Showing papers on "Fast Fourier transform published in 1975"

Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method.

Abstract: We present further calculations of the three-dimensional mode patterns and power outputs from a high-power gas-dynamic laser, including a nonuniform flowing, saturable gain medium plus index inhomogeneities (shocks) inside the laser. The calculations are carried out using a plane-wave or k-space expansion together with the fast Fourier transform. A new expanding-beam coordinate transform converts all diverging or converging sections of the resonator mode into equivalent collimated beam sections. The resulting FFT propagation code is significantly faster than earlier propagation codes using other eigenmode expansions.

Digital signal analysis

Abstract: Review of least squares, orthogonality and the Fourier series review of continuous transforms transfer functions and convolution sampling and measurement of signals the discrete Fourier transform the fast Fourier transform the z-transform non-recursive digital systems digital and continuous systems simulation of continuous systems analogue and digital filter design review of random functions correlation and power spectra least-squares system design random sequences and spectral estimation.

Number theoretic transforms to implement fast digital convolution

Abstract: Transforms using number theoretic concepts are developed as a method for fast and error-free calculation of finite digital convolution. The transforms are defined on finite fields and rings of integers with arithmetic carried out modulo an integer and it is shown that under certain conditions this gives the same results as conventional digital convolution. Because of these characteristics they are ideally suited to digital computation by taking into account quantization of amplitude as well as time in their definition. When the modulus is chosen as a Fermat number a transform results that requires only on the order of N log N additions and word shifts but no multiplications. In addition to being efficient, they have no roundoff error and do not require storage of basis functions. There is a restriction on sequence length imposed by word length and a problem with overflow but methods for overcoming these are presented. Results of an implementation on an IBM 370/155 are presented and compared with the fast Fourier transform showing a substantial improvement in efficiency and accuracy. Variations on the basic number theoretic transforms are also presented.

Automated analysis of the electrical activity of the human brain (EEG): A progress report

Abstract: Clinical evaluation of electroencephalographic (EEG) recordings is based on complex subjective processes of data reduction and feature extraction. The high dimensionality of the EEG signal, its variability, and the lack of standard population values have retarded development of automated systems. An interactive, real-time analysis system (ADIEEG) has been implemented to develop features to simplify visual interpretation and facilitate automated classification. It uses a 40 000 word PDP15-PDP11 dual processor computer. Resident code occupies approximately 11 000 locations, while a maximum of 12 000 locations are used for buffers. The system performs 1) continuous spectral analysis using the fast Fourier transform to produce estimates of power and coherence, 2) parallel time domain analysis to detect sharp transients significant to diagnosis, 3) several forms of graphics, 4) simple algorithms to reject noncortical and instrumental artifact, 5) interactive parameter alteration and on-line feedback to adjust decision thresholds when necessary, and 6) extraction of diagnostically helpful features using heuristics based on clinical EEG. The ADIEEG system resides in the University of California, San Francisco Medical Center, and Langley Porter Neuropsychiatric Institute.

Scattering analysis of nonlinearly loaded antennas

Abstract: A novel technique-Volterra series analysis-is applied to the analysis of a nonlinearly loaded antenna. The electromagnetic field problem is first reduced to a network problem by application of the method of moments. The nonlinear network problem is then solved using the Volterra technique. A procedure with sinusoidal inputs for obtaining a time domain solution from the frequency domain solution without using fast Fourier transform techniques is demonstrated. The i-v characteristic of the nonlinear load is approximated from scattered power measurements. The derived i-v characteristic is then used to predict scattered power levels at different intermodulation responses of the loaded antenna.

A Discussion of Digital Processing in Synthetic Aperture Radar

Abstract: This is a summary paper describing the processing of synthetic aperture radar (SAR) data using digital correlation algorithms. Fundamental SAR theory as it applies to the various SAR modes, namely, strip mapping, spotlight mapping, and Doppler beam sharpened mapping, is described and a baseline design applicable to all SAR modes is presented. Digital processor design is developed, starting with a simple single filter mechanization and proceeding through more complex processing algorithms. Prefilter design is discussed, as is the more advanced processing algorithms, namely, multiple parallel prefilters, two-stage correlation, and FFT processing. The primary processor tradeoff is increased functional complexity versus reduced arithmetic and memory requirements. For high-resolution applications, the arithmetic requirements can be reduced by an order of magnitude or more by implementing the more advanced processing algorithms.

