Fractional Fourier Transform Research Articles

Joint Clutter Suppression and Moving Target Indication in 2-D Azimuth Rotated Time Domain for Single-Channel Bistatic SAR

Moving target indication (MTI) in bistatic synthetic aperture radar (BiSAR) is a promising task in both civilian and military fields. However, it suffers severe challenges under the influence of clutter. MTI in BiSAR mainly faces three challenges: 1) the range cell migration (RCM) of BiSAR is larger than that of monostatic synthetic aperture radar; 2) the clutter range-Doppler spectrum is severely extended; and 3) the clutter characteristic is closely related to geometrical configuration, with nonhomogeneity and nonstationarity. To solve these problems, based on single-channel BiSAR, a joint clutter suppression and MTI method is proposed. First, to ensure that the energy of an arbitrary target is concentrated in one range cell, RCM correction (RCMC) is completed by the first-order keystone transform (KT) and high-order RCMC. Then, an important step named increased dimension rotation (IDR) is applied, which mainly consists of two stages. One is increased dimension processing, which introduces a 2-D azimuth rotated time domain (ARTD). The other is rotation processing, which makes signals in one range cell rotated to 2-D ARTD. After that, the distribution characteristic between the moving target and stationary clutter in 2-D ARTD is analyzed. Next, an optimal filter for clutter suppression is designed according to the characteristics of signal distribution in 2-D ARTD. Furthermore, the corresponding inverse IDR processing and fractional Fourier transform (FrFT) are performed, and finally, MTI can be realized accurately. Generally, the proposed method has good robustness and low system complexity, and its effectiveness is proven by numerical simulations.

Airborne SAR Suppression of Blanket Jamming Based on Second Order Blind Identification and Fractional Order Fourier Transform

Airborne SAR Suppression of Blanket Jamming Based on Second Order Blind Identification and Fractional Order Fourier Transform

Synchrosqueezing-Based Short-Time Fractional Fourier Transform

A novel synchrosqueezing transform to the time-frequency representation based on short-time fractional Fourier transform for strong time-varying signals is proposed in this paper. To finely describe the local characteristics of nonstationary signal in the short-time fractional Fourier transform domain, a new instantaneous frequency estimator, a new instantaneous fractional frequency estimator and the corresponding energy reassignment operators are given by time-fractional frequency basis. The discrete implementation and signal reconstruction of the proposed method are investigated. Furthermore, the calculation of the optimal fractional order, computational complexity and limitation of the proposed method are discussed. Various strong time-varying signal experiments and time-frequency spectrum energy attenuation gradient extraction experiments of real synthetic aperture radar (SAR) images are conducted to verify the effectiveness and superiority of the proposed method over some advanced time-frequency analysis methods in energy concentration, time-frequency localization, signal reconstruction and clutter suppression.

3D palatal rugae recognition based on Fractional Fourier Transform

Following a major disaster, most of the biological characteristics of the victims are typically destroyed, which increases the difficulty of forensic identification. However, palatal rugae cannot be easily destroyed owing to its unique location characteristics, owing to which it has become a new type of biological feature used in forensic identification under special environments. This study proposes a new method of 3D palatal rugae recognition which uses fractional Fourier transform (FRFT). The isometric slicing operation is first used to simplify the complexity of the algorithm and improve the efficiency of the 3D data storage and expression. FRFT is the used to extract the fractional domain features of the profile images obtained by slicing the 3D palatal rugae, and the support vector machine recursive feature elimination (SVM-RFE) method is used for feature selection in the fractional domain. The use of FRFT increases the number of the palatal rugae features extracted from the time-frequency domain, and can improve the effectiveness of feature extraction. The recognition accuracy rate of the proposed 3D palatal rugae recognition system can reach up to 97.8%. The proposed method meets the requirements of 3D palatal rugae recognition, and can help in performing rapid forensic identification. The method proposed in this study is useful in reducing the dependence on human experts and promoting the digitization of a forensic identification system based on palatal rugae.

