Fractional Fourier theory has provided a generalization of the classical Fourier transform, and as a result has become a rich area of new concepts and applications. For instance, the implicit relationship that exists between the fractional Fourier transform (FrFT) and time-frequency representations has revealed a continuum of time-frequency (T-F) rotated domains of which the well-known frequency domain is simply a special case. Consequently, the existence of such domains allows for the generalization of Fourier filtering in ways that make it possible to easily realize various time-varying operators. This can in turn lead to more effective signal processing approaches for a range of practical applications. The main focus of this thesis is on the novel concept of fractional Fourier-based filtering. Particularly the work looks into the design of single, as well as multi-stage, systems for the restoration of both simulated and real-world signals. The thesis starts by first examining some of the essential properties of the fractional Fourier transform which relate to filtering. Precisely, the concept of rotated domains in the joint time-frequency plane is elaborated and further exploited for filtering. Results and improvements achieved are demonstrated and discussed through different application examples over the chapters of this thesis.
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