Order Of Fractional Fourier Transform Research Articles

Fixed Windows in Fractional Fourier Domain

In this study, some mathematical relations have been derived for the useful parameters of fixed window functions in fractional Fourier transform (FRFT) domain. These reported expressions are also verified with the simulation studies. The FRFT provides an important extension to conventional Fourier transform with an additional degree of freedom by which these parameters of window functions can be controlled while inherent time domain behavior of the windows remains intact. The behavior of fixed windows on time- frequency plane has been varied by varying the FRFT order. The obtained variability in the window functions has been applied in the designing of FIR filters.

Optimal Transform Order of Fractional Fourier Transform for Decomposition of Overlapping Ultrasonic Signals

The separation time-overlapping ultrasound signals is necessary to obtain accurate estimate of transit time and material properties. In this letter, a method to determine the optimal transform order of fractional Fourier transform (FRFT) for decomposition of overlapping ultrasonic signals is proposed. The optimal transform order is obtained by minimizing the mean square error (MSE) between the output and the reference signal. Furthermore, windowing in FRFT domain is discussed. Numerical simulation results show the performances of the proposed method in separating signals overlapping in time.

Analysis of Chirp as windowing function through fractional Fourier transform

This article presents a new mathematical model for obtaining the fractional Fourier transform (FrFT) of Chirp function. FrFT is a parameterised transform having an adjustable transform parameter which makes it more flexible and superior over ordinary FT in several applications. It is an important tool used in signal processing for spectral analysis. The closed-form expression derived for FrFT of finite duration Chirp establishes the dependence of FrFT of Chirp on the order of FrFT and the Chirp parameter. The mathematical model obtained shows that the Chirp function can be used as an adjustable window function in the fractional Fourier domain. The main-lobe width, side-lobe level and side-lobe fall-off rate of a Chirp window can be controlled by changing the adjustable transform parameter to different values. For some particular values of fractional angle, Chirp can give better spectral parameters than the existing window functions. By varying the order of FrFT, the variations in spectral parameters of the Chirp window are obtained and studied. The performance of the Chirp window in fractional domain is also compared with some of the existing windows.

Double image encryption based on phase-amplitude hybrid encoding and iterative phase encoding in fractional Fourier transform domains

A new method for double image encryption is proposed that is based on amplitude-phase hybrid encoding and iterative random phase encoding in fractional Fourier transform (FrFT) domains. In the iterative random phase encoding operation, a binary random matrix is defined to encode two original images to a single complex-valued image, which is then converted into a stationary white noise image by the iterative phase encoding with FrFTs. Compared with the previous schemes that uses fully phase encoding, the proposed method reduces the difference between two original images in key space and sensitivity to the FrFT orders. The primitive images can be retrieved exactly by applying correct keys with initial conditions of chaotic system, the pixel scrambling operation and the FrFT orders. Computer simulations demonstrate that the encryption method has impressively high security level and certain robustness against data loss and noise interference.

Fractional Fourier transform of Airy beams

An analytical expression of an Airy beam passing through a fractional Fourier transform (FRFT) system is presented. The effective beam size of the Airy beam in the FRFT plane is also derived. The influences of the order of FRFT, the modulation parameter, and the transverse scale on the normalized intensity distribution and the effective beam size of an Airy beam in the FRFT plane are examined. The order of FRFT controls the effective beam size and the orientation of the beam spot. The modulation parameter affects the effective beam size and the number of lateral side lobes. The normalized intensity distribution and the effective beam size are both proportional to the transverse scale.

Fractional Fourier Transform for Ultrasonic Chirplet Signal Decomposition

A fractional fourier transform (FrFT) based chirplet signal decomposition (FrFT-CSD) algorithm is proposed to analyze ultrasonic signals for NDE applications. Particularly, this method is utilized to isolate dominant chirplet echoes for successive steps in signal decomposition and parameter estimation. FrFT rotates the signal with an optimal transform order. The search of optimal transform order is conducted by determining the highest kurtosis value of the signal in the transformed domain. A simulation study reveals the relationship among the kurtosis, the transform order of FrFT, and the chirp rate parameter in the simulated ultrasonic echoes. Benchmark and ultrasonic experimental data are used to evaluate the FrFT-CSD algorithm. Signal processing results show that FrFT-CSD not only reconstructs signal successfully, but also characterizes echoes and estimates echo parameters accurately. This study has a broad range of applications of importance in signal detection, estimation, and pattern recognition.

A Novel Multiuser SISO-BOFDM Systems with Group Fractional Fourier Transforms Scheme

Existing multiuser (MU) schemes are mainly accomplished by MIMO techniques. In this letter, we propose a novel group fractional Fourier transform based MU SISO Biorthogonal Frequency Division Multiplexing system, in which multiple independent data streams from co-channel mobile stations (MSs) can be transmitted in the same frequency and time slot. Because the chirp bases generated by different orders based fractional Fourier transform (FrFTs), at same subchannel, are mutually approximately orthogonal, when FrFT orders are effectively allocated to different co-channel MSs, the co-channel interference can be perfectly suppressed by simple correlative detection in FrFT domain with respective optimal order, at the receiver of each MS. Finally, system simulations show the essential advantages over conventional MU schemes.

