Techniques for arbitrary sampling in two-dimensional Fourier transforms

The Discrete Fourier Transform (DFT), which can be calculated efficiently by the Fast Fourier Transform (FFT), is one of the most commonly used tools for frequency estimation of a multi-frequency exponential signal. However, the performance of the FFT is inherently affected by spectral leakage when non-coherent sampling occurs. The Chirp-Z Transform (CZT), which can refine the spectrum to the narrowband range of the source signal accurately, can be generalized to a special form of DFT. This paper proposes a CZT based algorithm which can estimate frequencies of the multi-frequency exponential signal accurately. In our method, the CZT bin, as a nonlinear equation, is firstly used to describe the correspondence among the frequencies, amplitudes, phases and the observed spectrum. Then, the above nonlinear equation is rewritten as a linear equation with unknown frequencies. We further extend the proposed method to the Discrete-time Fourier Transform (DTFT), zero-padding FFT and DFT, which are the special cases of the CZT. We also give the derivation procedure of theoretical variance of our method. In our method, the spectrum leakage is fully considered as useful information to establish the linear equation set. Therefore, it can estimate the frequencies of multi-frequency exponential signal with high accuracy. Simulation results verify the performance of the proposed method.

ECG compression cut down on the amount of space needed to develop a telecardiology system for cardiac diagnosis and analysis. In this paper, ECG signal compression is based on different compression techniques, such as Discrete Cosine Transform (DCT), DCT2, and DCT3, Fast Fourier Transform (FFT), Fast Fourier Transform (FFT2), Discrete Cosine Transform (DCT), Chirp Z Transform (CZT), and Complex Cepstral Analysis. After the ECG signal has been compressed, each method is compared one by one. From the MIT-BIH arrhythmia database, different ECG signals are used to test these techniques. In terms of CR, PRD, QS, and SNR, the experimental results show that FFT signal compression is better.

Density Functional Theory calculations with plane waves and pseudopotentials represent one of the most important simulation techniques in high performance computing. Together with parallel linear algebra (ZGEMM and matrix diagonalization), the most important bottleneck results from the Fast Fourier Transform (FFT), required, for example, when the local potential is applied to the wavefunction. In these calculations, the existence of a cutoff on the plane waves is reflected on a spherical domain for the FFT. After a 1D FFT is performed on pencils distributed among processors, data is transposed with a MPI_Alltoall and a 2D FFT is executed [2]. Typically, the workload of the FFT is not particularly high, since grid sizes do not exceed (103 102)3. However, the load distribution is crucial and the consequent impact of collective communications becomes a critical factor for achieving a high parallel efficiency. Quantum ESPRESSO [3] is one of the most used codes based on plane-wave DFT in the community of material science. It has been successfully ported and optimized on a large number of HPC infrastructures all over the world. The parallel structure of Quantum ESPRESSO is mainly based on several layers of MPI communicators, plus a finer grain OpenMP parallelization. Recently, the parallelization structure of the FFT was deeply refactored. The combination of two different data distributions, i.e. bands and taskgroups, allow the underlyinghardware to be hierarchically filled and two different layers of communications to be tuned. In particular, with sufficient memory, by tuning the number of taskgroups one can fit all the data required to perform a single 3D FFT reducing the impact of the MPI_Alltoall between the 1D and 2D FFTs. In order to better check the results of the parametrization of the parallel distributions, a miniapp [1] containing only the FFT kernel was extracted from the Quantum ESPRESSO distribution. This miniapp is also important for the future activity of code design of novel architectures. We present and discuss the profiling data obtained from the QE-FFT miniapp and the impact on the communication pattern deriving from the choice of the parallelization parameters.

Nondestructive testing (NDT) methods for identifying stress levels in materials mostly rely on the theory of acoustoelasticity. However, the sensitivity and the accuracy of acoustoelasticity are affected by several factors such as the (1) type, (2) propagation, and (3) polarization directions of the used signals. This paper presents the results of an experimental investigation of longitudinal waves propagating perpendicular to the applied uniaxial tensile stresses in structural steel specimens. The changes in four ultrasonic signal characteristics were investigated with increasing stress levels ranging from below to above the yield stress of steel. The considered signal characteristics were the peak amplitudes and signal energy in the time domain, and the fast Fourier transform (FFT) and chirp-Z transform (CZT) in the frequency domain. Even though the acoustoelastic effect on the type ultrasonic signal used is very small, clear distinctions between prior to and postyielding are observed for all investigated parameters. The results are presented with a detailed statistical and receiver operating characteristics (ROCs) analyses. The results show that identifying damage to steel structures due to local yielding is possible using the simple ultrasonic signal classification.