Analysis of photoplethysmogram signal to estimate heart rate during physical activity using fractional fourier transform – A sampling frequency independent and reference signal-less method

Background and objectiveAcquiring accurate and reliable health information using a PPG signal in wearable devices requires suppressing motion artifacts. This paper presents a method based on the Fractional Fourier transform (FrFT) to effectively suppress the motion artifacts in a Photoplethysmogram (PPG) signal for an accurate estimation of heart rate (HR). MethodsBy analyzing various PPG signals recorded under various physiological conditions and sampling frequencies, the proposed work determines an optimal value of the fractional order of the proposed FrFT. The proposed FrFT-based algorithm separates the motion artifacts component from the acquired PPG signal. Finally, the HR estimation accuracy during the strong motion artifact-affected windows is improved using a post-processing technique. The efficacy of the proposed method is evaluated by computing the root mean square error (RMSE). ResultsThe performance of the proposed algorithm is compared with methods in recent studies using test and training datasets from the IEEE Signal Processing Cup (SPC). The proposed method provides the mean absolute error of 1.88 beats per minute (BPM) on all twenty-three recordings. ConclusionsThe proposed method uses the Fourier method in the fractional domain. A noisy signal is rotated into an intermediate plane between the time and frequency domains to separate the signal from the noise. The algorithm incorporates FrFT analysis to suppress motion artifacts from PPG signals to estimate HR accurately. Further, a post-processing step is used to track the HR for accurate and reliable HR estimation. The proposed FrFT-based algorithm doesn't require additional reference accelerometers or hardware to estimate HR in real-time. The noise and signal separation is optimum for a fractional order (a) value in the vicinity of 0.6. The optimized value of fractional order is constant irrespective of the physical activity and sampling frequency.

Fractional Fourier Transform and Distributions in the Ray Space: Application for the Analysis of Radio Occultation Data

The concept of the phase space plays a key role in the analysis of oscillating signals. For a 1-D signal, the coordinates of the 2-D phase space are the observation time and the instant frequency. For measurements of propagating wave fields, the time and instant frequency are linked to the spatial location and wave normal, defining a ray. In this case, the phase space is also termed the ray space. Distributions in the ray space find important applications in the analysis of radio occultation (RO) data because they allow the separation of interfering rays in multipath zones. Examples of such distributions are the spectrogram, Wigner distribution function (WDF), and Kirkwood distribution function (KDF). In this study, we analyze the application of the fractional Fourier transform (FrFT) to the construction of distributions in the ray space. The FrFT implements the phase space rotation. We consider the KDF averaged over the rotation group and demonstrate that it equals the WDF convolved with a smoothing kernel. We give examples of processing simple test signals, for which we evaluate the FrFT, KDF, WDF, and smoothed WDF (SWDF). We analyze the advantages of the SWDF and show examples of its application to the analysis of real RO observations.

Direction of arrival estimation based on slope fitting of wideband array signal in fractional Fourier transform domain

AbstractTraditional direction of arrival (DOA) estimation methods of wideband linear frequency modulated (LFM) signal based on the fractional Fourier transform (FRFT) highly depend on the peak value of LFM signal in the FRFT domain. However, the peak value can be affected by various factors, such as a low signal‐to‐noise ratio (SNR), a small number of snapshots, or the large incident angle of signals, which can lead to a decrease in DOA estimation accuracy. To tackle this problem, based on the energy‐concentrated property of LFM signal in the FRFT domain, the DOA estimation is accomplished by adopting the approach of slope fitting for the wideband array signal in the FRFT domain in this study. The method uses the correspondence between the peak position in the FRFT domain and the delay of the received signal on different uniform linear array elements. The least square method based on threshold outlier detection is used for slope fitting. Compared to conventional methods, the method of this study has high accuracy and relatively low complexity in the case of low SNR or large incident angle of the signal without relying on the peak value. The effectiveness of this method is verified by theoretical analysis and Monte‐Carlo simulations. The proposed method provides a new path for DOA estimation of LFM signal.

