The concise fractional Fourier transform and its application in detection and parameter estimation of the linear frequency-modulated signal

A concise fractional Fourier transform(CFRFT) is proposed to detect the linear frequency-modulated(LFM) signal with low signal to noise ratio(SNR).The frequency axis in time-frequency plane of the CFRFT is rotated to get the spectrum of the signal in different angles using chirp multiplication and Fourier transform(FT).For LFM signal which distributes as a straight line in time-frequency plane,the CFRFT can gather the energy in the corresponding angle as a peak and improve the detection SNR,thus the LFM signal of low SNR can be detected.Meanwhile,the location of the peak value relates to the parameters of the LFM signal.Numerical simulations and experimental results show that,the proposed method can be used to efficiently detect the LFM signal masked by noise and to estimate the signal's parameters accurately.Compared with the conventional fractional Fourier transform(FRFT),the CFRFT reduces the transform complexity and improves the real-time detection performance of LFM signal.     

简明分数阶傅里叶变换及其对线性调频信号的检测和参数估计

针对低信噪比下线性调频信号的检测问题,提出了一种简明分数阶傅里叶变换方法。该变换借助chirp相乘和傅里叶变换对时频平面上的频率轴进行旋转,以获取信号在各个角度下频率轴上的频谱分布。对时频分布呈直线状的线性调频信号,简明分数阶傅里叶变换能在特定角度上将信号能量聚集成尖锐的强能量峰,从而提高信噪比,实现对线性调频信号的可靠检测和参数估计。数值仿真和实验验证结果表明,简明分数阶傅里叶变换可对较低信噪比的线性调频信号实现有效检测,并由变换域峰值的位置对信号参数进行准确估计。相比于传统的分数阶傅里叶变换方法,简明分数阶傅里叶变换的复杂度更低,离散计算效率更高,在对噪声掩盖下的线性调频信号进行检测和参数估计时能更好地满足实时处理的要求。      

Acceleration target detection based on LFM radar

In radar systems, the echo signal caused by an accelerated target can be similarly considered as linear frequency modulation (LFM) signal. In high signal-to-noise ratio (SNR), discrete polynomial-phase transform (DPT) algorithm can be used to detect the echo signal, as it has low computation complexity and high real-time performance. However, in low SNR, the DPT algorithm has a large mean square error of the rate of frequency modulation and a low detection probability. In order to detect LFM signal in low SNR, this paper proposes a detection method, segment discrete polynomial-phase transform (SDPT), which means, at first, dividing the whole echo pulses into several segments with same duration in time domain, and then, using coherent accumulation method of DFT to segments, at last, processing this signal with DPT in intra-segment. In the case of a large number of segments, the SDPT can improve the output SNR. In addition, in a certain SNR, to the target signal with big sampling interval, large acceleration and less segments, this paper proposes an algorithm to detect the LFM signal generated from the combination of an improved DPT (IDPT) and fractional Fourier transform (FRFT). The output SNR of this algorithm is connected with the length of time delay. In the simulation, when the length of the time delay is 0.2 N, the output SNR is 2.5 dB more than that which results from directly using DPT. Finally, the detection performance and algorithm complexity of the proposed algorithm were analyzed, and the simulated and measured data verify the effectiveness of the algorithm.     

基于多项式调频Fourier变换的信号分量提取方法

为了从含有噪声的混合信号中有效提取各个信号分量, 提出一种基于多项式调频Fourier变换的分量提取方法. 通过研究Fourier变换和分数阶Fourier变换的信号能量积累方式及变换基函数的时频表示, 提出利用时频平面上的多项式调频曲线族代替Fourier变换和分数阶Fourier 变换的调频直线族, 将变换的适用范围扩展到非线性调频信号. 采用粒子群智能优化算法搜索调频曲线族的最优多项式参数, 使混合信号中的某一分量在多项式调频Fourier域上能量谱集中. 最后对能量谱集中的分量进行窄带滤波, 并利用多项式调频逆Fourier变换重构信号分量. 仿真实验结果表明, 该方法不仅能够提取混合信号中的线性调频分量, 还能够实现非线性调频分量的能量谱集中、信号分离和时频特征提取.     

