Showing papers on "Fast Fourier transform published in 1992"

Digital Signal Processing: Principles, Algorithms, and Applications

Abstract: 1. Introduction. 2. Discrete-Time Signals and Systems. 3. The Z-Transform and Its Application to the Analysis of LTI Systems. 4. Frequency Analysis of Signals and Systems. 5. The Discrete Fourier Transform: Its Properties and Applications. 6. Efficient Computation of the DFT: Fast Fourier Transform Algorithms. 7. Implementation of Discrete-Time Systems. 8. Design of Digital Filters. 9. Sampling and Reconstruction of Signals. 10. Multirate Digital Signal Processing. 11. Linear Prediction and Optimum Linear Filters. 12. Power Spectrum Estimation. Appendix A. Random Signals, Correlation Functions, and Power Spectra. Appendix B. Random Numbers Generators. Appendix C. Tables of Transition Coefficients for the Design of Linear-Phase FIR Filters. Appendix D. List of MATLAB Functions. References and Bibliography. Index.

Computational Frameworks for the Fast Fourier Transform

Abstract: 1. The Radix-2 Frameworks. Matrix Notation and Algorithms The FFT Idea The Cooley-Tukey Factorization Weight and Butterfly Computations Bit Reversal and Transposition The Cooley-Tukey Framework The Stockham Autosort Frameworks The Pease Framework Decimation in Frequency and Inverse FFTs 2. General Radix Frameworks. The General Radix Ideas Index Reversal and Transposition Mixed-Radix Factorizations Radix-4 and Radix-8 Frameworks The Split-Radix Frameworks 3. High Performance Frameworks. The Multiple DFT Problem Matrix Transposition The Large Single-Vector FFT Problem Multi-Dimensional FFT Problem Distributed Memory FFTs Shared Memory FFTs 4. Selected Topics. Prime Factor FFTs Convolution FFTs of Real Data Cosine and Sine Transforms Fast Poisson Solvers Bibliography Index.

Frequency-domain and multirate adaptive filtering

Abstract: An overview is presented of several frequency-domain adaptive filters that efficiently process discrete-time signals using block and multirate filtering techniques. These algorithms implement a linear convolution that is equivalent to a block time-domain adaptive filter, or they generate a circular convolution that is an approximation. Both approaches exploit the computational advantages of the FFT. Subband adaptive filtering is also briefly described. Here the input data are first processed by a bank of narrowband bandpass filters that are approximately nonoverlapping. The transformed signals are then decimated by a factor that depends on the degree of aliasing that can be tolerated, resulting in a large computational savings. Several performance issues are considered, including convergence properties and computational complexities of the adaptive algorithms and the effects of circular convolution and aliasing on the converged filter coefficients. >

Simulation of 3-D random rough surface by 2-D digital filter and fourier analysis

Abstract: 2-D FIR filters are applied to the generation of 3-D random surfaces in this paper. The circularly symmetric low-pass filters are firstly employed to generate the isotropic random surfaces with no restrictions on the shape of ACF. To simulate real rough surfaces, however, one has to generate the surfaces having an expected autocorrelation function and height distribution, which requires determining the filter coefficients corresponding to the specified ACF. A spectrum analysis approach is developed in this paper to compute the filter coefficients and the FFT procedure provides an efficient way to implement the filter. The Johnson translator system of distribution is introduced into the generation procedure of non-Gaussian surface with a required ACF. The analysis indicates that the 2-D FIR filters used in this paper are mathematically identical with the 2-D MA time series model which simulates the ACF within whole correlation regions.

Convolution backprojection image reconstruction for spotlight mode synthetic aperture radar

Abstract: Convolution backprojection (CBP) image reconstruction has been proposed as a means of producing high-resolution synthetic-aperture radar (SAR) images by processing data directly in the polar recording format which is the conventional recording format for spotlight mode SAR. The CBP algorithm filters each projection as it is recorded and then backprojects the ensemble of filtered projections to create the final image in a pixel-by-pixel format. CBP reconstruction produces high-quality images by handling the recorded data directly in polar format. The CBP algorithm requires only 1-D interpolation along the filtered projections to determine the precise values that must be contributed to the backprojection summation from each projection. The algorithm is thus able to produce higher quality images by eliminating the inaccuracies of 2-D interpolation, as well as using all the data recorded in the spectral domain annular sector more effectively. The computational complexity of the CBP algorithm is O(N/sup 3/). >

Optimal meshes for integrals in real- and reciprocal-space unit cells.

