Showing papers on "Discrete Fourier transform published in 1983"

Signal estimation from modified short-time Fourier transform

Abstract: In this paper, we present an algorithm to estimate a signal from its modified short-time Fourier transform (STFT). This algorithm is computationally simple and is obtained by minimizing the mean squared error between the STFT of the estimated signal and the modified STFT. Using this algorithm, we also develop an iterative algorithm to estimate a signal from its modified STFT magnitude. The iterative algorithm is shown to decrease, in each iteration, the mean squared error between the STFT magnitude of the estimated signal and the modified STFT magnitude. The major computation involved in the iterative algorithm is the discrete Fourier transform (DFT) computation, and the algorithm appears to be real-time implementable with current hardware technology. The algorithm developed in this paper has been applied to the time-scale modification of speech. The resulting system generates very high-quality speech, and appears to be better in performance than any existing method.

Transform domain LMS algorithm

Abstract: The concept of transform domain adaptive filtering is introduced. In certain applications, filtering in the transform domain results in great improvements in convergence rate over the conventional time-domain adaptive filtering. The relationship between several existing frequency domain adaptive filtering algorithms is established. Applications of the discrete Fourier transform (DFT) and the discrete cosine transform (DCT) domain adaptive filtering algorithms in the areas of speech processing and adaptive line enhancers are discussed.

Interpolation Algorithms for Discrete Fourier Transforms of Weighted Signals

Abstract: A new scheme is presented for the determination of the parameters that characterize a multifrequency signal. The essential innovation is that the signal is weighted before the discrete Fourier transform (DFT) is calculated from which the frequencies and complex amplitudes of the various components of the signal are obtained by interpolation. It is shown that by using the Hanning window for tapering substantial improvements are achieved in the following respects: i) more accurate results are obtained for interpolated frequencies, etc., ii) harmonic interference is much less troublesome even if many tones with comparable strengths are present in the spectrum, iii) nonperiodic signals can be handled without an a priori knowledge of the tone frequencies. The stability of the new method with respect to noise and arithmetic roundoff errors is carefully examined.

Fast Transforms Algorithms, Analyses, Applications

Abstract: Preface. Acknowledgments. List of Acronyms. Notation. Introduction. Fourier Series and Fourier Transform. Discrete Fourier Transforms. Fast Fourier Transform Algorithms. FFT Algorithms That Reduce Multiplications. DFT Filter Shapes and Shaping. Spectral Analysis Using the FFT. Walsh-Hadamard Transforms. The Generalized Transform. Discrete Orthogonal Transforms. Number Theoretic Transforms. Appendix. References. Index.

Coherent Doppler tomography for microwave imaging

Abstract: A tomographic extension of the type of microwave Doppler imaging typified by synthetic aperture radar has recently been developed and shown experimentally to exhibit a high degree of spatial resolution. When CW irradiation is used, the sidelobes in the pointspread function are inherently high and tend to limit the dynamic range of the reconstructed images. The point-spread function of a system using CW irradiation and an aperture that completely surrounds the object has a central lobe of width of λ/5, but the first sidelobe is only 8 dB below the central peak. The limitation due to the high sidelobes can be partially overcome by using wide-band signals or bistatic diversity. One of the steps in reconstructing a coherent Doppler tomogram is to perform a two-dimensional Fourier transform. The ordinary two-dimensional discrete Fourier transform (DFT) produces points in the transform space on a Cartesian raster. In coherent Doppler tomography (CDT), however, the data are sampled on a polar raster. To diminish the computational burden associated with converting to the Cartesian raster and interpolating, we have developed an alternative algorithm which requires no interpolation and is based on interpreting the two-dimensional Fourier transform as a one-dimensional circular convolution integral. The quality of the images computed in this fashion compares favorably with that for the old method and the computational burden is greatly reduced.

Signal reconstruction from signed Fourier transform magnitude

Abstract: In this paper, we show that a one-dimensional or multidimensional sequence is uniquely specified under mild restrictions by its signed Fourier transform magnitude (magnitude and 1 bit of phase information). In addition, we develop a numerical algorithm to reconstruct a one-dimensional or multidimensional sequence from its Fourier transform magnitude. Reconstruction examples obtained using this algorithm are also provided.