Hyperspectral anomaly detection with local correlation fractional Fourier transform and vector pulse coupled neural network

Hyperspectral anomaly detection has attracted widespread attention because of its significant practical application value. However, the existence of noise and the lack of target spatial shape and texture features in hyperspectral image (HSI) limit the performance of hyperspectral anomaly detection. Here, we present a novel hyperspectral anomaly detection method by local correlation fractional Fourier transform (FrFT) and vector pulse coupled neural network (VPCNN). First, combined with the correlation between local pixels, the HSI is transformed into the fractional Fourier domain through the proposed local correlation FrFT, which can suppress noise and background while preserving anomaly targets features in the HSI. Second, the VPCNN, which can effectively deal with the lack of spatial shape and texture features of the target, is proposed to segment the transformed HSI into different regions to obtain the segmentation image. Finally, anomaly detection result is extracted from the segmentation image. Experimental results on four real-world datasets show that the proposed method has better superiority and advancement compared with other state-of-the-art alternative methods. We hope that our research will inspire the application of segmentation methods in related fields.

Fractional Fourier transforms of circular cosine-hyperbolic-Gaussian beams

The propagation properties of a circular cosh-Gaussian beam (CiChGB) in a fractional Fourier transform (FRFT) optical system are investigated theoretically. The analytical expressions for a CiChGB propagating through apertured and unapertured FRFT systems are derived based on the Collins formula and the expansion of the hard-aperture function into a finite sum of Gaussian functions. From the obtained expressions, the evolution of the intensity distribution at the output plane is analysed with illustrative numerical examples. It is shown that the intensity distribution of the CiChGB propagating in FRFT is closely dependent on the decentred beam parameter and the aperture size besides the fractional order of the FRFT system. The results obtained may be beneficial to applications in laser beam shaping, optical trapping, and micro-particle manipulation.

Approximation Theorems Associated with Multidimensional Fractional Fourier Transform and Applications in Laplace and Heat Equations

In this paper, we establish two approximation theorems for the multidimensional fractional Fourier transform via appropriate convolutions. As applications, we study the boundary and initial problems of the Laplace and heat equations with chirp functions. Furthermore, we obtain the general Heisenberg inequality with respect to the multidimensional fractional Fourier transform.

Research on Speech Emotion Recognition Based on the Fractional Fourier Transform

Speech emotion recognition is an important part of human–computer interaction, and the use of computers to analyze emotions and extract speech emotion features that can achieve high recognition rates is an important step. We applied the Fractional Fourier Transform (FrFT), and then constructed it to extract MFCC and combined it with a deep learning method for speech emotion recognition. Since the performance of FrFT depends on the transform order p, we utilized an ambiguity function to determine the optimal order for each frame of speech. The MFCC was extracted under the optimal order of FrFT for each frame of speech. Finally, combining the deep learning network LSTM for speech emotion recognition. Our experiment was conducted on the RAVDESS, and detailed confusion matrices and accuracy were given for analysis. The MFCC extracted using FrFT was shown to have better performance than ordinal FT, and the proposed model achieved a weighting accuracy of 79.86%.

Photonic Fractional Fourier Transformation Based on Discrete-Frequency Processing

Photonic fractional Fourier transformation (FRFT) based on discrete-frequency processing is proposed to overcome the limitations of processing latency, resolution, and tunability. Typical applications of FRFT, such as de-chirp processing, separation of two linear frequency modulated waveforms (LFMWs) in fractional domain and filtering of a LFMW from a noisy background, are experimentally demonstrated. The operation bandwidth and spectral resolution are 4 GHz and 60 MHz, respectively. Both the phase and amplitude can be measured by the proposal without relying on complex algorithms, while the processing latency is close to the minimum delay required in the FRFT. All optical devices employed in our system shows great potential in integration.

Generalized pseudo-differential operators associated with symbol classes involving fractional Fourier transform

Generalized pseudo-differential operators \(A(x,\Delta'_x)\) and \(\mathcal{A}(x,\Delta'_x)\) involving fractional Fourier transform (FrFT) associated with symbol-classes are defined and respective symbol classes are introduced. Product and commutators of the generalized pseudo-differential operators are investigated.

