Short-time Fractional Fourier Transform Research Articles

The Synchrosqueezed Method and Its Theory-Analysis-Based Novel Short-Time Fractional Fourier Transform for Chirp Signals

Time–frequency analysis is an important tool used for the processing and interpretation of non-stationary signals, such as seismic data and remote sensing data. In this paper, based on the novel short-time fractional Fourier transform (STFRFT), a new modified STFRFT is first proposed which can also generalize the properties of the modified short-time Fourier transform (STFT). Then, in the modified STFRFT domain, we derive the instantaneous frequency estimator for the chirp signal and present a new type of synchrosqueezing STFRFT (FRSST). The proposed FRSST presents many results similar to those of the synchrosqueezing STFT (FSST), and it extends the harmonic signal to a chirp signal that offers attractive new features. Furthermore, we provide a detailed analysis of the signal reconstruction, theories, and some properties of the proposed FRSST. Several experiments are conducted, and all of the results illustrate that the proposed FRSST is more effective than the FSST. Finally, based on the linear amplitude modulation and frequency modulation signal, we present a derivation for analyzing the limitations of the FRSST.

Interference Suppression Algorithm Based on Short Time Fractional Fourier Transform.

Narrowband interference and wideband interference are both common jamming signals against synthetic aperture radar, which can degrade the signal severely. To suppress interference effectively, an interference suppression method based on short-time fractional Fourier transform (STFrFT) is proposed. After transforming the signal into the time-frequency domain through STFrFT, an adaptive gain coefficient is determined for the instantaneous frequency spectrum at every certain time. The gain coefficient can be preserved while suppressing the interference. Finally, we obtain the useful signal by inverse STFrFT. The simulation and performance analysis show the effectiveness and validity of the proposed algorithm for measured data.

Multisynchrosqueezing short-time fractional Fourier transform and its application in rolling bearing instantaneous frequency estimation

Multisynchrosqueezing transform (MSST) enhances the time-frequency energy concentration by using iterative reassignment operations in time-frequency analysis (TFA). However, its effectiveness is limited for signals with rapidly changing instantaneous frequency. To address this issue, this paper presents a novel time-frequency representation (TFR) method called multisynchrosqueezing short-time fractional Fourier transform, which offers improved TF concentration for strongly frequency-modulated signals. Firstly, a high-resolution TFR of the signal is obtained by locally optimized short-time fractional Fourier transform (STFrFT). Secondly, iterative synchrosqueezing operations are introduced to further enhance the STFrFT energy concentration, with a termination strategy relying on Rényi entropy proposed to ascertain the optimal number of iterations. Finally, the ideal TFA with high energy concentration is achieved. The proposed method was validated using multi-scene simulated signals and variable-speed bearing signals. The results show that the proposed method exhibits superior time-frequency energy concentration and instantaneous frequency estimation accuracy. The estimation error of the method is consistently at least 40% lower than that of the compared short-time Fourier transform-based methods, as assessed through the evaluation criteria of maximum relative error, mean square error and symmetric mean absolute percentage error.

Phase retrieval from short-time fractional Fourier measurements using alternating direction method of multipliers

The phase retrieval (PR) problem is to reconstruct real/complex functions from the magnitudes of their Fourier/frame measurements in classical computational imaging. In this paper, we consider phase retrieval of complex vectors/images from the magnitudes of short-time fractional Fourier transform (STFrFT). In our setting, the above problem is solved by minimizing a least square ReLu loss function and a novel algorithm by alternating direction method of multipliers (ADMM) is presented. As shown in the numerical simulations on complex signals/images, our proposed PR-ADMM algorithm from STFrFT has a better recovery performance with flexible window functions and appropriate fractional orders. It demonstrates to have satisfactory performance from mixed phaseless STFrFT measurements. Compared with several six other main algorithms, the proposed algorithm explicitly recovers the phase of image with higher the peak signal-to-noise ratio. Meanwhile, the proposed algorithm is robust to noise. These also generalize some of about phase retrievals with Fourier measurements.

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.

