Estimating frequency by interpolation using Fourier coefficients

The periodogram of a time series that contains a sinusoidal component provides a crude estimate of its frequency parameter, the maximizer over the Fourier frequencies being within O(T^-1) of the frequency as the sample size T increases. In the paper, a technique for obtaining an estimator that has root mean square error of order T ^-3/2 is presented, which involves only the Fourier components of the time series at three frequencies, The asymptotic variance of the estimator varies between, roughly, the asymptotic variance of the maximizer of the periodogram over all frequencies (the Cramer-Rao lower bound) and three times this variance. The advantage of the new estimator is its computational simplicity     

台阶光栅衍射效率的傅里叶分析

用傅里叶方法对台阶光栅衍射效率进行了分析,得出任意台阶数,任意台阶高度的二元光栅任意级次的衍射效率公式.针对二台阶光栅分析了台阶高度,台阶宽度,不同波长对+1级衍射效率的影响,理论图像清晰,结果简单明了.     

Iterative frequency estimation by interpolation on Fourier coefficients

The estimation of the frequency of a complex exponential is a problem that is relevant to a large number of fields. In this paper, we propose and analyze two new frequency estimators that interpolate on the Fourier coefficients of the received signal samples. The estimators are shown to achieve identical asymptotic performances. They are asymptotically unbiased and normally distributed with a variance that is only 1.0147 times the asymptotic Crame/spl acute/r-Rao bound (ACRB) uniformly over the frequency estimation range.     

Coarse frequency estimation using the discrete Fourier transform (Corresp.)

The discrete Fourier transform (DFT) is applied as a coarse estimator of the frequency of a sine wave in Gaussian noise. Probability of anomaly and the variance of the estimation error are determined by computer simulation for several DFT block sizes as a function of signal energy-to-noise density ratiomathcal{E}/N_0. Several data windows are considered, but uniform weighting gives the best performance.     

A unified approach to short-time Fourier analysis and synthesis

Two distinct methods for synthesizing a signal from its short-time Fourier transform have previously been proposed. We call these methods the filter-bank summation (FBS) method and the overlap add (OLA) method. Each of these synthesis techniques has unique advantages and disadvantages in various applications due to the way in which the signal is reconstructed. In this paper we unify the ideas behind the two synthesis techniques and discuss the similarities and differences between these methods. In particular, we explicitly show the effects of modifications made to the short-time transform (both fixed and time-varying modifications are considered) on the resulting signal and discuss applications where each of the techniques would be most useful The interesting case of nonlinear modifications (possibly signal dependent) to the short-time Fourier transform is also discussed. Finally it is shown that a formal duality exists between the two synthesis methods based on the properties of the window used for obtaining the short-time Fourier transform.     

一种改进的消零频傅里叶变换轮廓术

为提高傅里叶变换轮廓术(FTP)的测量精度和范围,提出了一种改进的消零频FTP。将两幅随机相移正弦条纹通过DLP投影仪投射到待测物体上,由CCD相机采集受物体形貌调制的变形光栅条纹图,将采集的两幅条纹图相减得到无零频的条纹差图。对条纹差图进行快速傅里叶变换(FFT)后,通过简易滤波器提取基频成分,将其与彩色条纹或复合光栅投影相结合,可实现单次投影。理论和实验证明了本方法的可行性。     

On discrete short-time Fourier analysis

Weyl-Heisenberg frames are a principal tool of short-time Fourier analysis. We present a comprehensive study of Weyl-Heisenberg frames in l²(Z), with a focus on frames that are tight. A number of properties of these frames are derived. A complete parameterization of finite-length windows for tight Weyl-Heisenberg frames in l²(Z) is described. Design of windows for tight Weyl-Heisenberg frames requires optimization of their frequency characteristics under nonlinear constraints. We propose an efficient design method based on expansions with respect to prolate spheroidal sequences. The advantages of the proposed method over standard optimization procedures include a reduction in computational complexity and the ability to provide long windows that can be specified concisely using only a few parameters; these advantages become increasingly pronounced as the frame redundancy increases. The resilience of overcomplete Weyl-Heisenberg expansions to additive noise and quantization is also studied. We show that manifestations of degradation due to uncorrelated zero-mean additive noise are inversely proportional to the expansion redundancy, whereas the quantization error is for a given quantization step inversely proportional to the square of the expansion redundancy     

Fourier analysis using spectral observers

New techniques for Fourier spectral analysis are reported, for which ongoing spectral estimates are generated in real time. The algorithms are recursive, expressable in state variable form, and involve real number computations. The ability of these spectral observers to perform one-step-per-sample updating is demonstrated with numerical examples.     

