基于EDFT的非均匀欠采样叶尖定时信号分析

针对叶尖定时信号因严重的非均匀采样和欠采样导致的谱分析难题,提出了基于扩展离散傅里叶变换(EDFT)的分析方法。不同于传统的傅里叶变换,EDFT是以傅里叶积分变换为目标,在扩展的频率范围内通过构造和优化变换基函数替代传统FFT变换中的指数基来实现非均匀和欠采样信号的谱分析,在分析中利用原始数据迭代和近似拟合保证了分析精度。该方法突破了奈奎斯特采样定理的限制,扩大了分析频率范围,分析谱线数也不再受限于采样点数,提高了频率分辨率。为了验证所提方法的可行性和可靠性,采用所构建的叶尖定时系统采样数学模型生成数据和试验台测试数据分别进行了方法应用分析,结果显示无论是仿真数据或试验数据,当传感器数量大于等于3时,基于该方法能够准确分析得到叶片振动的谱信号,表明该方法可以有效地解决非均匀欠采样叶尖定时信号的谱分析的难题,且具有良好的抗干扰性。     

Frequency de-aliased FFT analysis of step-like functions

The authors present a method for decreasing aliasing error in the fast Fourier transform (FFT) of step-like functions. This technique substantially decreases the sole remaining significant error in the extended function FFT (EF-FFT) method and is implemented by multiplying the EF-FFT spectral results with a simple de-aliasing function based on a piecewise linear model for the shape of the original function between data points. The attractive features of the aliasing error reduction method introduced, compared to increasing the sampling rate, are that data reacquisition is not required, computer requirements are small, and the spectra are of high accuracy up to the Nyquist frequency. The applicability is limited to functions that can be modeled with linear transitions between data points, as opposed to step transitions unless the precise timing and shape of the step transitions are known. Since most data sets are comprised of samples from slowly varying analog signals, the de-aliasing procedure provides enhanced spectral accuracy with minor additional mathematical complexity     

Application of autoregressive spectral analysis to cepstral estimation of mean scatterer spacing

The problem of estimation of mean scatterer spacing in an object containing regularly spaced structures is addressed. An autoregressive (AR) spectral estimation method is compared with a conventional fast Fourier transform (FFT)-based approach for this task. Regularly spaced structures produce a periodicity in the power spectrum of ultrasonic backscatter. This periodicity is manifested as a peak in the cepstrum. A phantom was constructed for comparison of the two methods. It contained regularly spaced nylon filaments. It also contained randomly positioned glass spheres that produced incoherent backscatter. In an experiment in which this target was interrogated using broadband ultrasound, the AR spectral estimate offered considerable improvement over the FFT when the analysis gate length was on the order of the structural dimension. Advantages included improved resolution, reduction in bias and variance of scatterer spacing estimates, and greater resistance to ringing artifacts. Data were also acquired from human liver in vivo. AR spectral estimates on human data exhibited a decreased dependence on gate length. These results offer promise for enhanced spatial resolution and accuracy in ultrasonic tissue characterization and nondestructive evaluation of materials.     

Precise and fast phase wraps reduction in fringe projection profilometry

Phase wraps in a 2D wrapped phase map can be completely eliminated or greatly reduced by frequency shifting. But it usually cannot be optimally reduced using conventional fast Fourier transform (FFT) because the spectrum can be shifted only by a integer number in the frequency domain. In order to achieve a significant phase wrap reduction, we propose a fast and precise two-step method for phase wraps reduction in this paper, which is based on the iterative local discrete Fourier transform (DFT). Firstly, initial estimate of the frequency peak is obtained by FFT. Then sub-pixel spectral peak with high resolution is determined by iteratively upsampling the local DFT around the initial peak location. Finally, frequency shifting algorithm that operates in the spatial domain is used to eliminate phase wraps. Simulations and experiments are conducted to demonstrate the superb computing efficiency and overall performance of the proposed method.     