Mutual-induced fractional Fourier transform in strongly nonlocal nonlinear medium

The effect of mutual-induced fractional Fourier transform (FRFT) between a weak signal beam and a strong pump beam in a strongly nonlocal nonlinear medium is described. The signal beam is an FRFT during propagation and the order of FRFT is dependent on the propagation distance and the power of the pump beam. When the propagation distance is fixed, the order of FRFT of the signal beam is proportional to the square root of the power of the pump beam. Mutual-induced FRFT is another method of light controlling light, and its properties contribute to the development of new type FRFT devices, optical information processing and optical imaging.

Image Encryption With Multiorders of Fractional Fourier Transforms

The original information in the existing security system based on the fractional Fourier transform (FRFT) is essentially protected by only a certain order of FRFT. In this paper, we propose a novel method to encrypt an image by multiorders of FRFT. In the image encryption, the encrypted image is obtained by the summation of different orders inverse discrete FRFT of the interpolated subimages. And the original image can be perfectly recovered using the linear system constructed by the fractional Fourier domain analysis of the interpolation. The proposed method can be applied to the double or more image encryptions. Applying the transform orders of the utilized FRFT as secret keys, the proposed method is with a larger key space than the existing security systems based on the FRFT. Additionally, the encryption scheme can be realized by the fast-Fourier-transform-based algorithm and the computation burden shows a linear increase with the extension of the key space. It is verified by the experimental results that the image decryption is highly sensitive to the deviations in the transform orders.

An optical waveform pre-distortion method based on time domain fractional Fourier transformation

An optical waveform pre-distortion method is proposed in reducing the temporal broadening of pulse based on the optical implementation of time domain fractional Fourier transform (FRFT). Moreover, a new analysis of the dispersion and self-phase modulation effects is investigated. The simulation results show that prechirp coefficient and proper fractional orders of FRFT can be beneficial in mitigating pulse broadening.

Robust Hash for Detecting and Localizing Image Tampering

In order to improve the sensitive to illegal manipulations of image hashing, a novel robust image hashing algorithm based on fractional Fourier transform (FRFT) for detecting and localizing image tampering is proposed in this paper. The framework of generating an image hashing includes three steps: preprocessing, feature extracting and post processing. The robust hashing sequence is obtained by FRFT coefficients of image blocks. The security of proposed algorithm is totally depended on the orders of FRFT which are saved as secret keys. Experimental results show that the proposed scheme is robust against perceptually acceptable modifications to the image such as JPEG compression, mid-filtering, and rotation, more importantly the tampered place can be identified accurately.

Order selection in fractional Fourier transform based beamforming

Traditionally, beamforming using fractional Fourier transform (FrFT) involves a trial-and-error based FrFT order selec- tion which is impractical. A new numerical order selection scheme is presented based on fractional power spectra (FrFT moment) of the linear chirp signal. This method can adaptively determine the optimum FrFT order by maximizing the second-order central FrFT moment. This makes the desired chirp signal substantially concentrated whereas the noise is rejected considerably. This improves the mean square error minimization beamformer by re- ducing effectively the signal-noise cross terms due to the finite data length de-correlation operation. Simulation results show that the new method works well under a wide range of signal to noise ratio and signal to interference ratio.

Soliton-like dispersion-free optical transmission based on optical time-domain fractional Fourier transform

In the present article, implementation of the optical time-domain fractional Fourier transform (FRFT) is achieved and a new method for soliton-like dispersion-free optical transmission is proposed. With this method, the pulses’ envelope will keep unchanged and non-broadened after transmission under the combined effects of chromatic dispersion and self-phase modulation in the fiber link without any dispersion compensation. Numerical analysis shows how a pulse evolves for different orders of FRFT.

Teaching stable two-mirror resonators through the fractional Fourier transform

We analyse two-mirror resonators in terms of their fractional Fourier transform (FRFT) properties. We use the basic ABCD ray transfer matrix method to show how the resonator can be regarded as the cascade of two propagation–lens–propagation FRFT systems. Then, we present a connection between the geometric properties of the resonator (the g parameters) and those of the equivalent FRFT systems (the FRFT order and scaling parameters). Expressions connecting Gaussian beam q-transformation with FRFT parameters are derived. In particular, we show that the beam waist of the resonator's mode is located at the plane leading to two FRFT subsystems with equal scaling parameter which, moreover, coincides with the mode Rayleigh range. Finally we analyse the resonator's stability diagram in terms of the fractional orders of each FRFT subsystem, and the round trip propagation. The presented analysis represents an interesting link between two topics (optical resonators and Fourier optics) usually covered in optics and photonics courses at university level, which can be useful to teach and connect the principles of these subjects.