Deceptive jamming against synthetic aperture radar (SAR) receives intensive interests during the past decade. However, it is still a challenging task to design a jamming method that is competent both in focus capability and in computational efficiency, especially in the case where jammer is confronted with SAR with significant squint angle and long synthetic aperture. In this paper, we propose an inverse omega-K algorithm and present an accurate and an approximate implementation scheme of the algorithm. The accurate scheme can achieve full focus with no regard to the squint angle and synthetic aperture of radar. Its computations include fast Fourier transform (FFT), Stolt interpolation, and complex multiply. Advantage of computational efficiency can be achieved under assumption that the support region of radar can be crudely evaluated a priori so that the most time-consuming Stolt interpolation can be done offline. The support region is determined by carrier frequency and bandwidth of radar signal, pointing direction, and azimuth beam width of radar antenna. For the case in which the support region of radar is not available to jammer beforehand, the approximate scheme is a remedy. By substituting the Stolt interpolation with Chirp-Z transform (CZT), the approximate scheme is readily fit for parallel computation and hence appealing for its high efficiency. However, the focus criterion exerts a limitation on range scale of electromagnetic deception when the squint angle of SAR is large. Both implementation schemes are verified by simulation results.

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.

A decimation-in-time radix-2 fast Fourier transform (FFT) algorithm is considered here for implementation in multiprocessors with shared bus, multistage interconnection network (MIN), and in mesh connected computers. Results are derived for data allocation, interprocessor communication, approximate computation time, and speedup of an N point FFT on any P available processing elements (PE's). Further generalization is obtained for a radix-r FFT algorithm. An N X N point two-dimensional discrete Fourier transform (DFT) implementation is also considered when one or more rows of the input data matrix are allocated to each PE.

This research report describes a methodology whereby multi-spectral and hyperspectral imagery from remote sensing, is used for deriving predicted field maps of selected plant growth attributes which are required for precision cropping. A major task in precision cropping is to establish areas of the field that differ from the rest of the field and share a common characteristic. Yield distribution f maps can be prepared by yield monitors, which are available for some harvester types. Other field attributes of interest in precision cropping, e.g. soil properties, leaf Nitrate, biomass etc. are obtained by manual sampling of the filed in a grid pattern. Maps of various field attributes are then prepared from these samples by the "Inverse Distance" interpolation method or by Kriging. An improved interpolation method was developed which is based on minimizing the overall curvature of the resulting map. Such maps are the ground truth reference, used for training the algorithm that generates the predicted field maps from remote sensing imagery. Both the reference and the predicted maps are stratified into "Prototype Plots", e.g. 15xl5 blocks of 2m pixels whereby the block size is 30x30m. This averaging reduces the datasets to manageable size and significantly improves the typically poor repeatability of remote sensing imaging systems. In the first two years of the project we used the Normalized Difference Vegetation Index (NDVI), for generating predicted yield maps of sugar beets and com. The NDVI was computed from image cubes of three spectral bands, generated by an optically filtered three camera video imaging system. A two dimensional FFT based regression model Y=f(X), was used wherein Y was the reference map and X=NDVI was the predictor. The FFT regression method applies the "Wavelet Based", "Pixel Block" and "Image Rotation" transforms to the reference and remote images, prior to the Fast - Fourier Transform (FFT) Regression method with the "Phase Lock" option. A complex domain based map Yfft is derived by least squares minimization between the amplitude matrices of X and Y, via the 2D FFT. For one time predictions, the phase matrix of Y is combined with the amplitude matrix ofYfft, whereby an improved predicted map Yplock is formed. Usually, the residuals of Y plock versus Y are about half of the values of Yfft versus Y. For long term predictions, the phase matrix of a "field mask" is combined with the amplitude matrices of the reference image Y and the predicted image Yfft. The field mask is a binary image of a pre-selected region of interest in X and Y. The resultant maps Ypref and Ypred aremodified versions of Y and Yfft respectively. The residuals of Ypred versus Ypref are even lower than the residuals of Yplock versus Y. The maps, Ypref and Ypred represent a close consensus of two independent imaging methods which "view" the same target. In the last two years of the project our remote sensing capability was expanded by addition of a CASI II airborne hyperspectral imaging system and an ASD hyperspectral radiometer. Unfortunately, the cross-noice and poor repeatability problem we had in multi-spectral imaging was exasperated in hyperspectral imaging. We have been able to overcome this problem by over-flying each field twice in rapid succession and developing the Repeatability Index (RI). The RI quantifies the repeatability of each spectral band in the hyperspectral image cube. Thereby, it is possible to select the bands of higher repeatability for inclusion in the prediction model while bands of low repeatability are excluded. Further segregation of high and low repeatability bands takes place in the prediction model algorithm, which is based on a combination of a "Genetic Algorithm" and Partial Least Squares", (PLS-GA). In summary, modus operandi was developed, for deriving important plant growth attribute maps (yield, leaf nitrate, biomass and sugar percent in beets), from remote sensing imagery, with sufficient accuracy for precision cropping applications. This achievement is remarkable, given the inherently high cross-noice between the reference and remote imagery as well as the highly non-repeatable nature of remote sensing systems. The above methodologies may be readily adopted by commercial companies, which specialize in proving remotely sensed data to farmers.