Multi component LFM signal detection and parameter estimation based on EEMD–FRFT

This paper studied multi component LFM signal detection and parameter estimation under the noise circumstance of various signal-to-noise ratios. Based on the analysis of fractional Fourier transform detection and parameter estimation on simple component LFM signal, this paper proposed the method of multi component LFM signal detection and parameter estimation based on EEMD–FRFT (Ensemble Empirical Mode Decomposition–Fractional Fourier transform), and this method was that with the EEMD algorithm, from the frequency domain decompose the analyzable signal to narrow-bandwidth components, whose center frequency changed from high to low, then accurately estimate the parameter and detect the signal of each component out of the pseudo-component with FrFT. This method solved the problem of mode aliasing of signal decomposition; meanwhile, the problem of detecting the multi component LFM signal would be simplified as the problem of one-dimensional search in small scope, which could reduce the amount of operation and improved the detection accuracy. A simulation computation for multi component LFM signal of various SNR (signal-to-noise ratios) was made and the result showed that the error of parameter estimation was less than 5% in the case of SNR not less than −10 dB.     

低信噪比线性调频信号目标的方位估计

线性调频(LFM)信号目标的方位估计是水声探测研究的重要内容,在进行方位估计时,若存在强干扰信号源与强背景噪声,阵元接收信号的信噪比会显著降低,严重影响LFM信号目标方位估计结果的准确性.针对该问题,提出了一种简明分数阶滤波方法,并将其与常规波束形成方法(CBF)相结合来实现低信噪比条件下LFM信号目标的方位估计.简明分数阶傅里叶变换能在正交角度上将LFM信号的能量聚集在特定频点处并形成明显的能量峰,利用该特性,可对阵列各阵元接收的低信噪比LFM信号在简明分数阶域聚集的能量峰进行最佳滤波,以滤除干扰信息及背景噪声.对滤波输出进行逆简明分数阶傅里叶变换可得到增强信干比和信噪比的阵元域信号,进一步用于目标方位估计,就能获得更加准确的目标方位。数值仿真结果和海试实验数据处理结果验证表明,本文所提出的方法可有效抑制干扰和背景噪声,并对低信噪比LFM信号进行准确、稳健的方位估计。      

Fractional Gabor transform

A fractional Gabor transform (FRGT) is proposed. This new transform is a generalization of the conventional Gabor transform (GT) based on the Fourier transform to the windowed fractional Fourier transform (FRFT). The FRGT provides analyses of signals in both the real space and the FRFT frequency domain simultaneously. The space-FRFT frequency pattern can be rotated as the fractional order changes. The FRGT has an additional freedom, compared with the conventional GT, i.e., the transform order. The FRGT may offer a useful tool for guiding optimal filter design in the FRFT domain in signal processing.     

An echo detection algorithm for underwater continuous wave active detection

The model of linear frequency modulation continuous wave （LFMCW） applied in underwater detection and the method for the detection of echo signal and the estimation of target parameters were studied. By analyzing the heterodyne signal, an algorithm with the structure of heterodyne-Practional Fourier Transform （FRFT） was proposed. To reduce the computation of searching targets in a two-dimensional FRFT result, the heterodyne signal would be processed by FRFT at a specific order, after Radon-Ambiguity Transform （RAT） was applied to estimate the sweep rate of the signal. Simulations proved that the algorithm can eliminate the coupling phenomenon of distance and velocity of LFMCW, and estimate targets＇ parameters accurately. The lake trial results showed that the processing gain of LFMCW processed by the algorithm in this paper was 13 dB better than that of the LFM processed by matched filter. The research results indicated that the algorithm applied in LFMCW underwater detection was feasible and effective, and it could estimate targets＇ parameters accurately and obtain a good detection performance.     

Improved space time prewhitener for linear frequency modulation reverberation using fractional Fourier transform

This letter presents an improved space time prewhitening method for linear frequency modulation (LFM) reverberation. The proposed method transforms the reverberation to fractional Fourier domain to whiten using fractional Fourier transform. The linear varying frequency in LFM reverberation is focused on a stationary frequency, and the adjacent block signal is used as the reference signal of prewhitening. Finally, experiment results with real reverberation data verify that the proposed method improves the detection performance of active sonar in shallow sea significantly.     

