Showing papers on "Fractional Fourier transform published in 1976"
TL;DR: In this article, a new class of apodizing functions suitable for Fourier spectrometry (and similar applications) is introduced, and three specific functions are discussed in detail, and the resulting instrumental line shapes are compared to numerous others proposed for the same purpose.
Abstract: A new class of apodizing functions suitable for Fourier spectrometry (and similar applications) is introduced. From this class, three specific functions are discussed in detail, and the resulting instrumental line shapes are compared to numerous others proposed for the same purpose.
291 citations
TL;DR: This paper discusses a digital formulation of the phase vocoder, an analysis-synthesis system providing a parametric representation of a speech waveform by its short-time Fourier transform, designed to be an identity system in the absence of any parameter modifications.
Abstract: This paper discusses a digital formulation of the phase vocoder, an analysis-synthesis system providing a parametric representation of a speech waveform by its short-time Fourier transform. Such a system is of interest both for data-rate reduction and for manipulating basic speech parameters. The system is designed to be an identity system in the absence of any parameter modifications. Computational efficiency is achieved by employing the fast Fourier transform (FFT) algorithm to perform the bulk of the computation in both the analysis and synthesis procedures, thereby making the formulation attractive for implementation on a minicomputer.
240 citations
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: A particularly simple way to control fast Fourier transform (FFT) hardware that allows parallel organization of the memory such that at any stage the two inputs and outputs of each butterfly belong to different memory units, hence can always be accessed in parallel.
Abstract: A particularly simple way to control fast Fourier transform (FFT) hardware is described. The method produces the indices both for inputs of each butterfly operation and for the appropriate W. In addition, this method allows parallel organization of the memory such that at any stage the two inputs and outputs of each butterfly belong to different memory units, hence can always be accessed in parallel.
108 citations
TL;DR: The scale invariance of the Mellin transform and its optical synthesis in real-time are discussed in this article, where a scale invariant correlation and a combined Fourier-Mellin transform that is both scale and shift invariant are discussed.
Abstract: The scale invariance of the Mellin transform and its optical synthesis in real-time are discussed. An initial off-line demonstration of an optical Mellin transform is presented. A scale invariant correlation and a combined Fourier-Mellin transform that is both scale and shift invariant are discussed. Applications of these transforms in optical data processing and optical pattern recognition are emphasized.
87 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: In this paper, it was shown that multicomponent exponential decays can be analyzed using a technique which produces a spectrum whose peaks correspond in amplitude and position to the various exponential components present in the data.
Abstract: It is shown that multicomponent exponential decays can be analysed using a technique which produces a spectrum whose peaks correspond in amplitude and position to the various exponential components present in the data. This technique is analogous to the Fourier Transform which provides a spectrum of the frequency components (complex exponentials) of a signal. Three techniques—the Orthonormal Exponential Transform, the Inverse Laplace Transform and the Gardner Transform are examined and their relative effectiveness in producing the desired spectra from theoretically generated and experimental data is discussed. It is shown that the updated Gardner Transform can be Ilsed to analyse experimental multicomponent exponential decays.
46 citations
TL;DR: Some of the important features of the fast Fourier transform which are relevant to its increasing application to biomedical data are reviewed and a distinction is made between the power spectrum of ergodic signals, computed from the autocorrelation function, and the frequency spectrum of nonstationary biomedical signals.
Abstract: The fast Fourier transform (f.f.t.) is a powerful technique which facilitates analysis of signals in the frequency domain. This paper reviews some of the important features of the fast Fourier transform which are relevant to its increasing application to biomedical data. A distinction is made between the power spectrum of ergodic signals, computed from the autocorrelation function, and the frequency spectrum of nonstationary biomedical signals. The major practical pitfalls that are encountered in applying the f.f.t. technique to biomedical data are discussed, and practical hints for avoiding such pitfalls are suggested.
44 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
01 Mar 1976
TL;DR: In this paper, an efficient algorithm for the DFT of N point symmetric real-valued series with time samples taken as odd multiples of half the sampling period T/2 and frequency samples are taken as 1/2NT is presented.
Abstract: An efficient algorithm is obtained for the DFT of N point symmetric real-valued series if time samples are taken as odd multiples of half the sampling period T/2 and frequency samples are taken as odd multiples of 1/2NT.
41 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.
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: In this paper, the transfer of the partial coherence by an optical system is studied, within the framework of a polychromatic light, where the coherence is transformed by a linear filter univocally connected to the filter acting on the electric field.
Abstract: The transfer of the partial coherence by an optical system is studied, within the framework of a polychromatic light. The method relies on the fundamental fact that the coherence is transformed by a linear filter univocally connected to the filter acting on the electric field. It results a particular relationship of structure between the transformations assumed by the field and the coherence: space-time filtering as much as spatial Fourier transform, which are successively analysed in the case of transfer by direct diffraction. The results obtained are specified in the particular case of a narrow-band radiation or yet an incoherent object; which allows, among others, to situate Zernike theorem in a quite general context.
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.
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.
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.