On the frequency weighted least-square design of finite duration filters

Abstract: The frequency weighted least-square design of finiteduration filters, in one and two dimensions, continuous and discrete is considered. Some new theoretical results and some practical design techniques for conventional and unconventional filters are presented. In some cases optimum discrete filters can be found conveniently by matrix inversion. In many cases a simple, iterative approximation technique using FFT can be used to carry out the design or to adjust the frequency response of filters.

Quantization errors in the fast Fourier transform

Abstract: When a fast Fourier transform (FFT) is implemented on a digital machine, quantization errors will arise due to finite word lengths in the digital system. The magnitudes and characteristics of these errors must be known if an FFT is to be designed with the minimum word lengths needed for acceptable performance. Two forms of FFT quantization, coefficient rounding and floating point arithmetic quantization, are analyzed in this paper. A theory is presented from which several new results can be obtained. The error characteristics of FFT's using exact and truncated values for the coefficients 1 and -j are found to be roughly equivalent. The accuracy of the theory is tested by computer simulations. Using the models introduced in this paper, new and accurate models can be derived to model quantization errors in high-speed convolution filters.

On the application of the number theoretic methods of high-speed convolution to two-dimensional filtering

Abstract: The Fermat number transform has been proposed as an aid to the rapid and precise computation of convolutions for digital filtering applications, but because it is less flexible than the fast Fourier transform, while more efficient, its applicability to one-dimensional filtering situations is somewhat limited. The purpose of this letter is to call attention to the fact that the constraints associated with filtering by Fermat number transforms are far less restrictive for two-dimensional filtering, and that, in fact, the Fermat number transform is well matched to the requirements of two-dimensional linear filtering.

A Comparative Study of 3-D Image Reconstruction Algorithms with Reference to Number of Projections and Noise Filtering

Abstract: A comparative study of four popular 3-D image reconstruction algorithms has been made. Particular attention was given to artifacts generated and noise sensitivity. The methods considered include two spatial domain convolution algorithms, the Linear Superposition with Compensation (LSC) and a Fourier Convolution Method (FCM), a direct Fast Fourier Transform method (FFT), and an algebraic technique, the Simultaneous Iterative Reconstruction Technique (SIRT). The methods were compared by computing reconstructed images for an identical input phantom image. The phantom image contains several edges and a 2% contrast object. Variations, artifacts and noise sensitivity are easily visualized by perspective plots of the reconstructed images. Considerations as to the optimum algorithm for a particular application are discussed.

Digital matched filtering with pipelined floating point fast Fourier transforms (FFT's)

Abstract: A special floating point arithmetic technique for fast Fourier transform (FFT) processors has been developed. The implementation of a high-speed pipeline FFT matched filter using the method employs significantly fewer components than a fixed-point processor with an equivalent performance level. The special floating point process avoids the nonlinear effects of fixed-point processors while achieving performance levels of traditional floating point arithmetic. Computer simulations were used to examine the performance of the system for linear FM pulse compression under a variety of conditions. In a specific case, with 8 bits plus sign quantization of the complex I- and Q-mantissa words in the processor and a 12-stage (4096-coefficient) FFT convolver, the mean square error (MSE) of the sidelobes relative to the peak from a single point target was less than -70 dB. Changes in waveform duration and sampling rate had negligible effect on the error level. This error characteristic can be treated as a computational or self noise, added to the input thermal noise of the radar receiver. Quantization artifacts or noise peaks which occur at levels consistent with the mantissa quantization are below levels (-50 dB for 8 bits plus sign) which would normally cause difficulty in an operational system.

