Chirp Z-transform Research Articles

Flexible time domain averaging technique

Time domain averaging(TDA) is essentially a comb filter, it cannot extract the specified harmonics which may be caused by some faults, such as gear eccentric. Meanwhile, TDA always suffers from period cutting error(PCE) to different extent. Several improved TDA methods have been proposed, however they cannot completely eliminate the waveform reconstruction error caused by PCE. In order to overcome the shortcomings of conventional methods, a flexible time domain averaging(FTDA) technique is established, which adapts to the analyzed signal through adjusting each harmonic of the comb filter. In this technique, the explicit form of FTDA is first constructed by frequency domain sampling. Subsequently, chirp Z-transform(CZT) is employed in the algorithm of FTDA, which can improve the calculating efficiency significantly. Since the signal is reconstructed in the continuous time domain, there is no PCE in the FTDA. To validate the effectiveness of FTDA in the signal de-noising, interpolation and harmonic reconstruction, a simulated multi-components periodic signal that corrupted by noise is processed by FTDA. The simulation results show that the FTDA is capable of recovering the periodic components from the background noise effectively. Moreover, it can improve the signal-to-noise ratio by 7.9 dB compared with conventional ones. Experiments are also carried out on gearbox test rigs with chipped tooth and eccentricity gear, respectively. It is shown that the FTDA can identify the direction and severity of the eccentricity gear, and further enhances the amplitudes of impulses by 35%. The proposed technique not only solves the problem of PCE, but also provides a useful tool for the fault symptom extraction of rotating machinery.

InSAR Adaptive Extended Kalman Filter Phase Unwrapping Algorithm

Phase Unwrapping (PU) is the key step in the image processing for Interferometric Synthetic Aperture Radar (InSAR). In the Extended Kalman Filter (EKF) model of PU, due to the state space model is not taken into account the terrain factors, it is often resulted in unwrapping error delivery as the pixel to the next when the state changes rapidly in steep terrain. The observation equation is nonlinear and usually applied in PU through linear processing, requiring the system model and noise statistics known. But in fact the mathematical model or statistical noise is completely or partially unknown; the results have been inevitably lead to the declining of valuation accuracy and filter divergence. If directly applied in phase unwrapping, it is made impossible to retrieve surface deformation. In order to solve this problem and fully consider the terrain effect and model error, an adaptive EKF PU algorithm (AEKFPU) for InSAR is presented. On the one hand, it is achieved local adaptive estimation of image fringe frequency through 2D FFT and Chirp-Z Transform (CZT) joint method, by considering the impact of terrain factors on unwrapping results; On the one hand, the fading factor is calculated by innovation covariance and adaptively adjusted with the error covariance so as to suppress the memory length of the filter, compensating the effect of incomplete information on unwrapping. The experimental results are proved the proposed method is effective, it can be dealt with phase unwrapping and filtering simultaneously, and can be adaptively considered terrain factors in state space model and compensated for model error in observation equation model, ultimately improving the accuracy of phase unwrapping.

Generalized Goertzel algorithm for computing the natural frequencies of cantilever beams

The paper proposes a fast algorithm for accurate estimation of frequency within a specified, narrow range. The algorithm is useful for the identification of the natural frequencies of cantilever beams for damage detections purposes. The procedure is based on the generalized Goertzel algorithm combined with apriori knowledge of the natural frequencies intervals for cantilever beams given their physical characteristics. We compare our approach with the Chirp Z-transform and several frequency or time-frequency methods to illustrate its advantages for online damage detection.

AN EXTENDED INVERSE CHIRP-Z TRANSFORM ALGORITHM TO PROCESS HIGH SQUINT SAR DATA

This paper proposes an Extended Inverse Chirp-Z Transform (EICZT) algorithm to handle the high squint FMCW SAR data, where the conventional Inverse Chirp-Z Transform (ICZT) cannot work due to the failure in dealing with the range-variance of second- and higher-order range-azimuth coupling terms. A pre- processing operation is implemented in the azimuth-Doppler and range-time (Doppler-time) domain, where a perturbation function consisting of second-order and third-order range time variables is implemented to compensate the range variance of the second order range terms. Moreover, a new scaling factor is formulated to represent the Range Cell Migration (RCM), and further corrected by the presented EICZT approach. The proposed approach is analyzed and compared with the conventional ICZT. The simulated high squint SAR scene with nine targets is well focused by the proposed approach and the quality is greatly improved with respect the conventional ICZT. The proposed algorithm is also validated by the X-band high-resolution real SAR data.

