Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Radar applications <s> An autofocusing technique is proposed, based on the modelling of the instantaneous phase, induced by the relative radar/target motion, as a generic order polynomial and on the estimation of the polynomial coefficients by the so-called product high-order ambiguity function. The proposed estimation method is able to deal with multicomponent signals and is robust against model order mismatching, provided that the selected order is greater than the true order. The proposed method is compared with the phase gradient autofocusing method, and applications are presented of focusing of extended scenes and detection and imaging of moving targets. <s> BIB001 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Radar applications <s> As a useful signal processing technique, the fractional Fourier transform (FrFT) is largely unknown to the radar signal processing community. In this correspondence, the FrFT is applied to airborne synthetic aperture radar (SAR) slow-moving target detection. For airborne SAR, the echo from a ground moving target can be regarded approximately as a chirp signal, and the FrFT is a way to concentrate the energy of a chirp signal. Therefore, the FrFT presents a potentially effective technique for ground moving target detection in airborne SAR. Compared with the common Wigner-Ville distribution (WVD) algorithm, the FrFT is a linear operator, and will not be influenced by cross-terms even if multiple moving targets exist. Moreover, to solve the problem whereby weak targets are shadowed by the sidelobes of strong ones, a new implementation of the CLEAN technique is proposed based on filtering in the fractional Fourier domain. In this way strong moving targets and weak ones can be detected iteratively. This combined method is demonstrated by using raw clutter data combined with simulated moving targets. <s> BIB002 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Radar applications <s> The fractional Fourier transform (FrFT), which is a generalized form of the well-known Fourier transform, has only recently started to appear in the field of signal processing. This has opened up the possibility of a new range of potentially promising and useful applications. In this letter, we develop a new FrFT-based chirp scaling algorithm (CSA) and compare its performance with the classical CSA based on the fast Fourier transform (FFT). Simulation results show that the FrFT-based CSA can offer significantly enhanced features compared to the classical FFT-based approach. <s> BIB003 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Radar applications <s> The adaptive local polynomial Fourier transform is employed for improvement of the ISAR images in complex reflector geometry cases, as well as in cases of fast maneuvering targets. It has been shown that this simple technique can produce significantly improved results with a relatively modest calculation burden. Two forms of the adaptive LPFT are proposed. Adaptive parameter in the first form is calculated for each radar chirp. Additional refinement is performed by using information from the adjacent chirps. The second technique is based on determination of the adaptive parameter for different parts of the radar image. Numerical analysis demonstrates accuracy of the proposed techniques. <s> BIB004 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Radar applications <s> Inverse synthetic aperture radar (ISAR) imaging systems produce electromagnetic images of targets in the range-Doppler domain. In order to rescale the image in a homogeneous range-cross range domain (meters by meters), the modulus of the target effective rotation vector must be known. Although in some cases it can be retrieved by means of ancillary data, in most cases the modulus of the target effective rotation vector must be estimated. A blind technique is proposed for estimating the modulus of the target effective rotation vector that exploits information carried by the chirp rate of scattering centres. A technique based on image segmentation, local polynomial Fourier transform (LPFT), and image contrast (IC) maximisation is used in order to extract the scattering centres and estimate their chirp rate. Simulated and real data analyses are provided to confirm the effectiveness of the proposed algorithm. <s> BIB005 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Radar applications <s> The polynomial Fourier transform (FT) is employed as a tool for the SAR imaging of moving targets. An efficient algorithm is proposed that can be used for radar images containing both moving and stationary targets. The proposed algorithm can be used in the form of the second-order polynomial FT, but it can also be extended to higher-order and order-adaptive polynomial FT forms. <s> BIB006 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Radar applications <s> This paper presents the quantitative analysis of signal-to-noise ratio (SNR) for the local polynomial Fourier transform (LPFT) used for application of inverse synthetic aperture radar (ISAR) imaging. The relationship between the LPFT and Wigner-Ville distribution (WVD) is derived and theoretical analysis on the SNR achieved by using the LPFT is given. Comparisons on the SNR performances achieved by using the LPFT, the short-time Fourier transform (STFT) and the Fourier transform (FT) are presented to illustrate the merits of using the LPFT. Measures are also taken to minimize the required computational complexity by reducing the overlap length between adjacent segments of input data. Simulation results of processing radar data show the performance improvements achieved by using the LPFT. <s> BIB007 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Radar applications <s> In the past 10 years the Time-Frequency (TF) techniques have found significant importance for improvement and refinement of radar images (both SAR and ISAR) and in extraction of objects' features from radar images. The main contributions in this direction are given by V. Chen (Chen and Ling, 2002). Here, we will concentrate our attention on the radar imaging and two novel strategies: S-Method (SM) imaging and Local Polynomial Fourier Transform (LPFT) imaging. The application of these ideas to radar imaging, as well as a procedure for setting design parameter of the algorithms, will be presented. Transformations will be compared in terms of calculation complexity and quality of the resulted