Convolution Theorem of Fractional Fourier Transformation Derived by Representation Transformation in Quantum Mechancis

Based on our previous paper (Commun.Theor.Phys.39 (2003) 417) we derive the convolution theoremof fractional Fourier transformation in the context of quantum mechanics,which seems a convenient and neat way.Generalization of this method to the complex fractional Fourier transformation case is also possible.

Convolution theorems for the linear canonical transform and their applications

As generalization of the fractional Fourier transform （FRFT）, the linear canonical transform （LCT） has been used in several areas, including optics and signal processing. Many properties for this transform are already known, but the convolution theorems, similar to the version of the Fourier transform, are still to be determined. In this paper, the authors derive the convolution theorems for the LCT, and explore the sampling theorem and multiplicative filter for the band limited signal in the linear canonical domain. Finally, the sampling and reconstruction formulas are deduced, together with the construction methodology for the above mentioned multiplicative filter in the time domain based on fast Fourier transform （FFT）, which has much lower computational load than the construction method in the linear canonical domain.

CONVOLUTION THEOREM AND ASYMPTOTIC EFFICIENCY IN TWO SIDE TRUNCATED DISTRIBUTION FAMILY——THE NORMAL CASE

In the distribution family with common support and the one side truncated distribution family, Bickle, I. A. Ibragimov and R. Z. Hasminskii proved two important convolution theorems. As to the two-side truncated case, we also proved a convolution theorem, which plays an extraordinary role in the efficiency theory. In this paper, we will study another kind of two-side truncated distribution family, and prove a convolution result with normal form. On the basis of this convolution result, a new kind of efficiency concept is given; meanwhile, we will show that MLE is an efficient estimate in this distribution family.

Studying the Two New Convolutions of Fractional Fourier Transform by Using Dirac Notation

Based on quantum mechanical representation and operator theory,this paper restates the two new convolutions of fractional Fourier transform(FrFT)by making full use of the conversion relationship between two mutual conjugates:coordinate representation and momentum representation.This paper gives full play to the efficiency of Dirac notation and proves the convolutions of fractional Fourier transform from the perspective of quantum optics,a field that has been developing rapidly.These two new convolution methods have potential value in signal processing.

Design and simulation of digital channelized receivers in fractional Fourier domain

An approach is proposed to realize a digital channelized receiver in the fractional Fourier domain （FRFD） for signal intercept applications. The presented architecture can be considered as a generalization of that in the traditional Fourier domain. Since the linear frequency modulation （LFM） signal has a good energy concentration in the FRFD, by choosing an appropriate fractional Fourier transform （FRFT） order, the presented architecture can concentrate the broadband LFM signal into only one sub-channel and that will prevent it from crossing several sub-channels. Thus the performance of the signal detection and parameter estimation after the sub-channel output will be improved significantly. The computational complexity is reduced enormously due to the implementation of the polyphase filter bank decomposition, thus the proposed architecture can be realized as efficiently as in the Fourier domain. The related simulation results are presented to verify the validity of the theories and methods involved in this paper.