented architecture with the spline function, several novel adaptive nonlinear spline adaptive filters (SAFs) have been introduced such as Wiener spline filter, Hammerstein spline filter, cascade spline filter and IIR spline adaptive filter (IIRSAF). These spline adaptive models can be implemented by different connections of the spline function and linear timeinvariant (LTI) model. The nonlinearity in this kind of structure is modeled by the spline function, which can be represented by the adaptive lookup table (LUT) interpolated by a local loworder spline curve. The SAFs achieve improved performance in modeling the nonlinearity. Furthermore, in each iteration, only a portion of the control points is tuned depending on the order of the spline function and the nonlinear shape is slightly changed. Consequently, this local behavior of the spline function results in the considerable saving in the computation complexity.
Note that in all spline filters mentioned above, their update rules are based on the mean square error (MSE) criterion in additive white Gaussian noise (AWGN) environment which considers the cost function J=E[e^{2}(n)], where E[·] denotes the mathematical expectation and e(n) is the output error. However, in some cases of nonGaussian noise such as underwater acoustic signal processing [13], radar signal processing [14], and communication systems [15], the SAFs may suffer from performance deterioration or failure to be robust against nonGaussian noises. To address this problem, a least Mestimate scheme [16, 17] has been proposed by using the least Mestimator as the cost function which achieves the satisfactory performance when the input and desired signals are corrupted by nonGaussian impulsive noises.
In this paper, extending the least Mestimate idea into the IIRSAF, a normalized least Mestimate adaptive filtering algorithm based on IIR spline adaptive filter (IIRSAFNLMM) is proposed for nonlinear system identification. The update rule is based on the modified Huber Mestimate function, thus yielding a good effectiveness in suppressing nonGaussian impulsive noises. To further improve the convergence performance of the IIRSAFNLMM, we incorporate the setmembership framework into the IIRSAFNLMM and propose a setmembership IIRSAFNLMM (IIRSAFSMNLMM) algorithm. It is derived by minimizing a new Mestimatebased cost function associated with a robust setmembership error bound. Due to the combination of the robust setmembership error bound and threshold parameter used to reject the outliers, the IIRSAFSMNLMM provides faster convergence rate and robustness against nonGaussian impulsive noise compared with the conventional SAF algorithms.
The paper is organized as follows. The IIRSAF structure is reviewed in Section 2. In Section 3, we derive the IIRSAFNLMM and IIRSAFSMNLMM algorithms. The computational complexity is given in Section 4, and convergence properties of the IIRSAFSMNLMM are analyzed in Section 5. Some simulation results are demonstrated in Section 6. Finally, Section 7 concludes the paper.
IIRSAF structure
The structure of the IIRSAF is shown in Fig. 1, which consists of an adaptive infinite impulse response (IIR) filter followed by a nonlinear network. In the nonlinear network, the spline interpolater, connected behind the adaptive LUT, determines the number and spacing of points (knots) contained in the LUT. The output of the adaptive IIR is given by:
where w(n) is the weight vector of the IIR filter which is defined as w(n)=[b_{0}(n),b_{1}(n),⋯,b_{M−1}(n),a_{1}(n),⋯,a_{N}(n)]^{T}, b_{l}(n)(l=0,1,⋯,M−1), and a_{k}(n)(k=1,2, ⋯,N) denote the lth coefficient of the MA part and kth coefficient of the AR part in the IIR adaptive filter respectively. \(\bar {\mathbf {x}}(n)\,=\, \left [x(n),x(n  1), \cdots, x(n  M + 1),s(n  1), \cdots,s(n  N)\right ]^{T}\) is the input vector of the IIR filter.
The local parameter u_{n} and span index i can be computed as:
where Q is the number of the control point, Δx is the uniform space between two adjacent control points, and ⌊·⌋ denotes the floor operator.
The output of the whole system is given as:
where, in this paper considering the cubic spline interpolation scheme, thus the control point vector q_{i,n} can be defined q_{i,n} =[q_{i,n},q_{i+1,n},q_{i+2,n},q_{i+3,n}]^{T} with length 4, and the vector u_{n} is defined as \(\mathbf {u}_{n} = \left [u_{n}^{3},u_{n}^{2},u_{n},1\right ]^{T}\). The superscript T denotes the transposition operation. C is the spline basis matrix whose dimension is selected to be 4×4. Two suitable types of spline basis matrix are CatmulRom (CR) spline and Bspline matrices which are given by:
According to the Lagrange multiplier method presented in [18], two recursive equations of the tap weights and control points of the normalized least mean square algorithm based on IIRSAF (IIRSAFNLMS) can be formulated as;
where μ_{w} and μ_{q} are the stepsizes in the linear network and nonlinear network, respectively; the small positive constant ε is used for avoiding zero division. The vector g_{n} is defined as g_{n}= [ ∂s(n)/∂b_{0}(n),⋯, ∂s(n)/∂b_{M−1}(n),∂s(n)/∂a_{1}(n),⋯,∂s(n)/∂a_{N}(n)]^{T}, and \(\dot {\mathbf {u}}_{n} = \left [3u_{n}^{2},2u_{n}^{},1,0\right ]^{T}\), e(n) is the output error which can be expressed as \(e (n) = d(n)  y(n) = d(n)  \mathbf {u}_{n}^{T} {\mathbf {Cq}}_{i,n}\), where d(n) is the desired signal which contains nonGaussian impulsive noises.
Proposed IIRSAFNLMM and IIRSAFSMNLMM algorithms
IIRSAFNLMM algorithm
In the nonGaussian impulsive noise environment, the desired signal d(n) is commonly contaminated by impulsive noises. Then, the performances of the SAFLMS [6] and SAFNLMS [18] algorithms based on the MSE criterion can be affected severely by large elements in the output error. Instead of the MSE cost function, the least Mestimate scheme makes use of a robust Mestimate cost function to suppress the adverse effect caused by the outliers in output errors. In this paper, the cost function can be expressed as:
where ρ[e(n)] is the modified Huber Mestimate function which gives:
where ξ is a threshold parameter for rejecting the outliers which is computed as \(\xi = 2.576 \hat \sigma _{e} (n)\), and \( \hat \sigma _{e}^{2} (n)\) is the variance estimate of the impulsivefree error [17], which is given by:
where λ_{0} is the forgetting factor close to but smaller than 1, a_{1}=1.483(1+5/(N_{w}−1)) is a finite correction factor, and N_{w} is the data window. med[·] denotes the median operator and C_{e}(n)=[e^{2}(n),e^{2}(n−1),⋯,e^{2}(n−N_{w}+1)].
Note that in [17], the threshold parameter ξ is evaluated with the assumption of Gaussian distribution of the output error. However, even in the case that e(n) is subject to other distribution, we also can compute the threshold value which is used to reject the impulse in output errors.
Taking the derivative of the cost function J(n) with respect to the IIR weight vector w_{n} and applying the steepest descent method, the update equation of the weight vector can be obtained by:
where the function ψ[ e(n)] is given as:
In a similar way, taking the derivative of the cost function J(n) with respect to q_{i,n} and using the steepest descent method, the recursive equation of the control point vector is expressed by:
It can be seen in (12) and (14) that the output error is replaced by the score function ψ[ e(n)], resulting into the freezing on the update of the IIR weight vector and control point vector when the output error is larger than the threshold parameter. This way helps the IIRSAFSNLMM algorithm to suppress the adverse effect of the nonGaussian impulsive noise.
IIRSAFSMNLMM algorithm
As we know, in the case of linear adaptive filters, the setmembership scheme chooses the specified bound γ∈R_{+} to find an appropriate data set S containing all possible inputdesired data pairs (x,d) which satisfy [19]
where Θ is the feasibility set in which all the tapweight vectors are available for d−w^{T}x≤γ, and L denotes the linear filter length.
In the case of the IIRSAF, we consider both the IIR adaptive filter tapweight vector and control point vector, and the feasibility set can by given by:
The spline adaptive filter updates IIR tap weights and control points by using the inputdesired data pairs [x_{n},d(n)] at time instant n, and then we define the constraint set H_{n} with all the combined vectors (w,q) for which the output error is upper bounded by γ and is mathematically expressed by:
The exact membership set which is interpreted as the intersection of the constraint sets H_{k} over all time instants k=1,2,⋯,n is given as:
Note that the membership set Λ_{n} is the minimal set estimate of Θ_{0} at time n, if we choose the magnitude of the error upper bound γ properly, the membership set is nonempty. Thus, the setmembership adaptive scheme can be incorporated into the IIRSAFNLMM to seek the valid estimates of combined vectors (w,q) which lie in the membership set at the steadystate.
Employing the setmembership framework in the IIRSAFNLMM and using the setmembership constraint value g(n), the modified Mestimatebased cost function can be set as:
Then, the modified Huber Mestimate function associated with the constraint value is given by:
where θ is a constant, g(n)=γsgn[e(n)], γ≥0 is the setmembership error bound, and sgn[·] is the sign function.
Applying the steepest descent method, the update equation of the IIR tapweight vector can be obtained as:
For 0≤e(n)<ξ, the derivative of the cost function (19) with respect to w_{n} is derived as;
where \(\varphi ^{\prime }_{i} (u_{n}){\mathrm {= }}\dot {\mathbf {u}}_{n}^{T} {\mathbf {Cq}}_{i,n}\). Substituting (13) into (22), the derivative of \(\bar {J}(n)\) with respect to w_{n} can also be expressed as:
where the constant 2 is absorbed by θ and the parameter α_{n} is defined as:
Hence, the recursive relation of the IIR tapweight vector is given as:
where ε_{0} is a small regular parameter for preventing from zero division. For the special case e(n)=0 and ψ[ e(n)]=0, the weight updating is suspended.
For the updating of the control point vector, taking the derivative of the cost function (19) with respect to q_{i,n}, we have:
Using the steepest descent method, the learning rule of the control point vector is given as:
It is noted that in (25) and (27), θα_{n} is equivalent to the step size in the IIRSAFNLMM, i.e., the step sizes μ_{w}=μ_{q}=θα_{n} for the IIRSAFSMNLMM are not constants any more. Furthermore, the IIRSAFSMNLMM algorithm can be viewed as the variable step size IIRSAFNLMM algorithm. When the upper bound γ is set to be 0, then resulting into α_{n}=1, the SAFSMNLMM algorithm degenerates into the SAFNLMM.