Abstract: We present a detailed method to construct uniform meshes for fast Fourier transforms in electronic-structure calculations. We show that a drastic reduction in the mesh size can be frequently achieved by using an unconventional set of primitive lattice vectors. The same method can be applied also to obtain optimum sets of special points for Brillouin-zone integrals, generalizing previous schemes.

Discrete multiple tone modulation with coset coding for the spectrally shaped channel

Abstract: A discrete approach to multiple tone modulation is developed for digital communication channels with arbitrary intersymbol interference (ISI) and additive Gaussian noise. Multiple tone modulation is achieved through the concatenation of a finite block length modulator based on discrete Fourier transform (DFT) code vectors, and high gain coset or trellis codes. Symbol blocks from an inverse DFT (IDFT) are cyclically extended to generate ISI-free channel-output symbols that decompose the channel into a group of orthogonal and independent parallel subchannels. Asymptotic performance of this system is derived, and examples of asymptotic and finite block length coding gain performance for several channels are evaluated at different values of bits per sample. This discrete multiple tone technique is linear in both the modulation and the demodulation, and is free from the effects of error propagation that often afflict systems employing bandwidth-optimized decision feedback plus coset codes. >

Method and apparatus for public key exchange in a cryptographic system

Abstract: The present invention is an elliptic curve cryptosystem that uses elliptic curves defined over finite fields comprised of special classes of numbers. Special fast classes of numbers are used to optimize the modulo arithmetic required in the enciphering and deciphering process. The class of numbers used in the present invention is generally described by the form 2 q -C where C is an odd number and is relatively small, for example, no longer than the length of a computer word (16-32 bits). When a number is of this form, modulo arithmetic can be accomplished using shifts and adds only, eliminating the need for costly divisions. One subset of this fast class of numbers is known as "Mersenne" primes, and are of the form 2 q -1. Another class of numbers that can be used with the present invention are known as "Fermat" numbers of the form 2 q +1. The present invention system whose level of security is tunable. q acts as an encryption bit depth parameter, such that larger values of q provide increased security. Inversion operations normally require an elliptic curve algebra can be avoided by selecting an inversionless parameterization of the elliptic curve. Fast Fourier transform for an FFT multiply mod operations optimized for efficient Mersenne arithmetic, allow the calculations of very large q to proceed more quickly than with other schemes.

Frequency Domain Analysis of Undamped Systems

Abstract: A numerical tool commonly used in digital signal processing, the exponential window method, is briefly reviewed in this paper and applied to problems in structural dynamics. This method allows one to carry out analyses of undamped structures in the frequency domain, and yields highly accurate results for both discrete and continuous systems. In essence, the solution involves: (1) Finding both the transfer function and the forward Fourier transform of the excitation for complex frequencies; (2) performing a standard inverse Fourier transformation into the time domain; and (3) removing the effect of the complex frequencies by means of an exponential factor (or window). Excellent results are obtained when this factor is chosen so that the power of the excitation and response signals at the end of the window are attenuated by some three orders of magnitude. In such case, it is found that a quiet zone (a tail of trailing zeroes) is not needed for accurate computations, and that temporal aliasing (folding) is n...

