Studies on Convolution and Correlation Theorems for the Linear Canonical Transform and Their Applications in Signal Processing

Abstract

The main aim of the proposed work is to provide an inclusive approach towards the introduction to the principles and applications of the product and convolution theorem in the linear canonical transform (LCT) domain. As a generalization of fractional Fourier transform (FRFT), Fresnel transform (FST) and Fourier transform (FT), the LCT is a three variable class of integral transform and has been used in many fields of optics and signal processing. The LCT 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 applications, where FT and fractional domain concepts are used, the performance can be enhanced through the use of LCT because of its three extra degrees of freedom as compared to one degree of freedom for FRFT and no degree of freedom for FT. Many properties of the LCT are currently well known, including sampling, uncertainty principle, product and convolution theorems, which are generalization of the corresponding properties of the FT and FRFT. The product and convolution theorems for the LCT available in the literature, however these do not generalize very nicely to the classical result for the FT and FRFT. The proposed work can be divided into two broader segments. The first segment includes, the efforts made in establishing the LCT a complete integral transform by developing and deriving the weighted convolution and correlation identities. Also the proposed definitions of these theorems are compared with existing ones and their superiority has been determined with the help of some newly devised performance metrics. The second segment comprises of the applications of proposed identities along with some new application areas of the LCT. In the first phase, a comprehensive closed-form analytical expression of the behavior of Dirichlet, Generalized Hamming and triangular window functions is established, utilizing various special mathematical functions in the LCT domain. It has been shown that the LCT of Dirichlet, Generalized Hamming and triangular window functions is directly dependent on the LCT variables , thus exhibiting the flexibility of various applications in signal processing. Based upon the window analysis, by selecting different LCT variables as tuning parameter in the convolution operation between LCT of window function and ideal frequency response, variability in the transition band of the resulting Hamming window based low pass FIR filter response has been achieved. Then the closed form analytic expression of the pass-stop band filter has been established in the LCT domain. The beneficial role of the LCT in the filtering application lies in the capability of the LCT in localizing the non-stationary (chirp) signals in time-frequency plane. This ascertains the superiority of LCT domain filtering over frequency domain filtering and fractional domain filtering in the case of overlapping bandlimited signal and noise. To establish the legacy of the LCT under such circumstances, the proposed weighted convolution theorem has been used to carry out the multiplicative filtering. It has been noticed that in the proposed weighted convolution theorem and even in the theorems available in the literature, consist of an undesirable extra chirp function in the derived results. To eradicate the effect of this chirp function, the LCT takes the advantage of its extra three degrees of freedom as compare to one degree of freedom for FRFT and none for FT. With the help of simulation, it has been shown that in LCT domain filtering, mean square error (MSE) is minimum for different values of signal-to-noise ratio (SNR) as compared to fractional domain filtering and frequency domain filtering. Hence the proposed convolution theorem is best suitable for filtering action as compare to fractional domain filtering. Based upon the proposed methodology used to derive the convolution theorem, an improved correlation theorem has been developed and for different values of LCT variables. The superiority of the proposed correlation theorem has been shown based upon the computational complexity and simulation comparison. The simulation comparison shows that the plot of theorem derived in the literature is more oscillatory because of more chirp functions are included to derive the correlation integral. Thereafter, the proposed weighted auto-correlation theorem for LCT has been applied in power spectral density analysis of the frequency modulated (FM) wave and it has been found that for different values of LCT variables, the bandwidth of the FM wave shrinks and concludes that the same FM signal may be transmitted with less bandwidth requirement. Finally, a through introduction has been given to a more powerful integral transform, known as offset LCT (OLCT) or Special Affine Fourier Transform (SAFT). As a generalization of LCT, offset FRFT (OFRFT), FRFT, FST and FT, the OLCT is a six variable class of integral transform and has been used in many fields of optics and signal processing. Apart from the LCT variables, it has two extra variables called time-shifting and frequency-modulation variables. These parameters are helpful to move the time-frequency representation in horizontal direction, vertical direction or combination of both. Further it has been clearly shown from the simulation results that time-shifting and frequency-modulation variables play an important role to approach the required results. Finally, the proposed theorems have proven the efficacy of the LCT and motivated to develop more application in future.