JOURNAL OF BEIJING INSTITUTE OF TECHNOLOGY

Research Progress of the Sampling Theorem Associated with the Fractional Fourier Transform

Jinming Ma, Ran Tao

2021, 30(3): 195-204. doi:10.15918/j.jbit1004-0579.2021.041

[Abstract](548) [FullText HTML](292) [PDF 661KB](82)

Abstract:
Sampling is a bridge between continuous-time and discrete-time signals, which is important to digital signal processing. The fractional Fourier transform (FrFT) that serves as a generalization of the FT can characterize signals in multiple fractional Fourier domains, and therefore can provide new perspectives for signal sampling and reconstruction. In this paper, we review recent developments of the sampling theorem associated with the FrFT, including signal reconstruction and fractional spectral analysis of uniform sampling, nonuniform samplings due to various factors, and sub-Nyquist sampling, where bandlimited signals in the fractional Fourier domain are mainly taken into consideration. Moreover, we provide several future research topics of the sampling theorem associated with the FrFT.

Research Progress on Discretization of Linear Canonical Transform

Yannan Sun, Bingzhao Li, Ran Tao

2021, 30(3): 205-216. doi:10.15918/j.jbit1004-0579.2021.036

[Abstract](349) [FullText HTML](195) [PDF 735KB](50)

Abstract:
Linear canonical transformation (LCT) is a generalization of the Fourier transform and fractional Fourier transform. The recent research has shown that the LCT is widely used in signal processing and applied mathematics, and the discretization of the LCT becomes vital for the applications of LCT. Based on the development of discretization LCT, a review of important research progress and current situation is presented, which can help researchers to further understand the discretization of LCT and can promote its engineering application. Meanwhile, the connection among different discretization algorithms and the future research are given.

Generalized Uncertainty Inequalities on Fisher Information Associated with LCT

Guanlei Xu, Xiaogang Xu, Xiaotong Wang

2021, 30(3): 217-227. doi:10.15918/j.jbit1004-0579.2021.025

[Abstract](396) [FullText HTML](289) [PDF 785KB](38)

Abstract:
Uncertainty principle plays an important role in multiple fields such as physics, mathematics, signal processing, etc. The linear canonical transform (LCT) has been used widely in optics and information processing and so on. In this paper, a few novel uncertainty inequalities on Fisher information associated with linear canonical transform are deduced. These newly deduced uncertainty relations not only introduce new physical interpretation in signal processing, but also build the relations between the uncertainty lower bounds and the LCT transform parametersa,b,canddfor the first time, which give us the new ideas for the analysis and potential applications. In addition, these new uncertainty inequalities have sharper and tighter bounds which are the generalized versions of the traditional counterparts. Furthermore, some numeric examples are given to demonstrate the efficiency of these newly deduced uncertainty inequalities.

A Random Nonstationary Pulse Train Model

Leon Cohen

2021, 30(3): 228-237. doi:10.15918/j.jbit1004-0579.2021.055

[Abstract](307) [FullText HTML](248) [PDF 1987KB](35)

Abstract:
A simple and mathematically tractable model of a nonstationary process is developed. The process is the sum of waves where the parameters of the waves are random. Explicit expressions for the mean and autocorrelation function at each position as a function of time are obtained. In the case of infinite time, the model evolves into a stationary process. The time-frequency distribution at each position is also obtained. An explicit example is given where the initial waves are Gaussian. The case where there is dispersion in the propagation is also discussed.

Uncertainty Principle for the Quaternion Linear Canonical Transform in Terms of Covariance

Yanna Zhang

2021, 30(3): 238-243. doi:10.15918/j.jbit1004-0579.2021.034

[Abstract](355) [FullText HTML](121) [PDF 551KB](27)

Abstract:
An uncertainty principle (UP), which offers information about a signal and its Fourier transform in the time-frequency plane, is particularly powerful in mathematics, physics and signal processing community. Under the polar coordinate form of quaternion-valued signals, the UP of the two-sided quaternion linear canonical transform (QLCT) is strengthened in terms of covariance. The condition giving rise to the equal relation of the derived result is obtained as well. The novel UP with covariance can be regarded as one in a tighter form related to the QLCT. It states that the product of spreads of a quaternion-valued signal in the spatial domain and the QLCT domain is bounded by a larger lower bound.

Existence and Uniqueness Analysis for Fractional Differential Equations with Nonlocal Conditions

Chun Wang

2021, 30(3): 244-248. doi:10.15918/j.jbit1004-0579.2021.027

[Abstract](274) [FullText HTML](104) [PDF 508KB](21)

Abstract:
Fractional differential equations and systems are gradually becoming an essential approach to real world applications in science and engineering technology. The boundary value problems of the fractional differential equations and systems were investigated by using several different methods. Recently, much attention has been paid to investigate fractional differential equations with nonlocal conditions by using the fixed point method. In this paper, by using the Banach fixed point theorem and Krasnoselkii fixed point theorem, the existence and uniqueness of the solutions to the initial value problem of the nonlinear fractional differential equations with nonlocal conditions are investigated, and some sufficient conditions are obtained. We extend some results that already exist. Finally, an example is given to show the usefulness of the theoretical results.

