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Wavelet theory is associated with building a model for a signal, system or processes with a set of special signals and is emerged as a powerful tool of signal de-noising. Earlier, the much celebrated fractional Fourier transform (FrFT) has been used to decompose the contaminated signals and obtain the desired signals on removing the noise. In optical data transmission, continuous signals are represented as functions of space or spatial frequency. Fractional Fourier transforms are closely related to chirp transforms, which in turn characterize and formulate optical transmission as the propagation in free space whereas fractional domains are the generalizations of conventional space and frequency domain. This leads to an interpretation of fractional Fourier or chirp transforms as wavelet transforms. The Wavelet Transform of one dimension has two parameters viz. scaling and shifting parameters. This makes possible to establish the correspondence between fractional Fourier transform and wavelet transform by choosing chirp function as the wavelet transform kernel. In this paper, a strong relationship between wavelet transform with fractional Fourier transform has been exploited to develop a full-fledged analytical framework in tempered distributional settings, which can be viewed as the extension of the FrFT. This interpretation of wavelet transform in terms of fractional Fourier transform is then used in filtering and separation of undesired noise and distortion from optical signals. The proposed model has far reaching applications especially in the field of signal processing and in particular, in the field of long range optical fiber transmission; which has been an active area of research ever since the introduction of multiresolution techniques in the fractal representation of modulated signals. Experimental results have shown that the wavelet based models have better performance over the other transform techniques ever applied for signal processing.

On Wavelet Transform: An extension of Fractional Fourier Transform and its applications in optical signal processing