Analysis of Chirp as windowing function through fractional Fourier transform

This article presents a new mathematical model for obtaining the fractional Fourier transform (FrFT) of Chirp function. FrFT is a parameterised transform having an adjustable transform parameter which makes it more flexible and superior over ordinary FT in several applications. It is an important tool used in signal processing for spectral analysis. The closed-form expression derived for FrFT of finite duration Chirp establishes the dependence of FrFT of Chirp on the order of FrFT and the Chirp parameter. The mathematical model obtained shows that the Chirp function can be used as an adjustable window function in the fractional Fourier domain. The main-lobe width, side-lobe level and side-lobe fall-off rate of a Chirp window can be controlled by changing the adjustable transform parameter to different values. For some particular values of fractional angle, Chirp can give better spectral parameters than the existing window functions. By varying the order of FrFT, the variations in spectral parameters of the Chirp window are obtained and studied. The performance of the Chirp window in fractional domain is also compared with some of the existing windows.     

An Efficient FPGA-based Implementation of Fractional Fourier Transform Algorithm

In this paper, we present an efficient FPGA implementation method of fractional Fourier transform (FrFT) algorithm. Firstly, a polyphase implementation of the FrFT computation algorithm base on the theory of multirate signal processing and filter banks is proposed. This polyphase implementation costs less computations and its parallel structure is suitable for FPGA realization. Then we present one computational method, which improve the resolution on any portion of fractional spectrum. Some realization details, such as the parity restrictions of signal length, special interval for transform order, interpolation filter, continuous frame computation and continuous order computation are also investigated. Finally, the efficiency of the novel method is verified by FPGA implementation results.     

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.     

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.     

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。     

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.     

一种基于分数阶傅里叶变换的OFDM系统及其均衡算法

在快速时变信道环境下,由于子载波间干扰(ICI)的影响,传统OFDM系统性能有较大下降.本文提出了一种基于分数阶傅里叶变换的OFDM系统,它用分数阶傅里叶变换代替傅里叶变换进行子载波调制与解调;同时,文中给出了最优分数阶傅里叶变换阶次的选取方法,并根据最小均方误差(MMSE)准则设计了分数阶傅里叶域乘性滤波器在接收端进行均衡.分析和数值仿真结果表明,最优分数阶傅里叶域的乘性滤波算法较频域方法有更好的均衡效果.     

分数阶傅立叶变换OFDM系统自适应均衡算法

由于子载波间干扰(ICI)的影响,传统OFDM系统均衡方法在快速衰落的信道环境下性能有较大下降.本文提出了一种基于分数阶傅立叶变换的OFDM系统自适应均衡方法,它用分数阶傅立叶变换代替傅立叶变换进行子载波调制与解调,同时在分数阶傅立叶域对接收信号进行自适应均衡.文中给出了最优分数阶傅立叶变换阶次的选取方法,和分数阶傅立叶域最小均方算法的步骤.分析和数值仿真结果表明,最优分数阶傅立叶域的自适应均衡算法较传统频域方法有更好的均衡效果,并且复杂度不高.     

An Efficient VLSI Architecture for Computation of Discrete Fractional Fourier Transform

Since decades, the fractional Fourier transform (FrFT) has attracted researchers from various domains such as signal and image processing applications. These applications have been essentially demanding the requirement of low computational complexity of FrFT. In this paper, FrFT is simplified to reduce the complexity, and further an efficient CORDIC-based architecture for computing discrete fractional Fourier transform (DFrFT) is proposed which brings down the computational complexity and hardware requirements and provides the flexibility to change the user defined fractional angles to compute DFrFT on-the-fly. Architectural design and working method of proposed architecture along with its constituent blocks are discussed. The hardware complexity and throughput of the proposed architecture are illustrated as well. Finally, the architecture of DFrFT of the order sixteen is implemented using Verilog HDL and synthesized targeting an FPGA device ”XLV5LX110T”. The hardware simulation is performed for functional verification, which is compared with the MATLAB simulation results. Further, the physical implementation result of the proposed design shows that the design can be operated at a maximum frequency of 154 MHz with the latency of 63-clock cycles.     

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.     

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.     

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.     

基于分数阶傅里叶变换的Chirp浅剖精细探测方法

实现弱回波信号检测和高信噪比(SNR)浅剖图像获取是浅剖精细探测的首要任务。该文在分析分数阶傅里叶变换(FrFT)解卷积原理,推导时间量纲化变换公式的基础上,提出一种基于FrFT的浅剖精细探测新方法。该方法通过FrFT解卷积实现分数阶傅里叶域(u域)沉积层冲激响应求解,采用u域加窗滤波技术对带内噪声进行有效抑制,经时间量纲化变换实现高信噪比u域沉积层冲激响应包络信号至时域浅剖包迹的直接变换,得到高质量的浅剖图像。仿真实验和实测数据处理验证了算法的精细探测能力,算法性能优于脉冲压缩和自回归(AR)预测滤波方法。     

LFM signal parameter estimation in the fractional Fourier domains: Analytical models and a high-performance algorithm

The index of the peak magnitude of the fractional Fourier transform (FrFT) of linear frequency modulated (LFM) signals has been widely used as a powerful chirp-rate estimator. In this paper, we propose analytical approximations for the peak FrFT magnitude (PFM) of mono and multicomponent LFM signals, both for noiseless and noisy cases. In addition, we present a novel coarse-to-fine FrFT-based algorithm designed specifically for chirp-rate estimation of multi-component LFM signals. Our approach entails an initial coarse estimation of the chirp-rates for each component by utilizing our proposed mathematical models. By leveraging these models, we achieve improved performance and a reduced signal-to-noise breakdown threshold. Furthermore, we incorporate a unique and efficient estimate-and-subtract strategy to refine the estimated parameters using our proposed models. Rather than removing the components from the LFM signal, we utilize the derived model to identify and remove peaks in the PFM. This strategy enhances the algorithm’s capability to handle challenging scenarios. Extensive simulation results demonstrate that our proposed algorithm performs very close to the Cramér–Rao lower bound. It effectively eliminates the leakage effect between signal components, avoids error propagation, and maintains an acceptable computational cost compared to other state-of-the-art methods.     

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。     

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。     

基于分数阶谱相减的语音增强法

该文提出了基于分数阶谱相减的语音增强法(FSS)。该方法通过对带噪语音信号作分数阶傅里叶变换(FRFT),将得到的分数阶语噪混合谱与估计的分数阶噪声谱相减,最后利用分数阶Fourier反变换获得去噪后的语音信号。理论分析表明,所提方法存在一个最佳分数阶阶数,使得语噪混合信号能在分数阶变换域得到最好的分离,从而有效地提高了增强语音的性能。计算机仿真表明,对于混有加性白噪声的男/女声发音信号,所提方法在信噪比提高量和Itakura距离减少量两个方面都优于传统的谱相减法(SS),并且增强语音中的音乐噪声得到了明显抑制。     

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。     

基于自适应分数阶傅里叶变换的线性调频信号检测及参数估计

该文提出了一种基于最小均方算法的自适应计算分数阶傅里叶变换的方法并将该方法应用到多分量chirp信号的检测与估计之中。该方法通过对连续型分数阶傅里叶反变换进行离散化采样,得到适合数值计算的离散形式,进而通过适当的选择输入向量和目标函数构造自适应滤波器,经过最小均方算法进行训练后所得的滤波器权系数即为分数阶傅里叶变换的结果。仿真实验表明,该方法可以用来计算分数阶傅里叶变换及对chirp信号进行检测和参数估计,且计算延时相对较小。     

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.