Vulnerability to known-plaintext attack of optical encryption schemes based on two fractional Fourier transform order keys and double random phase keys

From the perspective of optical information security, we demonstrate that conventional phase retrieval algorithms, such as the Gerchberg–Saxton algorithm, can be still valid when extended from the Fourier transform domain into the fractional Fourier transform (FRFT) domain. Based on this extension, we propose a method of known-plaintext attack (KPA) on double random phase encoding in the fractional Fourier transform domain (DRPE-FRFT). With this attack, an opponent can access two encryption phase keys with the help of the phase retrieval algorithm in the FRFT domain, and meanwhile obtain the two FRFT order keys by using an exhaustive search. This indicates that the optical encryption scheme based on the DRPE-FRFT scheme is vulnerable to KPA due to the nature of the linearity of FRFT. The validity of this attack is verified by numerical simulations.

ISAR Image Formation Based on Minimum Entropy Criterion and Fractional Fourier Transform

Conventional radar imaging systems use Fourier transform for image formation, but due to the target's complicated motion the Doppler spectrum is time-varying and thus the reconstructed image becomes blurred even after applying standard motion compensation algorithms. Therefore, sophisticated algorithms such as polar reformatting are usually employed to produce clear images. Alternatively, Joint Time-Frequency (JTF) analysis can be used for image formation which produces clear image without using polar reformatting algorithm. In this paper, a new JTF-based method is proposed for image formation in inverse synthetic aperture radars (ISAR). This method uses minimum entropy criterion for optimum parameter adjustment of JTF algorithms. Short Time Fourier Transform (STFT) and Fractional Fourier Transform (FrFT) are applied as JTF for time-varying Doppler spectrum analysis. Both the width of Gaussian window of STFT and the order of FrFT, α, are adjusted using minimum entropy as local and total measures. Furthermore, a new statistical parameter, called normalized correlation, is defined for comparison of images reconstructed by different methods. Simulation results show that α-order FrFT with local adjustment has much better performance than the other methods in this category even in low SNR.

Use of the fractional Fourier transform in π/2 converters of laser modes

The possibility of using the fractional Fourier transform (FrFT) in optical schemes for astigmatic π/2 converters of Hermite—Gaussian modes to donut Laguerre—Gaussian modes is considered. Several schemes of converters based on the FrFT of the half-integer and irrational orders are presented. The lowest FrFT order than can be used in astigmatic mode converters is found. The properties of converters based on the fractional and ordinary Fourier transforms are compared.

Short-time fractional Fourier methods for the time-frequency representation of chirp signals.

The fractional Fourier transform (FrFT) provides a valuable tool for the analysis of linear chirp signals. This paper develops two short-time FrFT variants which are suited to the analysis of multicomponent and nonlinear chirp signals. Outputs have similar properties to the short-time Fourier transform (STFT) but show improved time-frequency resolution. The FrFT is a parameterized transform with parameter, a, related to chirp rate. The two short-time implementations differ in how the value of a is chosen. In the first, a global optimization procedure selects one value of a with reference to the entire signal. In the second, a values are selected independently for each windowed section. Comparative variance measures based on the Gaussian function are given and are shown to be consistent with the uncertainty principle in fractional domains. For appropriately chosen FrFT orders, the derived fractional domain uncertainty relationship is minimized for Gaussian windowed linear chirp signals. The two short-time FrFT algorithms have complementary strengths demonstrated by time-frequency representations for a multicomponent bat chirp, a highly nonlinear quadratic chirp, and an output pulse from a finite-difference sonar model with dispersive change. These representations illustrate the improvements obtained in using FrFT based algorithms compared to the STFT.

BCD optical system and the fractional fourier transform

A general and systematic analysis about the relationship between ABCD optical systems and the fractional Fourier transform (FRT) is provided. It is shown that the FRT can be implemented with an ABCD system but usually different scaling factors for the input and output functions must be used. The requirement for the property of direct additivity of the FRT order is derived for a cascade system; and the method of finding the final order of the FRT for a general cascade ABCD system by using the similarity theorem is discussed. As an application example of the results, an approach to observation of the FRT of continuously variable orders with a scale invariant input is demonstrated.

Optical implementation of scale invariant fractional Fourier transform of continuously variable orders with a two-lens system

A simple two ordinary lens system is proposed to realize the fractional Fourier transform (FRT) of continuously variable order while keeping the scaling factors of both the input and output functions as the same constant independent of the geometric parameters of the system and the FRT orders. It can remove the necessity and inconvenience of using input masks of different scales or any other special devices, and then is helpful to those who want to observe the FRT but have no specific facilities.