Doppler angle estimation using AR modeling

The transit time spectrum broadening effect has long been explored for Doppler angle estimation. Given acoustic beam geometry, the Doppler angle can be derived based on the mean Doppler frequency and the Doppler bandwidth. Spectral estimators based on the fast Fourier transform (FFT) are typically used. One problem with this approach is that a long data acquisition time is required to achieve adequate spectral resolution, with typically 32-128 flow samples being needed. This makes the method unsuitable for real-time two-dimensional Doppler imaging. This paper proposes using an autoregressive (AR) model to obtain the Doppler spectrum using a small number (e.g., eight) of flow samples. The flow samples are properly selected, then extrapolated to ensure adequate spectral resolution. Because only a small number of samples are used, the data acquisition time is significantly reduced and real-time, two-dimensional Doppler angle estimation becomes feasible. The approach was evaluated using both simulated and experimental data. Flows with various degrees of velocity gradient were simulated, with the Doppler angle ranging from 20° to 75°. The results indicate that the AR method generally provided accurate Doppler bandwidth estimates. In addition, the AR method outperformed the FFT method at smaller Doppler angles. The experimental data for Doppler angles, ranging from 33° to 72°, showed that the AR method using only eight flow samples had an average estimation error of 3.6°, which compares favorably to the average error of 4.7° for the FFT method using 64 flow samples. Because accurate estimates can be obtained using a small number of flow samples, it is concluded that real-time, two-dimensional estimation of the Doppler angle over a wide range of angles is possible using the AR method     

Microprocessor-Based Walsh?Fourier Spectral Analyzer

A microprocessor-based Walsh-Fourier spectral analyzer is described. It includes the sampling of the incoming signal at 64 times the input signal frequency, using a special purpose frequency multiplier module (FMM), storing the digital data in a permuted sequence in the system memory under the control of a direct memory access (DMA) controller, and performing a fast Walsh-Hadamard transform (FWHT) of the permuted data sequence. The system uses a single board computer SBC80/10 in an Intel System 80/10 and a special purpose board which includes FMM, DMA, and A/D conversion circuits. Fourier coefficients are obtained via a Walsh to Fourier conversion algorithm; the total process is then faster than the Cooley-Tukey FFT algorithm for a data length of 64 or less.     

FFT‐based spectral element analysis for the linear continuum dynamic systems subjected to arbitrary initial conditions by using the pseudo‐force method

In this paper, a fast Fourier transform (FFT)‐based spectral element method (SEM) is developed for the linear continuum dynamic systems subjected to arbitrary, non‐null initial conditions. In the FFT‐based SEM, the original equations of motion subjected to arbitrary initial conditions are transformed into a new set of equations of motion subjected to completely null initial conditions by using the pseudo‐force method so that the conventional spectral element analysis can be applied to obtain desired dynamic responses. A simply supported beam and a cantilevered beam are considered as the illustrative problems to evaluate the FFT‐based SEM. The dynamic responses obtained by using the FFT‐based SEM are shown to be in good agreement with the analytical solutions obtained by using the mode superposition method. Copyright © 2007 John Wiley & Sons, Ltd.     

Measuring power system harmonics and interharmonics by an improved fast Fourier transform-based algorithm

The fast Fourier transform (FFT) has been widely used for the signal processing because of its computational efficiency. Because of the spectral leakage and picket-fence effects associated with the system fundamental frequency variation and improperly selected sampling time window, a direct application of the FFT algorithm with a constant sampling rate may lead to inaccurate results for continuously measuring power system harmonics and interharmonics. An improved FFT-based algorithm to measure harmonics and interharmonics accurately is proposed. In the proposed algorithm, a frequency-domain interpolation approach is adopted to determine the system fundamental frequency, and the interpolatory polynomial method is applied to reconstruct the sampled time-domain signal; it is followed by using the FFT to calculate the actual harmonic components. Then, the frequency-domain interpolation is again applied to find the interharmonic components. The performance of the proposed algorithm is validated by testing the actual measured waveforms. Results are compared with those obtained by directly applying a typical FFT algorithm and by the IEC grouping method. It shows that the solutions determined by the proposed algorithm are more accurate, and a reasonable computational efficiency is maintained.     

基于全相位FFT的油套环空中声速计算方法

油套环空中会产生各种噪声,使测得的接箍反射信号非常复杂,环隙中真实的声速则很难计算准确,可通过对原始接箍数据进行傅里叶变换的方法对声速进行估计,但是存在不可忽视的误差。全相位傅里叶变换是傅里叶变换的一种改进方法,能够获得更加准确的频率谱与相位谱。文章采用全相位快速傅里叶变换（all-phase Fast Fourier Transform,apFFT）得到原始接箍信号的频谱,然后通过该频谱进一步计算环隙声速,可得到更加准确的声速估计。通过对不同信噪比下的模拟接箍信号采用快速傅里叶变换（Fast Fourier Transform,FFT）和全相位快速傅里叶变换（apFFT）得到其频谱,可以验证apFFT具有很强的抑制频谱泄漏的能力,且抗噪性能比FFT更好。根据FFT谱和apFFT谱分别计算出声速并对比其精度,可以验证通过apFFT谱计算出的声速稳定性更好、精度更高。然后采用上述两种方法对不同深度井的实测接箍数据进行频谱分析与对比,验证了apFFT较之于FFT对谱峰位置的辨识能力更强,根据谱峰位置计算声速的准确性也将更高。     

