Linear canonical transformation (LCT) is a generalization of the Fourier transform and fractional Fourier transform. The recent studies have shown that LCT is widely used in optics, signal processing and applied mathematics, and the discretization of the LCT becomes vital for the applications of LCT. Since the discretization of LCT cannot be obtained by directly sampling in time domain and LCT domain, the discretization of the LCT becomes the focus of investigation recently. Based on the development history of LCT discretization, a review of important research progress and current situation of discretization of the LCT is presented in this paper. Meanwhile, the connection among different discretization algorithms and the future development direction are given. It is of great reference value for researchers to fully understand the LCT discretization method and can further promote its engineering applications.

The fractional Fourier transform (FRFT) is a generalization of the Fourier transform. The FRFT can characterize signals in multiple fractional domains and provide new perspectives for non-stationary signal processing and linear time variant system analysis, thus it is widely used in reality applications. We first review recent developments of the FRFT in theory, including discretization algorithms of the FRFT, various discrete fractional transforms, sampling theorems in fractional domains, filtering and parameter estimation in fractional domains, joint analysis in multiple fractional domains. Then we summarize various applications of the FRFT, including radar and communication signal processing in fractional domains, image encryption, optical interference measurement, medicine, biology, and instrument signal processing based on the FRFT. Finally we discuss the future research directions of the FRFT, including fast algorithm of the FRFT, sparse sampling in fractional domains, machine learning utilizing the FRFT, graph signal processing in fractional domains, and discrete FRFT based on quantum computation.

With the increase of data volume and the rapid development of modern radar, more requirements are put forward for radar target detection technology. There are both advantages and limitations of classical radar detection methods. Innovative methods are urgently needed to improve the radar target detection performance under complex background and limited radar resources. The main purpose of this paper is to illustrate the implementation of sparse fractional Fourier transform (SFRFT), which is developed on the basis of sparse Fourier transform (SFT). Besides, the SFRFT algorithm is applied to radar signal processing and a SFRFT-based fast and high resolution detection method is proposed to improve the detection performance of radar maneuvering target. It is expected that the method can provide a new way for radar moving target detection.

For the denoising problem of odd and even signals, a multiplicative filter design method based on the convolution theorem of the linear canonical sine and cosine transform is proposed. Two kinds of convolution theorems associated with the linear canonical sine and cosine transform based on the existing linear canonical transform domain convolution theory are derived. Using this two convolution theorems, two kinds of the multiplicative filtering models of the band-limited signal are designed by choosing an appropriate filter function in linear canonical sine and cosine transform domain. And the complexity of these schemes is analyzed. The results indicate that these filtering models are particularly suitable for handling odd and even signals, and can effectively improve computational efficiency by reducing computational complexity.

In order to improve the security of traditional optical image encryption and reduce the amount of data what needs to process, we propose a color image asymmetric optical encryption method based on compressed sensing and quantum logistic map, and use the compressive sensing theory and single-channel encrypted method to deal with the problem of large amount of data in the process of color image encryption. Aiming at the linear problem of the traditional optical cryptosystem, we use asymmetric optical encryption based on phase truncation fractional Fourier transform. We also use quantum logistic map to generate the random phase masks for the convenience of transmitting random phase masks. The results show that the proposed algorithm can obtain better image encryption and decryption results.

The existing NMR equipment is uneven to face the main magnetic field, mostly adopts the hardware method of magnetic field compensation, such as magnetic field compensation, but it brings bad effects such as image artifact and blurred image. In view of the problem of magnetic field inhomogeneous in magnetic resonance imaging, a fractional domain magnetic resonance imaging (fMRI) method under the main magnetic field inhomogeneous is proposed. First, select a layer of living tissue to be imaged, select several points on the layer and measure the intensity of the magnetic field on the layer. Based on the principle of magnetic resonance imaging, establish the model of the magnetic field intensity distribution in the imaging area, and then establish. The polynomial model of the magnetic field can be divided into the second-order polynomial model and the higher-order polynomial model according to whether there is a significant second-order component in the measured magnetic field. Then, the two models are respectively substituted into the free-induction decay (FID) signals of the magnetic resonance. For the second-order model, the fractional Fourier transform tool can be used to solve the spin density function on one layer of the imaged object. The order model needs to obtain the spin density function at a certain level of the imaging object by solving the algebraic equation, thus establishing the MR signal model with any non-uniform main magnetic field. Experimental results show that this method achieves the same effect as the uniform main magnetic field.

The propagation equation of the elastic wave is more complicated than that of the electromagnetic wave. It is difficult to design medium parameters when controlling the direction of the elastic wave. In order to obtain precise results of elastic wave propagation, the equations were simplified or approximated to achieve the design effect, depending on the actual situation (such as high frequency). Therefore, as impedance matching and lossless dielectric material requirements are difficult to meet, the scattering phenomenon appears in the design of elastic wave propagation in the process of the device. Usually, and the scattering wave is variable frequency signal. It is a way and a tool in the evaluation of elastic wave device design, the size of the scattering phenomenon marks the design effect. The fractional Fourier transform (FRFT), which has good focusing characteristic, is adaptive to analyze frequency variation signal. The frequency change rate provides a quantitative description method of elastic scattering wave propagation control. This reduces the blindness of scattering degree of cognitive, simplifying the dielectric become evaluation of design.