Detection and Relative Quantification of Neodymium in Sillai Patti Carbonatite Using Decision Tree Classification of the Hyperspectral Data.

A recent increase in the importance of Rare Earth Elements (REEs), proportional to advancements in modern technology, green energy, and defense, has urged researchers to look for more sophisticated and efficient exploration methods for their host rocks, such as carbonatites. Hyperspectral remote sensing has long been recognized as having great potential to identify the REEs based on their sharp and distinctive absorption features in the visible near-infrared (VNIR) and shortwave infrared (SWIR) electromagnetic spectral profiles. For instance, neodymium (Nd), one of the most abundant Light Rare Earth Elements (LREEs), has among the most distinctive absorption features of REEs in the VNIR part of the electromagnetic spectrum. Centered at ~580, ~745, ~810, and ~870 nm in the VNIR, the positions of these absorption features have been proved to be independent of the mineralogy that hosts Nd, and the features can be observed in samples as low in Nd as 1000 ppm. In this study, a neodymium index (NI) is proposed based on the 810 nm absorption feature and tested on the hyperspectral images of the Sillai Patai carbonatite samples to identify Nd pixels and to decipher the relative concentration of Nd in the samples based on the depth of the absorption feature. A preliminary spectral study of the carbonatite samples was carried out using a spectroradiometer to determine the presence of Nd in the samples. Only two of the absorption features of Nd, centered at ~745 and ~810 nm, are prominent in the Nd-rich samples. The other absorption features are either weak or suppressed by the featureless spectra of the associated minerals. Similar absorption features are found in the VNIR and SWIR images of the rock samples captured by the laboratory-based hyperspectral cameras that are processed through Minimum Noise Fraction (MNF) and Fast Fourier Transform (FFT) to filter the signal and noise from the reflectance data. An RGB false-color composite of continuum-removed VNIR reflectance bands covering wavelengths of 587.5, 747.91, and 810.25 nm efficiently displayed the spatial distribution of Nd-rich hotspots in the hyperspectral image. The depth of the 810 nm absorption feature, which corresponds to the concentration of Nd in a pixel, is comparatively greater in these zones and is quantified using the proposed NI such that the deeper the absorption feature, the higher the NI. To quantify the Nd-rich pixels in the continuum-removed VNIR images, different threshold values of NI are introduced into a decision tree classifier which generates the number of pixels in each class. The strength of the proposed NI coupled with the decision tree classifier is further supported by the accuracy assessment of the classified images generating the Kappa coefficient of 0.82. Comparing the results of the remote sensing data obtained in this study with some of the previously published studies suggests that the Sillai Patti carbonatite is rich in Nd and associated REEs, with some parts of the samples as high in Nd concentration as 1000 ppm.

An Improved Technique for Single-Channel Video-SAR Based on Fractional Fourier Transform

In the field of synthetic aperture radar (SAR), a lot of researchers have tried to study an improved application beyond the conventional two-dimensional image. Video-SAR (ViSAR) is one of the hot issue in the SAR application. The main purpose of ViSAR is to monitor various targets in the area of interest in time order. It can offer video images to detect not only the fixed target but also the moving target. For the signal processing of ViSAR, this article introduces the improved performance by the Doppler shifting technique with fractional Fourier transform. And also, it shows that the proposed method can separate the unexpected signal of the moving target within each frame of single-channel ViSAR. Furthermore, the overall algorithm can increase efficiency and reliability in ViSAR processing. To verify the improvement of our method, we carry out the practical experiments by an X-band chirp pulse SAR system, mounted on an airplane.