radar images. Moreover, it will be demonstrated that the LPFT-based algorithm has better robustness to additive noise than the SM or the adaptive SM. <s> BIB008 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Radar applications <s> The local polynomial Fourier transform (LPFT) based algorithm for auto-focusing SAR images has recently been proposed by the authors. It produces a well focused image of moving targets, without defocusing stationary targets or inducing undesired cross-terms. The drawback of this algorithm is its high computational burden caused by the large number of elements in the set of used chirp-rates. We propose an algorithm with decreased number of elements used for the LPFT-based SAR imaging. The product high-order ambiguity function (PHAF) is applied to estimate parameters of a radar signal. The estimated chirp-rate is used as an initial value for forming the set of chirp-rates. The proposed algorithm has significantly smaller set of chirp-rate values (tens comparing to several hundreds or thousands used in the previous algorithm version). In this manner, the calculation complexity is significantly reduced. The proposed procedure is fully automated, meaning that it follows the change of motion parameters. In addition, our procedure considers the third-order phase compensation. <s> BIB009
A couple of techniques related to the FRFT have been recently used for focusing SAR/ISAR images (e.g., BIB002 BIB003 ). This technique is commonly referred to as the LPFT in the related literature. Two forms of the adaptive LPFT-based technique for focusing ISAR images are proposed in BIB004 . The focusing is performed without assuming any particular model of a target's motion. Therefore, these versions of the LPFT can be applied for any realistic motion. The first technique is based on knowing that, for monocomponent and multicomponent signals with similar chirp-rates, a single chirp-rate parameter can be estimated for all components. The ISAR image is then focused by using the estimated value of signal parameters obtained through the LPFT calculation. For multicomponent signals with different chirp-rates, a sum of the weighted LPFT is used. If the signal's components have significantly different chirp-rates, the estimation of these parameters should be performed separately for each component. Moreover, the obtained estimate is additionally refined by combining estimations obtained for close reflectors. For targets with very complex motion patterns, segmentation of the radar image in regions-of-interests is performed. The adaptive LPFT is calculated for each region in order to form focused ISAR image. Adaptive parameters are obtained by using a simple concentration measure. Here, we demonstrate test image of the B727 plane given in Fig. 7(a) (the figure is taken from BIB004 ). Focusing of the radar image is performed for each cross-range parameter and the focused image is given in Fig. 7(b) . Estimated chirp-rate parameters (angles in the FRFT terminology) are depicted with dashed lines in Fig. 7(c) . Additional improvements have been achieved when these chirp-rates are filtered with median filter (thick line). Then, the radar images focused with the filtered chirp-rate as depicted in Fig. 7(d) . A similar approach is proposed in BIB005 , but the chirprate is estimated by using contrast of the LPFT-based ISAR image. Moreover, a modulus of the target effective rotation vector is calculated from the obtained estimation. A quantitative analysis of the signal-to-noise ratio (SNR) for the LPFT applied in the ISAR imaging is presented in BIB007 . It has been shown that the LPFT-based methods for focusing ISAR images can achieve a significantly higher output SNR than those based on the STFT. Focusing of the SAR images is usually performed by estimating parameters of a received signal. A technique for applying the product higher-order ambiguity function (PHAF) for estimating parameters of the radar signal and focusing SAR images is proposed in BIB001 . Moving target focusing can be performed by this technique for some specific scenarios. More precisely, the obtained SAR image will be focused only when one target exists in a range bin, or when all targets in one range bin have similar motion parameters. Otherwise, this technique fails to achieve high concentration of each target simultaneously. In order to overcome this drawback, an algorithm for separating signal components corresponding to targets with different motion parameters is applied in BIB006 where the LPFTbased technique is used for focusing each component. The drawback of the LPFT-based technique, applied in this manner, is high computational complexity needed for selection of chirp-rate that produces the best concentration. Therefore, an algorithm for the automated selection of phase parameters used for the LPFT calculation is proposed in BIB009 . In this algorithm, an adaptive set of chirp-rates is formed for each unfocused target. When one or more unfocused targets are detected in a range, the PHAF is evaluated and the position of its maximum is used for the coarse estimation of chirp-rate. Further improvement of the obtained estimate is performed by using a fine search in the region around the chirp-rate obtained from the coarse search. The number of chirp-rates in the fine search stage is rather small (usually not larger than several tens) which yields a significant decrease in the computational complexity with respect to the technique proposed in BIB006 (which requires hundreds or thousands). In addition, a procedure for the third-order phase compensation is applied in the proposed algorithm without a significant increase of the calculation burden. In BIB006 BIB009 BIB008 it has been shown that the LPFT-based methods for focusing SAR images are more robust to the additive noise influence than the STFT radar imaging techniques. Here, we have borrowed figure from BIB009 where the SAR image of several moving and stationary targets is shown ( Fig. 8(a) ). It can be seen that the moving targets have seriously spread components. Alternative tools such as a technique called the S-method  are unable to separate overlapping targets (Fig. 8(b) ). The LPFT technique with predefined set of the chirp-rates gives significantly improved results as shown in Fig. 8(c) . However, the procedure proposed in BIB009 based on the LPFT with automatic selection of the chirp-rate parameters, produces highly focusing of all considered targets (Fig. 8(d) ).