A novel solution for improved performance of Time-frequency concentration

To improve the time–frequency (TF) concentration performance of the short-time fractional Fourier transform (STFrFT), and solve the multi-order matching problem of multi-component signals, this paper introduces a new algorithm that is referred to as the Variational mode decomposition-short-time fractional Fourier transform- Synchroextracting transform (VSSTFrFT). This work employs the Variational mode decomposition (VMD) algorithm to decompose multi-component signals into single-component sets, then the STFrFT algorithm matches the optimal rotation order of each component separately to solve the multi-order matching problem. Finally, this paper utilizes the Synchroextracting transform (SET) to extract the TF coefficient of ridgeline position in STFrFT distribution, improving the concentration performance. For different types of signals, the VSSTFrFT algorithm can get a higher TF concentration than the traditional TFA methods. In the application of engineering measured data, the VSSTFrFT algorithm can extract the modalities of the signals and display the frequency curve clearly, which can be used for structural complexity and time-varying characteristics analysis in practical engineering applications.

Sliding Short-Time Fractional Fourier Transform

The short-time fractional Fourier transform (STFRFT) has been shown to be a powerful tool for processing signals whose fractional frequencies vary with time. However, for real-time applications that require recalculating the STFRFT at each or several samples, the existing discrete algorithms are not suitable. To solve this problem, a new sliding algorithm is proposed, termed as the sliding STFRFT. First, the sliding STFRFT algorithm with the sliding step 1 is proposed. Then, it is derived to the circumstance when the sliding step turns to <inline-formula><tex-math notation="LaTeX">$\bm {p\;(p &gt; 1)}$</tex-math></inline-formula>. The proposed sliding STFRFT algorithm directly computes the STFRFT at the time <inline-formula><tex-math notation="LaTeX">$\bm {m+1}$</tex-math></inline-formula> or <inline-formula><tex-math notation="LaTeX">$\bm {m+p}$</tex-math></inline-formula> using the STFRFT output result at the time <inline-formula><tex-math notation="LaTeX">$\bm {m}$</tex-math></inline-formula>, which greatly reduces the computation complexity. The theoretical analysis demonstrates that the proposed algorithm has the lowest computational cost among existing STFRFT algorithms.

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

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.

Fractional frequency band entropy for bearing fault diagnosis under varying speed conditions

Fault diagnosis under varying speed conditions has become a research hotspot in rotating machinery monitoring. Due to the energy concentration property of linear frequency-modulated signals, fractional Fourier transform (FrFT) is helpful to bearing fault detection with varying speed. However, its performance depends on FrFT filter parameters, which are difficult to determine. So fractional frequency band entropy (FrFBE) is proposed to solve this problem, constructing optimized FrFT filter to extract time-varying fault characteristics. First, short-time fractional Fourier transform (STFrFT) is performed on the demodulated signal. Then, FrFBE is developed based on the obtained fractional time-frequency matrix to locate aggregation positions of fault harmonics. Next, a global/local minimum FrFBE criterion is proposed to determine FrFT filtering centers. Finally, the fault characteristics are extracted by computed order tracking (COT) from the filtered results. Simulation and experiment cases show good diagnosis result with different operating conditions, with recognition error kept within 1%.

Novel Method for Non-stationary Signals Via High-Concentration Time–Frequency Analysis Using SSTFrFT

The short-time fractional Fourier transform (STFrFT) is beneficial for addressing non-stationary signals in many application settings. However, the STFrFT algorithm fails to obtain a high time–frequency (TF) concentration because of the uncertainty principle. To resolve these problems, we introduce a new algorithm that is referred to as the SSTFrFT, which is a combination of the synchroextracting transform and STFrFT. The main principle of this algorithm is to establish a synchroextracting operator based on the STFrFT and then to extract the TF coefficient of the ridgeline position in the TF distribution to improve the concentration. Using numerical simulations with two examples (linear frequency modulation signal and nonlinear frequency modulation signal), we illustrate how the algorithm can be useful in improving concentration. The instantaneous frequency estimation and energy distribution description are more accurate than traditional methods, such as the short-time Fourier transform, Wigner Ville distribution, synchrosqueezed transform, and STFrFT. Furthermore, we apply the algorithm to identify the frequency curve generated by the target’s motion from Ice Multiparameter Imaging X-Band radar echo data from the sea clutter background. The test results of the SSTFrFT method that we developed can accurately distinguish moving targets and sea clutter, which suggest the possible utility of this approach for detection and motion characteristics of marine moving targets.

