Showing papers on "Fast Fourier transform published in 1977"

Random Decrement Technique for Modal Identification of Structures

Abstract: An algorithm is developed to obtain the free responses of a structure from its random responses, due to some unknown or known random input or inputs, using the random decrement technique without changing time correlation between signals. The algorithm is tested using random responses from a ' 'generalized pay load" model and from the "space shuttle" model. The resulting free responses are then used to identify the modal characteristics of the two systems. The method is limited to structures that are linear or have small nonlinearities. N general, the experimental identification of structural modes of vibration is carried out by measuring the input (or inputs) to the structure under test and the resulting responses due to this input. Some vibration testing techniques, in order to simplify the identification procedure, use the free responses of structures. In such cases, although the input excitation need not be measured, some initial excitation is applied to the structure, and free responses are measured immediately after the initial exciting force is removed. There are situations where controlled excitation or initial excitation cannot be used. For example, if the structure to be tested is in operation, applying any kind of external force may cause undesirable interruption. Another example is the case of in-flight response measurements, where a complete knowledge of the excitation is usually not available. In such cases, the use of the "random decrement signature" technique (a special averaging procedure that is used to determine the step and/or impulse response from the random response) to obtain the free responses is a promising technique. The random decrement signature technique1 has been used successfully for failure detection and damping measurement of structures in single-statio n, single-mode response cases. Application of the random decrement signature technique to a multiple of signals changes the time correlation between the individual signals. If the resulting responses are to be used to identify several modes of a structure, the random decrement signature technique must be modified to keep the time correlation between signals unchanged. In this paper, an algorithm is developed to obtain the free responses of a linear structure from its random responses, due to some unknown or known random input or inputs, using the random decrement technique without changing time correlation between signals. The algorithm is tested by ap- plying it to random responses obtained from two real structures. The first structure is a generalized payload model previously tested using sine sweep method and analyzed by NASA Structural Analysis (NASTRAN). The second structure is the V& -scale space shuttle model with modal parameters previously determined using sine sweep method and fast Fourier transform (FFT). Only responses from four stations on the solid rocket boosters (SRB's) were considered in the case of the space shuttle model. The filtered random responses from these two structures were recorded and

A prime factor FFT algorithm using high-speed convolution

Abstract: Two recently developed ideas, the conversion of a discrete Fourier transform (DFT) to convolution and the implementation of short convolutions with a minimum of multiplications, are combined to give efficient algorithms for long transforms Three transform algorithms are compared in terms of the number of multiplications and additions Timing for a prime factor fast Fourier transform (FFT) algorithm using high-speed convolution, which was programmed for an IBM 370 and an 8080 microprocessor, is presented

An introduction to programming the Winograd Fourier transform algorithm (WFTA)

Abstract: Recently, Dr. Shmuel Winograd discovered a new approach to the computation of the discrete Fourier transform (DFT). Relative to fast Fourier transform (FFT), the Winograd Fourier transform algorithm (WFTA) significantly reduces the number of multiplication operations; it does not increase the number of addition operations in many cases. This paper introduces the new algorithm and discusses the operations comparison problem. A guide for programming is included, as are some preliminary running times.

Index mappings for multidimensional formulation of the DFT and convolution

Abstract: The mapping of one-dimensional arrays into two- or higher dimensional arrays is the basis of the fast Fourier transform (FFT) algorithms and certain fast convolution schemes. This paper gives the general conditions for these mappings to be unique and cyclic, and then considers the application to discrete Fourier transform (DFT) and convolution evaluation.

Vector radix fast Fourier transform

Abstract: A new radix-2 two-dimensional direct FFT developed by Rivard is generalized in this paper to include arbitrary radices and non-square arrays. It is shown that the radix-4 version of this algorithm may require significantly fewer computations than conventional row-column transform methods. Also, the new algorithm eliminates the matrix transpose operation normally required when the array must reside on a bulk storage device. It requires the same number of passes over the array on bulk storage as efficient matrix transpose routines, but produces the transform in bit-reversed order. An additional pass over the data is necessary to sort the array if normal ordering is desired.

Symbolic analysis of analog and digital circuits

Abstract: An efficient computer method for symbolic analysis of linear analog and digital circuits is described. The tableau formulation is used in the derivations but the mixed nodal tableau method is used in actual computations of analog circuits. One triangular factorization of the system matrix, followed by m + 1 forward and back substitutions, is sufficient to generate all partial derivatives of the numerator and denominator of the immitance function in terms of m variable elements. In the case of frequency dependent elements, the fast Fourier transform algorithm is used to obtain the polynomial coefficients. The computational cost is discussed and compared with that of other well-known symbolic analysis algorithms. Difficult programming sections are given. Little software is needed beyond that if a program for frequency analysis is available.

