Showing papers on "Fast Fourier transform published in 1970"

A linear filtering approach to the computation of discrete Fourier transform

Abstract: It is shown in this paper that the discrete equivalent of a chirp filter is needed to implement the computation of the discrete Fourier transform (DFT) as a linear filtering process. We show further that the chirp filter should not be realized as a transversal filter in a wide range of cases; use instead of the conventional FFT permits the computation of the DFT in a time proportional to N \log_{2} N for any N, N being the number of points in the array that is transformed. Another proposed implementation of the chirp filter requires N to be a perfect square. The number of operations required for this algorithm is proportional to N^{3/2} .

A Pipeline Fast Fourier Transform

Abstract: This paper describes a novel structure for a hardwired fast Fourier transform (FFT) signal processor that promises to permit digital spectrum analysis to achieve throughput rates consistent with extremely wide-band radars. The technique is based on the use of serial storage for data and intermediate results and multiple arithmetic units each of which carries out a sparse Fourier transform. Details of the system are described for data sample sizes that are binary multiples, but the technique is applicable to any composite number.

A Generalized Technique for Spectral Analysis

Abstract: A technique is presented to implement a class of orthogonal transformations on the order of pN log p N operations. The technique is due to Good [1] and implements a fast Fourier transform, fast Hadamard transform, and a variety of other orthogonal decompositions. It is shown how the Kronecker product can be mathematically defined and efficiently implemented using a matrix factorization method. A generalized spectral analysis is suggested, and a variety of examples are presented displaying various properties of the decompositions possible. Finally, an eigenvalue presentation is provided as a possible means of characterizing some of the transforms with similar parameters.

Speech spectrograms using the fast Fourier transform

Abstract: An important aid in the analysis and display of speech is the sound spectrogram, which represents a time-frequency?intensity display of the short-time spectrum.1-3 With many modern speech facilities centering around small or medium-size computers, it is often useful to generate spectrograms digitally, online. The fast Fourier transform algorithm provides a mechanism for implementing this efficiently.

Accumulation of Round-Off Error in Fast Fourier Transforms

Abstract: The fast Fourier transform (FFT) is an algorithm to compute the discrete Fourier coefficients with a substantial time saving over conventional methods. The finite word length used in the computer causes an error in computing the Fourier coefficients. This paper derives explicit expressions for the mean square error in the FFT when floating-point arithmetics are used. Upper and lower bounds for the total relative mean square error are given. The theoretical results are in good agreement with the actual error observed by taking the FFT of data sequences.

The inverse problem of radiography

Abstract: Radiography is a common diagnostic tool in the field of medicine. The physical process by which a radiograph is formed imposes fundamental limits on the resolution of a radiograph, however. The problem of increasing the resolution of a radiograph can be formulated as the solution of a convolution-type integral equation, but solving this integral equation is extremely difficult if noise is present in the data. A technique for solving integral equation problems was discovered several years ago, but it requires matrix inversion and is unwieldy for data sequences of hundreds or thousands of points. In this article it is shown how this existing technique can be adapted to solving convolution-type integral equations. By using the convolution properties of discrete Fourier transforms and the fast Fourier transform, it is possible to solve convolution-type integral equations even when the data sequences consist of hundreds or thousands of points.

Roundoff error analysis of the fast Fourier transform

Abstract: This paper presents an analysis of roundoff errors occurring in the floating-point computation of the fast Fourier transform. Upper bounds are derived for the ratios of the root-mean-square (RMS) and maximum roundoff errors in the output data to the RMS value of the input data for both single and multidimensional transformations. These bounds are compared experimentally with actual roundoff errors.

Fft processor with unique addressing

Abstract: A real time digital Fourier analyzer using the Cooley-Tukey algorithm for calculating the Fast Fourier Transform. An arithmetic unit simultaneously performs the two complex calculations Ar = Br + Wr Cr and A(r n/2) = Br - Wr Cr. The calculated results are simultaneously stored and retrieved in an in-place operation in dual buffers in successive iterations. Addressing of the buffers uses a single binary address counter and sequential bit complementing for simultaneously addressing both buffers. Parity checking of the binary counter output address controls multiplexing logic to selectively address the buffers to store the calculated results into the desired buffer storage locations.

