Showing papers on "Non-uniform discrete Fourier transform published in 1978"

On the use of windows for harmonic analysis with the discrete Fourier transform

Abstract: This paper makes available a concise review of data windows and their affect on the detection of harmonic signals in the presence of broad-band noise, and in the presence of nearby strong harmonic interference. We also call attention to a number of common errors in the application of windows when used with the fast Fourier transform. This paper includes a comprehensive catalog of data windows along with their significant performance parameters from which the different windows can be compared. Finally, an example demonstrates the use and value of windows to resolve closely spaced harmonic signals characterized by large differences in amplitude.

On computing the discrete Fourier transform

Abstract: New algorithms for computing the Discrete Fourier Transform of n points are described. For n in the range of a few tens to a few thousands these algorithms use substantially fewer multiplications than the best algorithm previously known, and about the same number of additions.

Fully parallel, high-speed incoherent optical method for performing discrete Fourier transforms.

Abstract: An incoherent optical data-processing method is described, which has the potential for performing discrete Fourier transforms of short length at rates far exceeding those afforded by both special-purpose digital hardware and representative coherent optical processors.

On the Computation of the Discrete Cosine Transform

Abstract: An N -point discrete Fourier transform (DFT) algorithm can be used to evaluate a discrete cosine transform by a simple rearrangement of the input data. This method is about two times faster compared to the conventional method which uses a 2N -point DFT.

A new least-squares refinement technique based on the fast Fourier transform algorithm

Abstract: A new atomic-parameters least-squares refinement method is presented which makes use of the fast Fourier transform algorithm at all stages of the computation. For large structures, the amount of computation is almost proportional to the size of the structure making it very attractive for large biological structures such as proteins. In addition the method has a radius of convergence of approximately 0.75 A making it applicable at a very early stage of the structure-determination process. The method has been tested on hypothetical as well as real structures. The method has been used to refine the structure of insulin at 1.5 A resolution, barium beauvuricin complex at 1.2 A resolution, and myoglobin at 2 A resolution. Details of the method and brief summaries of its applications are presented in the paper.

The Discrete Fourier Transform Over Finite Rings with Application to Fast Convolution

Abstract: Necessary and sufficient conditions for a direct sum of local rings to support a generalized discrete Fourier transform are derived. In particular, these conditions can be applied to any finite ring. The function O(N) defined by Agarwal and Burrus for transforms over ZN is extended to any finite ring R as O(R) and it is shown that R supports a length m discrete Fourier transform if and only if m is a divisor of O(R) This result is applied to the homomorphic images of rings-of algebraic integers.

Discrete Fourier transform computation via the Walsh transform

Abstract: This paper presents a new computational algorithm for the discrete Fourier transform (DFT). In an algorithm proposed here, DFT coefficients are computed via the Walsh transform (WT). The number of multiplications required by the new algorithm is approximately NL/6, where N is the number of data points and L is the number of Fourier coefficients desired. As such, it is superior to the fast Fourier transform (FFT) approach in applications where L is relatively small compared with N. It is also useful in cases where the Walsh and Fourier coefficients are both desired.

Fourier transforms of Gaussian orbital products

Abstract: An expression for the Fourier transform of two-centre Gaussian orbital products is obtained which is identical in form with expressions for overlap integrals. The one-centre transform is a special case, and is obtained in a trivial way from the two-centre expression. Explicit expressions of the transform for all combinations up to ff products are given.