Heisenberg Uncertainty Principle for n-Dimensional Linear Canonical Transforms

Yonggang Li, Chuan Zhang, Huafei Sun

2021, 30(3): 249-253. doi:10.15918/j.jbit1004-0579.2021.032

[Abstract](235) [FullText HTML](91) [PDF 582KB](17)

Abstract:
The uncertainty principle proposed by German physicist Heisenberg in 1927 is a basic principle of quantum mechanics and signal processing. Since linear canonical transformation has been widely used in various fields of signal processing recently and Heisenberg uncertainty principle has been endowed with new expressive meaning in linear canonical transforms domain, in this manuscript, an improved Heisenberg uncertainty principle is obtained in linear canonical transforms domain.

A New Image Segmentation Method Based on Fractional-Varying-Order Differential

Yanshan Zhang, Yuru Tian

2021, 30(3): 254-264. doi:10.15918/j.jbit1004-0579.2021.028

[Abstract](1044) [FullText HTML](493) [PDF 3260KB](17)

Abstract:
In order to solve the problem of image segmentation with intensity inhomogeneity, a new partial differential equation image segmentation model based on fractional-varying-order differential is proposed. This model introduces an adaptive coefficient to set disparate differential order intervals for pixel with different gray values and use fractional-varying-order differential to process the input image combined with the CV model, then use a variety of image segmentation evaluation indicators, such as true positive (TP) rate, false positive (FP) rate, precision (P), Jaccard similarity (JS) rate, and Dice coefficient (DC) rate to measure the pros and cons of our model. The experimental results show that our method is improved on the original basis, which is more conducive to us to obtain more image details and obtain better segmentation results.

Adaptive Short-Time Fractional Fourier Transform Based on Minimum Information Entropy

Bing Deng, Dan Jin, Junbao Luan

2021, 30(3): 265-273. doi:10.15918/j.jbit1004-0579.2021.033

[Abstract](342) [FullText HTML](136) [PDF 993KB](36)

Abstract:
Traditional short-time fractional Fourier transform (STFrFT) has a single and fixed window function, which can not be adjusted adaptively according to the characteristics of frequency and frequency change rate. In order to overcome the shortcomings, the STFrFT method with adaptive window function is proposed. In this method, the window function of STFrFT is adaptively adjusted by establishing a library containing multiple window functions and taking the minimum information entropy as the criterion, so as to obtain a time-frequency distribution that better matches the desired signal. This method takes into account the time-frequency resolution characteristics of STFrFT and the excellent characteristics of adaptive adjustment to window function, improves the time-frequency aggregation on the basis of eliminating cross term interference, and provides a new tool for improving the time-frequency analysis ability of complex modulated signals.

A New Tensor Factorization Based on the Discrete Simplified Fractional Fourier Transform

Xinhua Su, Ran Tao

2021, 30(3): 274-279. doi:10.15918/j.jbit1004-0579.2021.037

[Abstract](277) [FullText HTML](103) [PDF 597KB](101)

Abstract:
Tensor analysis approaches are of great importance in various fields such as computation vision and signal processing. Thereinto, the definitions of tensor-tensor product (t-product) and tensor singular value decomposition (t-SVD) are significant in practice. This work presents new t-product and t-SVD definitions based on the discrete simplified fractional Fourier transform (DSFRFT). The proposed definitions can effectively deal with special complex tenors, which further motivates the transform based tensor analysis approaches. Then, we define a new tensor nuclear norm induced by the DSFRFT based t-SVD. In addition, we analyze the computational complexity of the proposed t-SVD, which indicates that the proposed t-SVD can improve the computational efficiency.

Adaptive Turbo Equalization for Probabilistic Constellation Shaped Underwater Acoustic Communications

Xisheng Wu, Yanbo Wu, Min Zhu, Dong Li

2021, 30(3): 280-289. doi:10.15918/j.jbit1004-0579.2021.030

[Abstract](339) [FullText HTML](132) [PDF 1118KB](14)

Abstract:
To increase the spectral efficiency of the underwater acoustic (UWA) communication system, the high order quadrature amplitude modulations (QAM) are deployed. Recently, the probabilistic constellation shaping (PCS) has been a novel technology to improve the spectral efficiency. The PCS with high-order QAM is introduced into the UWA communication system. A turbo equalization scheme with PCS was proposed to cancel the severe inter-symbol interference (ISI). The non-zero a priori information is available for the equalizer and decoder before turbo iteration. A priori hard decision approach is proposed to improve the detection performance and the equalizer convergence speed. At the initial turbo iteration, the relation between the a priori information and the probability of the amplitude of 16QAM symbols in one dimension is given. The simulation results verified the efficiency of the proposed method, and compared to the uniform distribution (UD), the PCS-16QAM had a significant improvement of the bit error rate (BER) performance with PCS-adaptive turbo equalization (PCS-ATEQ). The UWA communication experiments further verified the performance superiority of the proposed method.