Conflict free memory addressing for dedicated FFT hardware

Abstract: A multibank memory address assignment for an arbitrary fixed radix fast Fourier transform (FFT) algorithm suitable for high-speed single-chip implementation is developed. The memory assignment is 'in place' to minimize memory size and is memory-bank conflict-free to allow simultaneous access to all the data needed for calculation of each of the radix r butterflies as they occur in the algorithm. Address generation for table lookup of twiddle factors is also included. The data and twiddle factor address generation hardware is shown to have small size and high speed. >

Algorithms for Parallel Memory II: Hierarchical Multilevel Memories

Abstract: In this paper we introduce parallel versions of two hierarchical memory models and give optimal algorithms in these models for sorting, FFT, and matrix multiplication. In our parallel models, there are $P$ memory hierarchies operating simultaneously; communication among the hierarchies takes place at a base memory level. Our optimal sorting algorithm is randomized and is based upon the probabilistic partitioning technique developed in the companion paper for optimal disk sorting is a two-level memory with parallel block transfer. The probability of using $\ell$ times the optimal running time is exponentially small in $\ell$(log $\ell$)log $P$.

Fast adaptive algorithms for micromagnetics

Abstract: Evaluation of the long-range magnetostatic field is the most time-consuming part in a micromagnetic simulation. In a magnetic system with N particles, the traditional direct pairwise summation method yields O(N/sup 2/) asymptotic computation time. An adaptive fast algorithm fully implementing the multipole and local expansions of the field integral is shown to yield O(N) computation time. Fast Fourier transform techniques are generalized to entail finite size magnetic systems with nonperiodic boundary conditions, yielding O(N log/sub 2/ N) computation time. Examples are given for calculating domain wall structures in Permalloy thin films. The efficiency of the fast Fourier transform makes it almost always the faster method for any large-size system, while the multipole algorithm remains effective for more complex geometries and systems with highly irregular or nonuniform particle distributions. >

The efficient memory-based VLSI array designs for DFT and DCT

Abstract: Efficient memory-based VLSI arrays and a new design approach for the discrete Fourier transform (DFT) and discrete cosine transform (DCT) are presented. The DFT and DCT are formulated as cyclic convolution forms and mapped into linear arrays which characterize small numbers of I/O channels and low I/O bandwidth. Since the multipliers consume much hardware area, the designs utilize small ROMs and adders to implement the multiplications. Moreover, the ROM size can be reduced effectively by arranging the data in the designs appropriately. The arrays outperform others in the architectural topology (local and regular connection), computing speeds, hardware complexity, the number of I/O channels, and I/O bandwidth. They benefit from the advantages of both systolic array and the memory-based architectures. >

Orthogonal multiple carrier data transmission

Abstract: In this paper, the idea of data transmission by essentially using the discrete Fourier transform (DFT) algorithm, the idea of overlapped but orthogonally sampled multiple data transmission channels, and the idea of subband signal processing by means of DFT polyphase filter banks are combined. The main result in a novel orthogonal multiple carrier (OMC) data transmission system with high computational efficiency and high bandwidth efficiency. Intersymbol interference is avoided by choosing each channel as a Nyquist system. The problem of unavoidable crosstalk between adjacent channels is solved by sampling the equally spaced zeros of the crosstalk impulse response. It is shown that in a critically sampled DFT filter bank the space between the zeros exactly equals one symbol period. The time offset of half a symbol period between the zeros of the real part and the zeros of the imaginary part of the crosstalk impulse response leads to a special staggering technique with time offset. Finally the structure of the polyphase filter banks is optimized resulting in fast Fourier transforms of half of the original length.

Computation of resonant frequencies and quality factors of open dielectric resonators by a combination of the finite-difference time-domain (FDTD) and Prony's methods

Abstract: Open dielectric resonators using the FD-TD (finite-difference time-domain) method are analyzed. Resonant frequencies and quality factors are calculated using Prony's method instead of the classical fast Fourier transform. In this way, reductions of up to two orders of magnitude are achieved in computation time. The results obtained are in good agreement with those reported by other authors. >

A Spectral Element Approach to Wave Motion in Layered Solids

Abstract: A matrix methodology similar to that of the finite element method is developed for the analysis of stress waves in layered solids. Because the mass distribution is modeled exactly, the approach gives the exact frequency response of each layer. The fast Fourier transform and Fourier series are used for inversion to the time/space domain. The impact of a structured medium with multiple layers is used to demonstrate the method. Comparison with existing propagator and direct global matrix methods show the present approach to be computationally more efficient.