Doppler‐vision

Chinese researchers propose an m-D signal separation system, capable of distinguishing between different people walking. AlexNet was used to test the system, where it performed well with different group sizes, achieving nearly 90% accuracy for differentiating between 10 people: a considerable improvement over conventional radar systems. Radar m-D is a frequency modulation that can be applied over the main Doppler shift, caused by tiny movements by the target. When the target is a human, due to our bipedal stance, our arms, legs and torsos can give rise to unique m-D signatures, allowing for differentiation between humans and animals, or to distinguish between different activities: for example, running, crawling or walking. Incredibly, m-D is so sensitive it can be used to differentiate between different humans, based on idiosyncrasies in body movement. This sensitivity makes m-D visualisation suitable for surveillance use in closed-circuit television cameras or other visual sensors. Authors Tao Shan (left) Xingshuai Qiao (centre) and Ran Tao (right). Spectrograms of two human subjects used in the study. Experimental scene used to record movement of each subject Shan et al. identified that there was a dearth of research around using m-D to distinguish between different people doing the same task or activity, especially with regards to large (∼10 individuals) groups of people. They used the deep convolutional neural network (DCNN), one of the most successful deep learning algorithms for image recognition, to analyse limb movements of targets in order to differentiate them. Use of limb movements presents its own set of problems, however, as the radar cross section of a torso is much larger than that of a limb. The elaborate features associated with limb movement are obscured by echoes from the much bulkier torso, meaning that in order to use limb movement as an indicator, the authors needed to develop a method of improving the resolution of the limb's m-D component. Fortunately for the authors, they had already developed a method of separating signals from m-D radar, using a short-time fractional Fourier transform (STFrFT). Two STFrFTs were used with different orders and window lengths to sparsely characterise echoes from limbs and torsos. This is then combined with the so-called morphological component analysis (MCA), a method allowing separation of features contained in an image when these features present different morphological aspects. This combination results in an optimisation problem which, when solved, allows successful separation of echoes from limbs and torsos. The authors used 10 volunteers of mixed gender and varying heights and weights in their experiment. The volunteers were asked to walk normally in front of the radar equipment and were scanned, separating the signals from limbs and torso as outlined previously. The data was fed into a DCNN, AlexNet, and trained. The DCNN performed well at identifying individuals, reaching a success rate of almost 90% for distinguishing between 10 people, 91.25% for 8 individuals, 92.67% for 6 and 97.5% for 4. The high accuracy, say the authors, can be primarily attributed to the separation of limb and torso signals.

Novel Short-Time Fractional Fourier Transform: Theory, Implementation, and Applications

As a generalization of the classical Fourier transform (FT), the fractional Fourier transform (FRFT) has proven to be a powerful tool for signal processing and analysis. However, it is not suitable for processing signals whose fractional frequencies vary with time due to a lack of time localization information. A simple method to overcome this limitation is the short-time FRFT (STFRFT). There exist several different definitions of the STFRFT in the literature. Unfortunately, these existing definitions do not well generalize the classical result of the conventional short-time FT (STFT), which can be interpreted as a bank of FT-domain filters. The objective of this paper is to propose a novel STFRFT that preserves the properties of the conventional STFT and can be implemented easily in terms of FRFT-domain filter banks. We first present the novel STFRFT and then derive its inverse transform and basic properties. The time-fractional-frequency analysis of this transform is also presented. Moreover, the implementation of the proposed STFRFT is discussed. Finally, we provide several applications for the proposed STFRFT.