分数域滤波的分数傅里叶变换光学成像性能分析

将分数域滤波同光学成像相结合,提出了一种基于Lohmann Ⅰ型的带有滤波孔径的两级联分数傅里叶变换光学成像系统.根据分数傅里叶变换和菲涅耳衍射之间的关系以及分数傅里叶变换的分数阶可加性,结合分数域滤波分析了分数傅里叶变换光学成像的基本理论.以点扩散函数和调制传递函数作为评价成像质量的准则,详细分析了不同分数阶光学系统...     

On scaling properties of fractional Fourier transform and its relation with other transforms

Several properties of fractional Fourier transform (FRFT) have been studied recently and many are being investigated at present. In this article, scaling property of the FRFT is generalized and some of its applications are suggested. Some extensions of the sampling relations in the FRFT domain are also presented. The issues related to connections between the FRFT and other signal transforms such as scale transform, fractional Mellin transform, and chirp z-transform, are also investigated.     

An optical waveform pre-distortion method based on time domain fractional Fourier transformation

An optical waveform pre-distortion method is proposed in reducing the temporal broadening of pulse based on the optical implementation of time domain fractional Fourier transform (FRFT). Moreover, a new analysis of the dispersion and self-phase modulation effects is investigated. The simulation results show that prechirp coefficient and proper fractional orders of FRFT can be beneficial in mitigating pulse broadening.     

一种基于激光调制的高精度位移测量方法

激光准直技术在位移测量及相关应用中具有重要作用,其核心器件是位置敏感探测器,环境杂散光、电源波动、光电信号处理电路噪声等是影响测量精度的主要因素。针对以上问题,提出一种基于激光调制的高精度位移测量方法。将激光器由直流驱动的连续输出模式改为交流驱动的调制模式输出,优化设计光电探测器微弱光电信号的调理与采集系统,通过编写Labview软件对信号进行频谱分析、带通滤波、均值处理等,对激光连续输出和激光调制输出2种情况分别进行了实验测试与分析,并对比了位移测量的稳定性。实验结果表明:其测量值波动范围从7 μm减少到了2.6 μm,验证了基于激光调制方法的有效性,可以通过滤波更加有效地消除噪声,测量精度提高至3 μm以内。     

利用分数傅里叶变换相关实现散斑相关测量

提出了一种在匹配滤波相关中利用分数傅里叶变换代替傅里叶变换实现散斑相关测量的方法.利用分数傅里叶变换的指数性质,采用把移变后的散斑图像先进行相位调制,再进行分数傅里叶变换的方法,有效地解决了分数傅里叶变换的移变性带来的谱移问题.编程模拟和拉伸试件位移场测量的结果表明,只要选择合适的分数傅里叶变换级次和相位调制函数,可以得到优于傅里叶变换相关的理想输出.     

Reconstruction of the first-derivative EPR spectrum from multiple harmonics of the field-modulated continuous wave signal

Selection of the amplitude of magnetic field modulation for continuous wave electron paramagnetic resonance (EPR) often is a trade-off between sensitivity and resolution. Increasing the modulation amplitude improves the signal-to-noise ratio, S/N, at the expense of broadening the signal. Combining information from multiple harmonics of the field-modulated signal is proposed as a method to obtain the first derivative spectrum with minimal broadening and improved signal-to-noise. The harmonics are obtained by digital phase-sensitive detection of the signal at the modulation frequency and its integer multiples. Reconstruction of the first-derivative EPR line is done in the Fourier conjugate domain where each harmonic can be represented as the product of the Fourier transform of the 1st derivative signal with an analytical function. The analytical function for each harmonic can be viewed as a filter. The Fourier transform of the 1st derivative spectrum can be calculated from all available harmonics by solving an optimization problem with the goal of maximizing the S/N. Inverse Fourier transformation of the result produces the 1st derivative EPR line in the magnetic field domain. The use of modulation amplitude greater than linewidth improves the S/N, but does not broaden the reconstructed spectrum. The method works for an arbitrary EPR line shape, but is limited to the case when magnetization instantaneously follows the modulation field, which is known as the adiabatic approximation.     

