Genesis of the Solitary Wavelet

A solitary wavelet, based on the hyperbolic secant function is proposed, characterized and applied to real-time data. Numerical analysis of the solitary wavelet reveals that it has a huge number of vanishing higher order moments, tending rapidly towards zero with a negative logarithmic slope. It is seen that the wavelet has a very low number of oscillatory sub-lobes, thus making it the ideal candidate to perform signal analysis of burst-type phenomena without undergoing multiple levels of filtering and approximation, and this concept is illustrated by effectively detecting the QRS complex of an ECG cycle without undergoing multiple filtering levels. The ability of the proposed wavelet to perform analysis of a diverse variety of real time data without multiple levels of decomposition and reconstruction forms the novelty of the present work. One Sentence Summary: A Solitary Wavelet with vanishing higher moments yielding effective detection of bursts with minimal filtering is proposed and investigated. Introduction: The efficient understanding of the rich dynamics of most real-time signals and waveforms, both natural and man-made, requires thorough analysis at multiple levels of temporal and spectral resolution [1,2]. It is along these lines of thought that the past couple of decades have witnessed extensive research and application of the wavelet transform – a comprehensive extension to the frequency-localized Fourier Transform by adding time localization as well [2,3,4]. One of the most significant applications of the wavelet based analysis is in detecting, identifying, characterizing and predicting trends and features in real-time signals [2,3]. The key clues in such feature detections are discontinuities and bursts in the time series data, which usually tend to be extremely compact and localized in time [2,3]. It has been a constant challenge to formulate various mathematical functions as bases for wavelets, which are able to capture such bursts with the least possible level of decomposition, reconstruction and filtering [1,2,3,4]. The ability of a wavelet to capture such bursts effectively translates analytically to the wavelet function, also called the „Mother Wavelet‟ having the most possible number of zero higher order central moments [3,4,5]. In, literature, many wavelet families have been formulated such as the Daubechies, Biorthogonal, Reverse Biorthogonal and Coiflets, to try and achieve this criteria of maximum number of vanishing moments [4,5,6,7,8,9,10]. Of particular mention are the Daubechies wavelets, which exhibit the highest number A of vanishing moments for a given „support‟ N=2A [9]. Also of significant mention is the Discrete Meyer Wavelet which exhibits vanishing moments up to very large orders [10]. However, most of the wavelets mentioned above invariably exhibit increasing moments of higher orders, as a result of which, extensive wavelet based decomposition and reconstruction at multiple levels need to be performed for most real time data before effective burst detections can be carried out [1,5,8]. At a hardware level, this almost invariably and inevitably translates to higher amount of power dissipation. In the present work, a new kind of wavelet, termed the „Solitary Wavelet‟ is proposed based on the hyperbolic secant function, which is known to possess an extremely smooth and compact structure. It is seen that the higher order moments of this wavelet is closer to zero and vanishes at a rate logarithmically faster than existing wavelets mentioned above. The primary application targeted for this wavelet is the effective detection of bursts in signals [5]. In order to understand this, a case study pertaining to the automated detection of the QRS complex in Electrocardiogram Time Series without filtering/reconstruction is presented [11,12]. The extreme compactness and smoothness of the hyperbolic secant based solitary wavelet function, as seen by the rapidly vanishing higher moments, yields in very effective detection of bursts and peaks with minimal amount of filtering, and this forms the significant novelty of the present work, as a consequence of which, the solitary wavelet finds extensive application in analyzing a diverse variety of real time data effectively. Formulation of the Solitary Wavelet: 1. The Father and Mother Wavelets The primary inspiration for the concept of solitary wavelet arises from the hyperbolic secant (sech) function. This function is popular in the field of photonics as a solution to various nonlinear differential equations, and the waveform of the sech function is known to be compact and extremely smooth [13]. Based on these properties, the solitary wavelet is formulated according to the following procedure: 1. The first step is to define the Scaling Function, also called the „Father Wavelet‟ φ in continuous time, based on the hyperbolic secant as follows [14]: ( ) ( ) (1) 2. The Solitary Father Wavelet φ thus defined is used as the basis to form the Solitary „Mother Wavelet‟ ψ, such that the following criteria are satisfied [1,2,5,9,10]: a. ψ(t) belongs to a subspace of the space ( ) ( ), the space of absolutely and square integrable measurable functions. b. φ(t) and ψ(t) are orthogonal to each other. c. ψ(t) has zero mean, i.e. the following holds: