Showing papers on "Non-uniform discrete Fourier transform published in 1986"

The Hartley transform

Abstract: The author describes the fast algorithm he discovered for spectral analysis and indeed any purpose to which Fourier Transforms and the Fast Fourier Transform are normally applied.

Fringe-pattern analysis using a 2-D Fourier transform

Abstract: A refinement of the Fourier transform fringe-pattern analysis technique which uses a 2-D Fourier transform is described. The 2-D transform permits better separation of the desired information components from unwanted components than a 1-D transform. The accuracy of the technique when applied to real data recorded by a system with a nonlinear response function is investigated. This leads to simple techniques for optimizing an interferogram for analysis by these Fourier transform methods and to an estimate of the error in the retrieved fringe shifts. This estimate is tested on simulated data and found to be reliable.

Modeling the noise influence on the Fourier coefficients after a discrete Fourier transform

Abstract: An analysis is made to study the influence of time-domain noise on the results of a discrete Fourier transform (DFT). It is proven that the resulting frequency-domain noise can be modeled using a Gaussian distribution with a covariance matrix which is nearly diagonal, imposing very weak assumptions on the noise in the time domain.

Novel Properties Of The Fourier Decomposition Of The Sinogram

Abstract: The double Fourier decomposition of the sinogram is obtained by first taking the Fourier transform of each parallel-ray projection and then calculating the coefficients of a Fourier series with respect to angle for each frequency component of the transformed projections. The values of these coefficients may be plotted on a two-dimensional map whose coordinates are spatial frequency w (continuous) and angular harmonic number n (discrete). For |w| large, the Fourier coefficients on the line n=kw of slope k through the origin of the coefficient space are found to depend strongly on the contributions to the projection data that, for each view, come from a certain distance to the detector plane, where the distance is a linear function of k. The values of these coefficients depend only weakly on contributions from other distances from the detector. The theoretical basis of this property is presented in this paper and a potential application to emission computerized tomography is discussed.

Fast computation of discrete cosine transform through fast Hartley transform

Abstract: A relationship between the discrete cosine transform (DCT) and the discrete Hartley transform (DHT) is derived. It leads to a new fast and numerically stable algorithm for the DCT.

Fast algorithms for the multidimensional discrete Fourier transform

Abstract: In this paper, the prime factor algorithm for the evaluation of a one-dimensional discrete Fourier transform is generalized to the evaluation of multidimensional discrete Fourier transforms defined on arbitrary periodic sampling lattices. It is shown that such an algorithm is equivalent in computational complexity to the evaluation of a rectangular discrete Fourier transform. As a sidelight to the derivation of the algorithm, a Chinese remainder theorem is derived for integer lattices.

Conversion of frequency-domain data to the time domain

Abstract: The advantages and disadvantages of three different algorithms for transforming frequency-domain data to the time domain are reviewed The algorithms are a direct computation of the Fourier series, the fast Fourier transform, and the chirp-z transform It is concluded that the fast Fourier transform still has the advantage of speed, but the chirp-z transform offers some additional flexibility that makes it more useful in many applications

Calculation of geometric moments using Fourier plane intensities.

Abstract: A new system for calculating the geometric moments of an input image is presented. The system is based on a mathematical derivation that relates the geometric moments of the input image to the intensity of the Fourier transform of the image. Since optical systems are very efficient at obtaining Fourier transforms, an optical system is used to provide the transform of the input image in this design. An array of detectors is then used to sample the Fourier plane, and a digital postprocessor performs a differentiation process on these Fourier magnitude samples to obtain a vector of values which are combined in a predetermined fashion to provide the geometric moments of the original input function.

Device for computing a sliding and nonrecursive discrete Fourier transform and its application to a radar system

Abstract: A device for computing a nonrecursive and sliding discrete Fourier transform as applicable in particular to processing of a pulse compression radar signal has N identical and parallel stages (E k ) for receiving in each case samples of the input signal (e m+N ). Each stage comprises two complex rotation operators, two adder-subtracters and two delay circuits and delivers a signal X k m+1 obtained from the following equations: ##EQU1##

Solution to some electromagnetic problems using fast Fourier transform with conjugate gradient method

Abstract: Using the fast Fourier transform (FFT) to compute the convolution integrals that appear in the conjugate-gradient method (CGM), an efficient numerical procedure to solve electromagnetic problems is obtained. In comparison with the method of moments (MM), the proposed FFT-CGM avoids the storage of large matrices and reduces the computer time by orders of magnitude.

Fast computational algorithms for the discrete cosine transform

Abstract: Fast algorithms for computation of the discrete cosine transform (DCT) are evaluated. Implementation via the fast Fourier transform and also by the direct method are considered. DCT algorithms for arbitrary sequence lengths are also included.

