Showing papers on "Discrete Fourier transform published in 1982"

A Complete Set of Fourier Descriptors for Two-Dimensional Shapes

Abstract: A set of Fourier descriptors for two-dimensional shapes is defined which is complete in the sense that two objects have the same shape if and only if they have the same set of Fourier descriptors. It also is shown that the moduli of the Fourier coefficients of the parameterizing function of the boundary of an object do not contain enough information to characterize the shape of an object. Further a relationship is established between rotational symmetries of an object and the set of integers for which the corresponding Fourier coefficients of the parameterizing function are nonzero.

Signal reconstruction from the short-time Fourier transform magnitude

Abstract: This paper presents various conditions that are sufficient for reconstructing a discrete-time signal from samples of its short-time Fourier transform magnitude. For applications such as speech processing, these conditions place very mild restrictions on the signal as well as the analysis window of the transform. Examples of such reconstruction for speech signals are included in the paper.

Continuous and discrete Fourier transforms of steplike waveforms

Abstract: A steplike waveform which has attained its final value is converted into a duration-limited one which preserves the spectrum of the original waveform and is suitable for discrete Fourier transform (DFT) computations. The method, which is based upon the response of a time-invariant linear system excited by a rectangular pulse of suitable duration, is first applied to continuous waveforms and then to discrete (sampled) waveforms. The difference (errors) between the spectra of a continuous waveform and a discrete representation of it are reviewed.

Signal processing algorithms and architectures

Abstract: The advent of the Very Large Scale Integration (VLSI) technology has provided the ability to construct large systems on a single silicon chip. This dissertation is concerned with exploiting this ability to design a powerful signal processing chip capable of efficiently implementing such popular algorithms as the discrete Fourier transform, ladder filters and associated matrix algebra operations. The latter include Givens rotations and Cholesky factorization. The goal of the present work is to efficiently map algorithms onto architectures by maintaining a close link with the theoretical basis of a particular signal processing method. It is shown that all of the algorithms considered can be cast into a mathematical framework involving generalized vector rotations. Such rotation operations provide a natural description of the algorithms and the computational complexity measured in terms of these elementary operations is much lower than in terms of the usual measure of total number of multiplications. Thus, unlike present day signal processing computers which emphasize rapid multiplication, the signal processing architectures in this thesis are based on the ability to perform vector rotations in generalized coordinate systems. It is shown that the CORDIC algorithm of Volder provides a convenient implementation of vector rotations with only simple components such as adders, registers and shifters. Unfortunately, throughput is severely compromised owing to the need for performing special operations to account for the limited region of convergence and spurious scale constants inherent to the method. New techniques to circumvent these problems with no additional hardware and only a marginal speed penalty are described. Further speed enhancements through the use of a newly developed method known as hybrid CORDIC are discussed. Additionally, floating point CORDIC (FLORDIC) algorithms that are conceptually simpler than their fixed point counterparts are developed and the connection of CORDIC to the convergence computation methods is shown. The architecture of a dual CORDIC block chip is described for a target application of real time speech analysis. The resulting chip is shown to have a higher throughput per area than conventional chips based on fast multiplications. This is attributed to the close match of the present chip to the algorithms. Large mesh connected processor architectures for matrix factorization are developed which are also closely matched to the algorithms. Individual processing elements in the mesh are based on CORDIC operations, in fact on the aforementioned signal processing chip. Finally, a new technique for signal detection in additive Gaussian noise is developed with a view towards ease of implementation. It is based on ladder filters and may be implemented using the signal processing chip mentioned above.

The Fast Fourier Transform

Abstract: The object of this chapter is to briefly summarize the main properties of the discrete Fourier transform (DFT) and to present various fast DFT computation techniques known collectively as the fast Fourier transform (FFT) algorithm. The DFT plays a key role in physics because it can be used as a mathematical tool to describe the relationship between the time domain and frequency domain representation of discrete signals. The use of DFT analysis methods has increased dramatically since the introduction of the FFT in 1965 because the FFT algorithm decreases by several orders of magnitude the number of arithmetic operations required for DFT computations. It has thereby provided a practical solution to many problems that otherwise would have been intractable.

On recursive discrete Fourier transformation

Abstract: It is shown that the recent deadbeat state-observer approach to discrete Fourier transform evaluation is exactly equivalent to the nonrecursive frequency-sampling filters of finite impulse response digital filter design. The approach is then extended to consider the Kalman filter derived from the signal model and this is demonstrated to be nonrobust and pathological. The application of state observers to this problem is then brought into question and it is shown that they have a novel application to signal processing between the extremes of deadbeat and Kalman observers.