A parallel architecture especially designed for a synthetic-aperture-radar (SAR) processing algorithm based on an appropriate two-dimensional fast Fourier transform (FFT) code is presented. The algorithm is briefly summarized, and the FFT code is given for the one-dimensional case, although all results can be immediately generalized to the double FFT. The computer architecture, which consists of a toroidal net with transputers on each node, is described. Parametric expressions for the computational time of the net versus the number of nodes are derived. The architecture allows drastic reduction of the processing time, preserving elaboration accuracy and flexibility. >

Series of thermodiffusion experiments using optical digital interferometry (ODI) have been conducted onboard the International Space Station. Conventionally, the two-dimensional (2D) fast Fourier transform (FFT) fringe analysis technique has been applied as a fast and reliable technique to extract data. In this study, for the first time, the windowed Fourier transform (WFT) method is used to analyse the same experiments. In this method, a Fourier transformation is applied on the fringes at two different stages: initially, during the filtration of the non-zero peaks and then on the wrapped phase image. We provide a detailed comparison between FFT and WFT results of binary and ternary mixtures for ODI thermodiffusion experiments. The substantial enhancements of this method are presented and discussed for different experiments conducted for both binary and ternary mixtures. We show that while disturbances in the phase fringe pattern can cause significant error in FFT techniques, if the windowed Fourier filtration (WFF) parameters are properly chosen this type of noise can be eliminated during WFF analysis. The importance of replacing the FFT method becomes more pronounced for the ternary system, as this method fails to reconcile reliable concentration profiles. The results of this chapter can show that the application of the windowed Fourier transform in optical digital interferometry investigations show improved results over the same experiments analysed using FFT methods, especially for experiments involving very small heat and mass fluxes such as the Soret effect in multicomponent mixtures.

Efficient implementation of the Fourier dual reciprocity boundary element method using two-dimensional fast Fourier transforms

Spectrum analysis is very important in geological hazards of infrasonic signal observation systems. The spectrum of infrasonic signal is a dense spectrum which can leads to potential erroneous spectrum analysis. Hereby we propose a dense spectrum analysis algorithm combining all phase Fast Fourier Transform (apFFT) and Chirp Z-transform (CZT) to analyse dense low frequency signal. This is called all phase Chirp Z transform (apCZT). The apFFT spectrum analysis can reduce spectrum leakage, but does not enhance resolution while the CZT vice versa. The novel algorithm apCZT can suppress spectral leakage and improve the resolution at the same time. Simulation results demonstrate that the apCZT algorithm can distinguish the frequencies whose intervals are less than the ordinary frequency resolving power of Discrete Fourier Transform (DFT) and apFFT. The apCZT it is not only suitable for infrasonic signals but also in other dense spectrum analysis applications, such as voice, vibration, noise, electrocardiography, radar signals, power system harmonics and other engineering practice.

Compared with traditional Fast Fourier Transform (FFT) algorithms, FFT pruning is more computationally efficient in those cases where some of the input values are zero and/or some of the output components are not needed. In this letter, a novel pruning scheme is developed for mixed-radix and high-radix FFT pruning. The proposed approach is applicable over a wide range of FFT lengths and input/output pruning patterns. In addition, it can effectively employ the benefits of high-radix FFT algorithms that have lower computational complexity.

Kern's plane-wave scattering-matrix formulation is extended to treat the case of two interfaces plus a scatterer imbedded in the second region. This formulation can treat different shapes for the scatterer and only requires two different evaluations depending upon the location of the observer: 1) in the near field, a two-dimensional (2-D) fast Fourier transform (FFT) (one-dimensional (1-D) FFT is treated by an analytical integration), and 2) in the far-field an asymptotic evaluation of the integral. These two regimes are in contrast to the Sommerfeld integral approximations where different approximations are required for various parameter ranges (quasistatic, saddle-point evaluation for the radiation field, saddle point evaluation for the surface field when the saddle point is near a pole, and the lateral wave field evaluated using a uniform asymptotic evaluation for the integrals).

A system for obtaining two-dimensional images of an object using multifrequency microwave holography is presented. The phase variation of the reflected signal from an object as a function of frequency of illumination and angle of incidence is used to achieve resolution in range and cross-range. The processing of the reflected signal is accomplished by a two-dimensional discrete Fourier transform implemented by fast Fourier transform (FFT) algorithms. Examples of images obtained by processing simulated and experimental data are presented. The processing technique for obtaining focused three-dimensional images of an object is also discussed. >