Rapid estimation of spectra from irregularly sampled records

Abstract: Records of physical quantities often arise as continuous electrical signals. Spectral estimates may be formed either by analogue means or from digitised samples that are then processed on a computer, When the samples are provided at regularly spaced time instants, this can be achieved very quickly with the aid of the f.f.t (fast Fourier transform) algorithm. There are situations, however, where the data is known only at random time instants, and the paper is concerned with the computation of spectral estimates from such data. When the sample times are Poisson distributed, it has been shown, in previous papers, that unbiased alias-free estimates can be formed, either through the correlation function or by a direct Fourier transform of short blocks of data. Random sampling introduces additional variability in these spectral estimates, and it is consequently necessary to process a large amount of data in order to achieve stable results. Unfortunately, this is very time consuming, most of the computer effort being spent evaluating sine and cosine functions which are then multiplied by the data samples. Here, two methods that can be used to simplify this operation are discussed. It is shown that when the sine and cosine functions are replaced by their equivalent rectangular waveforms, the resulting estimates can be related to spectral estimates through the Fourier expansion for the rectangular waves. A second way of speeding up the processing of Gaussian signals can be achieved by quantising the data to a sign bit and using the ‘arc-sine’ rule to transform the autocorrelation function to that of the full signal. It is shown that when both techniques are used together, and the processing reduced to 1-bit logical operations, valid spectral estimates can indeed be formed. These ideas are tested on various simulated sets of data.

Detection performance of the smooth coherence transform (SCOT)

Abstract: When energy from a radiating source is received at two physically separated sensors, it is possible to detect the presence of that source and estimate signal travel time difference (τ d ) to the sensors using the Smoothed Coherence Transform (SCOT). The purpose of the SCOT is to enhance travel time difference (τ d ) estimation, however, to estimate (τ d ) a SCOT peak must first be detected. This detection performance is the subject which will be considered in this paper. Detection performance (probability of detection) is evaluated when the SCOT is obtained using the Inverse Discrete Fourier Transform (IDFT) of the Complex Coherence Function (CCF) estimate. Results are presented for both wideband and narrow-band signals assuming zero-mean, stationary, Gaussian signal and noise processes.

Method and apparatus for suppression of error accumulation in recursive computation of a discrete Fourier transform

Abstract: To limit the accumulation of recursive computation errors while recursively calculating the discrete Fourier transform of an input signal in response to moving window sample sets of that signal, the dependent variables for the recursive calculations are periodically refreshed. One embodiment generates an error limited continuous discrete Fourier transform of the input signal through the use of duplicate channels which are periodically reinitialized in time staggered relationship and alternately switched on-line and off-line in time synchronism with the reinitialization process. Each of the channels of that embodiment is equipped to recursively calculate the discrete Fourier transform of the input signal so that while one channel is on line feeding recursively calculated transform coefficients to the outputs, the other channel or channels are off-line being reinitialized and then computing fresh sets of transform coefficients for subsequent on-line recursive calculations. Another embodiment performs essentially the same function through the use of an on-line main processing channel which recursively calculates the discrete Fourier transform coefficients of the input signal on the basis of successive sets of dependent variables which are computed in and supplied by a periodically reinitialized off-line auxiliary processing channel.

Phase error model for simple Fourier transform lenses

Abstract: The effects of amplitude nonuniformities and phase errors on the accuracy of the optical Fourier transform are considered for the case of a simple lens. A quadratic phase model is derived, analyzed, and compared to experimental and point-by-point optical path difference data. The goal is to determine the lens specifications needed to produce an optical Fourier transform of given accuracy rather than the design of the lens itself.

Towards a discrete hanke! transform and its applications

Abstract: A discrete transform with a Bessel function kernel is defined, as a finite sum, over the zeros of the Bessel function. The approximate inverse of this transform is derived as another finite sum. This development is in parallel to that of the discrete Fourier transform (DFT) which lead to the fast Fourier transform (FFT) algorithm. The discrete Hankel transform with kernel Jo, the Bessel function of the first kind of order zero, will be used as an illustration for deriving the discrete Hankel transform, its inverse and a number of its basic properties. This includes the convolution product which is necessary for solving boundary problems. Other applications include evaluating Hankel transforms, Bessel series and replacing higher dimension Fourier transforms, with circular symmetry, by a single Hankel transform

More accurate interpolation using discrete Fourier transforms

Abstract: It is shown that substantial errors in interpolation by means of existing discrete Fourier transform (DFT) techniques are generally present for finite sequences, even if the function sampled is band-limited. A modified approach is proposed which provides significantly more accurate information.