In (20), the outlier rejection depends on the choice of ξ; improper choice of ξ leads to the presence of a part of the impulsive noise in e(n). This makes α_{n} in (24) nonoptimal. Here, we use the impulsivefree estimation of E[e(n)] [20] instead of e(n) in (24) which is derived as:
where A_{e}(n)=[e(n),e(n−1),⋯,e(n−N_{w}+1)], and λ_{1} is the forgetting factor approaching but smaller than one.
Hence, (24) can be approximated as:
Computational complexity
The computational burdens of the IIRSAFLMS, IIRSAFNLMS, IIRSAFNLMM, and IIRSAFSMNLMM algorithms per iteration are compared in Table 1. For the spline output calculation and adaptation, we take into account of the terms \(\mathbf {u}_{n}^{T} {\mathbf {Cq}}_{i,n}\), \(\dot {\mathbf {u}}_{n}^{T} {\mathbf {Cq}}_{i,n}\), and C^{T}u_{n}; it only needs 4K_{p} multiplications plus 4K_{q} additions, where K_{p} and K_{q} (less than 16) are the constants which can be defined with reference to the implementation spline structure in [21]. Due to the normalized operation, extra four multiplications, four additions, and two divisions are required for the IIRSAFNLMS algorithm. Compared to the IIRSAFNLMS algorithm, the proposed IIRSAFSMNLMM algorithm needs extra eight multiplications and three additions caused by (25)–(29). If M+N≫4, the proposed algorithms only require more O(N_{w} log2N_{w}) median operations per iteration than the other two cited algorithms.
Convergence properties
In this section, we study the convergence properties based on the energy conservation relation. The identification scheme is shown in Fig. 2; w_{0} and q_{0} represent the IIR filter and the nonlinear network of the target nonlinear system, respectively. It is reasonable to suppose that the adaptation of the variables w_{n} and q_{i,n} is in two separate phases, e.g., only the adaptation of linear filters is considered in the first phase of learning and then it is optimal in the second one [7]. To make the analysis tractable, the following assumptions are given:
Assumption 1
The ambient noise η(n)=η_{G}(n)+η_{I}(n), where η_{G}(n) is white Gaussian background noise with zeromean and variance \(\sigma _{G}^{2}\) and η_{I}(n) is the impulsive noise, modeled by an independent and identically distributed (i.i.d) random variable. The sequence η(n) whose variance is \(\sigma _{\eta }^{2}\) is independent of x(n) and s(n).
Assumption 2
For sufficient long IIR weight error vector, the output error e(n) is independent of \({\varphi ^{\prime }_{i} (u_{n})}\), ∥g_{n}∥^{2} and ∥Cu_{n}∥^{2} and the parameter α_{n} involved with e(n) in (24) is also independent of \({\varphi ^{\prime }_{i} (u_{n})}\), ∥g_{n}∥^{2} and ∥Cu_{n}∥^{2}.
In the first phase, we define the IIR weight error vector as Δw_{n}=w_{0}−w_{n}; the iteration of Δw_{n} can be written as:
Setting the regularization parameter ε_{0} to zero and taking the mathematical expectation of the squared Euclidean norm of both sides of (30), we have:
where D(n)=E[Δw_{n}^{2}] denotes the mean square deviation (MSD); ξ_{w}(n) is defined as a noisefree priori error associated with the IIR weight error vector Δw_{n} which can be expressed by [22]:
where c_{3} is the third row of the matrix C.
In addition, in this phase, the control point vector is assumed to be optimal; thus, the output error is given as:
Considering that ξ_{w}(n) is not corrupted by the impulsive noise, based on the features of the function (13) which rejects the outliers, ψ[ e(n)] can be approximated as [23]:
Assuming ξ_{w}(n) is independent of ψ[η(n)], we substitute (34) into (31) and apply the Assumptions 1 and 2, we have:
where \(\sigma _{\psi (\eta)}^{2}\) denotes the variance of ψ[η(n)].