Some aspects of band-limited signal extrapolation: Models, discrete approximations, and noise

Abstract: We present some theoretical results on the band-limited signal extrapolation problem. In Section I we describe four basic models for the extrapolation problem. These models are useful in understanding the relationship between the continuous extrapolation problem and some discrete algorithms given in [1] and [2]. One of these models was shown to approximate the continuous band-limited extrapolation problem [3]. Another model is obtained when the discrete Fourier transform (DFT) is used to implement the well-known iterative algorithm given in [4] and [5] which was designed for solving the continuous extrapolation problem; in Section II this model is related to the continuous model by means of an interesting approximation theorem. Also, an important conjecture is presented. Section III shows some approximation results. Specifically, we prove that some discrete-discrete and discrete-continuous extrapolations of noisy signals converge to solutions of a certain continuous-continuous noisy extrapolation problem when the noise η is bounded by a known number, max \eta(x)| leq \epsilon . This convergence is obtained by using normal families of entire functions in ¢nand some other complex analysis tools. We also show that the extrapolation problem is very sensitive to noise even in cases where only small amounts of extrapolation are desired. This result indicates that in the presence of noise, extrapolation techniques should be used judiciously in order to obtain reasonable results.

The design of optimal DFT algorithms using dynamic programming

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

Fast Fourier Transform Algorithms with Applications

Abstract: After introduction of the discrete Fourier transform and a short description of its main properties we concentrate on a discussion of some of the various methods which have been used for the derivation of fast Fourier transform (FFT) algorithms. The paper closes with applications of FFT procedures in the numerical calculation of Fourier coefficients, fast multiplication of large integers and computations which involve circulant matrices.

The finite fourier transform of a stationary process

Abstract: Publisher Summary The Fourier transform has proved of substantial use in most fields of science. It has proved of special use to statisticians concerned with stationary process data or concerned with the analysis of linear time-invariant systems. This chapter describes some of the uses and properties of Fourier transforms of stochastic processes. The Fourier transform turns up in the problems of functional approximation and interpolation. In seismic engineering, the Fourier transforms of observed strong motion records are taken as design inputs and corresponding responses of structures evaluated prior to construction. There are various classes of functions that may be viewed as subject to a harmonic analysis. Quite a different class of functions is provided by the realizations of stationary stochastic processes. Fourier transforms at distinct frequencies and based on nonintersecting data stretches may be approximated by independent normals. The variance of the approximating normal is proportional to the power spectrum of the series.

Processor Architectures for Two-Dimensional Convolvers Using a Single Multiplexed Computational Element with Finite Field Arithmetic

Abstract: This paper describes the theory, simulation, and construction of a two-dimensional number theoretic transform (NTT) convolver. The convolver performs indirect convolution by using the cyclic convolution property of a class of generalized discrete Fourier transforms (DFT's) defined over rings isomorphic to direct sums of Galois fields. The paper first presents the theoretical development of the computational element required for computing the generalized discrete Fourier transform (GDFIT) and its inverse. The theory extends the use of base fields to second degree extension fields and provides efficient choices for transform parameters to minimize hardware. The paper next presents results of recent work in multidimensional transform memory structures, and extends this work to the complete convolution process. The two theories are then "married" to produce efficient, very high speed convolution architectures. Simulation results are presented for a second degree extension field image convolver and constructional details are presented for a fast image convolver using 2 base fields and designed to operate as a peripheral to a fast 32 bit minicomputer.

Means and method for measuring power system frequency

Abstract: Samples from a first and second cycle of a system signal are used to generate discrete Fourier transforms for the respective first and second cycle. Multiplication of the discrete Fourier transform of the second cycle by the complex conjugate of the discrete Fourier transform of the first cycle produces a frequency deviation phasor whose angle with respect to the real axis is representative of the frequency deviation of the system signal from a predetermined reference frequency. Actual system frequency may be determined by obtaining the inverse tangent of the angle of the frequency deviation phasor with respect to the real axis. Apparatus for determining the frequency and frequency deviation of a system signal is disclosed as is a method and apparatus for determining frequency and frequency deviation of a system signal comprising a plurality of phases.