The classical Shannon sampling theorem has a profound influence on signal processing and communication. With the increasing contradiction between high rate sampling and conversion accuracy, the traditional analog to digital conversion technology, which is based on the Shannon sampling theorem, is facing a great challenge, especially for the bottleneck effect on reducing the sampling rate. In recent years, the analog-to-information conversion (AIC) technology, which is based on the theory of compressive sensing, provides an effective method to solve this problem. However, the signal model of the existing AIC is only suitable for sparse signals band-limited in the Fourier transform (FT) domain. It cannot be applied to non-bandlimited chirp signals which is widely used in electronic information systems, including radar and communications. Towards this end, we propose a new AIC based on the fractional Fourier transform (FRFT), which is not only the extension of the traditional AIC in the FRFT domain, but also can solve the problem as mentioned above. The theoretical derivation is presented, and the corresponding six mulation analysis is also given. The simulation results are consistent with the theoretical analysis.

Based on the virtual scene containing hundreds of movable sound sources, due to the high computational cost of clustering stage, the traditional spatial sound rendering schemes often take up too much computing resources, which have become a bottleneck in the development of VR audio rendering technology. In this paper, we use fractional Fourier transform (FRFT) as a tool in sound sampling to reduce the quantization noise during the ADC conversion stage. Moreover, we improve the processing speed of sound rendering and the operation efficiency of the entire system by adding the average angle deviation threshold in the clustering step. In addition, we design and implement a perceptual user experiment, and validates the notion that people are more susceptible to spatial errors in different types of sound sources, especially if it is visible. Based on this conclusion, this paper proposes an improved method of sound clustering, which reduces the possibility of clustering different types of sound sources.

Hartley transform is a generalization of Fourier transform and it transforms the real signal into real signal thereby reducing the amount of computation. In recent years, with the wide applications of fractional Fourier transform in signal processing, linear canonical transform has gradually been applied to signal processing. Hence, it is a valuable problem to generalize Hartley transform in linear canonical transform domain. In this paper, a kernel function with conjugate property is obtained by changing kernel function of Hartley transform in Fourier transform domain. After that, we obtain Hartley transform in linear canonical transform domain by using kernel function of linear canonical transform. Then, Hartley transform in linear canonical transform domain has the properties of real number and odd-even invariance. Finally, by using Heisenberg uncertainty principle in linear canonical transform domain, we obtain Heisenberg uncertainty principle of Hartley transform in linear canonical transform domain.

Fractional Fourier transformation (FRFT) is a very useful tool for signal processing and analysis, which can well represent the content of the image by projecting it to the time-frequency plane. The features extracted by 2D-FRFT have shown very promising results for face recognition. However, there is one problem when dealing with 2D-FRFT: it is hard to know that which order of 2D-FRFT (the angle of projection of time-frequency plane) is best for the specific task without prior knowledge. In spirit of multiple kernel learning in machine learning, we discuss the relations between the order selection in 2D-FRFT and kernel selection in multiple kernel learning. By treating the linear kernels over different features from 2D-FRFT with different orders as the input to multiple kernel learning framework, and also by applying support vector machines (SVM) on top of the learned kernels, we can update the weights in the multiple kernel learning framework and SVM parameters through alternative optimization. Therefore, the problem of order selection of 2D-FRFT is solved by the off-the-shelf algorithm of multiple kernel learning. The experiments of face recognition on ORL dataset and extended YaleB dataset show the effectiveness of the proposed algorithm.

The ship detection in infrared video has wide application value in fishery administration, port monitoring and other fields. Traditional background modeling methods, such as GMM(Gaussian mixture model), Codebook, and ViBe, will make more false detection in the ship detection from the infrared ocean video because of the impact of the waves. The paper proposes a new algorithm to detect ships in the infrared ocean video. The algorithm framework adopts the Top-Hat operation to preprocess the infrared image to filter the clutter effectively, then improves Vibe algorithm to detect the moving ship target. Experimental results show that the method can effectively suppress the background noise and get better detection results.

In fiber-optic Brillouin optical time-domain reflectometer (BOTDR) system, the estimation of the center frequency of Brillouin scattering spectrum in fiber is the key and the most time-consuming part of the measurement, which makes BOTDR system difficult to achieve fast response. In this paper, estimation of signal parameters via rotational invariance technique (ESPRIT) algorithm is proposed to estimate the center frequency of Brillouin scattering spectrum in BOTDR system. Due to fairly low requirement on data length, ESPRIT algorithm can obtain good frequency estimation over short data length, and makes it possible to increase measurement speed at high spatial resolution and measurement performance.