Semi-Fast Fourier Transforms over GF(2 m ).

Abstract: An algorithm which computes the Fourier transform of a sequence of length n over GF(2m) using approximately 2nm multiplications and n2+ nm additions is developed. The number of multiplications is thus considerably smaller than the n2multiplications required for a direct evaluation, though the number of additions is slightly larger. Unlike the fast Fourier transform, this method does not depend on the factors of n and can be used when n is not highly composite or is a prime.

Short-time spectral analysis with the conventional and sliding CZT

Abstract: Two sequential short-time spectral analysis techniques, amenable to nonrecursive filter implementation, are the conventional chirp-z-transform (CZT) realization of the discrete Fourier transform and the sliding CZT realization of the discrete sliding Fourier transform. This paper presents a comparative study of frame rate limitations, windowing, time and frequency resolution, spectral correlation, complexity, and inverse structures for these methods, with particular emphasis on the recently proposed sliding transform. The sliding transform and its CZT realization are viewed as skewed output samples of a filter bank, an approach which aids in understanding the relationship between the conventional and sliding schemes. Numerous forward and inverse CZT formulations are presented to improve resolution, frame rates, and compactness.

Spectrum analyzer overlap requirements and detectability using discrete Fourier transform and composite digital filters

Abstract: In spectrum analysis, a primary measure of performance is the minimum detectable signal (MDS). Detectability dictates the compute speed necessary to meet both time and frequency overlap requirements. It is becoming increasingly popular to compute the discrete Fourier transform (DFT), and these elemental filters are complex weighted or otherwise combined to form more complex filters. Applications of this include the proportional bandwidth filter comb and all time‐weighting functions such as Hanning, Hamming, and Taylor or their frequency equivalents. Overlap requirements are thereby changed and MDS must be defined so it can be extrapolated to these composite filters. The overlap requirement and MDS are so defined in this paper. Experimental performance is then given and compared to the predictions.

Two-dimensional signal processing from hexagonal rasters

Abstract: Two-dimensional signals are normally sampled and processed as rectangular arrays. For signals which are bandlimited with a circular region of support in the Fourier plane, however, it has been known for some time that a savings in the number of samples required for an exact reconstruction can be realized by sampling the signal on a hexagonal sampling raster. In this presentation we show that a substantial savings in computation can result as well. Included are methods for signal representation, linear system implementation, Fourier transform computation and FIR filter design. Some comparisons between systems defined over rectangular and hexagonal sampling rasters will also be given.

Adaptive Fourier Transformation by Using Acousto-Optic Convolution

Abstract: An implementation of the chirp transform algorithm for performing a real-time Fourier transform is described. The implementation is based upon an acousto-optic (AO) convolver with a large time-bandwidth product. An instantaneous bandwidth of about 80 MHz was measured for the A0 Fourier transform with a dynamic range in excess of 60 dB for a cw time-gated waveform. Sidelobe errors of less than 0.8 dB were measured for uniformlyweighted pulses, with a corresponding phase error of less than 5". The adaptability of this acoustooptic implementation is demonstrated by the ease with which both the sidelobe weighting function and the time rate of the transform can be varied.

The fast Fourier transform for general order

Abstract: An elementary and transparent representation of the fast Fourier transform is given. Instead of using the usual and highly algebraic approach it is shown how a Fourier transform of the ordern=p·m can be reduced top Fourier transforms of orderm by performing essentiallym Fourier transforms of orderp on the data. The resulting process is discussed in more detail forn=3q andn=5q. The problem of retrieval of the wanted coefficients from the final data is solved by a simple argument. The generalization for an ordern equal to a product of powers of prime numbers is immediate.