Performance of fixed-point FFT's: Rounding and scaling considerations

Abstract: The calculation of the discrete Fourier transform using a fast Fourier transform (FFT) algorithm with fixed-point arithmetic is considered. The input data is scaled to prevent overflow and to maintain accuracy. The implementation uses 16-bit fixed-point representation for the data and provides for double precision accumulation of sums and products. Algorithm variants as well as different rounding options are compared. Execution times for implementations based on a single chip signal processor are given. These show that a considerable increase in accuracy can be obtained with only a small penalty in execution time, by applying an alternating form of rounding rather than truncation.

Signals and Systems: Models and Behaviour

Abstract: Part 1 Introduction to signals and systems: signal classification signal processing systems linearity and time-invariance signal types and definitions signal symmetry and orthogonality signal sampling. Part 2 Time-domain models: discrete-time systems unit-sample response and convolution convolution for continuous systems. Part 3 Frequency-domain models: the frequency-domain approach the Fourier transform Fourier transforms of signals input-output relationships symmetry properties the inverse Fourier transform. Part 4 Laplace transforms: the Laplace integral Laplace model of signals properties of Laplace transforms the system transform function pole-zero models. Part 5 Z-transforms: the Z-transform the transfer function system response pole-zero models frequency response of a discrete-time system. Chapter 6 Periodic signals: strictly periodic signals the Fourier exponential series Fourier series and the Fourier integral input-output relationships band limited signals. Part 7 Fourier analysis of discrete-time signals and systems: the Laplace and Z-transforms the Fourier transform and the DTFT further properties of the DTFT signal sampling and aliasing frequency resolution the discrete Fourier transform. Appendices: A short table of Laplace transform pairs some Laplace transform properties some Z-transform pairs.

Fourier transform magnitudes are unique pattern recognition templates

Abstract: Fourier transform magnitudes are commonly used in the generation of templates in pattern recognition applications. We report on recent advances in Fourier phase retrieval which are relevant to pattern recognition. We emphasise in particular that the intrinsic form of a finite, positive image is, in general, uniquely related to the magnitude of its Fourier transform. We state conditions under which the Fourier phase can be reconstructed from samples of the Fourier magnitude, and describe a method of achieving this. Computational examples of restoration of Fourier phase (and hence, by Fourier transformation, the intrinsic form of the image) from samples of the Fourier magnitude are also presented.

A Transformation Algorithm for Estimating System Laplace Transform from Sampled Data

Abstract: The identification of continuous plants from sampled data is a technique commonly used. This method is normally divided into two parts: the determination of the discrete transfer function from the sampled data and the calculation of the continuous transfer function from the discrete one. An original method to perform the second part of this problem is described. The method, which establishes the minimum condition under which the continuous model can be obtained from the discrete one, has shown to have important advantages in the treatment of multiple poles. The method allows one to assume any temporal behavior of the input signal between sampling instants.

A method for computing electrical transients of transmission lines by numerical laplace transform

Abstract: This paper proposes a numerical method of solution for the transient phenomena on transmission line terminated by simple load, including a nonlinear element. The method is based on the discrete numerical Laplace transform and its inverse, utilizing the discrete Fourier transform. Problems and the solutions in the method are discussed. The feature of the method is that the voltage, current and surge impedance matrix of the transmission line are specified on the complex frequency (s) plane, while the boundary condition is given on the time (t) domain. Numerical solutions in the two regions are combined by Laplace transform and its inverse. The Laplace transform and its inverse by discrete Fourier transform have a drawback in that the accuracy of the computation deteriorates at t = 0 for stepwise change of the waveform. A method to solve this problem is described. For the latter half of the sampling point sequence in the Laplace and inverse transforms, the computation error is increased, which effectively halves the sampling points in the computation of the reflected wave. For this problem, a method is presented by which the accuracy of the computation is retained for each calculation of the reflected wave, keeping constant the number of sampling points. The convolution required in the calculation of the boundary condition is time-consuming, and a solution for this is proposed. As an example of the solution, the transient phenomenon in the multiconductor transmission-line terminated by the surge arrester is calculated, taking the skin effect into consideration.

Optimal algorithm for computing the two-dimensional discrete fourier transform.

Abstract: Since for each t ∈ [1, N/2], we have W t+N/2 = −W , one can also represent component (1) at the point (p, s) by the corresponding N/2-dimensional vector F̄ ′ p,s = (f ′ p,s,1, f ′ p,s,2, ..., f ′ p,s,N/2), whose components are calculated from the components of the corresponding initial vector F̄p,s by formula f ′ p,s,t = fp,s,t − fp,s,t+N/2, t = 1 ÷ N/2. (5) We call such representation of each component Fp,s of the spectrum in the form of the corresponding N/2-dimensional vector F̄ ′ p,s the paired vector representation, to distinct it from the original vector representation F̄p,s, and the constructed tensor of the 3rd order (f ′ p,s,t; p, s, = 1 ÷ N, t = 1 ÷ N/2 to be the paired tensor of the Fourier-spectrum. As for the original tensor representation of the spectrum of the signal, when for any p, s and k the following formula was valid [1]

Fast Fourier transform data address pre-scrambler circuit

Abstract: A fast Fourier transform (FFT) data address pre-scrambler technique and cuit for selectively generating pre-scrambled bit reversed, data address sequences needed to perform radix-2, radix-4 and mixed radix-2/4 fast Fourier transforms.

