Analysis of Dirichlet, Generalized Hamming and Triangular window functions in the linear canonical transform domain

Linear canonical transform is a four-parameter class of integral transform that plays an important role in many fields of optics and signal processing. Well-known transforms such as the Fourier transform, the fractional Fourier transform, and the Fresnel transform can be seen as the special cases of the linear canonical transform. This paper presents a new mathematical model for obtaining the linear canonical transforms of Dirichlet, Generalized “Hamming”, and Triangular window functions. The different window function parameters are also obtained from the simulations. By changing the value of four parameters and then changing the adjustable parameter, the main-lobe width, ?3 dB bandwidth, ?6 dB bandwidth and correspondingly, the minimum stop-band attenuation of the resulting window functions can be controlled. It has been shown that by using linear canonical transform, we are able to obtain all window parameters successfully as compared to fractional Fourier transform.     

Analysis of Dirichlet and Generalized “Hamming” window functions in the fractional Fourier transform domains

A new mathematical model for obtaining the fractional Fourier transforms of Dirichlet and Generalized “Hamming” window functions is presented. The different parameters for the window functions are also obtained with the help of simulations. The fractional Fourier transformation contains an adjustable parameter with which the main lobe width and correspondingly, the minimum stop band attenuation of the resulting window function can be controlled.     

Fractional Fourier transform of the Gaussian and fractional domain signal support

The fractional Fourier transform (FrFT) provides an important extension to conventional Fourier theory for the analysis and synthesis of linear chirp signals. It is a parameterised transform which can be used to provide extremely compact representations. The representation is maximally compressed when the transform parameter, /spl alpha/, is matched to the chirp rate of the input signal. Existing proofs are extended to demonstrate that the fractional Fourier transform of the Gaussian function also has Gaussian support. Furthermore, expressions are developed which allow calculation of the spread of the signal representation for a Gaussian windowed linear chirp signal in any fractional domain. Both continuous and discrete cases are considered. The fractional domains exhibiting minimum and maximum support for a given signal define the limit on joint time-frequency resolution available under the FrFT. This is equated with a restatement of the uncertainty principle for linear chirp signals and the fractional Fourier domains. The calculated values for the fractional domain support are tested empirically through comparison with the discrete transform output for a synthetic signal with known parameters. It is shown that the same expressions are appropriate for predicting the support of the ordinary Fourier transform of a Gaussian windowed linear chirp signal.     

Iterative Interpolation for Parameter Estimation of LFM Signal Based on Fractional Fourier Transform

This study introduces two iterative interpolation algorithms for the parameter estimation of linear frequency modulation (LFM) signal using fractional Fourier transform (FrFT). The estimated parameter of an LFM signal can be obtained by locating the peak of the periodogram in the FrFT domain. Two interpolation algorithms were proposed to improve the accuracy of parameter estimation by employing the FrFT coefficients relative to the true parameters and applying interpolation algorithms iteratively to refine the parameter estimation approach. The proposed algorithms can utilize more information from FrFT results, thereby achieving improvements in either accuracy or efficiency. Moreover, the simulation results revealed the validity and advantage of the proposed approach.     

基于双通道DFRFT互谱法的Chirp信号时延估计

针对脉冲Chirp类信号的时延估计问题,理论推导了基于离散分数阶Fourier变换的脉冲Chirp信号的特性,分析了当时延参量等效的分数阶Fourier域的频率大于采样率时,脉冲Chirp信号的分数阶Fourier域谱产生混叠,造成时延估计模糊的问题,并提出基于离散分数阶Fourier变换（DFRFT）双通道互谱法进行时延估计,给出两个通道采样率选取的原则及算法的性能分析,实验结果表明,在一定的采样率下,算法能够快速精确地估计脉冲Chirp信号的时延参数.     