Separation of Human Micro-Doppler Signals Based on Short-Time Fractional Fourier Transform

As an important signature associated with human movement, human micro-Doppler (m-D) signature can provide the basis for activity classification. In particular, the m-D signal of limbs can provide a highly distinctive feature for the activity with reduced limbs movement, which can be used to detect people who are carrying weapon or injured. Fully exploiting the elaborate m-D features that correspond to the motion of limbs can improve the classification accuracy of such activities. Therefore, it is significant to separate the limb-swing micro-Doppler signature from the torso signature and process them separately. In this paper, a novel separation method is proposed, which uses the short-time fractional Fourier transform (STFrFT) with different orders and window lengths to sparsely characterize the echoes from limbs and torso, respectively. Then STFrFT based sparse representation is combined with the morphological component analysis (MCA) to realize the m-D signal separation. Simulation and experimental results verify the effectiveness of the proposed algorithm, where the real data of different activities are utilized to demonstrate its adaptability.

Simultaneous OTDR Dynamic Range and Spatial Resolution Enhancement by Digital LFM Pulse and Short-Time FrFT

This paper proposes a novel optical time domain reflectometry (OTDR) method based on the digital linear frequency modulation (LFM) pulse, which can achieve a tradeoff between maximum measurable distance and spatial resolution. Direct modulation and detection are adopted at the transmitting and receiving ends, respectively, which is simple in construction and does not require additional optics. The short-time fractional Fourier transform (STFrFT) is introduced for the signal processing and noise filtering. The theoretical analysis of the working principle confirmed that the spatial resolution is determined by the sweep frequency range of the digital LFM signal rather than the pulse width. The influence of the STFrFT window on the peak sidelobe ratio of the reflection peak is also studied. By combining STFrFT and sidelobe suppression, the dynamic range and spatial resolution can be appreciably enhanced simultaneously. In the demo experiments testing the proposed method on a conventional OTDR development board for comparison, a 7-dB improvement in the dynamic range and an approximately 10-times improvement in the spatial resolution are simultaneously achieved.

A robust high-resolution time–frequency representation based on the local optimization of the short-time fractional Fourier transform

The Locally Optimized Spectrogram (LOS) defines a novel method for obtaining a high-resolution time–frequency (t,f) representation based on the short-time fractional Fourier transform (STFrFT). The key novelty of the LOS is that it automatically determines the locally optimal window parameters and fractional order (angle) for all signal components, leading to a high-resolution and cross-terms free time–frequency representation. This method is suitable for multicomponent and non-stationary signals without a priori signal information. Simulated signals, real biomedical applications, and various measures are used to validate the improved performance of the LOS and compare it with other state-of-the-art methods. The robustness of the LOS is also demonstrated under different signal-to-noise ratio (SNR) conditions. Finally, the relationship between the LOS and other time–frequency distributions (TFDs) is depicted and a recursive formulation is presented and shows the trade-off between the cross-terms suppression and auto-terms resolution.

Parameter Estimation of Wideband Underwater Acoustic Multipath Channels based on Fractional Fourier Transform

Because of relative motion between transmitter and receiver, the wideband underwater acoustic multipath channel can be more accurately described by a multi-scale multi-lag (MSML) model. The signal components received from different paths can be differentiated in terms of Doppler scales, time delays, and amplitudes. Estimation of these parameters is essential for many underwater applications. In this paper, by virtue of the fractional Fourier transform (FrFT), an accurate and efficient method for parameter estimation is proposed. The algorithm proceeds in an iterative manner, returning parameter estimates of the most dominant signal component successively. With the linear frequency modulation signal employed as the probe signal, the estimation of the Doppler scale factor of each dominant component can be converted into a process of searching for the optimal fractional angle/order of the received signal's FrFT. At each iteration, a sub-iteration is contained to adjust the optimal fractional order. The pulse compression techniques for LFM signal, in both time domain and FrFT domain, are utilized to obtain precise parameter estimates. Once the parameters of a multipath are estimated, the corresponding component will be separated from the received signal—through eliminating the resampled, delayed and attenuated version of the original probe signal. The sparsity of the underwater acoustic channel is considered to reduce the amount of calculation. Simulation results confrm that the proposed algorithm outperforms the existing FrFT domain filter and short-time FrFT domain filter in multicomponent separation, and that the proposed parameter estimation scheme surpasses the matching pursuit based method in accuracy.