High-accuracy Fourier analysis based on synchronous sampling techniques

Abstract: The lack of synchronization between the sampling rate and the signal frequency represents the main source of errors in the frequency analysis of periodic signals performed by means of digital techniques. Several algorithms have been proposed in the literature to reduce these errors, at the cost of an increment in the computational burden imposed by the instrument. The complete elimination of these errors can be achieved only when the sampling rate is synchronized to the fundamental frequency of the signal to be analyzed. It is shown how this can be attained, and the results of fast Fourier transforms (FFTs) performed on multifrequency signals under synchronous sampling conditions are indicated. The accuracy of these measurements is discussed. >

An overlapped block motion compensation for high quality motion picture coding

Abstract: An overlapped block motion compensation scheme, which is especially suitable for hybrid coding schemes combine with overlapped transforms, is described. Motion compensation is achieved using enlarged and overlapped blocks. When motion estimation is executed, a window function operates on the prediction error signals. The power of the resulting signal is calculated and used for motion vector evaluation. When motion-compensated prediction is executed, the same window function operates on the prediction signals. The total prediction signal is generated by summing up all the block signals obtained. With this scheme, prediction signals without blocking edges are obtained. The mean square error (MSE) of the prediction error signal is reduced about 19%, and FFT analysis results show that the coefficient power in the frequency domain is concentrated in the lower frequency part. This indicates higher coding efficiency for the overlapped transforms, and for conventional block based transforms as well. >

Calculation of Diffuse Scattering from Simulated Disordered Crystals: a Comparison with Optical Transforms

Abstract: An investigation is carried out on the feasibility of calculating the diffuse scattering from computer simulations of crystals containing substitutional and displacement disorder which have hitherto been used in conjunction with optical transform methods to aid in the interpretation of observed X-ray diffraction patterns. A straightforward method of calculation based on direct Fourier summation of the simulated lattice is used instead of fast Fourier transform (FFT) techniques, which are not in general suited to this type of calculation. This computational method provides a number of advantages over the optical method. It allows calculation in three dimensions, more flexibility in the assignment of atomic positions and scattering power of the individual atoms involved and the computation can be made in absolute units allowing for direct comparison with data scaled to electron units. Comparison of the two techniques is presented using, as an example, a simulation of planar disorder in a synthetic mullite. It is found that calculated patterns of comparable quality to ones obtained optically are feasible using the current generation of computers. Nevertheless, the transforms can still consume considerable computational resources particularly when the extension to three dimensions is required.

Regularized linear method for reconstruction of three-dimensional microscopic objects from optical sections.

Abstract: The inverse problem involving the determination of a three-dimensional biological structure from images obtained by means of optical-sectioning microscopy is ill posed. Although the linear least-squares solution can be obtained rapidly by inverse filtering, we show here that it is unstable because of the inversion of small eigenvalues of the microscope's point-spread-function operator. We have regularized the problem by application of the linear-precision-gauge formalism of Joyce and Root [J. Opt. Soc. Am. A 1, 149 (1984)]. In our method the solution is regularized by being constrained to lie in a subspace spanned by the eigenvectors corresponding to a selected number of large eigenvalues. The trade-off between the variance and the regularization error determines the number of eigenvalues inverted in the estimation. The resulting linear method is a one-step algorithm that yields, in a few seconds, solutions that are optimal in the mean-square sense when the correct number of eigenvalues are inverted. Results from sensitivity studies show that the proposed method is robust to noise and to underestimation of the width of the point-spread function. The method proposed here is particularly useful for applications in which processing speed is critical, such as studies of living specimens and time-lapse analyses. For these applications existing iterative methods are impractical without expensive and/or specially designed hardware.