Multi-path signal enhancing apparatus

Abstract: A plurality of processing channels each receive one of a number of direct d secondary signal energy rays. Each channel includes an A/D converter and computer the logic of which is controlled by a Fast Fourier Transform (FFT) algorithm to produce a plurality of data outputs each containing the amplitude and phase information of selected frequencies present in the channel input. To enhance signal detection and minimize signal strength fluctuations, each of the selected frequency outputs of each channel FFT is compared with each identical frequency output of other channel FFT's. Phase compensation means such as a shift register adjusts the phase relationship between all identical FFT outputs to permit their amplitudes to be constructively added. The individual subtotals so produced are totalled to provide an output representing the combined strength of the multi-path signal energy. In particular, medium interfaces such as an air-sea or air-ground interface, produce wide variations in the incoming direction of the signal energy to be detected.

A Special-Purpose Computer for Digital Signal Processing

Abstract: This paper describes a special-purpose processor for use in performing various operations on sampled signals. The system is fast, flexible, and programmable for performing, in real time, operations such as fast Fourier transformation (FFT) and digital filtering of acoustically derived signals.

Digital pulse compression via fast convolution

Abstract: The mathematical structure of the digital ambiguity function for a matched filtered linear FM (LFM) waveform is derived as a function of time-bandwidth product, sampling rate, and arbitrary delay and frequency shifts. It is found to be well behaved for sampling rates equal to or greater than the swept signal bandwidth, provided that time sidelobes are controlled using standard frequency domain weighting techniques. A digital convolution processor comprised of cascaded pipeline fast Fourier transforms (FFT's) is presented as a viable architecture for real-time filtering of moderately high bandwidth LFM signals, and tradeoffs among radix, pipeline clock rate, and convolutional efficiency are discussed. It is found that a modified floating-point computational scheme performs well in such a context and is especially useful if a large signal dynamic range must be accommodated. A radix-4 4096-point design example is considered and the effects of quantization and finite register length arithmetic upon the digital ambiguity function are demonstrated via simulation. It is found that input data, FFT coefficients, reference filter coefficients, and intermediate results can be represented with mantissas of modest bit length.

An efficient computer method for simulating transients on transmission lines with frequency dependent parameters

Abstract: A new technique for computer simulation of transients on transmission lines with frequency dependent parameters is presented. It is similar to the adaptation of Bergeron's method, which is used in the BPA transients program, but since it uses simpler shaped response functions in the required convolution process, the method inherently requires less storage and computational effort in a time-domain simulation. The main disadvantage of the method is the possible increase in programming effort necessary to handle certain types of boundary conditions. Accuracy of the method is verified by comparison with field tests. A new algorithm is also presented for the computation of transmission line response functions. For the functions evaluated in this paper, the method was found to be extremely accurate and computationally more efficient than the Fast Fourier Transform.

A Parallel Radix-4 Fast Fourier Transform Computer

Abstract: The organization and functional design of a parallel radix-4 fast Fourier transform (FFT) computer for real-time signal processing of wide-band signals is introduced.

A new hardware realization of high-speed fast Fourier transformers

Abstract: A new approach to the hardware implementation of high-speed processors dedicated to perform the fast Fourier transform (FFT) in real time is presented. This approach capitalizes on recent advances in semiconductor memory technology to eliminate conventional multipliers, and is shown to offer significant reductions in hardware complexity and power consumption. It also yields a highly modular hardware configuration. A modified floating-point arithmetic is incorporated to allow a wider dynamic range. Using standard available TTL integrated circuits, a throughput of complex data points at a 25 MHz word rate is possible.

Application of Fourier transforms for microwave radiometric inversions

Abstract: Existing microwave radiometer technology now provides a suitable method for remote determination of the ocean surface's absolute brightness temperature. To extract the brightness temperature of the water from the antenna temperature, an unstable Fredholm integral equation of the first kind is solved. Fourier transform techniques are used to invert the integral after it is placed into a cross correlation form. Application and verification of the methods to a two-dimensional modeling of a laboratory wave tank system are included. The instability of the ill-posed Fredholm equation is examined and a restoration procedure is included which smooths the resulting oscillations. With the recent availability and advances of fast Fourier transform (FFT) techniques, the method presented becomes very attractive in the evaluation of large quantities of data. Actual radiometric measurements of sea water are inverted using the restoration method, incorporating the advantages of the FFT algorithm for computations.