Machine learning-based fault diagnosis for research nuclear reactor medium voltage power cables in fraction Fourier domain

Fault diagnosis of Medium Voltage power Cables (MVCs) research nuclear reactor, incredibly inaccessible/remote ones, has to be carefully identified, located, and fixed within a short time. Therefore, this paper proposes a perfect simultaneous fault diagnosis scheme based on Multiclass Support-Vector Machine (MCSVM) in the fractional Fourier domain. First, the three-phase sending currents are simulated under different conditions then their features are extracted using Discrete Fractional Fourier Transform (DFRFT). Afterward, the features reduction process occurs via the Singular Value Decomposition (SVD) approach. MCSVM scheme is used to diagnose faults (i.e., discover, categorize, and trace) using reduced features outcome from DFRFT and SVD stages. Alternating Transient Program/Electromagnetic Transient Program (ATP/EMTP) simulations have been carried out for 22 kV unreachable MVC. Different kernels of SVM, i.e., linear, quadratic, or polynomial, and diverse factors of DFRFT, i.e., α, are investigated in simulations to obtain the optimum performance (i.e., best α and kernel pair). Hence, performance analysis of the proposed diagnosis method under different conditions (i.e., various fault resistances, locations, and inception angles) concluded two highest accuracy and lowest time settings, which were found at α = 0.5 (for both) quadratic kernel, and linear kernel, respectively. Moreover, the linear kernel achieves 99.8% accuracy rate, the lowest execution time (10 ms), and fault tracing error rate of 0.525789%, which is proper for real-time applications. Besides, our proposed method is more reliable and accurate against variable operating conditions (fault resistances, distances, and inception angles), leading to more reliable power production systems.

Low-Complexity and Highly-Robust Chromatic Dispersion Estimation for Faster-than-Nyquist Coherent Optical Systems

Faster-than-Nyquist (FTN) coherent optical transmission technology is considered to be an outstanding solution to achieve higher spectral efficiency (SE), larger capacity, and greater achievable transmission by using advanced modulation formats in concert with highly efficient digital signal processing (DSP) to estimate and compensate various impairments. However, severe inter-symbol interference (ISI) caused by tight FTN pulse shaping will lead to intractable chromatic dispersion (CD) estimation problems, as existing conventional methods are completely ineffective or exhibit unaffordable computational complexity (CC). In this paper, we propose a low-complexity and highly robust scheme that could realize accurate and reliable CD estimation (CDE) based on a designed training sequence (TS) in the first stage and an optimized fractional Fourier transform (FrFT) in the second stage. The training sequence with the designed structure helps us to estimate CD roughly but reliably, and it further facilitates the FrFT in the second stage to achieve accurate CDE within a narrowed searching range; it thereby results in very low CC. Comprehensive simulation results of triple-carrier 64-GBaud FTN dual-polarization 16-ary quadrature amplitude modulation (DP-16QAM) systems demonstrate that, with only overall 3% computational complexity compared with conventional blind CDE methods, the proposed scheme exhibits a CDE accuracy better than 65 ps/nm even under an acceleration factor as low as 0.85. In addition, 60-GBaud FTN DP quadrature phase shift keying (DP-QPSK)/16QAM transmission experiments are carried out, and the results show that the CDE error is less than 70 ps/nm. The advantages of the proposed scheme make it a preferable candidate for CDE in practical FTN coherent optical systems.

Fractional Fourier Transform and Ulam Stability of Fractional Differential Equation with Fractional Caputo-Type Derivative

In this paper, we study the Ulam-Hyers-Mittag-Leffler stability for a linear fractional order differential equation with a fractional Caputo-type derivative using the fractional Fourier transform. Finally, we provide an enumeration of the chemical reactions of the differential equation.

Fractional Fourier Transform, Signal Processing and Uncertainty Principles

Fractional Fourier Transform, Signal Processing and Uncertainty Principles

Superpositions of coherent states determined by Gauss sums

We describe a family of quantum states of the Schr\"odinger cat type as superpositions of the harmonic oscillator coherent states with coefficients defined by the quadratic Gauss sums. These states emerge as eigenfunctions of the lowering operators obtained after canonical transformations of the Heisenberg-Weyl algebra associated with the ordinary and fractional Fourier transformation. The first member of this family is given by the well known Yurke-Stoler coherent state.