By using the property tr[ AB]=tr[ BA] and inserting (32) into (35), where tr[·] denotes the trace operator for matrices. The relation (35) can be rewritten equaivalently as:
where \( {\text {cov}} \left (\Delta \mathbf {w}_{n} \right) = E\left (\Delta \mathbf {w}_{n} \Delta \mathbf {w}_{n}^{T} \right), A_{n}\! =\! \left [\left (\mathbf {c}_{3} \mathbf {q}_{i,n} \right)/\Delta x^{2} \right ] \)
\( E \left [\mathbf {u}_{n} ^{ 2} \varphi ^{\prime }_{i} (u_{n})\right ], B_{n}\!\! =\!\! \left [\!\left (\mathbf {c}_{3} \mathbf {q}_{i,n} \right)^{2} /\Delta x^{4} \right ] E\left [\varphi ^{\prime }_{i} (u_{n})^{2} \mathbf {g}_{n}^{2} \right. \)
u_{n}^{−4}], and \( C_{n} = \sigma _{\psi (\eta)}^{2} E\left [\varphi ^{\prime }_{i} (u_{n})^{2} \mathbf {g}_{n} ^{2} \mathbf {u}_{n} ^{ 4} \right ]. \)
Now by applying the unitary matrix Q, we have:
Assuming that Δw_{n+1} is independent of the filter inputs and using the Assumption 2, (37) can be rewritten as:
where Δw^{′}_{n+1}=Q^{T}Δw_{n+1}, Λ_{n} is a diagonal matrix whose elements are the eigenvalues of \(E\left (\mathbf {g}_{n} \mathbf {g}_{n}^{T} \right)\), denotes as λ_{l} for l=0,⋯,M+N−1. From (38), the algorithm is stable when \( \left  {1  2A_{n} \theta E(\alpha _{n})\lambda _{l} {{+ }}B_{n} \theta ^{2} E\left (\alpha _{n}^{2} \right)\lambda _{l}} \right  < 1\), which gives:
In the second phase, we define the control point error vector Δq_{i,n}=q_{0}−q_{i,n} and then obtain:
Taking the mathematical expectation of the energies of both sides of (40) and using Assumptions 1 and 2, again, we obtain:
where K(n)=E[Δq_{i,n}^{2}], ξ_{q}(n) is defined as the a noisefree priori error associated with the control point error vector Δq_{i,n} which is given by:
By using again the property tr[AB]=tr[BA] and inserting (42) into (41), we obtain:
In a similar way to cov(Δw_{n}), the relation (43) can be rewritten as:
where \({\text {cov}} \left (\Delta \bar {\mathbf {q}}_{i,n + 1} \right) = \mathbf {Q}^{T} {\text {cov}} \left (\Delta \mathbf {q}_{i,n + 1} \right) \), \(\bar { A}_{n} = E\left [\mathbf {u}_{n} ^{ 2} \right ]\), \(\bar { B}_{n} = E\left [\mathbf {u}_{n} ^{ 4} \mathbf {C}^{T} \mathbf {u}_{n} ^{2} \right ]\), \(\bar {\mathbf {\Lambda } }_{n}\) is a diagonal matrix whose elements \(\bar \lambda _{p}\) (p=0,1,2,3) are the eigenvalues of E[C^{T}u_{n}(C^{T}u_{n})^{T}]. The system is stable when \(\left  {1  2\bar A_{n} \theta E(\alpha _{n})\lambda _{p} {{+ }}\bar B_{n} \theta ^{2} E\left (\alpha _{n}^{2} \right)\lambda _{p}} \right  < 1\), which gives:
Note that in (39) and (45), the bound of the constant θ can be set by:
Results and discussion
In this section, several detailed experimental results are presented in the context of the IIRSAF nonlinear system identification as shown in Fig. 2. The mean square error (MSE) which is defined as 10 log10[e(n)]^{2} is used to evaluate the performance. All the following results are obtained by averaging over 100 Monte Carlo trials. The input signal is generated by the following relationship:
where a(n) is the white Gaussian noise signal with zeromean and unitary variance, and the parameter 0<ω<0.95 represents the level of correlation between the adjacent samples. The lengths of the MA and AR parts in the IIR adaptive filter are set to M=2 and N=3, respectively. The initial tapweight vector for the IIR adaptive filter is w_{−1}=[1,0,...,0] with length M+N=5, while the control point vector is initially set to a straight line with a unitary slope. Only the CRspline basis is applied in the simulations; however, similar results can also be obtained by using the Bspline basis.
The unknown IIRSAF nonlinear system is composed of an IIR filter whose transfer function is given by:
and the nonlinear spline function is implemented by a LUT q_{0} with 23 control points, Δx is set to 0.2 and q_{0} is defined by:
The ambient noise η(n)=η_{G}(n)+η_{I}(n), where η_{G}(n) is the white Gaussian background noise and η_{I}(n) is the impulsive noise. The background noise η_{G}(n) is the zeromean independent white Gaussian sequence with variance \(\sigma _{G}^{2}\), with 40 dB signaltonoise ratio (SNR) which is added to the input of the unknown system. The SNR is defined as \(\text {SNR} = 10\log _{10} \left (\sigma _{x}^{2} /\sigma _{G}^{2} \right)\), where \(\sigma _{x}^{2}\) is the variance of the system input x(n). The impulsive interference η_{I}(n) is modeled by the contaminated Gaussian (CG) process or the symmetric α−S distribution. The CG impulse can be represented by η_{I}(n)=z(n)b(n) with a signaltointerference ratio (SIR) of − 10 dB or − 20 dB, where z(n) is a white Gaussian process with zeromean and b(n) is a Bernoulli sequence with the probability mass function with P(b)=1−P for b=0 and P(b)=P for b=1, where P is the probability of the occurrence of the impulsive interference. The SIR is defined as \(\text {SIR} = 10\log _{10} \left (\sigma _{d}^{2} /\sigma _{z}^{2} \right)\), where \(\sigma _{z}^{2}\) and \(\sigma _{d}^{2}\) are the variances of z(n) and the desired signal \(\tilde d(n),\) respectively. The symmetric α−S distribution is characterized by the fractional order parameter p and characteristic exponent α, for which the fractionalorder signaltonoise ratio (FSNR) can be defined as \(\text {FSNR} = 10\log _{10} [E(\tilde d(n)^{p})/E(\eta _{I} (n)^{p})]\) and 0<p<α. The step sizes are set to μ_{w}=μ_{q}=0.01 for the IIRSAFLMS, IIRSAFNLMS, and proposed IIRSAFNLMM. For the proposed SAFSMNLMM, the constant θ is set to 0.06 except in Fig. 4. Other parameters are selected as follows: \(\gamma = \sqrt {\tau \sigma _{G}^{2}} /(\kappa + 1)\), τ=5, κ=0.6 except in Fig. 5, λ_{0}=λ_{1}=0.99, ε_{0}=ε=0.001, ω=0.7, α=0.8, and p=0.7.
The first experiment is to evaluate the performance of the proposed algorithms in the absence of impulsive noise. Figure 3 shows the MSE learning curves of the IIRSAFLMS, IIRSAFNLMS, proposed IIRSAFNLMM, and IIRSAFSMNLMM in the absence of impulsive noise. It can be noted that all the algorithms acquire the nearly identical steadystate MSEs. Compared with the cited algorithms, the IIRSAFNLMM suffers from the convergence performance deterioration due to the application of the modified Huber Mestimate function. However, the IIRSAFSMNLMM obtains the faster convergence rate than the other algorithms. The number of update ratio for the corresponding algorithms in the absence of impulsive noise is demonstrated in Table 2; we can see the proposed algorithms have lower update ratio over the other two cited algorithms, especially for the IIRSAFSMNLMM which involves in the setmembership error bound.
Figure 4 shows the MSE learning curves of the proposed IIRSAFSMNLMM for different values of θ in the absence of impulsive noise. It can be clearly seen that the larger value of θ leads to the faster convergence rate, and the proposed IIRSAFSMNLMM gets nearly similar steadystate MSEs for different values of θ. Besides, Table 3 displays the larger value of θ can decrease the number of the update ratio because of the faster convergence rate. Therefore, the parameter θ which is bounded by (45) can be set as large as possible in the application of the proposed IIRSAFSMNLMM. The performance of the IIRSAFSMNLMS algorithm for different values of κ in the absence of impulsive noise is shown in Fig. 5. It can be noted that the proposed IIRSAFSMNLMS holds similar convergence rate with respect to different values of κ. Moreover, the larger value of κ results in the lower steadystate MSE and a lager number of update ratio which is shown in Table 4.