High-Resolution Broadband Millimeter-Wave Astrophysical Spectrometer with Triple Product Acousto-Optical Processor

An advanced conceptual design of a high-bit-rate triple product acousto-optical processor is presented that can be applied in a number of astrophysical problems. We briefly describe the Large Millimeter Telescope as one of the potential observational infrastructures where the acousto-optical spectrometer can be successfully used. A summary on the study of molecular gas in relatively old (age > 10 Myr) disks around main sequence stars is provided. We have identified this as one of the science cases in which the proposed processor can have a big impact. Then we put forward triple product acousto-optical processor is able to realize algorithm of the space-and-time integrating, which is desirable for a wideband spectrum analysis of radio-wave signals with an improved resolution providing the resolution power of about 105 - 106. It includes 1D-acousto-optic cells as the input devices for a 2D-optical data processing. The importance of this algorithm is based on exploiting the chirp Z-transform technique providing a 2D-Fourier transform of the input signals. The system produces the folded spectrum, accumulating advantages of both space and time integrating. Its frequency bandwidth is practically equal to the bandwidth of transducers inherent in acousto-optical cells. Then, similar processor is able to provide really high frequency resolution, which is practically equal to the reciprocal of the CCD-matrix photo-detector integration time. Here, the current state of developing the triple product acousto-optical processor in frames of the astrophysical instrumentation is shortly discussed.

Efficient and precise frequency-modulated continuous wave synthetic aperture radar raw signal simulation approach for extended scenes

Frequency-modulated continuous wave (FMCW) synthetic aperture radar (SAR) system is widely used for earth observation and space detection, where short revisits time at low cost are desirable, or small size is required. Therefore a FMCW SAR raw signal simulator is required to quantitatively support the design of a SAR mission operating in the FMCW mode, to help mission planning, test processing algorithms and analyse jamming and noises. Time-domain SAR raw signal simulation can be easily conceived, but it is extremely time and memory consuming. In this study, an efficient FMCW SAR raw signal simulation approach for extended scene is proposed based on chirp-Z transform (CZT). This is the first time that a FMCW SAR simulator is proposed in the two-dimensional frequency domain. By taking advantage of fast Fourier transform, the proposed approach highly reduces the computational load with respect to the time domain approach. Furthermore, CZT algorithm can accurately simulate the range-variation of the range cell migration term, which guarantees the precision of the proposed approach. Wavenumber domain algorithm is applied to focus the conceived raw signal. Finally, point targets, the electromagnetic imaging and real FMCW image are used to validate the proposed simulation approach.

Space–time Radon–Fourier transform and applications in radar target detection

This study proposes the Space–time Radon–Fourier transform (STRFT) for wideband digital array radar (DAR). It is a novel coherent detection technique that jointly realise wideband digital beamforming (DBF), range compression and long-time coherent integration. The likelihood ratio detector (LRT) is first derived based on the three-dimensional (3D) signal model of moving targets, which is the output of the STRFT on the 3D range-compressed echoes. Some transform properties of the proposed STRFT are also discussed. This include the 3D impulse response, 3D translational invariance, multi-target linear additivity, linear SNR gain in additive white Gaussian noise (AWGN), as well as the 3D correlation function of transformed AWGN. Also, the fast implementation of STRFT is proposed in the element-pulse-range frequency domain, which is realised via the two dimensional Chirp-Z transform (CZT) based on fast Fourier transform (FFT) operators. The proposed method may remarkably reduce the computational burden for modern wideband DAR. Finally, some numerical experiments are provided to demonstrate the effectiveness of the proposed methods.