Closed-Form Analytical Expression of Fractional Order Differentiation in Fractional Fourier Transform Domain

In this paper, a closed-form analytical expression for fractional order differentiation in the fractional Fourier transform (FrFT) domain is derived by utilizing the basic principles of fractional order calculus. The reported work is a generalization of the differentiation property to fractional (noninteger or real) orders in the FrFT domain. The proposed closed-form analytical expression is derived in terms of the well-known confluent hypergeometric function. An efficient computation method has also been derived for the proposed algorithm in the discrete-time domain, utilizing the principles of the discrete fractional Fourier transform algorithm. An application example of a low-pass finite impulse response (FIR) fractional order differentiator in the FrFT domain has also been investigated to show the practicality of the proposed method in signal processing applications.     

宽带Chirp雷达回波相干积累方法

宽带Chirp雷达经去线性调频处理后的回波特性表明,运动目标回波脉间频率变化率与脉内剩余调频斜率一致。利用分数阶傅里叶变换(Fractional Fourier Transform,FrFT)处理Chirp信号的能量集聚性,提出基于FrFT的运动目标宽带Chirp雷达多周期回波相干积累方法。仿真结果表明该方法能实现匀速运动目标宽带雷达多次回波相干积累,并对机动目标的脉间能量相干积累具有较好的适应性。     

基于分数阶傅里叶变换的多分量Chirp信号的检测与参数估计

分数阶傅里叶变换是傅里叶变换的广义形式。利用Chirp信号在分数阶傅里叶变换域的特点,提出了对含有多个非平稳Chirp成分的信号在噪声中的检测与参数估计方法。理论分析和仿真结果表明,与现有的基于时间域、频率域和时频域的方法相比,该方法物理意义清楚,计算简便,无交叉项干扰,抗噪声性能强。     

Sampling and Reconstruction of Sparse Signals in Fractional Fourier Domain

Sampling theory for continuous time signals which have a bandlimited representation in fractional Fourier transform (FrFT) domain-a transformation which generalizes the conventional Fourier transform-has blossomed in the recent past. The mechanistic principles behind Shannon's sampling theorem for fractional bandlimited (or fractional Fourier bandlimited) signals are the same as for the Fourier domain case i.e. sampling (and reconstruction) in FrFT domain can be seen as an orthogonal projection of a signal onto a subspace of fractional bandlimited signals. As neat as this extension of Shannon's framework is, it inherits the same fundamental limitation that is prevalent in the Fourier regime-what happens if the signals have singularities in the time domain (or the signal has a nonbandlimited spectrum)? In this paper, we propose a uniform sampling and reconstruction scheme for a class of signals which are nonbandlimited in FrFT sense. Specifically, we assume that samples of a smoothed version of a periodic stream of Diracs (which is sparse in time-domain) are accessible. In its parametric form, this signal has a finite number of degrees of freedom per unit time. Based on the representation of this signal in FrFT domain, we derive conditions under which exact recovery of parameters of the signal is possible. Knowledge of these parameters leads to exact reconstruction of the original signal.     

分数阶Fourier变换对LFM信号的特征参数估计能力仿真评估

将分数阶傅里叶变换（FrFT）对线性调频（LFM）信号处理方法应用到雷达侦察信号分选与参数估计中,研究FrFT对LFM信号在低信噪比下的检测与参数估计能力。设定了雷达侦察信号背景,制定了仿真验证流程,考察了FrFT对LFM信号的调频率和中心频率的估计能力,对比了不同信噪比下的参数估计运算时间。仿真结果验证了该算法可以完成低信噪比条件下雷达侦察LFM信号特征参数的估计。     

An improved Hurst parameter estimator based on fractional Fourier transform

A fractional Fourier transform (FrFT) based estimation method is introduced in this paper to analyze the long range dependence (LRD) in time series. The degree of LRD can be characterized by the Hurst parameter. The FrFT-based estimation of Hurst parameter proposed in this paper can be implemented efficiently allowing very large data set. We used fractional Gaussian noises (FGN) which typically possesses long-range dependence with known Hurst parameters to test the accuracy of the proposed Hurst parameter estimator. For justifying the advantage of the proposed estimator, some other existing Hurst parameter estimation methods, such as wavelet-based method and a global estimator based on dispersional analysis, are compared. The proposed estimator can process the very long experimental time series locally to achieve a reliable estimation of the Hurst parameter.     