Very fast computation of the radix-2 discrete Fourier transform

Abstract: This paper develops and presents a radix-2 fast Fourier transform (FFT) algorithm that reduces the computational effort (as measured by the number of multiplications) to two-thirds of the effort required by most radix-2 algorithms. The resulting algorithm is similar to one obtained by applying a principle suggested by Rader and Brenner; however, its structure is particularly appealing when transforming pure real or imaginary sequences and/or symmetric or antisymmetric sequences; furthermore, memory requirements (other than those for storing the input data) do not grow with the size of the transform.

Iterative Procedures For Signal Reconstruction From Fourier Transform Phase

Abstract: Recently, a set of conditions has been developed under which a sequence is uniquely specified by the phase or samples of the phase of its Fourier transform. These conditions are distinctly different from the minimum or maximum phase requirement and are applicable to both one-dimensional and multi-dimensional sequences. Under the specified conditions, several numerical algorithms have been developed to reconstruct a sequence from its phase. In this paper, we review the recent theoretical results pertaining to the phase-only reconstruction problem, and we discuss in detail two iterative numerical algorithms for performing the reconstrucction.

Optical Fourier transform: what is the optimal setup?

Abstract: The two basic optical Fourier transform configurations are examined with respect to component complexity, aberrations, and optical noise. It is shown that the converging-beam illumination setup (CB-FT) is much simpler and works better than the classical parallel beam illumination setup within a restricted range of object size and lens aperture. This range corresponds to many practical cases. Therefore, the CB-FT should be preferred in ordinary cases whereas the classical setup with a special purpose Fourier lens should be used only for a large space–bandwidth product. It is probably never a good solution to use the parallel beam configuration with a general purpose lens as the Fourier lens.

Railway signalling receiver

Abstract: A railway signalling receiver in which a signal carried by the railway lines is sensed and the sensed signal (11) is sampled (14), digitised (16) and subjected to a discrete Fourier transform operation Figure 3) which transforms the time domain samples into the frequency domain. The frequency content of the signal may then be examined in order to identify the original signal or to decode the information carried. The signal may be a frequency shift keyed carrier signal. A technique of maintaining the safety of the system by discriminating against potentially confusingly similar ASK signals which can arise spontaneously in electric traction territories. A technique of digitally heterodyning down a carrier signal so that the transform operates on the lower sidebands frequencies in order to maximise efficiency of the transform calculations is also described (Figure 3). The receiver is useful as a jointless track circuit receiver (Figure 3) and also in a part of the train borne equipment of an automatic train protection system (Figure 4).

New algorithm for the calculation of the Fourier transform of discrete signals

Abstract: A new algorithm for the calculation of the Fourier transform of sampled time functions is described. The algorithm is especially applicable to the Fourier analysis of nonperiodic signals which are not band limited. The method is based on second‐degree polynomial interpolations between the sample points. The obtained continuous approximation of the signal allows the determination of the Fourier transform analytically. In the case of exponentially decaying functions the algorithm was found to be significantly more accurate than the conventionally used discrete Fourier transform (DFT). The computing time is only about twice the time required by the fast Fourier transform (FFT) algorithm.

Sampling spectrum analyzer

Abstract: A sampling spectrum analyzer provides instantaneous continuous wideband spectrum analysis to allow RF signals occurring simultaneously and spaced within the band to be displayed. An input signal is split and each resulting portion is passed down a tapped delay line, with samples at the taps of each line being processed by separate arithmetic units. The output of each arithmetic unit is applied to the taps of output delay lines which are similar to the input delay lines. Each arithmetic unit weights the input samples with a set of coefficients and sums the resulting signal in a prescribed manner, thus transforming the signal into the frequency domain, according to a discrete Fourier transform. The resulting sums are the real and imaginary terms of a signal which are vectorially added, resulting in a signal which is fed to the ordinate of a display having a swept time base representing frequency.

The design of optimal DFT algorithms using dynamic programming

Abstract: A broad class of efficient, discrete Fourier transform algorithms is developed by partitioning short DFT algorithms into factors. The factored short DFT's are combined into longer DFT's using a prime factor algorithm (PFA). By exploiting a property which allows some of the factors to commute, a large set of possible DFT algorithms is generated which contains both the prime factor algorithm and the Winograd Fourier Transform Algorithm (WFTA) as special cases. The problem of finding an algorithm from this class which is optimal with respect to the specific add, multiply, and data transfer characteristics of a partlcular implementation is posed, and a highly effective dynamic programming algorithm is presented as a solution.

Phase contrast flow visualization

Abstract: The properties and elementary theory of the method of phase contrast are discussed with respect to quantitative gas flow visualization. A more detailed diffraction theory of phase contrast that predicts the nonlinearities of the method such as image differentiation, halos, and fringes is outlined. This diffraction theory is implemented computationally using discrete Fourier transform techniques, and several examples are discussed. Some experimental results, obtained with a phase contrast system and several flows, are shown.