General purpose matrix processor with convolution capabilities

Abstract: Method and apparatus of computing a generalized convolution of values from two matrices of complex values Ao through Am and Bo through Bn respectively. The formula used in the computation of each complex vector element Ck of the generalized convolution is WHERE P and U specify the increment for each succeeding element involved in a single convolution from each sequence respectively, Q and V specify the increments between first elements of successive convolution coefficients, in each sequence, respectively, and R and W specify the first pair of elements used in forming Co. PC specifies the number of Ck's to be computed. This computation has wide applicability to such allied mathematical operations as vector and matrix algebra, linear programming and a wide variety of transformation weighting and skirting operations such as Bessel function weighting, Hanning windows, complex Kernal transformations, and fast Fourier transforms. In addition, the apparatus described has capability to compute various special cases of the generalized equation involving vectors of real values only.

Matrix description of the fast Fourier transform

Abstract: The fast Fourier transform is usually described as a factorization. Recently this has been done in matrix terms. In this paper we present these results in sufficient detail that interested nonexperts can obtain the computer algorithm, and the necessary label permutations. We also count the number of arithmetic operations required in the calculation and point out the well known utility of base 2, both because of mathematical and machine hardware considerations. A simple FORTRAN program based on these ideas is included.

Comparing Power Spectra Computed by Blackman-Tukey and Fast Fourier Transform

Abstract: Since controversy has arisen as to whether the Blackman-Tukey or the fast Fourier transform (FFT) technique should be used to compute power spectra, single and cross spectra have been computed by each approach for artificial data and real data to provide an empirical means for determining which technique should be used. The spectra were computed for five time series, two sets of which were actual field data. The results show that in general the two approaches give similar estimates. For a spectrum with a large slope, the FFT approach allowed more window leakage than the Blackman-Tukey approach. On the other hand, the Blackman-Tukey approach demonstrated a better window closing capability. From these empirical results it is concluded that the Blackman-Tukey technique is more effective than the FFT approach in computing power spectra of short historic time series, but for long records the fast Fourier transform is the only feasible approach.

Frequency spectrum analyzer with fft computer

Abstract: A broad band of signal frequencies, whose power spectrum is to be analyzed by the Fast Fourier Transform (FFT) technique, is sampled by heterodyning with various beat frequencies fb and passage of the modulation products through a low-pass filter of bandwidth b/2 where b represents the width of a subband substantially narrower than the overall band of width B. By concurrently performing the heterodyning operation in two parallel channels, with introduction of a 90 DEG phase shift between beat and input frequencies in one of the channels, the two sidebands fb +fx and fb -fx (where fx represents any frequency within the selected subband b) can be separated in the outputs of the two channel filters. Frequency limit b/2 is selected in conformity with the capacity of an associated computer to handle the data from the FFT analysis of the subband spectrum. The sampling may be preceded by a transposition of the entire band B to a higher frequency range, in order to prevent any possible cluttering of the spectrum by harmonics of fb .

Nonrecursive digital filtering using cascade fast Fourier transformers

Abstract: The organization of a special-purpose digital processor for performing nonrecursive digital filtering is described. The processor uses two complementary cascade fast Fourier transformers. Each transformer can simultaneously transform two independent data blocks of length N words using \log_{2} N arithmetic units and 3/2 N complex words of digital storage. Continuous filtering is achieved by sectioning the input signal, performing a fast transform on each section, multiplying by the frequency characteristics of the desired filter, and inverse transforming. The cascade organization of the processor allows processing at very high speeds. Word rates in excess of 3 MHz are possible with currently available hardware.

On the derivation of the fast Fourier transform

Abstract: The fast Fourier transform (FFT) provides an effective tool for the calculation of Fourier transforms involving a large number of data points. The paper presents new and simple derivations for the two basic FFT algorithms that provide an intuitive basis for the manipulations involved. The derivation for the "decimation in time" algorithm begins with a crude analysis for the zero frequency and fundamental components using only two data samples, one at the beginning and the second at the midpoint of the period of interest. Successive interpolations of data points midway between those previously used result in a refinement of the amplitudes already determined and a first value for the next higher order coefficients. The derivation of the "decimation in frequency" algorithm begins by resolving the original data set into two new data sets, one whose transform includes only even harmonic terms and a second whose transform includes only odd harmonic terms. Since the first of the two new data sets repeats after the midpoint, it can be transformed using only the first half of the data points. The second of the new data sets is multiplied by the negative fundamental function, thereby reducing its order by one and converting it into a data set that transforms into even harmonics only; in this form it can also be transformed using only the first half of the data set. Successive applications of this procedure result finally in reducing the operation to the calculation of a large number of simple two-data-point transforms.