Applications of chirp z transform and multiple modulation zoom spectrum to pulse phase thermography inspection

The fine spectrum structure is required when some key frequencies are estimated; however the implementation may be hindered due to insufficient frequency resolution. In this work, two types of transforms were introduced into Pulse Phase Thermography (PPT), the first was Chirp Z Transform (CZT), which was used for the badly sampled or short truncated data, and the other was Multiple Modulation Zoom spectrum Analysis (ZFFT), used for the long duration data. The CZT can refine the spectrum and improve the frequency resolution; consequently the subtle spectral structure of the selected local band can be observed for purchasing the precise frequency; furthermore time consumption for long point FFT calculation can be greatly alleviated by the ZFFT without worsening the resolution. The comparative test results demonstrate that the proposed CZT can obtain more accurate characteristic frequencies; in addition, the reconstructive images from the developed method are better than the ones by conventional technique, simultaneously ZFFT can save calculation time and storage space largely.

Influence of solar variability, CO2 and orbital forcing during the preindustrial part of the last millennium in the IPSLCM4 model

Linearly frequency modulation (LFM) pulse train and linearly stepped frequency (LSF) pulse train are mostly used in radar systems. However, the estimation performance of target motion parameters may be affected by the high recurrent lobe levels and the range–Doppler coupling phenomenon appearing in ambiguity function (AF). In multi-target scenarios, the estimation performance becomes even worse. The Costas frequency-modulation coded (CFMC) LFM pulse train has an ideal thumbtack-shaped AF, thus it can provide motion parameter estimation with high precision. However, the estimation of target motion parameter for the coherent CFMC LFM pulse train has not been investigated in depth. In this paper, we first analyze the properties of the AF of the CFMC LFM pulse train. Based on its convexity and narrow mainlobe width, a fast method to implement the maximum likelihood estimator (MLE) is proposed to estimate the motion parameters. To reduce the computation complexity, the Chirp-Z Transform (CZT) is introduced. Then, the Cramer–Rao lower bounds (CRLBs) on range, velocity and acceleration for frequency-modulation coded (FMC) pulse train are derived. It is shown that the CRLBs are relevant to the frequency coding patterns. Finally, Monte Carlo simulations are performed to verify the performance of the MLE. The results show that the performance of our proposed method can achieve the CRLBs when the signal-noise ratio (SNR) is higher than the threshold SNR.

UWB 안테나 링크 특성화를 위한 네트워크 분석기를 이용한 시간영역 측정 및 분석기술 연구

본 논문에서는 임펄스 신호를 방사하는 UWB 안테나의 링크 특성을 분석하기 위해 주파수영역에서 안테나 특성을 측정하기 위해 사용하는 네트워크 분석기를 이용한 시간영역 측정 및 분석기술에 대해 연구하였다. 이를 위해 네트워크 분석기에서 획득한 데이터를 줌-인(Zoom-In)이 가능한 시간영역으로 변환하기 위한 CZT(Chirp z-Transform) 알고리즘을 개발하였다. 개발된 CZT 알고리즘을 이용하여 주파수영역에서 각 안테나 링크에 대한 파형분석을 통해 표준이산푸리에 변환(Discrete Fourier Transform)보다 고속으로 변환이 가능하고 임의의 원하는 구간을 줌-인 함으로서 효율적으로 파형분석이 가능함을 증명하였다.

Diagnosis of Induction Motor Faults via Gabor Analysis of the Current in Transient Regime

Time-frequency analysis of the transient current in induction motors (IMs) is the basis of the transient motor current signature analysis diagnosis method. IM faults can be accurately identified by detecting the characteristic pattern that each type of fault produces in the time-frequency plane during a speed transient. Diverse transforms have been proposed to generate a 2-D time-frequency representation of the current, such as the short time Fourier transform (FT), the wavelet transform, or the Wigner-Ville distribution. However, a fine tuning of their parameters is needed in order to obtain a high-resolution image of the fault in the time-frequency domain, and they also require a much higher processing effort than traditional diagnosis techniques, such as the FT. The new method proposed in this paper addresses both problems using the Gabor analysis of the current via the chirp z-transform, which can be easily adapted to generate high-resolution time-frequency stamps of different types of faults. In this paper, it is used to diagnose broken bars and mixed eccentricity faults of an IM using the current during a startup transient. This new approach is theoretically introduced and experimentally validated with a 1.1-kW commercial motor in faulty and healthy conditions.