Discrete Fourier transform processor based on the prime-factor algorithm

Abstract: An architecture for a fast DFT processor is presented, which is well suited for implementation using custom or semicustom LSI techniques. A new indexing scheme, which results in an in-place and in-order prime-factor algorithm is first developed. An implementation of a set of four low-order DFT modules based on Winograd algorithms is then proposed. Serial arithmetic, parallel data streams and pipelined computations are used in these modules to give simple circuit configurations with high throughput rates. The low-order DFT modules can be combined without any arithmetic operations to compute DFTs with a wide range of transform lengths. For moderate transform lengths, these modules can be directly cascaded to give high throughput rates. In the case of long transform lengths, they can be combined using the new indexing scheme and intermediate memory. An efficient method of computing DFTs of real-valued data using this processor is also presented.

System for reprojecting images using transform techniques

Abstract: Systems and methods are presented for reprojecting images which comprise taking the two-dimensional Fourier transform of the image, interpolating the transform in order to obtain the values on radial lines, and taking the inverse one-dimensional Fourier transforms of the radial lines.

Fast algorithm for the computation of the zero-order Hankel transform

Abstract: The Hankel transform may be defined as the two-dimensional Fourier transform of a circularly symmetric function. A new Hankel-transform algorithm based on this definition is described. The proposed algorithm efficiently generates a rectangularly sampled two-dimensional output array by using the circular symmetry properties of the input array and two-dimensional vector radix fast-Fourier transform techniques. It accomplishes this by partitioning the input matrix into smaller and smaller processing blocks while removing redundant blocks from data manipulations. For applications that require the output data to be sampled on a two-dimensional rectangular raster, the convenience and the computational speed of the resulting algorithm offer advantages over the one-dimensional Hankel-transform algorithms currently available.

Performance Analysis of FFT Algorithms on Multiprocessor Systems

Abstract: A decimation-in-time radix-2 fast Fourier transform (FFT) algorithm is considered here for implementation in multiprocessors with shared bus, multistage interconnection network (MIN), and in mesh connected computers. Results are derived for data allocation, interprocessor communication, approximate computation time, and speedup of an N point FFT on any P available processing elements (PE's). Further generalization is obtained for a radix-r FFT algorithm. An N X N point two-dimensional discrete Fourier transform (DFT) implementation is also considered when one or more rows of the input data matrix are allocated to each PE.

Bit-level systolic array circuit for matrix vector multiplication

Abstract: A bit-level systolic array for computing matrix x vector products is described. The operation is carried out on bit parallel input data words and the basic circuit takes the form of a 1-bit slice. Several bit-slice components must be connected together to form the final result, and the paper outlines two different ways in which this can be done. The basic array also has considerable potential as a stand-alone device, and its use in computing the Walsh-Hadamard transform and discrete Fourier transform operations is briefly discussed.

Hardware-based Fourier transforms: algorithms and architectures

Abstract: Many signal processing systems rely on the efficient calculation of the discrete Fourier transform. The basic equations are manipulated to yield architectures to suit differing technologies and performance requirements. Finally a prime-radix transform algorithm is described offering very high performance that is ideally suited to present-day technology.

Receiver mode control for acquiring and tracking a signal

Abstract: A receiver is disclosed for acquiring and tracking a data signal in a highly stressed environment. The receiver comprises first and second I.F. sections, a mixer for translation from the first I.F. frequency to the second I.F. frequency, a 3 KHz bandpass filter at the second I.F. frequency, signal translator for synchronous translation of the signal at the second I.F. frequency to baseband, a digitizer for complex sampling operation on the baseband signal, a microprocessor for processing the digital samples, and a numerically controlled oscillator coupled to the mixer and controlled by the microprocessor. The microprocessor formulates matched digital discrete Fourier Transform filters which drive frequency, phase and symbol lock loops at the symbol rate. Each of the loop filters is formed by symbol-rate recursive, first-order equations. A novel mode control system is employed to implement an orderly transition through the receiver modes, comprising (i) out-of-band noise estimation, (ii) coarse frequency and time acquisition of the data signal employing a sequential probability ratio test and a handover process, (iii) frequency and symbol synchronization with the data signal, (iv) phase and symbol synchronization with the data signal, and (v) feedback loop lock confirmation. After loss of lock, the mode controller transfers the receiver operations back to the appropriate restart operation.

