Short-time Fractional Fourier Research Articles

Synchroextracting Transform Based on the Novel Short-Time Fractional Fourier Transform

As a generalization of the short-time Fourier transform (STFT), the novel short-time fractional Fourier transform (NSTFRFT) has been introduced recently. In order to improve the concentration of the time–frequency representation (TFR) generated by the NSTFRFT, two post-processing time–frequency analysis methods, two synchroextracting transforms based on the NSTFRFT with two different fractional Fourier transform (FRFT) angles, are proposed in this paper. One is achieved via an equation where the instantaneous frequency satisfies the condition where the FRFT angle takes π2, and the other one is obtained using the instantaneous frequency estimator in the case that the FRFT angle takes a value related to the chirp rate of the signal. Although the conditions of the two synchroextracting transforms are different, their implementation can be unified into the same algorithm. The proposed synchroextracting transforms supplement existing post-processing time–frequency analysis methods which are based on the NSTFRFT. Experiments are conducted to verify the performance and superiority of the proposed methods.

ESSENTIAL PROPERTIES RELATED TO SHORT-TIME FRACTIONAL FOURIER TRANSFORM

We start by defining the short-time fractional Fourier transform in this paper, which is a natural generalization of the fractional Fourier transform. We then investigate its essential properties and explore an uncertainty principle related to this proposed transformation.

A novel synchrosqueezing transform associated with linear canonical transform

Synchrosqueezing transforms have aroused great attention for its ability in time–frequency energy rearranging and signal reconstruction, which are post-processing techniques of the time–frequency distribution. However, the time–frequency distributions, such as short-time Fourier transform and short-time fractional Fourier transform, cannot change the shape of the time–frequency distribution. The linear canonical transform (LCT) can simultaneously rotate and scale the time–frequency distribution, which enlarges the distance between different signal components with proper parameters. In this study, a convolution-type short-time LCT is proposed to present the time–frequency distribution of a signal, from which the signal reconstruction formula is given. Its resolutions in time and LCT domains are demonstrated, which helps to select suitable window functions. A fast implementation algorithm for the short-time LCT is provided. Further, the synchrosqueezing LCT (SLCT) transform is designed by performing synchrosqueezing technique on the short-time LCT. The SLCT inherits many properties of the LCT, and the signal reconstruction formula is obtained from the SLCT. Adaptive selections of the parameter matrix of LCT and the length of the window function are introduced, thereby enabling proper compress direction and resolution of the signal. Finally, numerical experiments are presented to verify the efficiency of the SLCT.

The directional short-time fractional Fourier transform of distributions

The directional short-time fractional Fourier transform of distributions

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.

Target ranging via sparse non-uniform short-time fractional Fourier transformation for multi-parameter modulated signal

Multi-parameter modulated signal can improve the anti-interception ability yet makes the range extraction of the target inconvenient. To solve the problem, a novel scaling transformation combined with the non-uniform short-time fractional Fourier transformation is proposed to effectively acquire the target range information. Unlike the traditional methods, the influence of the varied carrier frequency is firstly mitigated by the proposed scaling transformation. Then, the error caused by the period-varying is relieved by the sampling conversion in the digital domain. Further, the corresponding dictionary is constructed to accomplish the ranging information searching and extracting. Simulation results show that the proposed method can mitigate the impact of multiple varied parameters on signal processing. The detection error could be reduced by 67.9% after using the proposed method.

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.

DRFM-Based Repeater Jamming Reconstruction and Cancellation Method with Accurate Edge Detection

Digital radio frequency memory (DRFM) based repeater jamming can create false targets, which can lead to a loss of situational awareness, misidentification of targets, and decreased overall performance of the radar system. Traditional jamming suppression methods do not give due importance to the accurate estimation of the jamming edge, resulting in jamming residual and poor anti-jamming performance. To tackle this issue, this paper explores the reason and impact of inaccurate jamming edge estimation and proposes a DRFM-based repeater jamming reconstruction and cancellation method with accurate edge detection. In the proposed method, firstly, multiple jamming parameters are obtained by computing the short-time fractional Fourier transformation (STFRFT) spectrogram of the received signal. To avoid jamming residue, the proposed method estimates the accurate jamming edges by the joint use of the difference of box (DOB) filters and time domain deconvolution (TDDC) curves. Numerical simulations and experiments are conducted to evaluate the algorithm’s effectiveness in countering smeared spectrum (SMSP) and interrupted sampling repeater jamming (ISRJ). The results demonstrate its superior jamming reconstruction and suppression performance than other methods.

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.

Seismic spectral decomposition using short-time fractional Fourier transform spectrograms

A time-frequency method with high resolution is introduced for seismic spectral decomposition. The classical short-time Fourier transform spectrogram has low resolution, limiting its usefulness in application such as stratigraphic visualization and gas reservoir detection for thin beds. This work expands the traditional short-time Fourier transform spectrogram to the fractional domain, leading to the proposed short-time fractional Fourier transform spectrogram (FrFTSG). The short-time Fourier transform spectrogram can be calculated using the 2-D Fourier transform of the product between the ambiguity functions (AF) of the signal and the window function, and using the fractional Fourier transform (FrFT) as the rotation operator of the window function, the window function could be rotated to the direction with the auto-term of the signal in the ambiguity domain. The FrFTSG could be obtained by the 2-D Fourier transform of the product between the ambiguity functions of the rotated window function and the signal. Analyses of synthetic test model and actual seismic data show that the method proposed in this study can characterize the signal with a higher accuracy. This allows more accurate characterization of reservoir location. The method proposed in this paper provides a novel technique for seismic spectral decomposition.

Quaternionic short-time fractional Fourier transform

Quaternionic short-time fractional Fourier transform

Efficient Rotational Angular Velocity Estimation of Rotor Target via Modified Short-Time Fractional Fourier Transform

As a special micro-motion feature of rotor target, rotational angular velocity can provide a discriminant basis for target classification and recognition. In this paper, the authors focus on an efficient rotational angular velocity estimation method of the rotor target is based on the combination of the time–frequency analysis algorithm and Hough transform. In order to avoid the problems of low time–frequency resolution and cross-term interference in short-time Fourier transform and Wigner–Ville distribution algorithm, a modified short-time fractional Fourier transform (M-STFRFT) is proposed to obtain the time-FRFT domain (FRFD)-frequency spectrum with the highest time–FRFD–frequency resolution. In particular, an orthogonal matching pursuit (OMP)-based algorithm is proposed to reduce the computational complexity when estimating the matched transform order in the proposed M-STFRFT algorithm. Firstly, partial transform order candidates are selected randomly from the complete candidates. Then, a partial entropy vector corresponding to partial transform order candidates is calculated from the FRFT results and utilized to reconstruct the complete entropy vector via the OMP algorithm, and the matched transform order can be estimated by searching minimum entropy. Based on the estimated matched transform order, STFRFT is performed to obtain the time–FRFD–frequency spectrum. Moreover, Hough transform is employed to obtain the energy accumulation spectrum, and the micro-Doppler parameter of rotational angular velocity can be estimated by searching the peak value from the energy accumulation spectrum. Both simulated data and measured data collected by frequency modulated continuous wave radar validate the effectiveness of the proposed algorithm.

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%.

Adaptive Generalised Fractional Spectrogram and Its Applications

The generalised time–frequency transform (GTFT) is a powerful tool to analyse a large variety of frequency-modulated signals. However, it is not adequate to represent the variation of frequency over time for non-stationary signals. To solve this problem, short-time GTFT and short-time GTFT-based adaptive generalised fractional spectrogram (AGFS) are proposed. The AGFS is capable of providing a high concentration, high resolution, cross-term-free time–frequency distribution for analysing multicomponent frequency-modulated signals. It is also a generalisation of the short-time Fourier transform-based spectrogram and the short-time fractional Fourier transform-based spectrogram. The uncertainty principle for short-time GTFT is derived, and its time-bandwidth product is compared with other time–frequency distributions. With the help of simulated data examples, the effectiveness of AGFS is demonstrated in comparison with other time–frequency distributions for resolving and extracting individual components of multicomponent quadratic chirps. Robustness of AGFS is demonstrated under different input signal-to-noise ratio conditions. A local spectrogram optimisation technique is adopted for AGFS to represent simulated and real chirp signals. Finally, an application of the AGFS is presented to resolve multiple ground moving targets in synthetic aperture radar data and obtain its focused synthetic aperture radar image.

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.