Noise Detection for Biosignals Using an Orthogonal Wavelet Packet Tree Denoising Algorithm

INTL JOURNAL OF ELECTRONICS AND TELECOMMUNICATIONS, 2016, VOL. 62, NO. 1, PP. 15-21 Manuscript received November 15, 2015; revised March, 2016. DOI: 10.1515/eletel-2016-0002 Noise Detection for Biosignals Using an Orthogonal Wavelet Packet Tree Denoising Algorithm Manuel Schimmack and Paolo Mercorelli Abstract—This article deals with the noise detection of discrete biosignals using an orthogonal wavelet packet. In specific, it compares the usefulness of Daubechies wavelets with different vanishing moments for the denoising and compression of the digitalised biosignals in case of surface electromyography (sEMG) signals. The work is based upon the discrete wavelet transform (DWT) version of wavelet package transform (WPT). A noise reducing algorithm is proposed to detect unavoidable noise in the acquired data in a model independent way. The noise of a signal sequence will be defined by a seminorm. This method was developed for a possible observation during a fracture healing period. The proposed method is general for signal processing and its design was based upon the wavelet packet. Keywords—noise detection, wavelet packet transform, wavelet analysis, Daubechies wavelet, linux-based embedded system, ARM processor platform I. INTRODUCTION W AVELETS are used in a wide range of applications for biomedical signal and image processing and in the fields of electrocardiography, electroencephalography, and electromyography, as well as for the algorithms for magnetic resonance imaging or positron emission tomography applied in biomedical image processing [1]. The control of active prostheses for human limbs with algorithms for classification and reliable signal pattern recognition is another important field of application [2]. This article presents a wearable embedded system to observe the activity of an injured forearm during rehabilitation, including the feedback which is not detectable by an accelerometer. The aim is to record the movements of the extremity, which is stabilized in a reusable orthosis. Therefore, a wearable measurement system was constructed to detect the biosignals [3] and [4]. In general, the sEMG signal provides information about the performance of muscles and nerves [5], [6], also in the context of neurological diagnosis of myopathies and neuropathies. Signal and data conditioning followed using wavelets for active denoising and data compression. This focuses upon on a portable Linux-based embedded system together with the use of Haars and Daubechies wavelets within the context of the digitalization of sEMG signals [7]. More specifically, it compares the usefulness of Daubechies wavelets with different vanishing moments for the denoising and Manuel Schimmack and Paolo Mercorelli are with the Institute of Product and Process Innovation, Leuphana University of Lueneburg, Volgershall 1, D-21339 Lueneburg, Germany, (e-mail: {schimmack, mercorelli}@uni.leuphana.de). compression of the digitalised biosignals. In [8], the calculation of the signal projection coefficients based on the signal interpolation is proposed by means of cubic B-splines. A noise reducing algorithm is proposed to detect unavoidable noise in the acquired data. With the help of a seminorm the noise of a sequence is defined. Using this norm it is possible to rearrange the wavelet basis, which can illuminate the differences between the coherent and incoherent parts of the sequence, where incoherent