Natural, Dyadic, and Sequency Order Algorithms and Processors for the Walsh-Hadamard Transform

Abstract: The Walsh-Hadamard transform has recently received increasing attention in engineering applications due to the simplicity of its implementation and to its properties which are similar to the familiar Fourier transform. The transform matrices found so far to possess fast algorithms are the naturally ordered and dyadically ordered matrices, whose algorithms are similar to the Cooley-Tukey algorithm, and to the machine-oriented algorithm of Corinthios [2], respectively.

Acoustical detection and tracking system

Abstract: A plurality of acoustical transducers such as microphones are placed in appropriate array so that they are capable of detecting sonic energy emanating from an acoustical source such as an aircraft or a ground vehicle. The outputs of the transducers are sequentially sampled and multiplexed together, the time multiplexed signals then being converted from analog to digital form in an analog/digital converter. The output of the analog/digital converter is fed to a fast Fourier transformer (FFT), which transforms these signals to Fourier transform coefficients represented as real and imaginary (cosine and sine) components. The output of the fast Fourier transformer is fed to a digital processor. In this processor, the power and phase of each frequency bin for each microphone output is determined and the phase differences between signals received by pairs of microphones for each frequency bin of interest are determined. Each of these phase difference signals is divided by the frequency of their associated bin to provide a "phase difference slope" for each frequency bin and for each microphone pair. Signals received by any pair of microphones from the same target (regardless of frequency) have a common phase difference slope. The processor groups all common phase difference slopes together, these individual phase difference slopes each identifying a separate target. The phase difference slopes for each target are used to compute the direction of that target. By using two pairs of microphones in a mutually orthogonal array, target direction in both azimuth and elevation can be computed.

The Estimation of the Prediction Error Variance

Abstract: Spectral methods are used to construct an estimate of the variance of the prediction error for a normal, stationary process. The estimate obtained is shown to be strongly consistent and asymptotically normally distributed. Some aspects of the computations with respect to the fast Fourier transform are considered. The latter half of the article consists of a number of simulations, based on both generated and real data, which illustrate the results obtained. The relation between the estimate and that obtained from a high order autoregression is discussed.

New algorithms for digital convolution

Abstract: It is shown how the Chinese Remainder Theorem (CRT) can be used to convert a one-dimensional cyclic convolution to a multi-dimensional convolution which is cyclic in all dimensions. Then, special algorithms are developed which, compute the relatively short convolutions in each of the dimensions. The original suggestion for this procedure was made in order to extend the lengths of the convolutions which one can compute with number-theoretic transforms. However, it is shown that the method can be more efficient, for some data sequence lengths, than the fast Fourier transform (FFT) algorithm. Some of the short convolutions are computed by methods in an earlier paper by Agarwal and Burrus. Recent work of Winograd, consisting of theorems giving the minimum possible numbers of multiplications and methods for achieving them, are applied to these short convolutions.

Beamformer for wideband signals

Abstract: A beamformer for wideband signals wherein the elemental signals available the sensors of a receiving array are subjected to Fast Fourier Transformations which decompose them into a plurality of narrowband signals. The narrowband signals, which consist of signals having the same Fourier coefficients, are subjected to appropriate phase shifts to form a plurality of narrowband beams having the directional characteristics desired. Reconstruction of the wideband signal is accomplished by subjecting the narrowband beams to an inverse Fast Fourier Transform. The beamforming is thus performed with narrowband signals, and only phase shift circuits are required.

Computational Complexity of Fourier Transforms over Finite Fields

Abstract: : This paper describes a method for computing the Discrete Fourier Transform (DFT) of a sequence of n elements over a finite field GF (p to the mth power) with a number of bit operations 0(nm log (nm). P(q)) where P(q) is the number of bit operations required to multiply two g-bit integers and g approx. = 2 log sub 2 + 4 log sub 2m + 4 log sub 2p. This method is uniformly applicable to all instances and its order of complexity is not inferior to that of methods whose success depends upon the existence of certain primes.

Power-Spectrum Estimation

Abstract: Spectral estimation often forms the basis for distinguishing and tracking signals of interest in the presence of noise and for extracting information from the received data. The application of Fourier techniques to the problem of estimating the properties of sinusoids in noise dates back as far as Shuster (1898), Fourier spectrum analysis is the basis for almost all spectral-estimation equipment, * including the common sweeping-filter spectrum analyzer, the parallel filter bank, the fast Fourier transform (FFT), the delay-line time compressor (Deltic), and the compressive spectrum analyzer (Microscan). A problem with Fourier spectrum analysis, however, is that it makes implicit assumptions concerning data outside the observation interval and, frequently, these physically unrealistic assumptions reduce the quality of the estimates. During the past decade, two radically different non-Fourier spectral-estimation techniques have emerged — maximum-entropy spectrum analysis (Burg, 1967) and spectral decomposition (Pisarenko, 1973). These techniques offer alternative and often more realistic data models which, in many cases, lead to better estimation performance. This paper reviews the Fourier methods and compares them to the new techniques in terms of signal models assumed by the three basic methods and their ability to distinguish multiple sinusoids in noise.