Feature extraction of SAR scattering centers using M-RANSAC and STFRFT-based algorithm

This paper introduces a modified random sample consensus (M-RANSAC) and short-time fractional Fourier transform (STFRFT)-based algorithm for feature extraction of synthetic aperture radar (SAR) scattering centers. In this algorithm, the range migration curve (RMC) of a scattering center is formulated as a parametric model. By estimating these parameters, the backscattering envelope of scattering center, corresponding to the backscattering variation in synthetic aperture time, is extracted directly from a time-domain range-compressed signal. The estimated parameters can also reconstruct the geographical location and along-track velocity of scattering centers. Thus, even without knowing explicit knowledge of platform velocity and forming a SAR image, this algorithm is capable of realizing feature extraction. To estimate parameters scatter by scatter, M-RANSAC approach is proposed as an implementary method with iterative procedure. In the iterations, fitting precision indicator (FPI) works cooperatively with construction fitness coefficient (CFC) to determine the optimal parameters of different scattering centers. Adapting this method to more general cases, STFRFT is introduced to separate the overlapped trajectories of RMCs of scattering centers. The root mean squared errors (RMSEs) of parameter estimation are close to their Cramer-Rao lower bounds (CRLB). The effectiveness of feature extraction based on the devised algorithm is validated by both simulated and real SAR data.

Detection and Extraction of Target With Micromotion in Spiky Sea Clutter Via Short-Time Fractional Fourier Transform

In order to effectively detect moving targets in heavy sea clutter, the micro-Doppler (m-D) effect is studied and an effective algorithm based on short-time fractional Fourier transform (STFRFT) is proposed for target detection and m-D signal extraction. Firstly, the mathematical model of target with micromotion at sea, including translation and rotation movement, is established, which can be approximated as the sum of linear-frequency-modulated signals within a short time. Then, due to the high-power, time-varying, and target-like properties of sea spikes, which may result in poor detection performance, sea spikes are identified and eliminated before target detection to improve signal-to-clutter ratio (SCR). By taking the absolute amplitude of signals in the best STFRFT domain (STFRFD) as the test statistic, and comparing it with the threshold determined by a constant false alarm rate detector, micromotion target can be declared or not. STFRFT with Gaussian window is employed to provide time-frequency distribution of m-D signals, and the instantaneous frequency of each component can be extracted and estimated precisely by STFRFD filtering. In the end, datasets from the intelligent pixel processing radar with HH and VV polarizations are used to verify the validity of this proposed algorithm. Two shore-based experiments are also conducted using an X-band sea search radar and an S-band sea surveillance radar, respectively. The results demonstrate that the proposed method not only achieves high detection probability in a low-SCR environment but also outperforms the short-time Fourier transform-based method.

Autofocusing of SAR images using STFRFT-based preprocessing

Prominent point processing (PPP) is a very popular synthetic aperture radar (SAR) autofocus algorithm, which can estimate errors caused by any type of unknown motion or other error source and has high accuracy under the assumption that a prominent point scatterer is available within an imaged scene. Proposed is an advanced PPP algorithm with a preprocessing technique based on the short-time fractional Fourier transform (STFRFT) to relax the limitation on the scene content. The performance of the proposed autofocus algorithm is validated using real radar data.

Short-Time Fractional Fourier Transform and Its Applications

The fractional Fourier transform (FRFT) is a potent tool to analyze the chirp signal. However, it fails in locating the fractional Fourier domain (FRFD)-frequency contents which is required in some applications. The short-time fractional Fourier transform (STFRFT) is proposed to solve this problem. It displays the time and FRFD-frequency information jointly in the short-time fractional Fourier domain (STFRFD). Two aspects of its performance are considered: the 2-D resolution and the STFRFD support. The time-FRFD-bandwidth product (TFBP) is defined to measure the resolvable area and the STFRFD support. The optimal STFRFT is obtained with the criteria that maximize the 2-D resolution and minimize the STFRFD support. Its inverse transform, properties and computational complexity are presented. Two applications are discussed: the estimations of the time-of-arrival (TOA) and pulsewidth (PW) of chirp signals, and the STFRFD filtering. Simulations verify the validity of the proposed algorithms.