Frequency Offset Estimation for Optical Coherent M-QAM Detection Using Chirp z-Transform

The chirp z-Transform (CZT) is introduced to replace the traditional gradient descent (GD) in frequency offset estimation for digital coherent quadrature amplitude modulation reception. Different from GD, CZT is not sensitive to the changes of the parameters and is free of uncertain latency that GD suffers. The implementation of frequency offset estimator based on CZT is presented and numeric simulations demonstrate that CZT can reach the same accuracy as GD. The computation complexity of CZT is analyzed and is roughly equivalent to GD with three iterations.

A Non-linear Scaling Algorithm Based on chirp-z Transform for Squint Mode FMCW-SAR

该文针对调频连续波合成孔径雷达(Frequency Modulated Continuous Wave Synthetic Aperture Radar, FMCW-SAR)在斜视模式下聚焦精度降低的问题，提出一种非线性尺度变换chirp-z 成像算法。该算法利用回波信号在多普勒域中距离向的非线性特性，通过引入一个非线性调频信号对回波信号进行基于chirp-z 的非线性尺度变换，降低距离压缩和距离徙动校正的误差，提高雷达图像分辨率。采用该成像算法对点目标模拟回波的成像结果与原始chirp-z 算法相比，距离向分辨率和图像对比度得到提高，方位向分辨率保持不变。

IMAGING THE EPOCH OF REIONIZATION: LIMITATIONS FROM FOREGROUND CONFUSION AND IMAGING ALGORITHMS

Tomography of redshifted 21 cm transition from neutral Hydrogen using Fourier synthesis telescopes is a promising tool to study the Epoch of Reionization (EoR). Limiting the confusion from Galactic and Extragalactic foregrounds is critical to the success of these telescopes. Instrumental response or the Point Spread Function (PSF) of such telescopes is inherently 3 dimensional with frequency mapping to the Line of Sight (LOS) distance. EoR signals will necessarily have to be detected in data where continuum confusion persists; therefore, it is important that the PSF has acceptable frequency structure so that the residual foreground does not confuse the EoR signature. This paper aims to understand the 3 dimensional PSF and foreground contamination in the same framework. We develop a formalism to estimate the foreground contamination along frequency, or equivalently LOS dimension, and establish a relationship between foreground contamination in the image plane and visibility weights on the Fourier plane. We identify two dominant sources of LOS foreground contamination-'PSF contamination' and 'gridding contamination'. We show that 'PSF contamination' is localized in LOS wavenumber space, beyond which there potentially exists an 'EoR window' with negligible foreground contamination where we may focus our efforts to detect EoR. 'PSF contamination' in this window may be substantially reduced by judicious choice of a frequency window function. Gridding and imaging algorithms create additional 'gridding contamination' and we propose a new imaging algorithm using the Chirp Z Transform (CZT) that significantly reduces this contamination. Finally, we demonstrate the analytical relationships and the merit of the new imaging algorithm for the case of imaging with the Murchison Widefield Array (MWA).

Fast vectorial calculation of the volumetric focused field distribution by using a three-dimensional Fourier transform

We show that the volumetric field distribution in the focal region of a high numerical aperture focusing system can be efficiently calculated with a three-dimensional Fourier transform. In addition to focusing in a single medium, the method is able to calculate the more complex case of focusing through a planar interface between two media of mismatched refractive indices. The use of the chirp z-transform in our numerical implementation of the method allows us to perform fast calculations of the three-dimensional focused field distribution with good accuracy.