基于短时傅立叶变换的跳频序列瞬时频率估计

采用了G函数的方法来模拟跳频序列，然后利用短时傅立叶变换对它进行分析与瞬时频率的估计。该方法能够有效地描绘出跳频信号的频率随时间跳变的规律（跳频图案），具有很高的时间一频率分辨率。计算机仿真验证了应用短时傅立叶变换分析跳频序列的有效性和实用性。     

A new LMS-based Fourier analyzer in the presence of frequency mismatch and applications

The performance of the conventional least mean square (LMS) Fourier analyzer may degenerate significantly, if the signal frequencies given to the analyzer are different from the true signal frequencies. This difference is referred to as frequency mismatch (FM). We first analyze the performance of the conventional LMS Fourier analyzer for a single sinusoid in the presence of FM. We derive the dynamics and steady-state properties of this analyzer as well as the optimum step size parameter which minimizes the influence of the FM. Extensive simulations reveal the validity of the analytical results. Next, a new LMS-based Fourier analyzer is proposed which simultaneously estimates the discrete Fourier coefficients (DFCs) and accommodates the FM. This new analyzer can very well compensate for the performance degeneration due to the FM. Applications to estimation/detection of dual-tone multiple frequencies (DTMF) signals and analysis of real-life noise signals generated by a large-scale factory cutting machine are provided to demonstrate the excellent performance of our new Fourier analyzer.     

Cavity mode analysis using the Fourier transform method

A novel and economical computer iterative method for cavity mode calculations is developed. The method involves the formulation of the cavity field in the form of Fourier integrals and involves the use of the fast Fourier transform as a computation tool for obtaining numerical results.     

License-exempt: Wi-Fi complement to 3G

In this paper we compare and contrast the development of Wi-Fi as a license-exempt wireless broadband technology to access the internet with the licensed regime of broadband cellular networks such as 3G. This exploration is based on an assessment of the two different innovation journeys which resulted from two different underlying communication paradigms and two different regulatory regimes, leading to different business models and different diffusions patterns. In concluding we compare the merits of the two cases and derive recommendations for policy and strategy formation.     

Fourier analysis of pseudorandom binary sequences

It is shown that, for a given-length msequence, the phase angles of the harmonic components of the sequence can take on only certain specified values. Variation of the generating polynomial causes an interchange of the phase angles among the harmonics in a manner which is related to the sampling properties of the sequence.     

A novel view of Fourier analysis

A different view of a familiar subject often provides insight that leads to innovations and new discoveries. A novel view of the Fourier transform, Fourier series, and discrete Fourier transform is presented in this letter.     

New approach to the frequency response analysis of an InGaAsavalanche photodiode

The frequency response of an avalanche photodiode has been calculated by solving the transport equations, taking the electric field profile into account. The relationship between the carrier concentration (Nb) in the multiplication layer and the frequency response has been obtained for the first time. This calculation has been carried out for a conventional In_0.53Ga_0.47As avalanche photodiode. The results explain well present experimental data. The saturation velocity of holes in InP is estimated to be 2×10⁶ cm/s. The upper limitation of the gain-bandwidth product is estimated to be 140 GHz at Nb=2×10¹⁷ cm^-3      

Short time Fourier analysis of the electromyogram: fast movements and constant contraction

We applied short-time Fourier analysis to surface electromyograms (EMG) recorded during rapid movements, and during isometric contractions at constant forces. We selected a portion of the data to be transformed by multiplying the signal by a Hamming window, then computed the discrete Fourier transform. Shifting the window along the data record, we computed a new spectrum each 10 ms. We displayed the transformed data in spectograms or "voiceprints." This short-time technique allowed us to see time-dependencies in the EMG that are normally averaged in the Fourier analysis of these signals. Spectra of EMG's during isometric contractions at constant force vary in the short (10-20 ms) term. Moments of the spectral distribution show this variability. Short-time spectra from EMG's recorded during rapid movements were much less variable. The windowing technique picked out the typical "three-burst pattern" in EMG's from both wrist and head movements. Spectra during the bursts were more consistent than those during isometric contractions. Furthermore, there was a consistent shift in spectral statistics in the course of the three bursts. Both the center frequency and the variance of the spectral energy distribution grew from the first burst to the second burst in the same muscle. We discuss this pattern with respect to the origin of the EMG bursts in rapid movement. We also extend the analogy between electromyograms and speech signals to argue for future applicability of short-time spectral analysis of EMG.     

Fourier decomposition method applied to mapped infinite domains:scalar analysis of dielectric waveguides down to modal cutoff

By transforming the infinite x-y plane onto a unit square and using two-dimensional Fourier series expansions, the modal fields and propagation constants of dielectric waveguides are accurately determined within the scalar (weak-guidance) regime. The new method is reliable down to modal cutoff and gives cutoff V-values directly. Numerical cutoff values for the LP₁₁ modes of square- and rectangular-core waveguides are determined as a function of core aspect ratio, and are found to agree with those obtained by the finite element method to within 0.1%