FFT organizations for high-speed digital filtering

Abstract: The subject of this paper concerns the analysis of high-radix FFT algorithms appropriate to hardware implementation of nonrecursive filters and the error propagation properties associated with these algorithms. Variations in the structure of high-radix fast Fourier transforms are illustrated by means of their Kronecker product expansions. The implications in hardware design of the different structures are discussed. These implications include tradeoffs between memory size and throughput rate as shown by Comparison of the logic organizations of serial and cascade fast Fourier processors. It is shown that in a high-radix cascade processor, significant reduction in memory can often be obtained with a modest increase in complexity of the arithmetic units. The relationship between the permutation matrices specified in a Kronecker factorization, and the structure and addressing of memory is brought out. As an example, the implementation of a nonrecursive digital filter is included using a radix-4 factorization which provides for an extremely simple memory organization. The permutation matrix is such that both data storage and addressing are provided using serial MOS shift registers. In order to compare the accuracy of high-radix algorithms relative to the base-2 algorithm, a fixed-point simulation study of the error propagation properties was conducted of a full radix-4 and a radix-16 FFT for N= 1024 and 256, respectively. The computational errors were compared with corresponding results obtained from the base-2 algorithm. The results of the simulations were also compared with a model which is a modification of that reported by P. D. Welch. The modification consists of the representation of roundoff error buildup in the radix-2 and radix-4 FFT in closed form by a nonhomogeneous difference equation. The illustrative example chosen is the lower bound discussed by Welch.

Synthesis of recursive digital filters using the FFT

Abstract: It has been shown that recursive digital filters can be synthesized using the fast Fourier transform. An algorithm for computer implementation has been developed and used in comparing the computation times and noise figures of filters synthesized in this manner with the computation times and noise figures of filters synthesized by recursion. A model has been proposed for analysis of noise in the two-pole filter. Predictions of this model have been found to be in good agreement with noise measurements.

Application of digital filtering in improving the resolution and the signal-to-noise ratio of nuclear and magnetic resonance spectra

Abstract: The digital filtering was proven to be useful in processing convolved signals. In this paper an application of this method in order to improve the resolution of nuclear and magnetic resonance spectra is presented, An emphasis is laid on the possibility of diminishing the noisy secondary effects. The processing was achieved using the FFT algorithm. Some practical results obtained are delivered.

Time domain waveform synthesizing method

Abstract: A method of synthesizing time domain waveforms from data in the frequency domain with improved resolution in the middle and lower frequencies Frequency data is broken down into groups and each group processed through an inverse Fourier or fast Fourier transform to obtain time domain data which is then added together in the proper time relationship The resulting composite digital information is then passed through a digital to analog converter to provide an analog time domain waveform

Fast fourier transform addressing system

Abstract: A fast Fourier transform addressing system using the DanielsonLanczos algorithm. The addressing system uses a basic relationship between addresses to allow implementation of the addressing system using a reduced number of address counters and reduced storage requirements for system constants.

A fast fourier transform algorithm for the radial distribution function

Abstract: A simple fast Fourier transformation (FFT) algorithm has been specifically adapted to calculate the experimental radial distribution function. The number of equi-spaced data points must be a power of two [N = 2n for integer n] and must be greater than the Nyquist frequency [N = 2(rmax) (smax)/2π]. When properly defined, the data set is expanded as an odd function. The greatest advantage of the FFT algorithm is its internal consistency—the ability to exactly transform back to the original domain.

Practical use of the fast fourier transform (fft) algorithm in time-series analysis.

Abstract: : The report serves as documentation for a collection of basic time-series analysis programs written for the CDC 3200 digital computers. These programs are predominantly written in FORTRAN and can be easily adapted to other digital computers. These programs are constructed around the fast Fourier transform (FFT) ALGORITHM. Rather than rediscuss the theory of the FFT algorithm, which is adequately described in the existing literature, this report deals with the practical aspects of the use of the FFT. The problems that can and in many instances do occur in computing spectral estimates are also addressed on the basis of an extensive literature review. (Author)

Nonlinear analysis of a balanced diode modulator

Abstract: Equations describing the performance of a balanced diode modulator are solved by an iterative process and the fast Fourier transform is used to obtain the spectrum of the output. The analysis provides an interesting example of the extension, to large-signal applications, of the method of solution recently described for quasilinear systems.