Bayesian versus Fourier spectral analysis of ion cyclotron resonance time-domain signals

The frequency-domain spectrum obtained by Fourier transformation (FT) of a time-domain signal is accurate only for a continuous noiseless time-domain signal of infinite duration. For discrete noisy truncated time-domain signals, non-FT (e.g., Bayesian analysis) methods may provide more accurate spectral estimates of time-domain signal frequencies, relaxation time(s), and relative abundances. In this paper, we show that Bayesian analysis of simulated and experimental ion cyclotron resonance (ICR) time-domain noisy signals can produce a spectrum with mass accuracy improved by a factor of 10 or more over that obtained from a magnitude-mode discrete fast Fourier transform (FFT) spectrum. Moreover, Bayesian analysis offers the useful advantage that it automatically estimates the precision of its iteratively determined spectral parameters. The main disadvantage of Bayesian analysis is its lengthy computation time compared to that of FFT (hours vs seconds on the same hardware for approximately 4K time-domain data points); the Bayesian computation time increases rapidly with the number of spectral peaks and (less rapidly) with the number of time-domain data points. Bayesian analysis should thus prove useful for those FT/ICR applications involving relatively few data points and/or requiring high mass accuracy.     

A programmable multistage half-band FIR decimator for input datarates up to 2.56 MSPS

A multistage half-band finite-impulse response (FIR) decimator is developed. It is implemented using a 40000-gate 1.5-μ CMOS gate array, which dissipates 1.5 W at a clock rate of 25.6 MHz (a sampling rate of 2.56 MHz). Its power-of-two decimation ratio is programmable within the range 1 through 2¹⁷ for frequency zooming in fast Fourier transform (FFT) spectrum analysis, and it is preceded by a digital multiplier for frequency shifting. The filter handles 20-b 2.56-megasamples/s (MSPS) input data. The frequency resolution is increased by up to 2¹⁷ times without aliasing resulting in frequency resolution on the order of 20 mHz. The decimator has a 96-dB dynamic range     

An induction motor speed measurement method based on current harmonic analysis with the chirp-Z transform

This paper presents a new method to measure motor speed by means of frequency estimation of rotor slot spectral components in the supply current of squirrel single-cage induction motors. The novelty of the method consists in the harmonic analysis of the supply current by means of the chirp-Z transform (CZT). The advantages are improved accuracy due to better spectral resolution and resolvability. Moreover, a shorter observation window is required, thus reducing errors related to nonstationary current signals. The experimental results are presented to validate the proposed method and to make a comparison with a similar method based on the fast Fourier transform (FFT).     

Superresolution of Fourier transform spectroscopy data by the maximum entropy method

The maximum entropy method (MEM) is applied to the interferogram data obtained using the technique of Fourier transform spectroscopy for estimating its spectrum with a resolution far exceeding the value set by the spectrometer. For emission line data, the MEM process is directly used with the interferogram data in place of the regular Fourier transformation process required in Fourier transform spectroscopy. It produces a spectral estimate with an enhanced resolution. For absorption data with a broad background spectrum, the method is applied to a modified interferogram which corresponds to the Fourier transform of the absorptance spectrum. Two results are presented to demonstrate the power of the technique: for the visible emission spectrum of a spectral calibration lamp and for the infrared chloroform absorption spectrum. Included in the paper is a discussion of the problems associated with practical use of the MEM.     

A simple adjustable window algorithm to improve FFT measurements

The Fourier spectrum of a periodic signal may be obtained by fast Fourier transform algorithms, but, as is well known, special care must be taken to avoid severe distortions introduced by the sampling process. The main problem is the leakage generated by the truncation required to obtain a finite length sampled data. The usual procedure to reduce leakage is to multiply the sampled signal by a weighting window. Several kinds of windows have been proposed in the literature, and today they are also included in many commercial instruments. A simple alternative procedure is proposed in this paper. It is implemented with a PC compatible data acquisition board (DAQ) and consists of an algorithm that uses decimation and interpolation techniques. This algorithm is equivalent to the use of an adjustable sampling frequency and correspondingly an adjustable window size. Results obtained by this method on both harmonic and polyharmonic signals are empirically analyzed and compared with those given by an instrument with built-in FFT capabilities     

Pure absorption-mode spectra from Bayesian maximum entropy analysis of ion cyclotron resonance time-domain signals

