Computational and Mathematical Methods in Medicine

Volume 2017 (2017), Article ID 3927486, 9 pages

https://doi.org/10.1155/2017/3927486

## Noise Attenuation Estimation for Maximum Length Sequences in Deconvolution Process of Auditory Evoked Potentials

^{1}School of Biomedical Engineering, Southern Medical University, Guangzhou, Guangdong, China^{2}Stephenson School of Biomedical Engineering, University of Oklahoma, Norman, OK, USA

Correspondence should be addressed to Xiaodan Tan

Received 15 December 2016; Revised 19 January 2017; Accepted 26 January 2017; Published 19 February 2017

Academic Editor: Anne Humeau-Heurtier

Copyright © 2017 Xian Peng 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 use of maximum length sequence (m-sequence) has been found beneficial for recovering both linear and nonlinear components at rapid stimulation. Since m-sequence is fully characterized by a primitive polynomial of different orders, the selection of polynomial order can be problematic in practice. Usually, the m-sequence is repetitively delivered in a looped fashion. Ensemble averaging is carried out as the first step and followed by the cross-correlation analysis to deconvolve linear/nonlinear responses. According to the classical noise reduction property based on additive noise model, theoretical equations have been derived in measuring noise attenuation ratios (NARs) after the averaging and correlation processes in the present study. A computer simulation experiment was conducted to test the derived equations, and a nonlinear deconvolution experiment was also conducted using order 7 and 9 m-sequences to address this issue with real data. Both theoretical and experimental results show that the NAR is essentially independent of the m-sequence order and is decided by the total length of valid data, as well as stimulation rate. The present study offers a guideline for m-sequence selections, which can be used to estimate required recording time and signal-to-noise ratio in designing m-sequence experiments.

#### 1. Introduction

Maximum length sequence (m-sequence) has been found useful in the study of linear and nonlinear responsive components in the auditory system [1, 2]. Convoluted auditory evoked potentials (AEPs) can be elicited by an m-sequence of stimuli with its interstimulus intervals (ISIs) varying pseudorandomly. The cross-correlation technique has been developed to deconvolute linear/nonlinear components in AEPs [3, 4] from overlapped responses. The linear component reflects evoked responses to individual stimuli independently, and the nonlinear component reflects the temporal interaction of two or more stimuli. Therefore, the m-sequence method provides a unique tool in characterizing the human auditory system.

Usually, AEPs are highly contaminated with background electroencephalograms (EEGs) from various sources of noise or artifacts. An ensemble averaging technique has to be applied to enhance the signal-to-noise ratio (SNR) before deconvolution. It is well-known that noise power level is attenuated inversely proportional to number of signal sweeps to be averaged. The noise property of AEPs obtained using m-sequence can be studied from different perspectives. For example, Marsh [5] presented an intuitive explanation of noise constraints for m-sequence to extract the linear components of auditory brain stem response (ABR) using a subaveraging technique and demonstrated that ABR elicited by an m-sequence was noisier than conventional ABR obtained with same number of stimuli. Thornton [6, 7] presented a simple estimation method for SNR improvement using m-sequence in acquisition of otoacoustic emissions (OAEs). This estimation is based on an assumption of no adaptation effect for OAEs and estimated 3 dB SNR improvement for an m-sequence eliciting OAEs. Late on, Van Veen and Lasky [8] provided a general matrix-based framework for the response to arbitrary stimulus sequences and derived estimated SNR formula for m-sequences. Inspired by the success of m-sequence AEP, other deconvolution techniques have been rapidly advanced for various application scenarios (e.g., [9–11]). Delgado and Ozdamar [12] proposed a deconvolution method called continuous loop average deconvolution (CLAD), which provided a computational efficient solution to the problem and a capability of SNR estimation in the frequency domain. Based on the similar idea, they then employed Parseval’s theorem to derive an SNR formulation for m-sequence and proved that m-sequence offers the highest SNR as compared with any CLAD sequences [13].

Conventionally, the raw EEG is epoched into EEG sweeps for averaging. A sweep of EEG is usually short in length equivalent to the ISI of the corresponding isochronic stimulus-sequence. The length of a signal sweep of m-sequence is much longer since EEGs to be averaged are in response to a full length of m-sequence containing a number of stimulus events, which is determined by the order of an m-sequence. The fact means that the number of EEG sweeps to be averaged has to be greatly reduced given a fixed EEG recording time, which gives rise to a problem of how to select the best m-sequence in terms of SNR. Although less number of sweeps will sacrifice SNR at the averaging step, the next cross-correlation step is expected to be able to attenuate more noise that may compensate its SNR loss. In the present study, we investigated the noise attenuation property of m-sequence with different orders using the cross-correlation technique. Based on the well-established noise attenuation relationship from the ensemble averaging process, we derived a noise attenuation ratio (NAR) metric for the m-sequence deconvolution procedure including both averaging and correlation processes. We then employed computer synthetic data and a real nonlinear AEP experiment to validate the proposed formula.

#### 2. Method

##### 2.1. Nonlinear m-Sequence Model

In general, a nonlinear system can be represented by a Volterra or Wiener series provided that the system is time-invariant with finite memory [14, 15]. The output of such a nonlinear system can be expressed by summations of multiorder convolutions of Volterra kernels: where , , and are the first, second, and th-order Volterra kernels of the system; is the system memory length; is the system input or the stimulation in this context. The Volterra kernels are equivalent to orthogonal Wiener kernels, which can be estimated by a method developed by Lee and Schetzen [16] using Gaussian white noise input. The Gaussian white noise input is unsuitable for transient AEPs that are usually elicited by individual short sound elements. Using binary m-sequence, Sutter [17] developed a computational efficient method to estimate the nonlinear kernels that are referred to as* binary kernels* based on the cross-correlation techniques [16]. Shi and Hecox [4] further extended it to m-pulse sequence which is in line with the linear application m-sequence firstly carried out by Eysholdt and Schreiner [3] in extracting the linear ABR at fast stimulus rate.

Mathematically, an m-sequence derived from a primitive polynomial is usually implemented by a number of shift-registers with different orders [18], say . And the number of binary values or the length of an m-sequence is . The m-pulse sequence proposed by Shi and Hecox [4] modified the m-sequence of binary element of to a pulse sequence of element, where the digit “1” is used to designate the occurrence of a transient stimulus, and digit “0” represents the silence of stimulation. In the discrete implementation of stimulations, the original m-sequence actually represents the most condensed stimulation rate that is practically unfeasible. Given the sampling rate in practice, we have to sparsify an m-sequence by padding zeros between the neighboring binary elements. The number of zeros denoted by is called sparse factor. In this case, the stimulation rates for an m-sequence can be derived from the reciprocal of the maximum ISI, the minimum ISI, and the mean ISI of an m-sequence. An instance of stimulation impulses derived from an order 5 m-sequence is shown in Figure 1. For every m-pulse sequence, a unique recovery sequence can be defined by an inverse operation on the original m-sequence (Figure 1). A unique mathematic property of m-sequence is that the cross-correlation function between the m-pulse sequence and the recovery sequence is an impulse function, which makes the deconvolution problem solvable and computational efficient.