Direct Fourier Transform NMR Tomography with Modified Kumar-Welti-Ernst(MKWE) Method

Abstract: Direct Fourier transform NMR tomographic method originally proposed by Kumar-Welti-Ernst(KWE) has been modified by double spin echo technique to improve the image quality. This new Modified-KWE(MKWE) method provides two quadrant data set in 2-D Fourier space which is essential for the accurate image representation of the NMR spin density. Further improvement of the MKWE method using the slice encoding technique and spin echo measurement time is also investigated.

A 32 point monolithic FFT processor chip

Abstract: The Discrete Fourier Transform (DFT) is used in a wide variety of digital signal processing applications. The algorithms used to implement this transform require intensive arithmetic computation as well as complex control and sequence functions. The designer of VLSI components is faced with the problem of identifying requirements and architectures for chips which directly support the DFT. Design goals of these chips include minimum chip count to implement an entire transform, very high speed and low power dissipation. This paper discusses a monolithic CMOS device that was fabricated to perform 32 point Fast Fourier Transforms at very high data rates. All data memory and arithmetic and control circuitry is contained on this single low power chip.

Generalized running discrete transforms

Abstract: This paper introduces a generalized running discrete transform with respect to arbitrary transform bases, and relates the generalized transform to the running discrete Fourier z and short-time discrete Fourier transforms. Concepts associated with the running and short-time discrete Fourier transforms such as 1) filter bank implementation, 2) synthesis of the original sequence by summation of the filter bank outputs, 3) frequency sampling, and 4) recursive implementations are all extended to the generalized transform case. A formula is obtained for computing the transform coefficients of a segment of data at time n recursively from the transform coefficients of the segment of data at time n - 1. The computational efficiency of this formula is studied, and the class of transforms requiring the minimum possible number of arithmetic operations per coefficient is described.

DFT-Vocoder using harmonic-sieve pitch extraction

Abstract: The DFT-vocoder is based on speech analysis and synthesis using the discrete Fourier transform (DFT). Analysis is done using hopping-DFT and spectral parameters are obtained by a piece-wise constant approximation of the amplitude spectrum. The harmonic-sieve technique for pitch extraction combines very well with this scheme because it is based on hopping-DFT as well. Synthesis is achieved by convolution of the generated excitation signal with the inverse-DFT of the reconstructed piece-wise constant amplitude spectrum. In this paper a 2400 bit/s implementation of the DFT-vocoder is discussed.

Discrete Fourier transforms of nonuniformly spaced data

Abstract: Time series or spatial series of measurements taken with nonuniform spacings have failed to yield fully to analysis using the Discrete Fourier Transform (DFT). This is due to the fact that the formal DFT is the convolution of the transform of the signal with the transform of the nonuniform spacings. Two original methods are presented for deconvolving such transforms for signals containing significant noise. The first method solves a set of linear equations relating the observed data to values defined at uniform grid points, and then obtains the desired transform as the DFT of the uniform interpolates. The second method solves a set of linear equations relating the real and imaginary components of the formal DFT directly to those of the desired transform. The results of numerical experiments with noisy data are presented in order to demonstrate the capabilities and limitations of the methods.

Selection criteria for efficient implementation of FFT algorithms

Abstract: The number of real operations and memory is presented for three efficient Fortran algorithms which compute the mixed radix discrete Fourier transform (DFT). It is shown that Singleton's mixed radix algorithm (MFFT) is the most flexible and uses the least memory, while the Winograd Fourier transform algorithm (WFTA) and Kolba-Parks prime factor algorithm (PFA) are the most efficient.

An Efficient Method for the Fourier Transform of a Neuronal Spike Train

Abstract: A spike train may be represented by a superposition of Dirac delta-functions. One of the simplest ways of converting such a comb function into a continuous function is to use a Fourier transform. In general there are two possibilities, both of which have their disadvantages: the direct transform which is extremely time-consuming, and the fast Fourier transform of the low pass filtered comb function; the latter method, although quicker, often requires a greater storage capacity than is readily available. In the present paper, therefore, a third possibility is suggested. Essentially, it is a direct Fourier transform which takes advantage of certain properties of a spike train. The corresponding algorithm works much faster than a common Fourier transform.

Analog scrambling by the general fast Fourier transform

Abstract: There are many different methods in use to scrample voice signals Two of them seem to be of special importance: band-splitting and time-division In existing devices for scrambling analog signals often only on of these methods is implemented However, newer equipment, which is realized by digital circuitry, allow us to use both methods, band splitting and time division, at the same time