Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Filtering <s> Fractional Fourier transforms, which are related to chirp and wavelet transforms, lead to the notion of fractional Fourier domains. The concept of filtering of signals in fractional domains is developed, revealing that under certain conditions one can improve upon the special cases of these operations in the conventional space and frequency domains. Because of the ease of performing the fractional Fourier transform optically, these operations are relevant for optical information processing. <s> BIB001 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Filtering <s> The ordinary Fourier transform is suited best for analysis and processing of time-invariant signals and systems. When we are dealing with time-varying signals and systems, filtering in fractional Fourier domains might allow us to estimate signals with smaller minimum mean square error (MSE). We derive the optimal fractional Fourier domain filter that minimizes the MSE for given non-stationary signal and noise statistics, and time-varying distortion kernel. We present an example for which the MSE is reduced by a factor of 50 as a result of filtering in the fractional Fourier domain, as compared to filtering in the conventional Fourier or time domains. We also discuss how the fractional Fourier transformation can be computed in O(N log N) time, so that the improvement in performance is achieved with little or no increase in computational complexity. <s> BIB002 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Filtering <s> A novel adaptive filtering scheme based on fractional Fourier transform (FrFT) is introduced and characterized. As a generalization of the ordinary Fourier transform, FrFT is a powerful analysis tool to describe signals with chirp-type components. We show that in case of linear frequency modulated (LFM) signals fractional Fourier domain adaptive filtering schemes provide less error and faster convergence. The adaptation algorithm is employed in an active noise control system application with various LFM signals and a real multi-component signal. Simulation results present that the adaptation performance in FrFT-domain adaptive filtering is superior compared to time-domain adaptation. <s> BIB003
The idea of using the FRFT for fundamental signal processing procedures such filtering, estimation and restoration is particularly interesting for applications involving optical information processing BIB001  . In several publications the concepts of filtering, estimation and restoration of signals in fractional domains were developed for these applications, revealing that under certain conditions one can improve upon the special cases of these operations in the conventional space and frequency domain. Furthermore, the FRFT can be applied to the problem of time-varying filtering of non-stationary, finite energy processes both in continuous-time and discretetime frameworks BIB002 . Filtering in fractional Fourier domains may enable significant reduction of the mean square error in comparison with ordinary Fourier domain filtering. In particular, the optimum multiplicative filter function that minimizes the mean square error in the a-th fractional Fourier domain was derived in BIB002 . Similarly, a novel fractional adaptive filtering scheme was introduced BIB003 , and simulation results showed that adaptive filtering in the fractional domain is superior in comparison to its time domain counterparts.