Multidimensional BIFORE transform

Abstract: The BIFORE transform is generalised to r dimensions. The corresponding power spectrum consists of Πi = 1r(1+ki) spectral points (Nt = 2ki), which are invariant under cyclic shifts of the r-dimensional input data. A physical interpretation of the BIFORE power spectrum is presented.

Fast Field Program

Abstract: The hydrodynamical field equations yield to three methods of solution. Two of these, the method of characteristics and that of separation of variables, have been widely used for both analytical and numerical investigation, but usually with either physical or mathematical approximations. The third, direct integration of the partial differential equations has always existed in principle but not until the advent of the fast Fourier transform could this approach be exploited in a systematic and economical manner. H. W. Marsh recognized this fact and introduced the concept of the fast field program (FFP). Several examples are presented and discussed which demonstrate that the FFP is not only feasible but a highly efficient algorithm for providing propagation estimations rapidly without undue approximations to the basic field equations. The results are compared with those predicted by both ray theory and normal‐mode theory. [This research was supported by the U. S. Underwater Sound Laboratory.]

Aspects of real-time digital spectral analysis

Abstract: In the field of control engineering there is a need to study the dynamic behaviour of systems which are subjected to random disturbances. A technique which is of great practical use is to describe the dynamic properties as a function of frequency. This involves determining the frequency content, or spectrum, of the disturbances, and the frequency response function of the system. There are many analogue and digital techniques which are designed for this type of spectral analysis. However, digital computer techniques are often avoided because they are slow, and data must be collected 'off-line'. A recently discovered computational method, termed the fast- Fourier-transform (FFT), enables digital spectral analysis to be carried- out in a much shorter time than was previously possible. In view of this discovery it was decided to develop digital computer programmes which would overcome the disadvantages of conventional digital spectral analysis. Using these programmes a computer would be connected, via an analogue to digital interface, to the signal source, and would process the data as it entered the computer. In the jargon of computing, the computer would be 'on-line' and analyzing the spectra in 'real-time'. The first part of the project consisted of an investigation of the FFP when programmed for an on-line digital computer. The results of this investigation showed that a rapid, accurate, and compact FFT could be programmed by using fixed-point arithmetic, and coding in an assembly language. The speed of the transform was sufficient to allow spectral analysis over a frequency range useful in control applications. Two on-line computer programmes based upon the YPP were then written; one for 'real-time' spectral analysis of a single record, and another for the 'real-time' estimation of the frequency response function relating two signals. In order that the results of these programmes could be sensibly interpreted, a statistical study was made of the spectral estimators used in the programmes. Arising from this study, several contributions to the field of digital spectra. analysis were made. These were : - 1) A more general covariance relationship for cross-spectral estimators. 2) An examination of aliasing in digital spectral estimators. 3) Some theoretical results concerning spectral estimators for closed loop systems with random disturbances inside the loop, Some experimental work was conducted with the real-time' spectral analysis programmes, and it was concluded that the tec: inique is more powerful than conventional digital. methods because it is on- line, and can provide estimates with improved resolution and statistical stability. Real-time digital spectral analysis methods also have the advantage that they may be simply and quickly modified to suit specific applications.

Digital voice processing with a wave function representation of speech

Abstract: Digital voice processing has advanced to a relatively high level of sophistication due to the development of the fast Fourier transform (FFT) algorithm. Complete vocoder systems have been developed around the FFT and with the advent of high-speed integrated circuits and read-only memories to implement sine and cosine tables for the FFT, actual real-time hardware processing has been accomplished.

Computation of short-pulse response from radar targets—An application of the fast Fourier transform technique

Abstract: The fast Fourier transform algorithm can be used as a highly efficient numerical means for the analysis of transient scattering phenomena. Examples illustrating the performance of this method are given for an incident pulse of Gaussian shape. It is noted that such a technique can best be exploited when additional characteristics of the radar system such as the actual transmitted waveform and the receiver transfer function are taken into account.

Application of a Modified Fast Fourier Transform to Calculate Human Operator Describing Function

Abstract: A modified fast Fourier transform (FFT) is used in a hybrid computer program to permit processing of tracking data during a run to yield the human operator's describing function almost immediately after the data-taking period. The computer processing time is substantially reduced at no cost in accuracy.