Fractional Fourier Transform Domain Research Articles

Enhanced SAR Compression through Multi-Look Doppler Compensation and Auto-Focusing Technique.

This paper presents a simple and streamlined compensation technique for improving the quality of synthetic aperture radar (SAR) images based on the Range Doppler Algorithm (RDA). Incorrect Doppler estimation in the space orbit, caused by unexpected radar motion errors, orbit mismatches, and other factors, can significantly degrade SAR image quality. These inaccuracies result in mismatches between the azimuth-matched filter and the received Doppler chirp signal. To address this issue, we propose a Doppler estimation method that leverages the Fractional Fourier Transform (FrFT) and cross-correlation techniques. The received signals are compared with the azimuth-matched filter based on the rotation angle in the FrFT domain, and the Doppler centroid is adjusted to achieve the optimal alignment. This process ensures high correlation values and enhanced resolution in the final SAR image. The efficacy of the proposed technique is validated through experiments using real spaceborne SAR data from the practical satellite. The results demonstrate significant improvements in image quality and resolution compared to conventional algorithms, highlighting the advantages of our approach for various remote sensing applications.

High-resolution multicomponent LFM parameter estimation based on deep learning

This paper addresses the complex challenge of parameter estimation in multi-component Linear Frequency Modulation (LFM) signals by introducing an innovative approach to high-resolution Fractional Fourier Transform (FrFT) parameter estimation, facilitated by convolutional neural networks. Initially, it analyzes the issues of peak shifts and the masking of weaker components due to spectral overlap in the FrFT domain of multi-component LFM signals. Convolutional neural networks are then employed to train and achieve high-resolution representations of FrFT parameters. Specifically, convolutional modules with residual structures are utilized to learn coarse features, while a weighted attention mechanism refines independent features across both channel and spatial dimensions. This approach effectively addresses the challenges posed by spectral peak overlap and frequency shifts in multi-component LFM signals, thereby enhancing the quality of high-resolution parameter estimation. Experimental results demonstrate that the proposed method significantly outperforms traditional methods in processing multi-component LFM signals. Moreover, it exhibits robust detection capabilities for both weak and compact components, thereby underscoring its potential applicability in the field of complex signal processing.

Enhanced target detection using fractional Fourier transform features with threshold‐modified normalization

AbstractFeature extraction from the normalized transformation domain is a key technique in target detection. Traditional normalization approaches assume that matrix elements follow a normal distribution, but any deviations from this assumption can lead to significant systematic errors. This article presents a novel method that modifies the normalization process in the fractional Fourier transform (FRFT) domain by incorporating a threshold mechanism to counteract the effects of non‐normal distributions. Three modified FRFT features are then extracted from this modified FRFT domain. Furthermore, a target detection method that utilizes these three adjusted features is proposed. Experimental results based on measured data indicate that the modified FRFT features exhibit superior classification capabilities for sea clutter and targets compared to the original ones. Additionally, the experiments also demonstrate that under the same conditions, the proposed detection method outperforms traditional FRFT feature detector and the tri‐feature based detector.

Denoising sparker seismic data with Deep BiLSTM in fractional Fourier transform

Sparkers are widely-used sources of high-resolution images for shallow sub-bottom seismic monitoring. Sparker seismic data contain high random noise levels caused by the high voltage and electrical current required to power the sparker source. Random noise is distributed over the entire frequency and wavenumber, and is thereby difficult to suppress. Numerous studies have been conducted to develop random noise suppression by using convolutional neural networks (CNNs). However, because CNNs learn common characteristics of random noise without considering the sequence while training, learning the sequence relationship between useful signal and random noise in time-series data is difficult. As an alternative, we proposed a recurrent neural network (RNN)-based denoising method that considers seismic data sequences. Effective denoising of random noise using RNNs requires features that distinguish between useful signals and random noise during network training. We proposed a denoising method for sparker seismic data that uses bidirectional long short-term memory RNN in fractional Fourier transform (FrFT). The time-frequency distribution in the FrFT domain was utilized for network training, and useful signals were effectively separated from random noise. The proposed method was applied to multi-channel sparker seismic data acquired in Yeongil, Pohang (offshore eastern Korea) by the Korea Maritime & Ocean University. Numerical results confirmed that the proposed method effectively suppressed random noise while maintaining a useful signal in sparker seismic data.

Single-frame reconstruction by fractional Fourier-transform domain filtering in off-axis digital holographic microscopy

We propose a single-frame zero-order-eliminated reconstruction method by fractional Fourier transform filtering for an off-axis digital hologram. The filtering in the fractional Fourier transform domain of the hologram can effectively improve the reconstruction resolution, but it is required to remove its zero-order term. With the zero-order-term elimination of the Laplacian hologram, the higher reconstruction resolution of a single-frame hologram is achieved by zero-padding the hologram and choosing the optimal option of the fractional-order number. The results demonstrate that the resolutions of reconstructed amplitude and phase images are obviously improved. It will have a promising application in real-time imaging for biological cells and moving objects.

Fractional Fourier Transform Receiver for Modulated Chirp Waveforms

A novel receiver structure for the reception of linearly frequency modulated (LFM) chirps carrying additional narrowband (phase) modulation is proposed. A linear relation between the time delay and the beat frequency shift of the target response in stretch processing is exploited to estimate the target range via correlation of the received signal with the replica in a Fractional Fourier Transform (FrFT) domain. According to numerical simulations, the proposed FrFT receiver demonstrates improved performance and computational efficiency over the state-of-the-art solutions for the moderate-to-large bandwidth of the information-carrying modulation. The receiver was integrated into the waveform-agile radar polarimetric agile radar in S band (PARSAX) and its performance has also been verified experimentally.

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.

Security-enhanced multiple-image encryption based on quick response codes and modified double random phase encoding in the fractional Fourier transform domain.

A security-enhanced multiple-image encryption method is proposed based on quick response (QR) codes and modified double random phase encoding (DRPE) in the fractional Fourier transform (FrFT) domain in this paper, where each plaintext is first converted into QR code, and then each QR code is employed to generate the corresponding binary key for decryption with the help of random binary plaintext (RBP). Subsequently, the used RBP is encrypted into noise-like ciphertext by using the modified DRPE in the FrFT domain. In the modified DRPE method, the first random phase mask is activated by the initial FrFT with chaotic phase, and the wavelength of light and the fractional orders as well as the focal lengths of lenses are all used as digital keys to expand the key space. Moreover, the sensitivities of these digital keys are extremely high because the digital keys are closely mapped with the initial values of the chaotic system in the encryption process, which contributes to an extremely high security of the multiple-image encryption method. Furthermore, the high feasibility and strong robustness of the proposed security-enhanced multiple-image encryption method are also demonstrated by using computational simulations.

Orthogonal projection constant false alarm rate algorithm in fractional domain for small surface targets

In this article, a constant false alarm rate (CFAR) detection algorithm based on the fractional Fourier transform (FRFT) and orthogonal projection (OP) is proposed. With a low signal-to-clutter ratio (SCR), the radar returns of sea-surface small targets are often submerged by sea clutter. The FRFT can accumulate target energy in the optimal order. Meanwhile, the OP suppresses the sea clutter in a projection space in the FRFT domain, thus enhancing the output SCR. The FRFT and the OP are in conjunction with the CFAR to realise target detection, which is referred to as the FRFT-OP-CFAR. For the FRFT-OP-CFAR detector, the most important is the selection of the optimal order p and the calculation of the OP operator ψ x ⊥ . The experimental results show that the proposed FRFT-OP-CFAR has a better clutter suppression and detection performance than some detectors in the authors’ comparison.

Automatic sleep stage classification: A light and efficient deep neural network model based on time, frequency and fractional Fourier transform domain features

This work proposed a novel method for automatic sleep stage classification based on the time, frequency, and fractional Fourier transform (FRFT) domain features extracted from a single-channel electroencephalogram (EEG). Bidirectional long short-term memory was applied to the proposed model to train it to learn the sleep stage transition rules according to the American Academy of Sleep Medicine's manual for automatic sleep stage classification. Results indicated that the features extracted from the fractional Fourier-transformed single-channel EEG may improve the performance of sleep stage classification. For the Fpz-Cz EEG of Sleep-EDF with 30 s epochs, the overall accuracy of the model increased by circa 1% with the help of the FRFT domain features and even reached 81.6%. This work thus made the application of FRFT to automatic sleep stage classification possible. The parameters of the proposed model measured 0.31 MB, which are 5% of those of DeepSleepNet, but its performance is similar to that of DeepSleepNet. Hence, the proposed model is a light and efficient model based on deep neural networks, which also has a prospect for on-device machine learning.

Opto-Video Encryption Based on Logistic Adjusted Sine map in FrFT

In the last few years, videos became the most common form of information transmitted over the internet, and a lot of the traffic is confidential and must be protected and delivered safely to its intended users. This introduces the challenges of presenting encryption systems that can encode videos securely and efficiently at the same time. This paper presents an efficient opto-video encryption system using Logistic Adjusted Sine map (LASM) in the Fractional Fourier Transform (FrFT). In the presented opto-video LASM-based FrFT scheme, the encoded video is split into distinct frames and transformed into optical signals utilizing an optical supply. Each of the developed optical video frames is ciphered by utilizing the LASM in optical FrFT system using two-phase modulation forms on the video frame, the first in the time-domain and the second in the FrFT domain. In the end, the ciphervideo frame is spotted utilizing a CCD digital camera and transformed into a digital structure that can be managed using a computer. We test the proposed opto-video LASM-based FrFT scheme using various security tools. The outcomes demonstrate that the presented scheme can effectively encrypt and decrypt video signals. In addition, it encrypts videos with a high level of encryption quality without sacrificing its resistance to noise immunity. Finally, the test outcomes demonstrate that the presented scheme is immune to known attacks.

Theoretical and experimental study of hybrid optical computing engine for arbitrary-order FRFT.

Fractional Fourier transform (FRFT) is the generalization of Fourier transform. It provides many significant advantages, such as fractional order as the new degree of freedom and high efficiency and great performance for non-stationary signal analysis/processing, that other operations including Fourier transform cannot. Here, we report a hybrid optical system for computation of arbitrary-order FRFT of temporal signals. In experiment, the fractional-domain information of input temporal signals could be directly acquired by detector. In addition, the optical computing results are in good agreement with numerical results. Then we apply the optical computing engine to demodulation of chirp spread spectrum signals. Using sub-Nyquist sampling, the proposed technology could greatly save the number of measurements in demodulation. The compression ratio could be as low as 0.4%, because of the high compression performance of chirp signals in FRFT domain. As a result, the proposed technology has unique advantages in analysis and information extraction for non-stationary signals, especially for chirp-like signals, and may become a powerful optical time-frequency analysis tool for temporal signals.

Asymmetric multiparameter encryption of hyperspectral images based on hybrid chaotic mapping and an equal modulus decomposition tree.

We present an asymmetric encryption scheme for hyperspectral images using hybrid chaotic maps (HCMs) and an equal modulus decomposition tree (EMDT) structure in a discrete multiple-parameter fractional Fourier transform (dmpFrFT) domain. The original hyperspectral image was scrambled by an HCM and then encrypted into asymmetric ciphertext using the EMDT. In the EMDT, each pair of the band images of the scrambled hyperspectral image were regarded as leaf nodes, while the encryption modules using chaotic random phase mask, dmpFrFT, and improved equal modulus decomposition were regarded as branch nodes, and the encryption process was implemented along the paths from the leaf nodes to the topmost branch node. The EMDT structure could provide multiparameter encryption, real-valued output, and different pairs of band images with different secret keys and encryption/decryption paths. Compared with the previous optical encryption approaches for hyperspectral images, our asymmetric cryptosystem had larger key space, less data amount of storage and transmission, and stronger resistance to statistical attacks. Various numerical simulations verified the performance of our proposed asymmetric cryptosystem.

Application of fractional Fourier transform in feature extraction from ELECTROCARDIOGRAM and GALVANIC SKIN RESPONSE for emotion recognition

Emotion recognition from physiological signals plays an essential role in human–computer interaction and affective computing. This paper aims to study the effectiveness of Fractional Fourier Transform (FrFT) as a novel feature extraction method in improving the accuracy of emotion recognition from physiological signals. Emotion detection is performed in two dimensions, of arousal and valence, using Electrocardiogram (ECG) and galvanic skin response (GSR) signals recorded on the ASCERTAIN database. Features extracted in the FrFT, time, and frequency domains are classified using two binary SVMs. The results suggest the usefulness of the phase information of the FrFT coefficients in arousal and valence detection and above-chance emotion recognition is achieved with both ECG and GSR signals. Comparison of the time domain features, frequency domain features, and their combination shows that FrFT are more distinct in emotion detection. The best recognition accuracy in both valence and arousal is achieved from the phase information of the FrFT coefficients using the ECG signal, which is equal to 78.32% and 76.83%, respectively.

Heisenberg’s uncertainty principle for [formula omitted]-dimensional fractional Fourier transform of complex-valued functions

The latest N-dimensional Heisenberg’s uncertainty principle associated with complex-valued functions’ uncertainty product in the Fourier transform domain is extended into two fractional Fourier transform (FRFT) domains. The result derived is sharper than the existing ones in the literature, giving rise to a tighter lower bound on the uncertainty product in time and N-dimensional FRFT domains, as well as that in two N-dimensional FRFT domains. Example and simulations also validate the correctness of theoretical analyses, and finally the effectiveness is illustrated by applications in the effective estimation of spreads in time–frequency analysis and optical systems analysis.

Sparse Representation Classification Method for Active Sonar Target Based on Multi-order Fractional Fourier Domain Feature Fusion

Marine environment noise and reverberation interference are serious, and the poor target separability is the bottleneck problem in active sonar target classification and recognition. In order to solve this problem, based on the echo signal model of active sonar target and the principle of FRactional Fourier Transform (FRFT), this paper deduces the multi-order FRFT domain feature representation form, establishes the FRFT domain sparse representation model, and proposes a method to classify the sparse representation of active sonar targets with multi-order FRFT domain feature fusion. The method achieves the purpose of suppressing noise and reverberation interference through the energy aggregation of FRFT and removing the residual of sparse decomposition; Through the fusion of multi-order FRFT domain features, the separability of targets is improved, and the active sonar target classification with low SNR is realized. Experimental results show that the classification accuracy of the proposed method can reach more than 90% when the SNR is about 0 dB.

Separation of Two-Dimensional Mixed Circular Fringe Patterns Based on Spectral Projection Property in Fractional Fourier Transform Domain

In this paper, a method for automatically separating the mixed circular fringe patterns based on the fractional Fourier transform (FrFT) analysis is proposed. Considering the mixed two-dimensional (2-D) Gaussian-based circular fringe patterns, detected by using an image sensor, we propose a method that can efficiently determine the number and parameters of each separated fringe patterns by using the FrFT due to the observed higher sparsity in the frequency domain than that in the spatial domain. First, we review the theory of FrFT and the properties of the 2-D circular fringe patterns. By searching the spectral intensities of the various fractional orders in the FrFT projected along both the frequency axes, the proposed method can automatically determine the total fringe number, the central position, binary phase, and the maximum fringe width of each 2-D circular fringe pattern. In the experimental results, both the computer-simulated and optically mixed fringe patterns are used to verify the proposed method. In addition, the additive Gaussian noise effects on the proposed method are investigated. The proposed method can still successfully separate the mixed fringe pattern when the signal-to-noise ratio is higher than 7 dB.

Cancelable Iris template for secure authentication based on random projection and double random phase encoding

Biometric recognition approaches have overwhelmed major issues in traditional authentication systems. Privacy invasion and security are the needed features in the effective development of biometric recognition systems. Recently some biometric techniques have been suffered to hacking trails also. Hence there is a despairing need to propose a novel cancelable iris recognition scheme. The prime objective of this research is to create a transformed cancelable biometric template through an encryption function and a one-way transformation function. The random projection matrix is employed here to generate the first feature vector. Then an encrypted cancelable iris code is generated using Double Random Phase Encryption (DRPE) in the Fractional Fourier Transform (FFT) domain. The proposed framework utilizes both the left and right iris image of a person and generates a single cancelable iris template. This feature guarantees the privacy-preserving cancelable iris code generation for secure authentication. The experiment is carried on two state-of-art datasets CASIA Iris V4 and IITD iris to affirm the efficacy of the proposed methodology. The result analysis witnessed a promising accuracy of 99.59%, a recognition rate of 99.88% with a lower Equal Error Rate (ERR) of 0.46%. It has also been proved that the proposed approach is computationally efficient as it recognizes the iris code with less recognition time of 7 ms with a maximum true positive and true negative rate of 100%.

The Property of Frequency Shift in 2D-FRFT Domain With Application to Image Encryption

The Fractional Fourier Transform (FRFT) has been playing a unique and increasingly important role in signal and image processing. In this letter, we investigate the property of frequency shift in two-dimensional FRFT (2D-FRFT) domain. It is shown that the magnitude of image reconstruction from phase information is frequency shift-invariant in 2D-FRFT domain, enhancing the robustness of image encryption, an important multimedia security task. Experiments are conducted to demonstrate the effectiveness of this property against the frequency shift attack, improving the robustness of image encryption.

Research on sparsity of frequency modulated signal in fractional Fourier transform domain

Frequency modulated (FM) signal is extensively applied in sonar, radar, laser and emerging optical cross-research, its sparsity is a common basic issue for the sampling, denoising and compression of FM signal. This paper mainly studies the sparsity of FM signal in the fractional Fourier transform (FRFT) domain, and a maximum singular value method (MSVM) is proposed to estimate the compact FRFT domain of FM signal. This method uses the maximum singular value of amplitude spectrum of FM signal to measure the compact domain, and WOA is used to search the compact domain, which effectively improves the shortcomings of the existing methods. Compared with MNM and MACF, this method gives a sparser representation of FM signal in the FRFT domain, which has less number of significant amplitudes. Finally, the primary application of this method in the FM signal filtering is given.