Detection of T-wave Alternans in ECG Signals Using FRFT and Tensor Decomposition

Chuanbin Ge, Shuli Zhao, Yi Xin

2021, 30(3): 290-294. doi:10.15918/j.jbit1004-0579.2021.035

[Abstract](176) [FullText HTML](63) [PDF 1053KB](16)

Abstract:
T-wave alternans (TWA) refers to the periodic beat-to-beat variation in the amplitude of T-wave in the electrocardiogram (ECG) signal in an ABAB-pattern. TWA has been proven to be a very important indicator of malignant arrhythmia risk stratification. A new method to detect TWA by combining fractional Fourier transform (FRFT) and tensor decomposition is proposed. First, the T-wave vector is extracted from the ECG of each heartbeat, and its FRFT amplitudes at multiple orders are arranged to form a T-wave matrix. Then, a third-order tensor is composed of T-wave matrices of several consecutive heart beats. After tensor decomposition, projection matrices are obtained in three dimensions. The complexity of the projection matrix is measured by Shannon entropy to obtain feature vector to detect the presence of TWA. Results show that the sensitivity, specificity, and accuracy of the algorithm on the MIT-BIH database are 91.16%, 94.25%, and 92.68%, respectively. This method effectively utilizes the fractional domain information of ECG, and shows the promising potential of the FRFT in ECG signal processing.

Speech Encryption in Linear Canonical Transform Domain Based on Chaotic Dynamic Modulation

Liyun Xu, Tong Zhang, Chao Wen

2021, 30(3): 295-304. doi:10.15918/j.jbit1004-0579.2021.038

[Abstract](271) [FullText HTML](93) [PDF 5003KB](19)

Abstract:
In order to transmit the speech information safely in the channel, a new speech encryption algorithm in linear canonical transform (LCT) domain based on dynamic modulation of chaotic system is proposed. The algorithm first uses a chaotic system to obtain the number of sampling points of the grouped encrypted signal. Then three chaotic systems are used to modulate the corresponding parameters of the LCT, and each group of transform parameters corresponds to a group of encrypted signals. Thus, each group of signals is transformed by LCT with different parameters. Finally, chaotic encryption is performed on the LCT domain spectrum of each group of signals, to realize the overall encryption of the speech signal. The experimental results show that the proposed algorithm is extremely sensitive to the keys and has a larger key space. Compared with the original signal, the waveform and LCT domain spectrum of obtained encrypted signal are distributed more uniformly and have less correlation, which can realize the safe transmission of speech signals.

Discrete Convolution Associated with Fractional Cosine and Sine Series

Xiuxiu Gao, Qiang Feng, Yinyin Mei, Yi Xiang

2021, 30(3): 305-310. doi:10.15918/j.jbit1004-0579.2021.040

[Abstract](302) [FullText HTML](115) [PDF 5176KB](28)

Abstract:
Fractional sine series (FRSS) and fractional cosine series (FRCS) are the discrete form of the fractional cosine transform (FRCT) and fractional sine transform (FRST). The recent studies have shown that discrete convolution is widely used in optics, signal processing and applied mathematics. In this paper, firstly, the definitions of fractional sine series (FRSS) and fractional cosine series (FRCS) are presented. Secondly, the discrete convolution operations and convolution theorems for fractional sine and cosine series are given. The relationship of two convolution operations is presented. Lastly, the discrete Young’s type inequality is established. The proposed theory plays an important role in digital filtering and the solution of differential and integral equations.

Key Frame Extraction of Surveillance Video Based on Fractional Fourier Transform

Yunzuo Zhang, Jiayu Zhang, Ran Tao

2021, 30(3): 311-321. doi:10.15918/j.jbit1004-0579.2021.058

[Abstract](467) [FullText HTML](157) [PDF 5261KB](31)

Abstract:
With the vigorous development of national infrastructure construction and public information construction, video surveillance systems have gradually penetrated various fields. The current key frame extraction technology has inadequate target details and inaccurate judgment of local actions. Addressing this problem, a key frame extraction method based on fractional Fourier transform is proposed. This method obtained the phase spectra information of different orders by performing fractional Fourier transform on the surveillance video frames. Next, the method designed an adaptive algorithm based on the golden section point to select the transformation order. Then, the phase spectrum information of two adjacent frames was used to characterize the changes in the global and local motion states of the target. The final step was to extract key frames based on this. Experimental results show that, compared with the previous methods, the key frames extracted by the method proposed in this paper can correctly capture the changes in the global and local motion states of the target.

2021 Vol. 30, No. 3