The discrete frequency Fourier transform

Abstract: In certain signal processing applications it may be required to reconstruct a spatial domain image form samples of its Fourier transform. For problems such as this it may be useful to use the dual of the well-known discrete time Fourier transform (DTFT) for purposes of analysis and design. In this paper, this dual concept, called the discrete frequency Fourier transform (DFFT), is shown to be a useful transform in its own right. In addition to being useful for certain physical problems, the DFFT fills a gap in the theory and aids the designer in understanding problems which have inherently sampled frequency domains.

A new method to locate sound sources by searching the minimum value of error function

Abstract: This paper describes a new method to locate sound sources using many sensors. Spectra of all the sensor outputs are calculated by discrete Fourier transform. In the proposed method, following parameters are assumed; (1) the number of sound sources, (2) a position of each sound source, and (3) the spectrum of the O-th sensor output due to each sound source. The spectra of all the sensor outputs are estimated using the parameters under the assumption of free field. Then, introduced is an error function which is the mean-square value of the difference between the calculated spectra and the estimated ones. By changing the values of the parameters, the minimum value of the error function is searched. When the error function takes the minimum value, the parameters represent the estimates; the number of sound sources, and the positions and spectra of the sound sources. By inverse Fourier transforming the estimated spectra, the wave form of a specified sound source can be obtained.

Fast algorithms for the discrete Fourier transform and for other transforms

Abstract: A new matrix factorization is proposed for DCT-IV, which is the basis of fast algorithms for many sinusoidal transforms. A new fast algorithm for complex-data DFT based on the new factorization requires the same number of multiplications and far fewer additions than the Preuss algorithm. A new fast algorithm for real-data DFT based on a new algorithm for the discrete Hartley transform is also proposed.

On the spectrum of pseudo noise

Abstract: Although the magnitude of the discrete Fourier transform of a maximal-length shift-register sequence is flat, except for its value at zero frequency, the higher resolution spectral content given by the Fourier-series transform is highly erratic. This little-known fact is described, and its ramifications on fast Fourier transforms of one-digit-extended pseudo noise and zero-padded pseudo noise are explained.

Phase-Only Image Reconstruction From Offset Fourier Data

Abstract: In a recent paper we investigated the problem of reconstructing the magnitude of a 2-D complex signal f from samples of the Fourier transform of f lying in a small region offset from the origin. The primary application of interest was synthetic aperture radar. We showed that high quality speckle reconstructions are possible so long as the phase of f is highly random. In this paper we explore the possibility of Fourier-offset reconstruction from just the phase of the Fourier transform. We provide and compare a large number of computer-simulated image reconstructions from phase plus magnitude, phase only (constant magnitude), magnitude only (zero phase), and magnitude plus quantized phase. A number of conclusions are drawn regarding Fourier offset phase-only reconstruction, and several topics are suggested for further research.

Two-dimensional transform domain decimation techniques

Abstract: Decimation is normally carried out in two passes; lowpass filtering and subsampling where the latter is normally performed in the time domain. This paper describes a technique whereby both the operations can be combined in the transform domain. The two-dimensional decimation scheme is first implemented in the discrete Fourier transform domain and then extended to the discrete cosine transform domain. It is further applied to a non-sinusoidal i.e. Hadamard transform domain. The effect of using ideal filter transfer function in transform domain decimation on the quality of the decimated images is investigated. This approach results in greater computational efficiency as the constraint of filter length to meet certain specifications is removed permitting the use of smaller transform block sizes.

Full Recursive Form of the Algorithms for Fast Generalized Fourier Transforms

Abstract: In this paper the full recursive forms of the discrete Fourier, Hadamard, Paley and Walsh transforms are developed. The algebraic properties and computational complexity of the GFT are investigated on the basis of a theoretical group approach and a matrix pseudoinversion. The approach considered reveals common and sometimes unexpected features of these transforms, the parallel realization of the algorithms becoming thus possible.

Measurement of time delay using the time shift property of the discrete Fourier transform (DFT)

Abstract: The time shift property of the discrete Fourier transform (DFT) is used in the measurement of time delay. The eccentricity of a screw rod has been experimentally measured, and delays of the ultrasonic wave have been known with the precision of one percent of the temporal sampling interval.