Recursive discrete Fourier transformation with unevenly spaced data

Abstract: New recursive techniques for Fourier spectral analysis are reported, for which ongoing spectral estimates are generated from unevenly spaced data in real time. The algorithms are robust and computationally efficient, and are well suited to state variable form involving real number calculations. These methods are particularly attractive in filtering and signal processing applications where signals are not necessarily sampled at a uniform rate.

Comments on "The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform"

Abstract: When one imposes a nonnegativity constraint, one usually can reconstruct a two-dimensional sequence of finite support from the modulus of its Fourier transform using an iterative algorithm, even when file initial estimate is an array of random numbers.

Data communications receiver operable in highly stressed environments

Abstract: A receiver is disclosed for acquiring and tracking a data signal in a highly stressed environment. The receiver comprises first and second I.F. sections, a mixer for translation from the first I.F. frequency to the second I.F. frequency, a 2 KHz bandpass filter at the second I.F. frequency, signal translator for synchronous translation of the signal at the second I.F. frequency to baseband, a digitizer for complex sampling operation on the baseband signal, a microprocessor for processing the digital samples, and a numerically controlled oscillator coupled to the mixer and controlled by the microprocessor. The microprocessor formulates matched digital discrete Fourier Transform filters which drive frequency, phase and symbol lock loops at the symbol rate. Each of the loop filters is formed by symbol-rate recursive, first-order equations. A novel mode control system is employed to implement an orderly transition through the receiver modes, comprising (i) out-of-band noise estimation, (ii) coarse frequency and time acquisition of the data signal employing a sequential probability ratio test and a handover process, (iii) frequency and symbol synchronization with the data signal, (iv) phase and symbol synchronization with the data signal, and (v) feedback loop lock confirmation. After loss of lock, the mode controller transfers the receiver operations back to the appropriate restart operation.

Magnitude-only reconstruction of two-dimensional sequences with finite regions of support

Abstract: In this paper, a conceptual algorithm for reconstructing a two-dimensional (2-D) complex-valued finite sequence from an adequate set of samples of the magnitude of its Fourier transform is presented. In particular, one obtains, at least theoretically, all solutions of the 2-D magnitude-only reconstruction problem, provided that the modulus of the DFT is available in a sufficiently large set of points. However, the practicability of this algorithm is limited to sequences with relatively small regions of support. The key for developing the method is shown to be an appropriate mapping of 2-D finite sequences into 1-D ones, such that 2-D discrete correlation can be formulated in terms of ordinary 1-D discrete correlation.

Polynomial transform implementation of digital filter banks

Abstract: This paper investigates the use of polynomial transforms for the implementation of uniform digital bandpass filter banks. The technique is based upon a decomposition of the N bandpass filters into a set of real polyphase filters followed by a DCT (discrete cosine transform) of size N. The DCT is converted into a DFT (discrete Fourier transform) of size N and the polyphase filters are evaluated by DFT's. This procedure yields a two-dimensional DFT which is computed by a polynomial transform and odd DFT's. We show that this technique reduces significantly the number of arithmetic operations when compared to conventional methods, and yields a regular structure in which most of the computations are performed with FFT-type algorithms.

Signal Reconstruction From Fourier Transform Amplitude

Abstract: In this paper, we show that a one-dimensional or multi-dimensional sequence is uniquely specified under mild restrictions by its Fourier transform amplitude (magnitude and one bit of phase information). In addition, we develop a numerical algorithm to reconstruct a one-dimensional or multi-dimensional sequence from its Fourier transform amplitude. Reconstruction examples obtained using this algorithm are also provided.© (1983) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

A 256-point discrete Fourier transform processor fabricated in a 2 /spl mu/m NMOS technology

Abstract: The authors describe a 2-/spl mu/m NMOS single-chip processor which computes a 256-point Fourier transform in 6.5 ms by means of the DFT algorithm. This chip is primarily intended for speech processing applications such as voice recognizers, pitch extractors, and DFT vocoders. The DFT algorithm has been implemented wih two parallel multipliers, an accumulator/shifter and two ROMs with separate address computation units. The processor contains over 20,000 transistors and dissipates ~400 mW when operating at a clock frequency of 10 MHz. The die size is only 12.5 mm/SUP 2/ as a result of the use of a full-custom design method and the incorporation of low-resistance implanted As/SUP +/ undercrossings within the logic circuitry.