Some new discrete biorthogonal wavelets constructed with Laguerre polynomials
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Some new discrete biorthogonal wavelets constructed with Laguerre polynomials Atemangoh Bruno Peachap1,2 · Daniel Tchiotsop2 · Valérie Louis‑Dorr3 · Didier Wolf3 Received: 12 March 2020 / Accepted: 24 July 2020 © Springer Nature Switzerland AG 2020
Abstract In this study, we present a new family of discrete wavelets which are constructed with the help of Laguerre polynomials and the Daubechies biorthogonal wavelets construction method. Our aim is to propose the discrete version of some previously constructed continuous Laguerre wavelets and also to present a method of discrete wavelets construction by several iterations. With this scheme, we use two different sets of finite impulse response filters for the signal decomposition and their duals for reconstruction. The quadruplet finite impulse response filters respect the anti-aliasing and the perfect reconstruction conditions, and at the same time, they resemble as much as possible the continuous Laguerre wavelets when using the cascade algorithm. We use the mean squared error, the maximum deviation, and the standard deviation to quantify the similarity between the continuous Laguerre wavelets and the constructed discrete Laguerre wavelets. The results show that, they are both the same wavelets due to the small nature of these parameters. Our method is important because, it can permit the determination of the finite impulse response filter coefficients corresponding to many other continuous wavelets. Keywords Laguerre wavelets · Biorthogonal wavelets · Cascade algorithm · FIR filters
1 Introduction Wavelets are a very vital tool in the field of signal processing and they are applied in several domains like medicine and engineering to perform tasks such as data compression, signal denoising, signal feature extraction for classification, etc. In recent years, many wavelets have been constructed by different researchers for various purposes since they have proven to be a better signal analysis tool than the Fourier transform. More so, they are popular due to the existence of algorithms that can compute wavelet coefficients fast such as the fast wavelet transform (FWT) algorithm [1, 2]. Fundamentally, we have the continuous wavelets (small waves of zero mean, that oscillate and varnish) and the discrete wavelets (finite impulse response
filters). The cascade algorithm is a technique whereby the continuous wavelets can be obtained from the FIR filters through several iterations. We are proposing a method whereby, the FIR filter coefficients of the discrete wavelets can be obtained from its continuous wavelets. It was long suggested that Laguerre functions could be used to construct wavelets [3]. Recently, some new continuous wavelets were constructed with Laguerre functions and applied in the classification of electroencephalogram (EEG) signals [4, 5]. The applications of the continuous wavelets are however limited because of the limitations of the continuous wavelet transform (CWT) algorithm (it is slower, redundant and requires more computational space and time) co
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