In the second experiment, the performances of the proposed algorithms are compared with those of the IIRSAFLMS and IIRSAFNLMS algorithms in the CG process impulsive noise or the symmetric α−S impulsive noise. Figures 6, 7, and 8 show the performance comparison at different SIR and probability of the occurrence of the impulsive interference for CG noises. Figures 9, 10, and 11 indicate the MSE learning curves of four algorithms in the symmetric noise environment at different FSNR. From these plots, the proposed algorithms provide the robust performance in the impulsive noise environment, whereas the other two cited algorithms fail to suppress the impulse. The proposed IIRSAFSMNLMS achieves the faster convergence rate. Besides, in Table 5, it can also be seen that the proposed algorithms have lower update ratios over the cited algorithms.
The third experiment evaluates the tracking ability of the proposed algorithms. The target system changes abruptly after 30,000 samples, i.e., (w_{0},q_{0})→(w_{1},q_{1}), where the system (w_{1},q_{1}) contains an IIR filter which is given as:
and a nonlinear spline network which is implemented by a LUT q_{1} with 23 control points and q_{1} is defined by:
Figures 12 and 13 show the MSE tracking curves of four algorithms in case of the CG noise and the symmetric α−S impulsive noise, respectively. It can be clearly seen that the proposed algorithms get better tracking ability and more robust against impulsive noise than the cited algorithms. The IIRSAFSMNLMM algorithm performs best.
Conclusions
In order to suppress the effect of the impulsive noise and decrease the computational burden, this paper combines the setmembership framework and leastM estimate scheme and proposes two variants based on the IIR spline adaptive filter. The proposed SAFIIRNLMM algorithm is derived by using a robust Mestimator as the cost function and the SAFIIRSMNLMM is characterized by the setmembership error bound leading into an evident decrease of the number of the update ratio. Moreover, the computational burdens and the convergence properties of the proposed SAFIIRSMNLMM algorithm are also given. Compared to the cited spline adaptive filtering algorithms, the proposed algorithms offer more robustness against impulsive noise, better tracking ability, and lower computational complexity.
Methods/Experimental
This paper studies the SAFIIRNLMM and SAFIIRSMNLMM algorithms aiming at suppressing the effect of the impulsive noise and decreasing the computational burden compared with the conventional nonlinear adaptive spline adaptive algorithms. The derivation of the algorithms are based on the modified Huber Mestimate function and setmembership framework. Besides, the convergence properties of the SAFIIRSMNLMM algorithm are analyzed by using the energy conversion relation. The numerical experiments are carried out by applying the white Gaussian noise signal and colored noise signal in the CG impulsive noise or symmetric α−S impulsive noise environment. The results demonstrated that the two proposed variants of the SAF are robust to the impulsive noise, and the SAFIIRSMNLMM algorithm obtains low updating ratio.
Abbreviations
 APA:

