Computational and Mathematical Methods in Medicine

Volume 2015, Article ID 454638, 9 pages

http://dx.doi.org/10.1155/2015/454638

## Identifying Odd/Even-Order Binary Kernel Slices for a Nonlinear System Using Inverse Repeat m-Sequences

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

Received 27 June 2014; Revised 9 September 2014; Accepted 1 October 2014

Academic Editor: Shengyong Chen

Copyright © 2015 Jin-yan Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

Living systems usually exhibit complex and nonlinear behaviors [1–3], which can be characterized by a mathematical model carefully tuned to represent the relationship between the input and output data. A linear system is capable of determining the input and output relationships through an impulse response function; however, for a nonlinear system, a higher order transfer function has to be used for this purpose. A nonlinear system can be typically modeled by Volterra or an equivalent Wiener series expansion, in which the Volterra or Wiener kernels to be estimated can fully define the system characteristics [4–6].

The kernel estimation for such nonlinear system usually requires the input signal to be a long Gaussian white noise to completely activate the underlying system. Under such conditions, Lee and Schetzen proposed a convenient cross-correlation method widely used to estimate the kernel functions [7, 8].

In several circumstances, particularly for a variety of living biosystems, input signals are constrained as a series of impulse trains instead of continuous signals, such as the Gaussian white noise [9, 10]. For instance, the auditory system is usually studied by stimulating the ear with a series of click sounds to activate the corresponding neurons in the cochlea and neural pathway to evaluate the hearing integrity [11, 12]. A well-studied impulse train for the input is a pseudorandom binary sequence called maximum length sequence (short for m-sequence), which has an important role in nonlinear system identification. The correlation property of an m-sequence is analogous to a Gaussian white noise such that to model the system by borrowing the idea of the cross-correlation method for Gaussian white noise input is possible. Hence, the binary kernels are defined using cross-correlation method for m-sequence inputs [3, 13–18].

Using the m-sequence approach, Sutter studied the binary kernel for identifying multifocal retinosystem through electroretinography and explained the visual function using the binary kernels [15]. Shi and Hecox transferred the m-sequence into an m-impulse sequence in a study on the nonlinear properties of the auditory system by measuring the electrical response from the scalp [16]. In a study on the dynamic characteristics of the primate retinal ganglion cell, Benardete and Victor developed a hybrid m-sequence allowing the summation of multiple m-sequences as input to estimate the main diagnostic kernel slice [17].

However, the binary kernel slices—derived by making use of the shift-and-product property of the m-sequence—are all laid in the first-order cross-correlation function between the m-sequence input and the system response, that is, the observed output. The specific location of any kernel slice in the cross-correlation function is determined through a complex shift function that cannot be explicitly determined. If the kernel slices are improperly arranged such that overlaps among neighboring slices occur, then the kernel estimation is inevitably distorted. A straightforward approach to solve this problem is to multiply the length of the input m-sequence, which is unfavorable for living systems with more or less time-varying property. Another approach to alleviate the overlap issue is to sparsify the impulse train of the m-sequence at risk of suffering the underestimation caused by the reduced number of available kernel slices [18].

In this study, we addressed the overlap problem through a new strategy using an inverse-repeat (IR) m-sequence. We will drive the estimation equations for the binary kernel slices corresponding to the IR m-sequence, through which the odd- and even-order kernel slices can be separately estimated and thus reduce the chance of slice overlapping. Last, a third-order nonlinear system is simulated to demonstrate the process of the proposed method.

#### 2. Binary Kernel Identification for m-Sequence

##### 2.1. The Properties of an m-Sequence

An m-sequence consisting of digit −1 and +1 that pseudorandomly occurred can be generated through the output of a linear circular shift register. The structure of which is determined through a primitive polynomial of degree , which is also the degree of the periodical m-sequence [19, 20]. And the period or length of the m-sequence is . An m-sequence is called a balanced sequence because the number of −1 is , which is just one more than the number of +1, that is, . Two crucial properties of the m-sequence used in the present study are as follows.

(i) Shift-and-product propertywhere is referred to as* shift function* representing the circular shift lag of the m-sequence . The exact value of a shift function depends on the shifting lags of the m-sequences to be multiplied. This property indicates that the product of the m-sequences with different circular shifts is also the same m-sequence circular shifting to a lag determined by a shift function that is unknown* a priori*. A straightforward approach to calculate the specific value of a shift function is to compare bit-by-bit the original m-sequence and the resulting -bit shifting version until , such that .

For , that is, the product of the same m-sequences, it yieldswhere denotes an all-one sequence—all members of the sequence are 1s.

Equation (2) provides an exception for the shift-and-product property that an all-one sequence instead of an m-sequence is produced under a certain condition when multiplying the same m-sequences. When more than two different m-sequences are multiplied, we have another exception for the shift-and-product property: Equation (3) implies that when dealing with higher-order kernels , the m-sequence should be selected with caution in case invalid results occur for the kernel estimation.

(ii) The autocorrelation function is a periodical real value function with the same minimal period of ; that is, In analogy to the Gaussian white noise method, this autocorrelation property of (4) is important to account for the derivation of the m-sequence in identifying a nonlinear system.

##### 2.2. From Volterra Kernels to Binary Kernels

The output of a general th-order nonlinear dynamic system in response to the input can be modeled by a Volterra series expansion [5, 21], where is called the th-order Volterra operator which is defined as where represents the* memory length* of the dynamic system, and represents the th-order* Volterra kernel*.

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 th-order binary kernel is given by where the th-order cross-correlation of input and output is Let and , and then (7) and (8) become According to the shift-and-product property, (10) becomeswhich transfers the multivariable correlation function into a single variable cross-correlation function . Substituting (11) to (9) yields Given , (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 , 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 th-order kernel slice must be in the range of the memory length ; hence,suggesting that is defined through the cross-correlation function between and . Therefore, if the shift functions of two neighboring slices satisfythat is, the interval between an arbitrary kernel slice of order and another kernel slice of order is less than the length of the prior slice, then a slice overlap occurs. The kernel slices overlapping condition is illustrated in Figure 1.