The zoom FFT using complex modulation

Abstract: A recent paper by Yip discussed the zoom transform as derived from the defining equation of the FFT. This paper simplifies the concepts and removes some of the restrictions assumed by Yip; ie., the total number of points need not be a power of 2. The technique is based on first specifying the desired center frequency, bandwidth, and frequency resolution. The signal is then sampled, modulated, and lowpass filtered. This result is purposely aliased, then transformed using an FFT algorithm. The result is an M-point frequency spectra of the desired bandwidth centered about the center frequency with a higher degree of resolution than could be directly obtained using an M-point transform.

Configurable parallel arithmetic structure for recursive digital filtering

Abstract: An FDM/TDM transmultiplexer uses sampling rate multiplication to increase the sampling rate for time division multiplexed (TDM) to frequency division multiplexed (FDM) conversion and decrease the sampling rate for FDM to TDM conversion. The rate multiplication filters are realized digitally in order to exploit the computational advantage of Fast Fourier Transform (FFT) algorithm, and channel filtering is implemented by a single time-shared sixth-order elliptic digital recursive filter. A novel FFT processor and recursive filter are disclosed which may be used in the system.

Digital filtering using pseudo fermat number transforms

Abstract: In this paper pseudo Fermat number transforms (FNT's) are discussed. These transforms are defined in a ring of integers modulo an integer submultiple of a pseudo Fermat number, and can be computed without multiplications while allowing a great flexibility in word length selection. Complex pseudo FNT's are then introduced and are shown to relieve some of the length limitations of conventional Fermat number transforms (FNT's). These transforms, which under certain conditions can be computed via fast transform algorithms allow the implementation of digital filters with better efficiency and accuracy than the fast Fourier transform (FFT).

Two‐dimensional maximum entropy reconstruction of radio brightness

Abstract: This paper describes a procedure for maximum entropy reconstruction of two-dimensional radio brightness maps from noisy interferometer measurements. The method defines a map that obeys the nonnegativity constraint and is, in a sense, the smoothest of all brightness distributions that agree with the visibility measurements within the errors of observation. This approach acknowledges the fact that signal-to-noise considerations have a strong influence on useful resolution; fine structure appears only to the extent justified by measurement accuracy. Iterative computing is needed to find the maximum entropy image. It is shown that the primary computational burden of maximum entropy reconstruction involves calculations that are efficiently performed by fast Fourier transform techniques. Different techniques are used depending on whether visibility data are irregularly distributed in the u,v plane or interpolated onto a rectangular lattice prior to reconstruction. The efficiency of the fast Fourier transform provides a tremendous computational advantage with the result that maximum entropy reconstruction on a moderately large grid (64×64) is practicable at reasonable cost. Several comparative examples are shown, and some of the limitations of the present theory of maximum entropy imaging are identified.

A Unified Treatment of Discrete Fast Unitary Transforms

Abstract: A set of recursive rules which generate unitary transforms with a fast algorithm (FUT) are presented For each rule, simple relations give the number of elementary operations required by the fast algorithm The common Fourier, Walsh-Hadamard (W-H), Haar, and Slant transforms are expressed with these rules The framework developed allows the introduction of generalized transforms which include all common transforms in a large class of “identical computation transforms” A systematic and unified view is provided for unitary transforms which have appeared in the literature This approach leads to a number of new transforms of potential interest Generalization to complex and multidimensional unitary transforms is considered and some structural relations between transforms are established

Spectra of PSK Signals with Overlapping Baseband Pulses

Abstract: This paper derives a new power spectrum formula for constant-envelope PSK signals. The only signal constraints are that the baseband pulses be of finite duration and the pulse amplitudes be mutually independent. The computer program used to implement the formula is made highly efficient by the use of fast Fourier transform (FFT) algorithms. The key contribution of the new result is the high degree of baseband pulse overlap permitted before the computation cost becomes excessive. In some computation approaches, this cost grows as LN, where L is the number of modulation levels and N is the number of data periods spanned by the pulse. In the approach developed here, the cost is independent of L and grows roughly linearly with N . The cost improvement, which is substantial for large L and N , is exploited here by using the new approach for a wide variety of baseband pulses. Results are given which demonstrate the potential value, in terms of narrower spectra and lower sidelobes, of using overlapping pulses.