Tuning of FIR filter transition bandwidth using fractional Fourier transform

The transition bandwidth of window-based FIR filters is proportional to the window main-lobe width, which in turn is proportional to the length of the window function. As such, transition bandwidth of FIR filters can be directly tuned by varying window length for on-line tuning applications. However, analysis of window functions in fractional Fourier domain, a generalization of Fourier domain, also establishes the dependence of window main-lobe width on the order of fractional Fourier transform (FRFT). Thus, an alternative methodology to tune the transition bandwidth, based on FRFT, is developed in this work. The proposed methodology is useful for frequency domain filtering and introduces a comparative ease in tuning by eliminating the need to re-compute the impulse response coefficients. Also, significant computational saving has been achieved using FRFT. However, it is observed that the direct approach can introduce a lot more adjustability in the transition bandwidth than the FRFT approach. Apart from Kaiser window, considered to be optimum for FIR filter design, another window with a high side-lobe fall-off-rate (SLFOR), viz, Parzen-cos⁶ (πt) (PC6), has also been used in the proposed on-line filter tuning. Better performance of windows with high SLFOR in on-line sharpening is illustrated with the aid of simulation results.     

基于频域遮隔及分数阶相关的反辐射导弹检测技术

该文提出一种基于频域遮隔及分数阶相关来检测反辐射导弹(ARM)的新方法。该方法利用雷达回波的频域(Fourier变换)数据,根据尖锋作遮隔处理后,直接运用基于分数阶Fourier变换(FrFT)和Fourier逆变换的分数阶相关方法,通过一维搜索来检测ARM。由于该方法利用了现有的Fourier数据,且分数阶Fourier变换可以通过Fourier变换实现,无需多通道相位补偿和二维搜索,故计算量小。仿真结果表明,该方法能够在ARM信噪比低达#61485;10dB,ARM与载机回波功率比为#61485;27dB的情况下,准确地检测出ARM,从而实现ARM发射的早期告警。     

一种改进的基于FrFT的SAR运动目标检测与成像方法

针对传统的基于FrFT的SAR地面运动目标检测方法的不足,该文结合分数阶Fourier域滤波、重排谱图以及CLEAN思想,提出了一种改进的方法。该方法能够以较高精度估计目标的多普勒起始频率和中心频率,并能分辨在时频平面位于同一直线的多个目标。最后给出了FrFT数值计算结果与有量纲的多普勒参数之间的转换公式。仿真结果表明了该方法的有效性。     

Propagation properties of controllable dark-hollow beams through fractional Fourier transform systems

Based on the definition of fractional Fourier transform(FrFT) in the cylindrical coordinate system,the propagation properties of a controllable dark-hollow beam(CDHB) are investigated in detail.An analytical formula is derived for the FrFT of a CDHB.By using the derived formula,the properties of a CDHB in the FrFT plane are illustrated numerically.The results show that the properties of the intensity of the beam in the FrFT are closely related to not only the fractional order but also initial beam parameter,beam order and the lens focal length of the optical system for performing FrFT.The derived formula provides an effective and convenient way for analyzing and calculating the FrFT of a CDHB.     

The discrete fractional Fourier transform based on the DFT matrix

We introduce a new discrete fractional Fourier transform (DFrFT) based on only the DFT matrix and its powers. Eigenvectors of the DFT matrix are obtained in a simple-yet-elegant and straightforward manner. We show that this DFrFT definition based on the eigentransforms of the DFT matrix mimics the properties of continuous fractional Fourier transform (FrFT) by approximating the samples of the continuous FrFT. By appropriately combining existing commuting matrices we obtain a new commuting matrix which performs better. We show the validity of the proposed algorithms by computer simulations comparing DFrFT points and continuous FrFT samples for various signals.     

