Wavelet Theory: Applications of the Wavelet

In this Chapter, continuous Haar wavelet functions base and spline base have been discussed. Haar wavelet approximations are used for solving of differential equations (DEs). The numerical solutions of ordinary differential equations (ODEs) and fractional differential equations (FrDEs) using Haar wavelet base and spline base have been discussed. Also, Haar wavelet base and collocation techniques are used to approximate the solution of Lane-Emden equation of fractional-order showing that the applicability and efficacy of Haar wavelet method. The numerical results have clearly shown the advantage and the efficiency of the techniques in terms of accuracy and computational time. Wavelet transform studied as a mathematical approach and the applications of wavelet transform in signal processing field have been discussed. The frequency content extracted by wavelet transform (WT) has been effectively used in revealing important features of 1D and 2D signals. This property proved very useful in speech and image recognition. Wavelet transform has been used for signal and image compression.

Wavelet Transform for Signal Processing in Internet-of-Things (IoT)

The primary contribution of this chapter is to provide an overview of different denoising methods used for signal processing in IoT networks from the perspectives of physical layer in the network. The chapter starts with the introduction to different kinds of noise that can be encountered in any kind of wireless communication networks, different kinds of wavelet transform and wavelet packet transform methods that can be used for denoising sensor signals in IoT networks and the different processing steps that are needed to be followed to accomplish wavelet packet transform for the sensor signals. Finally, a universal framework based on energy correlation analysis has been presented for denoising sensor signals in IoT networks, and such a framework can achieve considerable improvement in denoising performance reducing the effective noise correlation coefficient to 0.00001 or lower. Moreover, this method is found to be equally effective for Gaussian or impact noise or both.

Industrial IoT Using Wavelet Transform

For many years now, communication in the industrial sector has been characterized by a new trend of integrating the wireless concept through cyber-physical systems (CPS). This emergence, known as the Smart Factory, is based on the convergence of industrial trades and digital applications to create an intelligent manufacturing system. This will ensure high adaptability of production and more efficient resource input. It should be noted that data is the key element in the development of the Internet of Things ecosystem. Thanks to the IoT, the user can act in real time and in a digital way on his industrial environment, to optimize several processes such as production improvement, machine control, or optimization of supply chains in real time. The choice of the connectivity strategy is made according to several criteria and is based on the choice of the sensor. This mainly depends on location (indoor, outdoor, …), mobility, energy consumption, remote control, amount of data, sending frequency and security. In this chapter, we present an Industrial IoT architecture with two operating modes: MtO (Many-to-One) and OtM (One-to-Many). An optimal choice of the wavelet in terms of bit error rate is made to perform simulations in an industrial channel. A model of this channel is developed in order to simulate the performance of the communication architecture in an environment very close to industry. The optimization of the communication systems is ensured by error correcting codes.

Wavelet Theory and Application in Communication and Signal Processing

Wavelet analysis is the recent development in applied mathematics. For several applications, Fourier analysis fails to provide tangible results due to non-stationary behavior of signals. In such situation, wavelet transforms can be used as a potential alternative. The book chapter starts with the description about importance of frequency domain representation with the concept of Fourier series and Fourier transform for periodic, aperiodic signals in continuous and discrete domain followed by shortcoming of Fourier transform. Further, Short Time Fourier Transform (STFT) will be discussed to induce the concept of time frequency analysis. Explanation of Continuous Wavelet Transform (CWT) and Discrete Wavelet Transform (DWT) will be provided with the help of theoretical approach involving mathematical equations. Decomposition of 1D and 2D signals will be discussed suitable examples, leading to application concept. Wavelet based communication systems are becoming popular due to growing multimedia applications. Wavelet based Orthogonal Frequency Division Multiplexing (OFDM) technique and its merit also presented. Biomedical signal processing is an emerging field where wavelet provides considerable improvement in performance ranging from extraction of abnormal areas and improved feature extraction scheme for further processing. Advancement in multimedia systems together with the developments in wireless technologies demands effective data compression schemes. Wavelet transform along with EZW, SPIHT algorithms are discussed. The chapter will be a useful guide to undergraduate and post graduate who would like to conduct a research study that include wavelet transform and its usage.

Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory

Bessel functions form an important class of special functions and are applied almost everywhere in mathematical physics. They are also called cylindrical functions, or cylindrical harmonics. This chapter is devoted to the construction of the generalized coherent state (GCS) and the theory of Bessel wavelets. The GCS is built by replacing the coefficient zn/n!,z∈C of the canonical CS by the cylindrical Bessel functions. Then, the Paley-Wiener space PW1 is discussed in the framework of a set of GCS related to the cylindrical Bessel functions and to the Legendre oscillator. We prove that the kernel of the finite Fourier transform (FFT) of L2-functions supported on −11 form a set of GCS. Otherwise, the wavelet transform is the special case of CS associated respectively with the Weyl-Heisenberg group (which gives the canonical CS) and with the affine group on the line. We recall the wavelet theory on R. As an application, we discuss the continuous Bessel wavelet. Thus, coherent state transformation (CST) and continuous Bessel wavelet transformation (CBWT) are defined. This chapter is mainly devoted to the application of the Bessel function.