Sampling of fractional bandlimited signals associated with fractional Fourier transform

Fractional Fourier transform (FRFT) plays an important role in many fields of optics and signal processing. This paper considers the problem of reconstructing a fractional bandlimited signal with FRFT. We propose a novel reconstruction method for fractional bandlimited signals using the fractional Fourier series (FRFS). The advantage is that the sampling expansion can be deduced directly not based on the Shannon theorem. By utilizing the generalized form of Parseval’s relation for complex FRFS, we obtain the sampling expansion for fractional bandlimited signals with FRFT. We show that the sampling expansion for fractional bandlimited signals with FRFT is a special case of Parseval’s relation for complex FRFS.     

水下回波处理中分数阶傅里叶变换的带通采样实现方法

利用球形压电陶瓷自身所具有的耐压能力,采用径向极化空气背衬压电球壳换能器作为声学接收敏感元件,设计并制作了一种球形耐压水听器。首先对其低频开路接收灵敏度和谐振频率等声学特性进行了分析和有限元仿真,然后对其强度和稳定性等耐压性能进行了分析和有限元仿真,最后对其声学性能和耐压能力进行了测试。测试表明,该球形耐压水听器的直径为36 mm,工作频段为50 Hz10 kHz,低频接收灵敏度为􀀀198:4 dB(0 dB=1 V/Pa),等效自噪声谱级为46.5 dB@1 kHz,其耐压深度可达3000 m。该耐压水听器为大深度水听器设计提供了参考,在深水声学领域具有重要的应用价值。     

弹光调制非等相位干涉信号的快速光谱反演算法

在弹光调制傅里叶变换光谱仪(PEM-FTS)中,由于调制光程差的非线性,不能直接采用快速傅里叶变换(FFT)进行反演光谱,且直接计算开销过大。首先在Matlab软件中用非均匀快速傅里叶变换算法(NUFFT)对PEM非等相位干涉信号进行了软件仿真,其次设计了以TMS320C6713高性能浮点数字信号处理器(DSP)芯片为核心的光谱信息处理系统,实现了硬件上的光谱实时处理。研究表明,算法对光谱反演具有速度快、精度高等优点,1024点光谱反演的速度较直接运算的速度提高20多倍,反演精度可达0.78%。     

ns脉冲测量中的波形重建

为修正高电压测量探头寄生参数对测量ns脉冲波形的影响,对探头的数字补偿方法进行了研究。利用对低电压标定所得标准参考信号和探头信号做傅里叶变换,得到其各自的频谱密度,然后通过相除,得到探头灵敏度与频率的关系和信号相位与频率的关系。在实测时,再对测得信号做傅里叶变换,将得到的频谱密度乘以探头灵敏度及信号相位对频率的函数,通过傅里叶逆变换重建出待测信号。通过不同信号进行波形重建实验证明,此方法能够在短时间内很好地修正高电压测量探头寄生参数对ns脉冲波形的畸变。     

A flexible fast 3D profilometry based on modulation measurement

This paper proposes a flexible fast profilometry based on modulation measurement. Two orthogonal gratings through a beam splitter are vertically projected on an object surface, and the measured object is placed between the imaging planes of the two gratings. Then the image of the object surface modulated by the orthogonal gratings can be obtained by a CCD camera in the same direction as the grating projection. This image is processed by the operations consisting of performing the Fourier transform, spatial frequency filtering and inverse Fourier transform. Using the modulation distributions of two grating patterns, we can reconstruct the 3D shape of the object. In the measurement process, we only need to capture one fringe pattern, so it is faster than the MMP and remains the advantages of it. In the article, the principle of this method, the setup of the measurement system, some simulations and primary experiment results are given. The simulative and experimental result proves it can restore the 3D shape of the complex object fast and comparatively accurate. Because only one fringe pattern is needed in the testing, our method has a promising extensive application prospect in real-time acquiring and dynamic measurement of 3D data of complex objects.