Showing papers on "Non-uniform discrete Fourier transform published in 1976"
TL;DR: In this paper, an alternative form of the fast Fourier transform (FFT) is developed, which has the peculiarity that none of the multiplying constants required are complex-most are pure imaginary.
Abstract: An alternative form of the fast Fourier transform (FFT) is developed. The new algorithm has the peculiarity that none of the multiplying constants required are complex-most are pure imaginary. The advantages of the new form would, therefore, seem to be most pronounced in systems for which multiplication are most costly.
161 citations
TL;DR: In this article, the authors compared the effectiveness of the discrete cosine and Fourier transforms in decorrelating sampled signals with Markov-1 statistics, and showed that the DCT offers a higher (or equal) effectiveness than the discrete Fourier transform for all values of the correlation coefficient.
Abstract: This correspondence compares the effectiveness of the discrete cosine and Fourier transforms in decorrelating sampled signals with Markov-1 statistics. It is shown that the discrete cosine transform (DCT) offers a higher (or equal) effectiveness than the discrete Fourier transform (DFT) for all values of the correlation coefficient. The mean residual correlation is shown to vanish as the inverse square root of the sample size.
116 citations
TL;DR: This paper deals with two's complement arithmetic with either rounding or chopping with eitherRoundoff errors for radix-2 FFT's and mixed-radix FFTs.
Abstract: A statistical model for roundoff errors is used to predict the output noise of the two common forms of the fast Fourier transform (FFT) algorithm, the decimations in-time and in-frequency. This paper deals with two's complement arithmetic with either rounding or chopping. The total mean-square errors and the mean-square errors for the individual points are derived for radix-2 FFT's. Results for mixed-radix FFT are also given.
93 citations
TL;DR: This correspondence shows that the amount of work can be cut to doing two single length FFT's, which is equivalent to doing one double length fast Fourier transform.
Abstract: Ahmed has shown that a discrete cosine transform can be implemented by doing one double length fast Fourier transform (FFT). In this correspondence, we show that the amount of work can be cut to doing two single length FFT's.
77 citations
TL;DR: It is proposed that the bandwidth be constrained when shaping the image spectrum to reduce dynamic range and introduce aliasing error in the reconstructed image.
Abstract: The use of the discrete Fourier transform in digital holography introduces aliasing error in the reconstructed image. Spectrum shaping to reduce dynamic range may also result in a serious increase in aliasing error. The effect of aliasing in digital holography is analyzed. It is proposed that the bandwidth be constrained when shaping the image spectrum. Experimental results show the approach to be quite effective.
55 citations
54 citations
TL;DR: One-dimensional and two-dimensional generalized discrete Fourier transforms (GFTs) are introduced in this article, and the result holds also for the DFT, as it is a particular case of the GFT.
Abstract: One-dimensional and two-dimensional generalized discrete Fourier transforms (GFT) are introduced. If a one-dimensional vector A is fractured into a two-dimensional matrix B, a one-dimensional GFT on A and a two-dimensional GFT on B give the same result and require the same number of operations to be computed. The result holds also for the DFT, as it is a particular case of the GFT.
47 citations
TL;DR: In this article, the spectrum of a magnetic or a gravity anomaly due to a body of a given shape with either homogeneous magnetization or uniform density distribution can be expressed as a product of the Fourier transforms of the source geometry and the Green's function.
Abstract: The spectrum of a magnetic or a gravity anomaly due to a body of a given shape with either homogeneous magnetization or uniform density distribution can be expressed as a product of the Fourier transforms of the source geometry and the Green's function. The transform of the source geometry for any irregularly-shaped body can be accurately determined by representing the body as closely as possible by a number of prismatic bodies. The Green's function is not dependent upon the source geometry. So the analytical expression for its transform remains the same for all causative bodies. It is, therefore, not difficult to obtain the spectrum of an anomaly by multiplying the transform of the source geometry by that of the Green's function. Then the inverse of this spectrum, which yields the anomaly in the space domain, is calculated by using the Fast Fourier Transform algorithm. Many examples show the reliability and accuracy of the method for calculating potential field anomalies.
42 citations
34 citations
TL;DR: In this article, the linear inverse theory of Backus & Gilbert has been applied to the problem of calculating the Fourier transform of digitized data with the objective of assessing the effects of missingportions of the data series and of contamination of the signal by noise.
Abstract: Summary The linear inverse theory of Backus & Gilbert has been applied to the problem of calculating the Fourier transform of digitized data with the objective of assessing the effects of missingportions of the data series and of contamination of the signal by ' noise '. When ' noise ' in the data is of concern this method achieves a maximum decrease in the variance of the Fourier transform estimate for a minimum sacrifice in resolution, thereby optimizing the trade-off between resolution and accuracy. The effects of data gaps are easily treated and it is shown that it may sometimes be desirable to interpolate these gaps even though a large variance must be ascribed to the fabricated data. We also apply the Backus-Gilbert technique to the calculation of the reverse Fourier transform, and an application to the downward continuation of potential field data is given.
