Identifying odd/even-order binary kernel slices for a nonlinear system using inverse repeat m-sequences. Hu JY, Yan G, Wang T - Comput Math Methods Med (2015) Bottom Line: The study of various living complex systems by system identification method is important, and the identification of the problem is even more challenging when dealing with a dynamic nonlinear system of discrete time.In this study, we examine the relevant mathematical properties of kernel slices, particularly their shift-and-product property and overlap distortion problem caused by the irregular shifting of the estimated kernel slices in the cross-correlation function between the input m-sequence and the system output.We then derive the properties of the inverse repeat (IR) m-sequence and propose a method of using IR m-sequence as an input to separately estimate odd- and even-order kernel slices to reduce the chance of kernel-slice overlapping. View Article: PubMed Central - PubMed Affiliation: School of Biomedical Engineering, Southern Medical University, Guangzhou, Guangdong 510515, China. ABSTRACTThe study of various living complex systems by system identification method is important, and the identification of the problem is even more challenging when dealing with a dynamic nonlinear system of discrete time. A well-established model based on kernel functions for input of the maximum length sequence (m-sequence) can be used to estimate nonlinear binary kernel slices using cross-correlation method. In this study, we examine the relevant mathematical properties of kernel slices, particularly their shift-and-product property and overlap distortion problem caused by the irregular shifting of the estimated kernel slices in the cross-correlation function between the input m-sequence and the system output. We then derive the properties of the inverse repeat (IR) m-sequence and propose a method of using IR m-sequence as an input to separately estimate odd- and even-order kernel slices to reduce the chance of kernel-slice overlapping. An instance of third-order Wiener nonlinear model is simulated to justify the proposed method. Show MeSH MajorComputer Simulation*Nonlinear Dynamics*MinorAlgorithmsComputational BiologyNormal DistributionSoftwareSystems Biology Related in: MedlinePlus © Copyright Policy Related In: Results  -  Collection License getmorefigures.php?uid=PMC4385604&req=5 .flowplayer { width: px; height: px; } fig1: Illustration of overlapping condition for two arbitrary kernel slices. A piece of cross-correlation function contains qth- and sth-order kernel slices, that is, ws and wq, respectively. Corresponding shift functions fq and fs designate the beginnings of kernel slices. The duration of wq is dependent on memory length M and sum of shift lags (l1 + ⋯+lq−1). According to (14), nonoverlap and overlap cases are presented in (a) and (b), respectively. Mentions: To estimate the Volterra kernel for Gaussian white noise, input is not theoretically feasible for the difficulty of nonorthogonality. Instead, it is preferred to estimate the Wiener kernel after the Gram-Schmidt orthogonal process on the Volterra series expansion [21]. This method can be extended to deal with m-sequence input yielding the so-called binary kernel estimation [13, 16]. A pth-order binary kernel is given by (7)wpk1,k2,…,kp=1p!ϕbpyk1,k2…,kp,where the pth-order cross-correlation of input b[n] and output y[n] is (8)ϕbpy[k1,k2…,kp] =1L∑i=0L−1y[i]b[i−k1]b[i−k2]⋯b[i−kp].Let k = k1 and  li = ki+1 − ki, and then (7) and (8) become (9)  wp[k,k+l1,…,k+l1+⋯+lp−1] =1p!ϕbpy[k,k+l1…,k+l1+⋯+lp−1],(10)ϕbpy[k,k+l1…,k+l1+⋯+lp−1] =1L∑i=0L−1y[i]b[i−k]  ×b[i−k−l1]⋯b[i−k−l1−⋯−lp−1].According to the shift-and-product property, (10) becomes(11)ϕbpy[k,k+l1,…,k+l1+⋯+lp−1] =1L∑i=0L−1y[i]b[i−k−f(l1,…,lp−1)] =ϕbyk+fp,which transfers the multivariable correlation function ϕbpy[k, k + l1,…, k + l1 + ⋯+lp−1] into a single variable cross-correlation function ϕby[k + fp]. Substituting (11) to (9) yields (12)wpk,k+l1,…,k+l1+⋯+lp−1=1p!ϕbyk+fp.Given l1, l1,…, lp−1, (12) presents a portion of the kernel function values along the diagonal and subdiagonal dimensions and is called binary kernel slice. Considering the confinement for the independent variables for wp, the kernel slice is probably unable to completely cover the true binary kernel along this dimension. Given the memory length similar to (6), all variables for pth-order kernel slice must be in the range of the memory length M; hence,(13)lp−1∈[1,M);lp−2∈[1,M−lp−1);…;l1∈[1,M−l2−⋯−lp−1);k∈[0,M−l1−⋯−lp−1),suggesting that wp is defined through the cross-correlation function ϕby between fp and fp + (M − l1 − ⋯−lp−1). Therefore, if the shift functions of two neighboring slices satisfy(14)fs−fq≤M−l1−⋯−lq−1,that is, the interval between an arbitrary kernel slice of order s and another kernel slice of order q is less than the length of the prior slice, then a slice overlap occurs. The kernel slices overlapping condition is illustrated in Figure 1.

Identifying odd/even-order binary kernel slices for a nonlinear system using inverse repeat m-sequences.

Hu JY, Yan G, Wang T - Comput Math Methods Med (2015)

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fig1: Illustration of overlapping condition for two arbitrary kernel slices. A piece of cross-correlation function contains qth- and sth-order kernel slices, that is, ws and wq, respectively. Corresponding shift functions fq and fs designate the beginnings of kernel slices. The duration of wq is dependent on memory length M and sum of shift lags (l1 + ⋯+lq−1). According to (14), nonoverlap and overlap cases are presented in (a) and (b), respectively.
Mentions: To estimate the Volterra kernel for Gaussian white noise, input is not theoretically feasible for the difficulty of nonorthogonality. Instead, it is preferred to estimate the Wiener kernel after the Gram-Schmidt orthogonal process on the Volterra series expansion [21]. This method can be extended to deal with m-sequence input yielding the so-called binary kernel estimation [13, 16]. A pth-order binary kernel is given by (7)wpk1,k2,…,kp=1p!ϕbpyk1,k2…,kp,where the pth-order cross-correlation of input b[n] and output y[n] is (8)ϕbpy[k1,k2…,kp] =1L∑i=0L−1y[i]b[i−k1]b[i−k2]⋯b[i−kp].Let k = k1 and  li = ki+1 − ki, and then (7) and (8) become (9)  wp[k,k+l1,…,k+l1+⋯+lp−1] =1p!ϕbpy[k,k+l1…,k+l1+⋯+lp−1],(10)ϕbpy[k,k+l1…,k+l1+⋯+lp−1] =1L∑i=0L−1y[i]b[i−k]  ×b[i−k−l1]⋯b[i−k−l1−⋯−lp−1].According to the shift-and-product property, (10) becomes(11)ϕbpy[k,k+l1,…,k+l1+⋯+lp−1] =1L∑i=0L−1y[i]b[i−k−f(l1,…,lp−1)] =ϕbyk+fp,which transfers the multivariable correlation function ϕbpy[k, k + l1,…, k + l1 + ⋯+lp−1] into a single variable cross-correlation function ϕby[k + fp]. Substituting (11) to (9) yields (12)wpk,k+l1,…,k+l1+⋯+lp−1=1p!ϕbyk+fp.Given l1, l1,…, lp−1, (12) presents a portion of the kernel function values along the diagonal and subdiagonal dimensions and is called binary kernel slice. Considering the confinement for the independent variables for wp, the kernel slice is probably unable to completely cover the true binary kernel along this dimension. Given the memory length similar to (6), all variables for pth-order kernel slice must be in the range of the memory length M; hence,(13)lp−1∈[1,M);lp−2∈[1,M−lp−1);…;l1∈[1,M−l2−⋯−lp−1);k∈[0,M−l1−⋯−lp−1),suggesting that wp is defined through the cross-correlation function ϕby between fp and fp + (M − l1 − ⋯−lp−1). Therefore, if the shift functions of two neighboring slices satisfy(14)fs−fq≤M−l1−⋯−lq−1,that is, the interval between an arbitrary kernel slice of order s and another kernel slice of order q is less than the length of the prior slice, then a slice overlap occurs. The kernel slices overlapping condition is illustrated in Figure 1.

Bottom Line: The study of various living complex systems by system identification method is important, and the identification of the problem is even more challenging when dealing with a dynamic nonlinear system of discrete time.In this study, we examine the relevant mathematical properties of kernel slices, particularly their shift-and-product property and overlap distortion problem caused by the irregular shifting of the estimated kernel slices in the cross-correlation function between the input m-sequence and the system output.We then derive the properties of the inverse repeat (IR) m-sequence and propose a method of using IR m-sequence as an input to separately estimate odd- and even-order kernel slices to reduce the chance of kernel-slice overlapping.

View Article: PubMed Central - PubMed

Affiliation: School of Biomedical Engineering, Southern Medical University, Guangzhou, Guangdong 510515, China.

ABSTRACT
The study of various living complex systems by system identification method is important, and the identification of the problem is even more challenging when dealing with a dynamic nonlinear system of discrete time. A well-established model based on kernel functions for input of the maximum length sequence (m-sequence) can be used to estimate nonlinear binary kernel slices using cross-correlation method. In this study, we examine the relevant mathematical properties of kernel slices, particularly their shift-and-product property and overlap distortion problem caused by the irregular shifting of the estimated kernel slices in the cross-correlation function between the input m-sequence and the system output. We then derive the properties of the inverse repeat (IR) m-sequence and propose a method of using IR m-sequence as an input to separately estimate odd- and even-order kernel slices to reduce the chance of kernel-slice overlapping. An instance of third-order Wiener nonlinear model is simulated to justify the proposed method.

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