Fractional Fourier Transform Domain Research Articles

Image encryption based on extended fractional Fourier transform and digital holography technique

We present a new optical image encryption algorithm that is based on extended fractional Fourier transform (FRT) and digital holography technique. We can perform the encryption and decryption with more parameters compared with earlier similar methods in FRT domain. In the extended FRT encryption system, the input data to be encrypted is extended fractional Fourier transformed two times and random phase mask is placed at the output plane of the first extended FRT. By use of an interference with a wave from another random phase mask, the encrypted data is stored as a digital hologram. The data retrieval is operated by all-digital means. Computer simulations are presented to verify its validity and efficiency.

On scaling properties of fractional Fourier transform and its relation with other transforms

Several properties of fractional Fourier transform (FRFT) have been studied recently and many are being investigated at present. In this article, scaling property of the FRFT is generalized and some of its applications are suggested. Some extensions of the sampling relations in the FRFT domain are also presented. The issues related to connections between the FRFT and other signal transforms such as scale transform, fractional Mellin transform, and chirp z-transform, are also investigated.

Fractional Fourier transform of bandlimited periodic signals and its sampling theorems

In this correspondence, some features of the fractional Fourier transform (FRFT) of the bandlimited periodic signals are discussed. Two sampling relations in the context of FRFT domain bandlimited, periodic signals are also presented. More specifically, it is shown that only 2K+1 coefficients are sufficient to reconstruct any FRFT domain bandlimited periodic signal, where K is the order of the highest nonzero harmonic component in the particular FRFT domain in which the signal is bandlimited.

Fractional Fourier Transform based Beamforming for Next Generation Wireless Communication Systems

Today's mobile communication systems are primarily designed to provide cost efficient wide area coverage for a limited number of users with moderate bandwidth. The future telecommunication will provide the same services in a wireless fashion as provided by fixed network these days and thereby requiring high bandwidths. It is expected that future telecommunication users will be neither willing to sacrifice functionality for the added value of mobility nor to pay more for it—mainly because he will hardly be using any other stationary telecommunication devices. Third generation mobile systems proposals such as UMTS and IMT-2000 only partially address these requirements by providing mobile internet services, with up to 2 Mbps in certain areas. Thus future mobile and wireless applications will require significantly higher data rates and significantly reduced costs per transmitted bit as compared to third generation systems. These requirements on data rate, link quality, spectral efficiency, and mobility cannot be met with conventional beamforming. Therefore new beamforming techniques are essential to increase the spectral efficiency of wireless communication systems. This paper provides the results achieved through use of a new technique Fractional Fourier transform (FRFT) based beamforming which is especially useful in moving and accelerating source problem. The conventional Minimum Mean Squared Error (MMSE) beamforming in the frequency domain or spatial domain becomes a special case of optimal beamforming with FRFT. The advantage of the proposed optimum FRFT domain beamformer is the practical consideration that the synthesis or analysis of the FRFT can be implemented with the same complexity as the conventional Fast Fourier Transform (FFT).

Importance of the phase and amplitude in the fractional Fourier domain.

The importance of the amplitude and phase in the fractional Fourier transform (FT) domain is analyzed on the basis of the rectangular signal and the real-world image. The quality of signal restoration from only the amplitude or from only the phase of its fractional FT by applying the inverse fractional FT is considered. It is shown that the signal reconstructed from the amplitude of the fractional FT usually reveals the main features of the original signal only for relatively low fractional orders. On the basis of phase information in the fractional FT domains, significant details of the signal can be obtained for nearly all fractional orders.

An uncertainty principle for real signals in the fractional Fourier transform domain

The fractional Fourier transform (FrFT) can be thought of as a generalization of the Fourier transform to rotate a signal representation by an arbitrary angle /spl alpha/ in the time-frequency plane. A lower bound on the uncertainty product of signal representations in two FrFT domains for real signals is obtained, and it is shown that a Gaussian signal achieves the lower bound. The effect of shifting and scaling the signal on the uncertainty relation is discussed. An example is given in which the uncertainty relation for a real signal is obtained, and it is shown that this relation matches with that given by the uncertainty relation derived.

Beam shaping in the fractional Fourier transform domain

A new design approach for the diffractive phase elements (DPE’s) that implement beam shaping in the fractional Fourier transform (FRFT) domain is presented. The new algorithm can successfully achieve the design of DPE’s for beam shaping in both unitary and nonunitary transform systems. The unitarity transform condition of the FRFT domain is discussed. Modeling designs of the DPE’s are carried out for several fractional orders and different parameters of the beam for converting a Gaussian profile into a uniform beam. Our approach can realize beam shaping well for a nonunitary transform system in the FRFT domain.

Fractional Gabor transform

A fractional Gabor transform (FRGT) is proposed. This new transform is a generalization of the conventional Gabor transform (GT) based on the Fourier transform to the windowed fractional Fourier transform (FRFT). The FRGT provides analyses of signals in both the real space and the FRFT frequency domain simultaneously. The space-FRFT frequency pattern can be rotated as the fractional order changes. The FRGT has an additional freedom, compared with the conventional GT, i.e., the transform order. The FRGT may offer a useful tool for guiding optimal filter design in the FRFT domain in signal processing.