Real-time DBS imaging algorithm based on chirp z-transform

Real-time DBS imaging algorithm based on chirp z-transform

BISTATIC FMCW SAR RAW SIGNAL SIMULATOR FOR EXTENDED SCENES

By mounting the transmitter and receiver of Frequency Modulated Continuous Wave (FMCW) Synthetic Aperture Radar (SAR) system on separate platforms, bistatic FMCW SAR ofiers more considerable capabilities, reliability and ∞exibility while maintaining the small size, low cost and agile reaction. The bistatic FMCW SAR raw signal simulator is highly required to quantitatively support the design of bistatic FMCW SAR, to help mission planning, test processing algorithms, and analyze jamming and noises. Bistatic FMCW SAR raw signal can be exactly simulated target-by-target in time domain but with extremely time and memory consuming, especially when extended scenes are considered. In this paper, bistatic FMCW SAR signal model and Bistatic Point Target Reference Spectrum (BPTRS) is proposed, based on which a raw signal simulator is developed in the 2-D frequency domain for the flrst time, where Chirp-Z Transform (CZT) is used to formulate the range migration terms. By taking advantage of Fast Fourier Transform (FFT), the proposed raw signal simulator highly reduces the computational load with respect to the time domain approach. The simulated raw data is verifled by analyzing the corresponding images focused by Range Doppler Algorithm (RDA).

A FAST INVERSE POLYNOMIAL RECONSTRUCTION METHOD BASED ON CONFORMAL FOURIER TRANSFORMATION

A fast Inverse Polynomial Reconstruction Method (IPRM) is proposed to e-ciently eliminate the Gibbs phenomenon in Fourier reconstruction of discontinuous functions. The framework of the fast IPRM is modifled by reconstructing the function in discretized elements, then the Conformal Fourier Transform (CFT) and the Chirp Z-Transform (CZT) algorithms are applied to accelerate the evaluation of reconstruction coe-cients. The memory cost of the fast IPRM is also signiflcantly reduced, owing to the transformation matrix being discretized in the modifled framework. The computation complexity and memory cost of the fast IPRM are O(MN log2L) and O(MN), respectively, where L is the number of the discretized elements, M is the degree of polynomials for the reconstruction of each element, and N is the number of the Fourier series. Numerical results demonstrate that the fast IPRM method not only inherits the robustness of the Generalized IPRM (G-IPRM) method, but also signiflcantly reduces the computation time and the memory cost. Therefore, the fast IPRM method is useful for the pseudospectral time domain methods and for the volume integral equation of the discontinuous material distributions.

Different Zoom Approaches for Improving Spectral Resolution with Applications in Radar Signal Processing

Different approaches for a high-resolution analysis of narrow-band spectra are reviewed and compared. Partial-band algorithms are proved to be zoom-FFT's. In this contribution, three new modifications of the (Subband-FFT) SB-FFT are presented. In the first modification the chirp z-transform substitutes the small FFT which calculates the band of interest. In the second modification, the idea of zero-padding the input signal is applied to the SB-FFT with pruning at both input and output. Lastly zooming a small band of frequencies using a method of transforming by parts is applied for a narrow-band signal using the adaptive SB-FFT. A newly introduced version of the subband technique is included also in this work. In this version the subband decomposition technique is combined with the linear prediction method for higher spectrum resolution. Application of the SB-FFT and its modified versions and the new version in measuring the Doppler-frequency directly and indirectly for the purpose of vehicle-speed measurements is introduced in this paper. Comparison between all methods in terms of complexity and resolution is given. A new idea of channel test is included to keep the real-time successive measurements of Doppler frequency stable and consistent as well as simple.

A Chirp-Z Transform Imaging Algorithm for Missile-borne SAR with Diving Maneuver Based on the Method of Series Reversion

导弹下降段飞行速度快、飞行轨迹复杂，并且弹上成像实时性要求较高，给弹载SAR成像算法提出了新的要求。针对上述问题，该文提出了基于级数反演的CZT成像算法。文中首先给出了下降段点目标斜距表达式，并用泰勒级数对斜距进行高阶展开，接着用级数反演法得到了信号2维频谱高阶近似表达式，然后详细分析了频谱中各项的空变性对成像的影响，并在此基础上推导了改进的CZT成像算法并分析了其运算量。该算法能够精确校正空变的距离徙动，所有操作都由快速傅里叶变换和相位点乘完成，具有较高的效率。仿真结果证明，该文算法能够在下降段实现全孔径高精度成像。