TL;DR: In this paper, the Fourier transform noise was examined experimentally using specially prepared programs, and it was found that the dynamic range observable following a Fourier transformation is proportional to the computer's word length and inversely proportional to both the number of transformed words and the memory locations full at the outset of the transform.
TL;DR: An encoding figure of merit is established for a detector-noise limited Fourier transform spectrometer (FTS) and it is compared to the comparable figure for a Hadamard transform spectrumeter (HTS) to establish the mean square errors.
Abstract: We establish an encoding figure of merit for a detector-noise limited Fourier transform spectrometer (FTS) and compare it to the comparable figure for a Hadamard transform spectrometer (HTS). If N measurements are made to establish N spectral densities, the mean square errors obtained with the Fourier system are a factor of 2 greater than for the analogous Hadamard system. The limitation of the Fourier system is partly that it does not truly Fourier analyze the radiation. Instead a cosine squared modulation is imposed on the different spectral frequencies. An additional difficulty is that neither the cosine nor the cosine squared functions form an orthonormal set. This makes the Fellgett's advantage (root-mean-squared figure of merit) for a single detector Michelson interferometer a factor of (N/8)(1/2) greater than for a conventional grating instrument-rather than (N/2)(1/2) as maintained in standard texts. The theoretical limit, which may not be realizable with practical instruments, would be (N)(1/2).
Patent•
19 Jul 1976TL;DR: In this paper, an automatic equalizer for calculating the equalization transfer function and applying same to equalize received signals is presented, where the initial calculation as well as the equalisation proper are conducted entirely within the frequency domain.
Abstract: An automatic equalizer for calculating the equalization transfer function and applying same to equalize received signals. The initial calculation as well as the equalization proper are conducted entirely within the frequency domain. Overlapping moving window samplings are employed together with the discrete Fourier transformation and a sparse inverse discrete Fourier transformation to provide the equalized time domain output signals.
TL;DR: The problem of evaluating successively the discrete Fourier transform on ordered sets of N elements staggered of M is considered, and three procedures for solving such a problem are given, of which two are recursive and one nonrecursive.
Abstract: In this work the problem of evaluating successively the discrete Fourier transform (DFT) on ordered sets of N elements staggered of M is considered. Three procedures for solving such a problem are given, of which two are recursive and one nonrecursive. The complexity of each procedure, in number of complex multiplications, is about (N/2) \log_{2} 4M .
TL;DR: Techniques for dealing with signals having a high dynamic range in Fourier transform nmr are discussed, and the limitations imposed by the transform itself are pointed out.
TL;DR: In this paper, a formula to determine the characteristic function of N x N matrix by discrete Fourier series is given, which is based on the Fourier transform of the matrix.
Abstract: A formula to determine the characteristic function of N x N matrix by discrete Fourier series is given.
Patent•
01 Mar 1976
TL;DR: In this article, an arrangement for computing the discrete Fourier transform intended for converting N samples of a real signal in the time domain to N real Fourier coefficients is presented. But this device is implemented with a conventional Fourier transformer of the order N/4, to which an input computer unit and an output computer unit are connected in which a small number of multiplications of complex numbers is performed.
Abstract: An arrangement for computing the discrete Fourier transform intended for converting N samples of a real signal in the time domain to N real Fourier coefficients. This device is implemented with a conventional Fourier transformer of the order N/4, to which an input computer unit and an output computer unit are connected in which a small number of multiplications of complex numbers is performed.
TL;DR: This chapter discusses application of fast Fourier transform (FFT) in radio astronomy and it is shown how this algorithm is programmed on a digital computer.
Abstract: Publisher Summary This chapter discusses application of fast Fourier transform (FFT) in radio astronomy. The Fourier transform is a particularly useful computational technique in radio astronomy. The essence of the FFT technique is that it is possible to treat the one-dimensional DFT as though it were a pseudo-two-dimensional one, and then reduce the running time by performing the inner and outer summations separately. The basic idea behind the FFT is discussed and it is shown how this algorithm is programmed on a digital computer. Because of the requirement for computational speed, a number of programs are given. These include short, moderately efficient subroutines for the transform of one-dimensional, complex data (FOURG and FOURI). With the addition of a subroutine (FXRLI) to either of the above routines, real, one-dimensional data may be transformed in half the time with half the memory storage. Additional subroutines (CFFT2, RFFT2, and HFFT2) permit the transform of two-dimensional data. A program is also given for transforming real, symmetric data for which only the cosine (or sine) transform is desired (FORSI).
Patent•
19 Jul 1976TL;DR: In this paper, an automatic equalizer for calculating the equalization transfer function and applying same to equalize received signals is presented, where the initial calculation as well as the equalisation proper are conducted entirely within the frequency domain.