Shift- and rotation-invariant object reconstruction using the bispectrum

Abstract: Triple correlations and their Fourier transforms, called bispectra, have properties desirable for image-sequence analysis and reconstruction. Specifically, the triple correlation of a two-dimensional sequence is shift invariant, vanishes for symmetric probability-distribution-function processes including Gaussian random processes of unknown covariance, and can be used to recover the original sequence uniquely to within a linear phase shift. Discrete analysis is carried out for a deterministic signal in an additive random-noise field. This approach yields discrete algorithms for implementation and permits explicit treatment of the additive noise. Recursive and least-squares fast-Fourier-transform-based algorithms for reconstructing a two-dimensional discrete Fourier transform from the bispectrum are reviewed in detail. While phase retrieval using the least-squares algorithm requires phase unwrapping, the recursive method requires a more simple correction. The bispectrum is applied to the estimate of a randomly translating or rotating object from a sequence of noisy images. The technique does not require solution of the correspondence problem, which is the primary advantage. The method works in low signal-to-noise-ratio cases, when conventional solutions to estimating the object correspondence may fail. Experimental results presented include application of the method to a sequence of infrared images.

A new passive synthetic aperture technique for towed arrays

Abstract: An algorithm that synthesizes apertures in the beam domain using FFT transformations and performs coherent processing of subaperture signals at successive time intervals is presented. Experimental tests of the algorithm show that for ocean environments with spatial coherence longer than the synthetic aperture length and for signals with temporal coherence longer than the required acquisition time, a synthetic array gain is achieved which roughly corresponds to the length of an equivalent fully populated array. In the experiments, transducer generated CW with phase stability and pseudorandom signals were used. Limitations on the spatial and temporal coherence were introduced only by the medium, the temporal coherence of the pseudorandom signal, and the shape and stability of the line array used. >

Recursive fast algorithm and the role of the tensor product

Abstract: The use of the tensor product for modeling and designing FFT algorithms is addressed. The benefit of the tensor product approach lies in the strong connection between certain tensor product constructs and important computer architectures. The scope of the tensor product approach is generalized to include a much larger class of fast recursive algorithms. This greatly enhances the versatility of the tensor product technique and brings many different algorithms to the level of understanding and flexibility enjoyed by the FFT. >

Sensorless speed detection of inverter fed induction motors using rotor slot harmonics and fast Fourier transform

Abstract: A novel approach to sensorless speed detection for adjustable-speed AC drives is described. No a priori knowledge is required about the motor construction, electrical parameters, or load condition. In addition, no external tuning is needed for the system. The technique is based on instantaneous spectral estimation using the fast Fourier transform, whereby the speed-dependent slot ripple harmonic frequency is determined. For the assessment of this technique, an all-digital speed detector has been built around a general-purpose 386 microcomputer. The performance of this detector over a wide range of inverter frequencies and load conditions is discussed. >

Novel fast GPS/GLONASS code-acquisition technique using low update rate FFT

Abstract: A novel ‘differential’ decoding technique is proposed which enables pre-averaging instead of postintegration for a substantially low update rate ‘FFT-IFT’ correlation in sprcadspectrum (navigational) receivers. N-channel code acquisition can be performed to monitor the time dispersion with FFT time left to analyse frequency dispersion in highly reflective areas (e.g. an urban environment).

Computation of an odd-length DCT from a real-valued DFT of the same length

Abstract: The discrete cosine transform (DCT) is often computed from a discrete Fourier transform (DFT) of twice or four times the DCT length. DCT algorithms based on identical-length DFT algorithms generally require additional arithmetic operations to shift the phase of the DCT coefficients. It is shown that a DCT of odd length can be computed by an identical-length DFT algorithm, by simply permuting the input and output sequences. Using this relation, odd-length DCT modules for a prime factor DCT are derived from corresponding DFT modules. The multiplicative complexity of the DCT is then derived in terms of DFT complexities. >

An FFT-based novel approach to noninvasive speed measurement in induction motor drives

Abstract: A novel sensorless speed detection method based on the fast Fourier transform (FFT) spectral analysis is described. The main concern is the extraction of the speed information contained in the rotor slot-ripple harmonics created in the airgap of the induction motor using digital signal-processing techniques. A nonintrusive all-digital speed detector suitable for steady-state operation has been designed around an Intel/80386-based microcomputer equipped with a 80387 coprocessor. The detector performance with and without load over a wide range of speeds is described. >