The use of the Parabolic Equation Method in Sound Propagation Modelling

Abstract: : The parabolic equation method (PE method) is the latest major development in the field of acoustic modelling. A detailed description of the derivation of the parabolic equation from the wave equation is given together with a description of a marching solution technique based on Fast Fourier Transforms (FFT). Limitations on the method in terms of allowable step length in range, number of FFT points to be used, and restrictions on pressure field behaviour in order to obtain a stable solution, are mentioned. Furthermore, numerical examples of propagation predictions are given for both shallow and deep water cases, with and without bottom absorption, and for both range-dependent and range-independent environmental conditions.

Frequency−domain seismic deconvolution filtering

Abstract: Reverberation in shallow water seismic profiles is a frequent problem that sometimes makes the geophysical interpretation of the records difficult. This can be further complicated if the acoustic source emits a bubble pulse. This paper describes a frequency−domain approach to deconvolution of the seismic signals to remove these artifacts. The resulting computation speed was found to exceed that of a time−domain Wiener filter by a factor of 10, for the same length dereverberation operator. Guidelines are established for using the FFT approach to deconvolution of seismic signals. The guidelines established are for the statistical stability and deconvolution ’’reach’’ of the filter, and prevention of FFT wrap−around effects. A digital filtering technique for obtaining a replica of the source signal from the received data record is presented and utilized in processing continental shelf air−gun data.

Progress in Annular-Array Imaging

Abstract: An improved algorithm is presented for computing an image from the amplitude and phase on an annular array. The old algorithm is discussed first and the new one developed from it. Simulation data shows the new algorithm to be 40 times faster than the old for 128 × 128 element images.

Fast Fourier Transform

Abstract: The main objective of this chapter is to develop a fast algorithm for efficient computation of the DFT. This algorithm, called the fast Fourier transform (FFT), significantly reduces the number of arithmetic operations and memory required to compute the DFT (or its inverse). Consequently, it has accelerated the application of Fourier techniques in digital signal processing in a number of diverse areas. A detailed development of the FFT is followed by some numerical examples which illustrate its applications.

Fast beamforming techniques for circular arrays

Abstract: This note describes a technique for beamforming on circular arrays which exploits the array geometry to facilitate the use of FFT techniques, and this leads to substantial saving in computing effort. The algorithm described here is particularly advantageous for systems employing a large number of sensors.Subject Classification: 60.30.

Generalized transforms

Abstract: A recent class of continuous transforms is generalized to give an infinite number of new transforms. A frequency interpretation of the transform base functions gives fast generalized transform (FGT) matrices using standard digit reversal techniques. Examples of FGT factorizations are given. Advantages of the generalized transforms include continuous transform properties for system design and analysis and FGT algorithms for practical applications using digital hardware.

Efficient computation of the transient response of lumped-distributed linear active networks

Abstract: An efficient approach to the computer-aided transient analysis of lumped-distributed linear networks is presented. The method is based on obtaining the transfer function of the circuit by symbolic techniques and the transient response is computed using the fast Fourier transform (FFT), the shifting parameter a and the new two-dimensional weighting factors \omega_{n}, \varepsilon_{r} . Experimental observations give ample evidence of good agreement between the theoretical and computed results for different values of a and \omega_{n}, \varepsilon_{r} . The algorithm used provides a large saving in computing time and computer memory when compared with available techniques.

On approximating Fourier integral transforms by their discrete counterparts in certain geophysical applications

Abstract: In the past, fast Fourier transforms (FFT's) have been used in geophysical data analysis and, with some success, in theoretical analysis. However, in the general situation when neither the function nor its transform image is of compact support, the digital transforms inevitably introduce aliasing errors in both the domain and range space. A technique to assess and minimize this error is given for a restricted set of functions.

A fast Gaussian method for Fourier transform evaluation

Abstract: A Gaussian method for fast evaluation of approximations to Fourier integral transforms is presented. This method is faster than the FFT for transforms of functions that require considerable computer time to compute. It is especially useful when transforms of high accuracy are needed.