Abstract: An automatic equalizer for calculating the equalization transfer function and applying same to equalize received signals The initial calculation as well as the equalization proper are conducted entirely within the frequency domain Overlapping moving window samplings are employed together with the discrete Fourier transformation and a sparse inverse discrete Fourier transformation to provide the equalized time domain output signals
TL;DR: In this article, the authors examined the impact of finite register lengths on data acquisition, computation of the fast Fourier transform (FFT), and post-FFT spectral manipulations, and concluded that the minimum recommended register length is 27 bits.
Abstract: Finite registers used in computations act as additional noise sources in infrared Fourier transform spectroscopy The relationship between these noise sources and classical noise sources is examined The impact of finite register lengths on data acquisition, computation of the fast Fourier transform (FFT), and post-FFT spectral manipulations leads to the conclusion that the minimum recommended register length is 27 bits
TL;DR: This correspondence develops efficient fast Fourier transform programs to transform arrays of dimension N, where N can be written as a power of two possibly multiplied by arbitrary factors.
Abstract: This correspondence develops efficient fast Fourier transform (FFT) programs to transform arrays of dimension N, where N can be written as a power of two possibly multiplied by arbitrary factors. Two programs were developed which use radix-2 and radix-4 transformations for the binary factors. These programs call another subprogram to transform with respect to the arbitrary factors, if any. Since the sequential transformation is well known, the emphasis is on developing an efficient unscrambling procedure to follow the transformation.
TL;DR: The representation suggested in the paper is so rapidly convergent that an excellent approximation to the exact least-square optimum is achieved even if only a few terms are kept.
Abstract: Truncated series expansion is used to obtain discrete-time windows which are optimal in the least-square sense for a given number of terms. The representation suggested in the paper is so rapidly convergent that an excellent approximation to the exact least-square optimum is achieved even if only a few terms are kept. As a consequence, the resulting windows are easy to obtain and economical to implement in practical applications.
TL;DR: In this paper, the Laplace transform is approximated at exponentially spaced samples and analysis frequencies, and the ratio of the intervals between pairs of adjacent sampling positions is a constant greater than one.
Abstract: An estimate of the spectrum is based on the Laplace transform which is approximated at exponentially spaced samples and analysis frequencies. In this approximation the ratio of the intervals between pairs of adjacent sampling positions is a constant greater than one. The choice of this constant is influenced by the desired analysis bandwidth and by sampling effects. If analysis frequencies are spaced the same as sampling positions, this approximation becomes a discrete correlation. which can be computed by a fast Fourier transform (FFT) or a number theoretic transform. Except at low-analysis frequencies, the analysis bandwidth is "constant-Q," i.e., it is proportional to the analysis frequency. With a white noise input the noise in the computed spectrum is roughly constant at each analysis frequency. The numbers of samples and computations required for exponential spacing of samples and frequencies can be less than those required for equidistant spacing. Better performance at some (but not all) analysis frequencies is provided by a two-sided sampling arrangement consisting of a juxtaposition of the basic one-sided sampling arrangement and its mirror image.
12 Apr 1976
TL;DR: It is shown that the Discrete Fourier Transform, when used in the conventional manner with the frequency samples located at zero and integer multiples of 1/T, gives an inaccurate representation of the spectrum of certain frequencies that are located near the top and bottom end of the band.
Abstract: It is shown that the Discrete Fourier Transform (DFT), when used in the conventional manner with the frequency samples located at zero and integer multiples of 1/T, where T is the signal duration, gives an inaccurate representation of the spectrum of certain frequencies that are located near the top and bottom end of the band. It is further shown that this type of error can be eliminated by using the Odd Discrete Fourier Transform (ODFT) in which the frequency samples are located at odd multiples of 1/2T. An application of the ODFT in two dimensional filtering is also discussed.
01 Sep 1976
TL;DR: In this paper, a matrix formulation of the discrete Hilbert transform is presented, which has the advantage of reducing the number of multiplications by a factor of two, compared to the matrix formulation given by Burris.
Abstract: A matrix formulation of the discrete Hilbert transform, an alternative to that given by Burris [1], is presented. This has the advantage of reducing the number of multiplications by a factor of two.
Patent•
10 Mar 1976TL;DR: In this article, the Fourier coefficients of a finite block of bi-level bits are computed by formating the block of bits into an even function which is symmetrical about an origin such that both the data bits and a mirror image thereof are contained in a data block of N bits.
Abstract: Apparatus is provided for computing the Fourier coefficients of a finite block of bi-level bits. This is accomplished by formating the finite block of bits into an even function which is symmetrical about an origin such that both the data bits and a mirror image thereof are contained in a data block of N bits. This permits computation of a Fourier transform which will result in only cosine terms by the relatively simple Fourier transform generator structures disclosed herein. In one embodiment, a plurality of coefficients are generated simultaneously.