Affine projection algorithm
 AWGN:

Additive white Gaussian noise
 IIR:

Infinite impulse response
 LMS:

Least mean square
 LN:

Linearnonlinear
 LNL:

Linearnonlinearlinear
 LTI:

Linear timeinvariant
 LUT:

Lookup table
 MSE:

Mean square error
 NL:

Nonlinearlinear
 NLMM:

Normalized least Mestimate
 NLMS:

Normalized least mean square
 NLN:

Nonlinearlinearnonlinear
 NN:

Neural networks
 SAF:

Spline adaptive filter
 VAF:

Volterra adaptive filters
References
 1
S. Haykin, Adaptive filter theory, 4th edn (PrenticeHall, Englewood Cliffs, 2002).
 2
A. H. Sayed, Adaptive filters (Wiley, NJ, 2008).
 3
V. J. Mathews, Adaptive polynomial filters. IEEE Sig. Process. Mag. 8:, 10–26 (1991).
 4
S. Haykin, Neural networks and learning machines, 2th edn (PrenticeHall, Englewood Cliffs, 2008).
 5
E. W. Bai, F. Giri, Introduction to blockoriented nonlinear systems (Springer, London, 2010).
 6
M. Scarpiniti, D. Comminiello, R. Parisi, A. Uncini, Nonlinear spline adaptive filtering. Sig. Process.93:, 772–783 (2013).
 7
M. Scarpiniti, D. Comminiello, R. Parisi, A. Uncini, Hammerstein uniform cubic spline adaptive filtering: learning and convergence properties. Sig. Process.100:, 112–123 (2014).
 8
M. Scarpiniti, D. Comminiello, R. Parisi, A. Uncini, Novel cascade spline architectures for the identification of nonlinear systems. IEEE Trans. Circ. Syst.I: Regular Papares.62:, 1825–1835 (2015).
 9
M. Scarpiniti, D. Comminiello, R. Parisi, A. Uncini, Nonlinear system identification using IIR spline adaptive filters. Sig. Process.108:, 30–35 (2015).
 10
F. Lindsten, T. B. Schon, M. I. Jordanb, Bayesian semiparametric Wiener system identification. Automatica. 49:, 2053–2063 (2013).
 11
M. Rasouli, D. Westwick, W. Rosehart, Quasiconvexity analysis of the Hammerstein model. Automatica. 50:, 277–281 (2014).
 12
A. E. Nordsjo, L. H. Zetterberg, Identification of certain timevarying Wiener and Hammerstein systems. IEEE Trans. Signal Process.49:, 577–592 (2001).
 13
M. A. Chitre, J. R. Potter, S. H. Ong, Optimal and nearoptimal signal detection in snapping shrimp dominated ambient noise. IEEE Ocean. J. Eng. 31:, 497–503 (2006).
 14
K. J. Sangston, K. R. Gerlach, NonGaussian noise models and coherent detection of radar targets. IEEE Trans. Aerosp. Electron. Syst.30:, 330–340 (1992).
 15
A. Mahmood, M. A. Chitre, M. A. Armand, Detecting OFDM signals in alphastable noise. IEEE Trans. Commun.62:, 3571–3583 (2014).
 16
S. C. Chan, Y. X. Zou, A recursive least Mestimate algorithm for robust adaptive filtering in impulsive noise: fast algorithm and convergence performance analysis. IEEE Trans. Sig. Procecss.52:, 975–991 (2004).
 17
Y. Zou, S. C. Chan, T. S. Ng, Lesat mean Mestiamte algorithms for robust adaptive filtering in impulse noise. IEEE Trans. Circ. Syst.II: Analog. Digit. Sig. Process.47:, 1564–1569 (2000).
 18
S. Guan, Z. Li, Normalised spline adaptive filtering algorithm for nonlinear system identification. Neural Process. Lett.5:, 1–13 (2017).
 19
S. Gollamudi, S. Nagaraj, S. Kapoor, Y. F. Huang, Setmembership filtering and a setmembership normalized LMS algorithm with an adaptive step size. IEEE Sig. Process. Lett. 5:, 111–114 (1998).
 20
S Zhang, J Zhang, H Han, Robust shrinkage normalized sign algorithm in an impulsive noise environment. IEEE Trans. Circuits Syst.II: Express Briefs. 64:, 91–95 (2017).
 21
S. Guarnieri, F. Piazza, A. Uncini, Multilayer feedforward networks with adaptive spline activation function. IEEE Trans. Neural Netw.10:, 672–683 (1999).
 22
M. Scarpiniti, D. Comminiello, G. Scarano, R. Parisi, A. Uncini, Steadystate performance of spline adaptive filters. IEEE Trans. Sig. Process.64:, 816–828 (2016).
 23
Z. Zheng, H. Zhao, Affine projection Mestiamte subband adaptive filters for robust adaptive filtering in impulsive noise. Sig. Process.120:, 64–70 (2016).
Acknowledgements
The authors would like to thank National Natural Science Foundation of China for financially support.
Funding
This work was financially supported by the National Natural Science Foundation of China under Grant 61501119 and by the Fund for the Dongguan Municipal Science and Technology Bureau under Grant 2016508140
Availability of data and materials
Please contact author for data requests.
Ethic approval and consent to participate
Not applicable.
Author information
Author notes
Affiliations
Contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Corresponding author
Correspondence to Chang Liu.
Ethics declarations
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Received
Accepted
Published
DOI
Keywords
 Nonlinear adaptive filter
 IIR spline adaptive filter
 Setmembership
 Least Mestimate