Digital technique for generating synthetic aperture radar images

Abstract: This paper describes a digital processing method applicable to a synthetic aperture radar, to be carried by the space shuttle or by satellites The method uses an earth-fixed coordinate system in which corrective procedures are invoked to compensate for errors introduced by the satellite motion, earth curvature, and wavefront curvature Among the compensations discussed are those of the coordinate system, skewness, roll, pitch, yaw, earth rotation, and others The application of a Fast Fourier Transform in the numerical processing of the two-dimensional convolution is discussed in detail

Some Results in Fixed-Point Fast Fourier Transform Error Analysis

Abstract: This correspondence presents some results in fixed-point error analysis of fast Fourier transform algorithms. Two kinds of schemes for preventing overflow are considered in the analysis. The results, obtained for the decimation-in-frequency form of the algorithm, are compared with those of decimation-in-time. The results show that the error performance of the decimation-in-frequency algorithm is better than that of decimation-in-time.

X-Ray Reconstruction by Finite Field Transforms

Abstract: The reconstruction of two-dimensional density distributions, obtained by scanning a source of radiation through various angles, was suggested originally by Bracewell and Riddle. Recently Ramachandran and Lakshminarayanan proposed that a direct convolution technique might be used to reconstruct three-dimensional objects from projections obtained from x-ray scanning. They showed that the reconstruction of the sections of an object in the spatial domain was the average of the convolutions of an appropriate filter function with these projections over all possible projection angles. The best filter function presently known for this method was given by Shepp and Logan. The new tomograph machines are now using substantially larger sets of data than considered originally by Shepp and Logan. As a consequence, the fast Fourier traisform (FFT) method of convolution now competes favorably inspeed with the direct convolution method, described by Shepp and Logan. In this paper a fast transform method is used to perform the convolution of the Shepp and Logan filter with the projection data. To perform these convolutions at optimal computational speed and accuracy, a fast Fourier transform (FFT) algorithm is developed over a finite or Galois field GF(q) of integers instead of the usual field of complex numbers. GF(q) is the field of integers modulo q where q is a prime of the form k. 2n+ 1. Important features of this computational technique to tomography are both a high computational speed and an absence of round-off errors.

Hardware realization of digital signal processing elements using the residue number system.

Abstract: In the past, hardware realization of digital signal processing elements have been based upon binary arithmetic concepts. Because of the dependence between digits in binary arithmetic operations, the hardware required to construct arithmetic elements is cumbersome. In the residue number system, arithmetic operations can be performed with complete independence between digits and a corresponding reduction in hardware complexity. In fact, using current technology, arithmetic operations can be carried out using arrays of look-up tables placed in high density ROMs. This paper discusses the application of the residue number system to realizing digital signal processing elements using such arrays and advantages and disadvantages over conventional realizations are discussed. Examples are given of recursive filter and FFT butterfly element realization.

An algorithm for the two dimensional FFT

Abstract: Conventional two dimensional fast Fourier transforms become very slow if the size of the matrix becomes too large to be contained in memory. This is due to the transposition of the matrix that is required. This new algorithm is designed to remove the requirement for transposition, thereby, greatly increasing the speed of the process. This algorithm is extremely valuable on small disc based computers.

A Fast Z Transformation Algorithm for System Identification

Abstract: Algorithms for system identification and the computation of its mathematical model through a ``fast'' Z transformation of its sampled response in the presence of noise are introduced. It is shown that by iteratively applying constant-damping?and constant-frequency contour finite Z transforms a system's mathematical model?in the presence of noise can be efficiently evaluated. On line tracking of the poles and zeros of relatively rapidly time-variant systems such as a space shuttle or a jet aircraft are possible applications. An organization for a high-speed machine including a fast Fourier transform processor for on line identification of relatively rapidly time-variant system is suggested. Applications of the described algorithms include enhancement of poles in spectral analysis of signals, representation of signals by poles and zeros for signal classification, coding and recognition, filter synthesis, adaptive filtering, identification of parameters in curve fitting problems, in addition to system identification in the presence of noise. The procedure presented here is a transform domain approach that is distinct, to the knowledge of the author, when compared to known identification techniques in which a best fitting is made to an assumed mathematical model of the system. In addition to the smoothing obtained here through the computation of spectra in the Z plane of a time series including redundancy, no priori knowledge of the order of the system needs be assumed.

Spectral interpolation: zero fill or convolution

Abstract: Zero fill, or augmentation by zeros, is a method used in conjunction with fast Fourier transforms to obtain spectral spacing at intervals closer than obtainable from the original input data set. In the present paper, an interpolation technique (interpolation by repetitive convolution) is proposed which yields values accurate enough for plotting purposes and which lie within the limits of calibration accuracies. The technique is shown to operate faster than zero fill, since fewer operations are required. The major advantages of interpolation by repetitive convolution are that efficient use of memory is possible (thus avoiding the difficulties encountered in decimation in time FFTs) and that is is easy to implement.