Analysis and Design of Non Recursive Digital Differentiators in Fractional Domain for Signal Processing Applications

Abstract

There is a universe of mathematics lying in between the complete differentiations and integrations. — O. Heaviside.

THE PURPOSE OF THE REPORTED WORK is to provide a comprehensive and unified introduction to the principles and applications of the differentiation property in the fractional Fourier transform domain. The fractional Fourier transform (FrFT) is a generalized definition of the classical Fourier transform (FT). The FrFT has proved to be a powerful tool for the analysis of time–varying signals by representing rotation of a signal in the time–frequency plane. In the areas where FT and the frequency domain concepts are used, the performance can be enhanced through the use of FrFT. In addition, it has a close relationship with other signal transforms like wavelet, linear canonical transform. The well–known properties of the FrFT have been established in many of the rich literatures, which have many applications in the signal and image processing fields. In the reported work, the concept of fractional order differentiation property in the FrFT domain is established. Keeping an eye on the aspects of the fractional order calculus, the proposed work is carried out. The fractional order calculus (FOC) is a branch of mathematics is as old as the classical Newtonian calculus, and deals with the theory of differential and integral operators of non–integer order and to the differential equations containing such operators. The origin of the FOC theory can be traced back to the end of the seventeenth century, the time when Newton and Leibniz developed the foundation of the differential and integral calculus. Even though the concept of the FOC theory is established long back, yet the subject came practically over the last few decades. The new models of the systems have been developed by engineers and scientists based on the fractional differential equations approach. These models have been successfully applied in various research fields such as, mechanics (theory of viscoelasticity), electrical engineering (transmission of ultrasound waves), medicine (modeling of human tissue under mechanical loads) and the list continues. The applications of the FOC theory are limited in the fields of electronics and communication engineering and an attempt has been made to incorporate the concept of FOC theory in the signal and image processing regimes. In the proposed study, an idea of amalgamation comes into picture. To achieve the fractional order differentiation property in the FrFT domain, the concept of FOC is amalgamated with the FrFT theory and it exhibits two degrees of freedom. In the case of the traditional differentiation property in the FrFT domain, there is only one degree of freedom thus, having single varying parameter of interest. Due to two degrees of freedom in the reported work, now there are two varying parameters and thus, more design flexibility can be achieved. In order to be self–contained, the fundamentals of FrFT, FOC have been thoroughly studied before coming to the main theme of the subject. A particular goal of this reported work is to provide a solid foundation that may later be used for the construction of the efficient and reliable numerical methods of the results. We strongly believe that a successful development and a thorough understanding of such numerical schemes are not possible without such analytical background. Before laying down the foundations of the fractional order differentiation in the FrFT domain, the integer order differentiation is considered in the FrFT domain. The closed–form analytical expression of the integer–order differentiator is established in the FrFT domain utilizing the design method of window based technique. The window functions are successfully been used in diverse areas of filtering, beamforming, communication and signal processing therefore, its role is quite impressive and economical from the point of view of computational complexity and ease associated with its application. Thus, an attempt is made to analyze the window functions in the FrFT domain by establishing the closed–form analytical expression which shows its behavior with varying fractional Fourier transform (FrFT) parameter. The study of main side-lobe level, half main lobe width and side–lobe fall–off rate is also done for different values of the FrFT parameter. The analytical aspects of establishing the fractional order differentiation property in the FrFT domain is well–considered. Based on the fractional differintegral definitions of FOC theory namely, Riemann–Liouville (RL), Grünwald–Letnikov (GL) and Caputo, the approach begins. Thus, the formulation is established that has two varying parameters namely, the fractional derivative order parameter (μ)related to the FOC approach and the fractional Fourier transform parameter (φ)of the FrFT theory. The fractional order differentiating filter is designed based on the popular and familiar definitions of FOC and FrFT theories. Following the established analytical results, the one–dimensional and the two–dimensional applications are discussed. The signal filtering application is considered that utilizes the fractional order differentiation property in the FrFT domain based on the FOC concept. The signal of interest is made to corrupt with the high–frequency chirp noise and it is shown that the fractional order differentiating filter performs better to eliminate the noise as compared to the time–domain and the frequency–domain filtering. Thereby, the comparative understanding is also developed between the fractional order calculus differintegral definitions based on the example considered. Furthermore, the understanding so developed is inconformity with the already established mathematical results available in most of the mathematics literatures. The established analytical result is used for the two–dimensional signal processing application of edge detection on an image. Based on the FOC definitions, the fractional differential mask is used for the edge detection operation in the FrFT domain. Thus, the design flexibility of two varying parameters, μ and φ performs the edge detection operation in the FrFT domain. As stated in the rich literatures, the traditional approach of edge detection that utilizes various edge detection operators like Roberts, Prewitt, Sobel and second–derivative Laplacian method underperforms in the case of noisy environment. So, the study shows that by utilizing the fractional differential mask in the FrFT domain, the edge detection operation performs well in the noisy environment. The performance is judged through qualitative (visual perception) and quantitative analysis (performance metric parameters of peak signal to noise ratio and mean square error). Thus, the edge detection operation utilizing the fractional order operators exhibits good noise immunity and effective performance that opens up the horizon in the future image processing applications. Finally, this study has multidimensional facets in various research areas of electronics and communication engineering. The fractional order calculus has inherent advantages, therefore by amalgamating the concept of fractional order calculus with the signal processing transforms, many great future developments could come up that would lead to many suitable applications in various fields. We believe that the researchers, both new and old cannot remain within the boundaries of the integral order calculus and that too exhibiting only one degree of freedom for applications. So the need arises for the viable mathematical tool that will accomplish future endeavors of engineering developments and that the fractional order calculus is the future for the signal processing society. Thus, an era has come up that opens up the horizon of generalizing various science and engineering problems to get better results and opening a new paradigm of FRACTIONAL ORDER SIGNAL PROCESSING and the era of FRACTIONAL ORDER CALCULUS is the calculus of21st CENTURY.