Performance of Discrete Fractional Fourier Transform Classes in Signal Processing Applications

Abstract

Given the widespread use of ordinary Fourier transform in science and engineering, it is important to recognize this integral transform as the fractional power of FT. Indeed, it has been this recognition, which has inspired most of the many recent applications replacing the ordinary FT with FrFT (which is more general and includes FT as special case) adding an additional degree of freedom to problem, represented by the fraction or order parameter a. This in turn may allow either a more general formulation of the problem or improvement based on possibility of optimizing over a (as in optimal wiener filter resulting in smaller mean square error at practically no additional cost). The FrFT has been found to have several applications in the areas of optics and signal processing and it also lead to generalization of notion of time (or space) and frequency domains which are central concepts of signal processing. In every area where FT and frequency domain concepts are used, there exists the potential for generalization and implementation by using FrFT. With the advent of computers and enhanced computational capabilities the Discrete Fourier Transform (DFT) came into existence in evaluation of FT for real time processing. Further these capabilities are enhanced by the introduction of DSP processors and Fast Fourier Transform (FFT) algorithms. On similar lines, so there arises a need for discretization of FrFT. Furthermore, DFT is having only one basic definition and nearly 200 algorithms are available for fast computation of DFT. But when FrFT is analysed in discrete domain there are many definitions of Discrete Fractional Fourier Transform (DFrFT). These definitions are broadly classified according to the methodology of computation adopted. In the current study the various class of DFrFT algorithms are studied and compared on the basis of computational complexity, deviation factor, properties of FrFT retained by particular class and constraints on the order or fraction parameter a etc.

As discussed earlier, the FrFT has found a lot of applications in signal processing, so the DFrFT is used for some of the one-dimensional and two-dimensional applications in the present work. The one dimensional applications discussed include filtering using window functions, optimal filtering of faded signals, beamforming for the mobile antenna and optimal beamforming in faded channels. In the two dimensional applications the image processing for compression and encryption are discussed. Window functions have been successfully used in various areas of filtering, beam forming, signal processing and communication. The role of windows is quite impressive and economical from the point of view of computational complexity and ease associated with its application. An attempt has been made to evaluate the window functions in FrFT domain. The study of side lobe fall of rate (SLFOR), main side lobe level (MSLL) and half main lobe width (HMLW) for window functions are done for different values of a. A new FrFT based Minimum Mean Square Error (MMSE) filtering technique is also suggested and it is also established that the results are much better as compared to FT. The signals in real life are non-stationary random processes. This may be attributed to the randomness of signals due to variation in amplitude and phase and associated Doppler shift, delay spread etc (as in the case of mobile sources and receivers). The multipath environment of mobile communication also makes the signal non-stationary due to changing spatial position with time. In these type of applications, where signal and noise both are non-stationary (time-frequency varying) FrFT is a powerful tool in designing an optimal filter. The proposed filter is compared with time and frequency domain filtering. This algorithm is also used as an optimal beamformer for mobile and wireless communication, as in this is also an example of non-stationary signal and noise. This beamforming technique also works more efficiently for faded signals. The FrFT and FrCT are used for image compression and the results are much better than FT. It is also shown that in compression the FrFT gives better results than FrCT. The image encryption is also done using FrCT and phase masking. This technique gives an advantage of additional keys i.e. order parameter of the transform. The number of these additional keys can be further enhanced by using repetition the FrCT with various orders. The merits of FrFT are that it is not only richer in theory and more flexible in application but the cost of implementation is also low as it can be implemented with same complexity as that of conventional Fast Fourier transform. The FrFT provides additional degree of freedom to the problem as parameter a gives multidirectional applications in various areas of optics and signal processing in particular and physics and mathematics in general. The most important aspect of the FrFT is its use in time varying signals for which the FT fails to work. In past few years researchers are trying to fractionalize every transform so as to formulate a more general problem. It is obvious this that an era has been opened up for a generalization of the problems to get better results in every area of engineering by using Fractional Domains of a Transform opening up a new signal processing technique may be referred as FRACTIONAL SIGNAL PROCESSING. The advances in FrFT are multidimensional bus still it is interesting to note that the algebra of Fractional domain is far from complete at present and there are several unforeseen indentities an results to be derived.