refers to the part of the signal that has either no information or contradictory information. The structure of the contribution is the following. Section II devoided some background aspects of the orthogonal wavelets in context of the vanishing moment and the computation. Section III presents a wavelet based denoising algorithm and describes it in a graphical and in an analytical way. Before the conclusion, the measurement procedure is shown in Section IV. The implementation in a Linux-based embedded system and its validation in a case study through simulations is presented in V. MAIN NOMENCLATURE 𝑏: 𝑐𝑛 : 𝐷𝑏: 𝐷: 𝑑: 𝑒(𝑡): 𝐼: 𝑘: ℒ(𝜔): 𝑚0 (𝜔): 𝑁: 𝑛: 𝑦(𝑡): 𝑃𝑁 (𝜔): 𝑠(𝑑,𝑏,𝑘) : 𝑉: 𝑤𝑝(𝑑,𝑏,𝑘) : 𝜆: 𝜓 𝑛 (𝑡): 𝑧: 𝜔: frequency-dependent parameter wavelet coefficients Daubechies wavelet index wavelet tree depth index scale noise interval of time time-dependent parameter trigonometric polynomial generating function vanishing moment samples biosignal polynomial wavelet coefficient vector space wavelet coefficient tree scalar wavelet family discrete complex variable angular frequency 16 M. SCHIMMACK, P. MERCORELLI Fig. 1. Input-signal and the wavelet coefficients in a tree-like structure with the scale index d, the time translation parameter parameter k and different values of the phase parameter b II. ORTHOGONAL WAVELET PACKET TREE A. Haars wavelet One of the first orthogonal wavelets was the Haars wavelet. In this case very short discrete signals can been used, following the example of [9]. To give a concise overview on the Haar wavelet a function 𝜓(𝑑,𝑏,𝑘) (𝑡) = 𝜓𝑏 (2𝑑 𝑡 − 𝑛) (1) is considered with a support of size 2−𝑑 of the Nyquist frequency. For the two properties of the Haar basis follows:   2 any ℒ (ℝ) function 𝑓(𝑡) can be approximated, up to arbitrarily low precision, by a finite linear ℎ combination of the 𝜓(𝑑,𝑏,𝑘) (𝑡). The weighted coefficients 𝑤𝑝(𝑑,𝑏,𝑘) are calculated as follows 2 B. Daubechies wavelet An established methods for sEMG signal analysis is the Daubechies (Db) wavelet, which is used in [12], [13], [14]. The computation of the Daubechies wavelet require a polynomial with binomial coefficients as follows 𝑁−1+𝑘 𝑘 𝑃𝑁 (𝑡) = ∑ ( )𝑥 , 𝑘 where 𝜓 𝐷𝑏 has 𝑁 vanishing moments. For the response of time and frequency for the Daubechies wavelet, it also needs a trigonometric polynomial, as shown below 𝑛 (2) (3) where 𝐼 is the considered interval of time, 𝑓(𝑡) is the required ℎ signal, and 𝜓(𝑑,0,𝑘) (𝑡) is the mother function of Haars wavelet [10]. To conclude 𝑁 𝑚0 (𝜔) = ( 𝑏 (4) 𝑘 where 𝑠(𝑑,𝑘) = 𝑤𝑝(𝑑,0,𝑘) . The Haar functions are identified using the parameter tuple, (𝑑, 𝑏, 𝑘), here 𝑑 = 1 represents the highest degree of refinement with respect to time. The wavelet packets "MakeWaveletPacket", which comes from the 1 + 𝑒 −𝑗𝜔 ) ℒ(𝜔), 2 (7) being 𝑚(𝜔) defined as 𝑚(𝜔) = 𝑘 (6) 𝑘=0 The generating function, also known as transfer function is defined as follows 𝐼 ℎ ℎ (𝑡) + ∑ ∑ 𝑤𝑝(𝑑,𝑏,𝑘) 𝜓(𝑑,𝑏,𝑘) 𝑓(𝑡) = ∑ 𝑠(𝑑,𝑛) 𝜓(𝑑,0,𝑘) , (5) 𝑘=0 ℒ(𝜔) = ∑ 𝑏𝑘 ⋅ 𝑒 −𝑗𝑘𝜔 . For 𝑏 = 0 it is possible to define the following coefficients h (𝑡)𝑑(𝑡), 𝑠(𝑑,0,𝑘) = ∫ 𝑓(𝑡)𝜓(𝑑,0,𝑘) 2 denotes the coefficients on the first level on the left with time 𝑛 shifts 0 through − 1. 𝑁−1 ℎ the 𝜓(𝑑,𝑏,𝑘) (𝑡) are orthonormal; ℎ (𝑡)𝑑(𝑡). 𝑤𝑝(𝑑,𝑏,𝑘) = ∫𝑓(𝑡) 𝜓(𝑑,𝑏,𝑘) 𝐼 Wavelab Version 850 of Stanford University [11], is represented by the indixes (𝑑, 𝑏, 𝑘). Figure 1 shows the corresponding tree with the wavelet coefficients, described by 𝑤𝑝(𝑑,𝑏,𝑘) . It represents the contribution of each of the wavelets to the signal b (...truncated)