A Novel Multiuser SISO-BOFDM Systems with Group Fractional Fourier Transforms Scheme

Existing multiuser (MU) schemes are mainly accomplished by MIMO techniques. In this letter, we propose a novel group fractional Fourier transform based MU SISO Biorthogonal Frequency Division Multiplexing system, in which multiple independent data streams from co-channel mobile stations (MSs) can be transmitted in the same frequency and time slot. Because the chirp bases generated by different orders based fractional Fourier transform (FrFTs), at same subchannel, are mutually approximately orthogonal, when FrFT orders are effectively allocated to different co-channel MSs, the co-channel interference can be perfectly suppressed by simple correlative detection in FrFT domain with respective optimal order, at the receiver of each MS. Finally, system simulations show the essential advantages over conventional MU schemes.     

基于分数阶傅里叶变换的窄带雷达飞机目标回波特征提取方法

该文研究了常规窄带雷达体制下,利用分数阶傅里叶变换扩展特征域,从而解决直升机、螺旋桨飞机和喷气式飞机3类飞机目标回波分类中的特征提取问题。在现代战场中,直升机、螺旋桨飞机和喷气式飞机具有不同的机动性能,并各自承担着重要的任务。因此,实现这3类飞机的分类具有重大的意义。该文针对3类飞机目标分类的传统特征数目少,包含信息量有限,导致分类性能不够好的问题,基于现有的特征提取方法引入分数阶傅里叶变换(Fractional Fourier Transform, FrFT),在经过FrFT后的分数域提取3类飞机目标回波的分数阶特征,弥补传统特征的不足。并利用线性相关向量机(Relevance Vector Machine, RVM)的特征选择功能对提取的分数阶特征进行特征选择并分类。基于仿真和实测数据的实验结果证明该文提出的分数阶特征的分类性能较传统时域、多普勒域特征有较大提升。     

基于分数阶Fourier变换的并行组合扩频Chirp水声通信系统研究

Chirp信号具有较大的时宽带宽积,因此具有较强的抗多普勒扩展的能力,近年来在水下通信环境中得到广泛应用。但由于水声信道带宽非常有限,而Chirp信号又是一种宽带的信号,因此水下多载波Chirp系统中的的通信速率一般较低。利用Chirp信号在分数阶Fourier变换(Fractional Fourier Transform,FrFT)的最佳分数阶变换域的冲激特性,结合并行组合扩频(Parallel Combinatory Spread Spectrum,PCSS)技术,本文提出了一个基于FrFT的并行组合扩频Chirp水声通信系统。该系统的特点是:一方面充分利用Chirp信号的性质,在保证可靠性的同时,省去均衡环节,大大简化通信系统接收端结构;另一方面,通过并行组合多载波的选择方式有效提升了系统的速率,提高了系统的频带利用率。通过在计算机仿真和实验室水池实验验证了该系统的性能。     

A Novel Multi-carrier Order Division Multi-access Communication System Based on TDCS with Fractional Fourier Transform Scheme

In this paper, we propose a novel fractional Fourier transform (FrFT) based multi-carrier order division multi-access communication system, in which each user is uniquely identified by an FrFT order. Transform domain communication system (TDCS) with FrFT scheme is also proposed to synthesize the wide-band baseband waveforms in all FrFT domains with different users’ FrFT orders, which enhances the interference avoidance capability of this system under most of interference. Therefore, multiple independent data streams can be transmitted by using FrFT–OFDM in the same time and different FrFT domains. However, chirp bases, as the new kind of carriers with different modulated rates, are merely mutually approximately orthogonal. There is a problem of energy leakage between multiple chirp carriers, which possibly causes the multiple chirp carriers inter-shielding to influence the FrFT–OFDM demodulation performance. An efficient allocation algorithm of multiple chirp carriers by presetting the carrier parameters is proposed to solve this problem. Based on MC-CDMA, a variable bit rate system structure is proposed for TDCS with FrFT scheme under different channel environments. In order to simplify the process of modulation and demodulation of TDCS with FrFT scheme, a whole new cyclic shift key modulation mode in FrFT domain is also proposed. Both theories and simulations confirm strictly the validity of the proposed system.