Fourier transform ion cyclotron resonance (FT/ICR) mass spectra are normally reported in the (phase-independent) magnitude-mode format. In principle, the absorption-mode format offers spectral resolution enhanced by a factor ranging from square root of 3 to 2 over the corresponding magnitude-mode spectrum obtained by discrete FT of the same unapodized time-domain data. However, an absorption-mode display is generally unsuitable in practice because of the auxiliary spectral peaks (Gibbs oscillations) resulting from the relatively long time delay between excitation and detection. Although the resulting large phase variation (up to 100 pi rad or more across the Nyquist spectral bandwidth) can be corrected exactly for a continuous time-domain signal, phase correction of a discrete time-domain signal results in Gibbs oscillations even for a perfectly phased absorption-mode spectrum. In this paper, we show that a Bayesian "maximum-entropy" analysis of simulated and experimental ion cyclotron resonance time-domain noisy signals can recover a precisely phased absorption-mode frequency-domain spectrum that is devoid of Gibbs oscillations, is less sensitive to noise, and offers improved mass accuracy over that obtained from a conventional magnitude-mode discrete fast Fourier transform (FFT) spectrum.     

Enhanced spectral resolution FFT for step-like signals

A method is presented for obtaining a sampled Fourier transform of a steplike signal with a spectral resolution greater than previously reported. The method is related to zero padding, which is used for enhancing the spectral resolution of an FFT of a limited duration signal. The new method involves final value padding when the extension function outside the observation window is a step, or it involves padding with a sampled exponential function when the extension function is a decaying exponential     

Spectral velocity profiles for detailed ultrasound flow analysis

The operation of a novel ultrasound multigate instrument capable of computing in real-time the fast Fourier transform (FFT) of Doppler signals detected from 64 equally spaced range cells is presented. The new system provides up to 50 velocity profiles per second, which are displayed in such a manner that information about the full spectral content of Doppler signals at all the investigated depths is continuously monitored over a PRF-wide frequency range which can be set arbitrarily between -PRF and +PRF. Experimental results are presented, which demonstrate that the true velocity profile can be accurately detected through the computation of “local” maximum velocities obtained by properly correcting the maximum frequency of each spectrum. There is also a discussion on how the results of multigate analysis are influenced by the sample volume length, a parameter which can be usually set by modifying the duration of the transmitted burst. In particular, it is shown that, in regions close to the vessel walls, the shear rate can be measured with a spatial resolution related to the spacing between subsequent range cells and not to the sample volume length     

Signal-to-noise ratio trade-offs associated with coarsely sampled Fourier transform spectroscopy

We derive the spectral signal-to-noise ratio (SNR) trade-offs associated with coarsely sampled Fourier transform spectroscopy using a step-and-integrate measurement scheme. We show that there is no SNR penalty in the shot noise limit and a slight SNR benefit in the detector noise limit for the case of coarse sampling to achieve the same spectral resolution as a baseline Nyquist sampling scenario, where the total detector integration time is the same for both sampling cases.     

Application of Rader transforms to the analysis of nuclear spectral data

This paper describes a Rader transform method using a special arithmetic for the processing of nuclear spectral data. Rader transforms offer impressive computational savings vis-à-vis Fourier transform methods. Rader transforms require only integer additions and word shifts but no multiplications while Fourier transforms require complex arithmetic operations. Moreover, use of Rader transforms gives exact computations without any roundoff errors and does not require storage of basis functions. They are “the best transforms” for computer processing of nuclear spectral data. Rader transforms using a Fermat prime 65537 have been applied to deconvolve observed spectral data using a special filter function. A uniform improvement in resolution of 45% has been observed both in single and double spectral lines. A FORTRAN program GAMRAD is written to deconvolve spectral data using the special filter function.     

Enhancement of infrared spectral images for maximizing chemical information by minimizing baseline interferences

The popularity of spectral images in many areas of analysis has greatly increased during the last decade due to the development of charge-coupled device (CCD) and infrared sensitive cameras. Large amounts of spatial information can be obtained in short periods of time. The general goal in analytical chemistry is to convert spectral images into chemical images, which show the spatial locations of various chemical components. Self-modeling multivariate curve resolution methods can be used to extract pure component spectra from the mixture spectra in images and produce chemical images. However, there is a difficulty in processing infrared spectral images due to large pixel-to-pixel baseline variations. Herein, a method for minimizing baseline interferences using fast Fourier transform (FFT) filtering in both the spectral and spatial domains is discussed. The methodology is demonstrated on a microscopic sample of butter contaminated with non-pathogenic E. coli and on a cross-sectional sample of rabbit aorta containing plaque. The processing to reduce baseline effects improved the spatial resolution without compromising the spectral resolution.