Non-uniform discrete Fourier transform

About: Non-uniform discrete Fourier transform is a research topic. Over the lifetime, 4067 publications have been published within this topic receiving 123952 citations.

Fast Discrete Linear Canonical Transform Based on CM-CC-CM Decomposition and FFT

Abstract: In this paper, a discrete LCT (DLCT) irrelevant to the sampling periods and without oversampling operation is developed. This DLCT is based on the well-known CM-CC-CM decomposition, that is, implemented by two discrete chirp multiplications (CMs) and one discrete chirp convolution (CC). This decomposition doesn’t use any scaling operation which will change the sampling period or cause the interpolation error. Compared with previous works, DLCT calculated by direct summation and DLCT based on center discrete dilated Hermite functions (CDDHFs), the proposed method implemented by FFTs has much lower computational complexity. The relation between the proposed DLCT and the continuous LCT is also derived to approximate the samples of the continuous LCT. Simulation results show that the proposed method somewhat outperforms the CDDHFs-based method in the approximation accuracy. Besides, the proposed method has approximate additivity property with error as small as the CDDHFs-based method. Most importantly, the proposed method has perfect reversibility, which doesn’t hold in many existing DLCTs. With this property, it is unnecessary to develop the inverse DLCT additionally because it can be replaced by the forward DLCT.

Calculation of Fourier Transforms by the Backus-Gilbert Method

Abstract: Summary The linear inverse theory of Backus & Gilbert has been applied to the problem of calculating the Fourier transform of digitized data with the objective of assessing the effects of missingportions of the data series and of contamination of the signal by ' noise '. When ' noise ' in the data is of concern this method achieves a maximum decrease in the variance of the Fourier transform estimate for a minimum sacrifice in resolution, thereby optimizing the trade-off between resolution and accuracy. The effects of data gaps are easily treated and it is shown that it may sometimes be desirable to interpolate these gaps even though a large variance must be ascribed to the fabricated data. We also apply the Backus-Gilbert technique to the calculation of the reverse Fourier transform, and an application to the downward continuation of potential field data is given.

Lomb-Wech periodogram for non-uniform sampling

Abstract: The Lomb-Scargle transform has been proposed for the direct evaluation, namely without interpolation, of non-uniformly sampled signals. In its current form, it is suitable only for single transform evaluation due to the implicit normalization. Enhancements of this transform are proposed to allow the evaluation of shorter transforms, combined with windows and averaging of overlapped records. This requires a de-normalization of the transform by a factor of 2(sigma)/sup 2//N, the use of equal time duration records, and multiplication by windows sampled at corresponding non-uniform time instances. This results in a Welch-like periodogram for non-uniform sampling.

Recursive computation of the Fourier transform

Abstract: Efficient recursive methods and circuits for computing a continuously updated discrete Fourier transform (DFT) of an input-digital signal are considered. The Fourier transform (FT) is recomputed at each sample input time, with only O(N) operations being required to compute the transform, where N is the number of frequency bins. Various window functions are considered for windowing the input-wave form, namely a rectangular window, a triangular window, and an exponential window. The last type of window has not been widely considered in the past, partly due to its asymmetrical shape, and hence nonlinear phase response. Nevertheless, it is shown to have certain advantages in ease of computation and in flexibility. For the exponential window, a circuit that conveniently allows zooming in to particularly interesting parts of the frequency spectrum is shown. By appropriately loading a multiplier storage RAM, arbitrarily fine resolution may be achieved in any part of the spectrum, thus permitting closely adjacent peaks to be distinguished. The general approach is based on the interpretation of the FT as a set of simultaneous bandpass filters. Though these filters are generally finite-impulse-response filters, computational advantages are derived from formulating them as recursive filters. >