Showing papers on "Non-uniform discrete Fourier transform published in 1989"

Studies in astronomical time series analysis. III - Fourier transforms, autocorrelation functions, and cross-correlation functions of unevenly spaced data

Abstract: This paper develops techniques to evaluate the discrete Fourier transform (DFT), the autocorrelation function (ACF), and the cross-correlation function (CCF) of time series which are not evenly sampled. The series may consist of quantized point data (e.g., yes/no processes such as photon arrival). The DFT, which can be inverted to recover the original data and the sampling, is used to compute correlation functions by means of a procedure which is effectively, but not explicitly, an interpolation. The CCF can be computed for two time series not even sampled at the same set of times. Techniques for removing the distortion of the correlation functions caused by the sampling, determining the value of a constant component to the data, and treating unequally weighted data are also discussed. FORTRAN code for the Fourier transform algorithm and numerical examples of the techniques are given.

Algorithms for Discrete Fourier Transform and Convolution

Abstract: Contents: Introduction to Abstract Algebra.- Tensor Product and Stride Permutation.- Cooley-Tukey FFF Algorithms.- Variants of FFT Algorithms and Their Implementations.- Good-Thomas PFA.- Linear and Cyclic Convolutions.- Agarwal-Cooley Convolution Algorithm.- Introduction to Multiplicative Fourier Transform Algorithms (MFTA).- MFTA: The Prime Case.- MFTA: Product of Two Distinct Primes.- MFTA: Transform Size N = Mr. M-Composite Integer and r-Prime.- MFTA: Transform Size N = p2.- Periodization and Decimation.- Multiplicative Character and the FFT.- Rationality.- Index.

Fourier Transforms in NMR, Optical, and Mass Spectrometry: A User's Handbook

Abstract: 1. Spectral Line Shape Derived from the Motion of a Damped Mass on a Spring. 2. Fourier Transforms for Analog (Continuous) Waveforms. 3. Fourier Transforms of Digital (Discrete) Waveforms. 4. Fourier Transform Spectrometry: Common Features. 5. Noise. 6. Non-FT Methods for Proceeding from Time- to Frequency-Domain. 7. Fourier Transform Ion Cyclotron Resonance Mass Spectrometry. 8. FT/NMR. 9. FT/Interferometry. 10. Epilog: Fourier Transforms in Other Types of Spectroscopy. References. Problems (at the end of each chapter). Solutions to problems. Appendices. Index.

An introduction to digital signal processing

Abstract: Signals and Systems Sampled Data and the Z Transform Sinusoidal Response of LSI Systems Couplets and Elementary Filters The Discrete Fourier Transform The Continuous Fourier Integral Transform Application of the Fourier Transform to Digital Signal Processing Digital Filter Design Inverse Filtering and Deconvolution Spectral Factorization Power Spectral Estimation Multidimensional DSP References

General phase modulation method for stored waveform inverse fourier transform excitation for fourier transform ion cyclotron resonance mass spectrometry

Abstract: A method for reducing the dynamic range of FT-ICR signal generated by the SWIFT technique includes the step of time shifting wave packets corresponding to segments of the Fourier spectral magnitude function to prevent coherent summing of the various frequency components of the excitation signal.

Theory of Discrete and Continuous Fourier Analysis

Abstract: Basic mathematical background integration theory distribution theory the Fourier series: the Fourier transform Fourier transform of a distribution the discrete Fourier transform sampling theory. Appendix: Fourier transfer FORTRAN subroutine.

Nonlinear matched filtering

Abstract: A new class of nonlinear matched filters is discussed These filters involve the transformation of the signal spectrum and the filter transfer function through a nonlinearity before they are multiplied in the transform domain The resulting filter structures can be considered to be analogous to three-layer neural nets They have better performance in terms of signal discrimination and lack of false correlation signals and artifacts than previously known filters The matched filters are further subdivided into two major classes according to whether the filtering is based on a discrete Fourier transform (DFT) or a real discrete Fourier transform (RDFT) The DFT and the RDFT are approximations to the complex and real Fourier transforms, respectively The RDFT-based filtering gives better performance in terms of signal discrimination and lack of false correlation signals and artifacts than the DFT-based filtering

Efficient computation of the discrete pseudo-Wigner distribution

Abstract: A description is given of a novel algorithm, the fast Fourier transform in part (FFTP), for the computation of the discrete pseudo-Wigner distribution (DPWD). The FFTP computes the cosine and sine parts of the discrete Fourier transform (DFT) separately by employing real inverse sinusoidal twiddle factors. Unlike the conventional methods which directly utilize the complex DFT, the FFTP yields real output since the DPWD is always real. In addition, the new method reduces the computational cost by making full use of symmetries and removing redundancies in the FFTP computation. The authors also describe a simple algorithm for computing the discrete Hilbert transform (DHT) to produce the nonaliased DPWD. A pipeline structure for real-time and a bulk processing technique for offline implementations of the method are presented. >

Computation of discrete Hilbert transform through fast Hartley transform

Abstract: A fast algorithm is proposed to compute the discrete Hilbert transform via the fast Hartley transform (FHT). Instead of the conventional fast Fourier transform (FFT) approach, the processing is carried out entirely in the real domain. Also, since many efficient FHT algorithms exist, the computation complexity can be greatly reduced from two complex FFTs into two real FHTs. >

Estimation of Fourier coefficients

Abstract: It is shown that the maximum-likelihood estimation or robust estimation of the Fourier coefficients may be preferable to Fourier transformation if the noise contains outliers or is otherwise not normally distributed. The reason is that, in that case, these estimators produce Fourier coefficient estimates and, therefore, system parameter estimates having a smaller variance. >

Conjugate pair fast Fourier transform

Abstract: A new algorithm for the fast computation of the discrete Fourier transform is introduced. The algorithm, called the conjugate pair FFT (CPFFT), is used to compute a length-2m DFT. The number of multiplications and additions required by the CPFFT is less than that required by the SRFFT algorithm.

Standard deviation of instantaneous frequency

Abstract: Explicit expressions are derived for the standard deviation of instantaneous frequency (local bandwidth) at a particular time using the short-time Fourier transform and the general class of bilinear distributions. Examples are given. Application to the characterization and description of a multicomponent signal is discussed. The theorem of Fink (1966) and Mandel (1974), which states that the bandwidth of a signal is always greater than the global deviation of the derivative of the phase from the average frequency, is generalized for the short-time Fourier transform. The difference in the two standard deviations is found to be precisely the average of the local bandwidth as calculated with the bilinear distributions. >

A Fast Recursive Two Dimensional Cosine Transform

Abstract: This paper presents a recursive, radix two by two, fast algorithm for computing the two dimensional discrete cosine transform (2D-DCT). The algorithm allows the generation of the next higher order 2D-DCT from four identical lower order 2D-DCT's with the structure being similar to the two dimensional fast Fourier transform (2D-FFT). As a result, the method for implementing this recursive 2D-DCT requires fewer multipliers and adders than other 2D-DCT algorithms.

Pitch synchronous spectral analysis scheme for voiced speech

Abstract: The problem of spectrum analysis of harmonic signals which are periodic or at least quasi-periodic, such as human voice, is addressed. An efficient and accurate pitch-synchronized spectral analysis scheme for obtaining the Fourier coefficients of a harmonic signal, sampled at an arbitrary rate above the Nyquist critical rate, is outlined. The pitch is derived from the sampled signal prior to the spectral analysis. The rationale behind the scheme is based on an interpolation of the signal with an upsampling rate that is synchronized with the pitch period of the signal. It is shown that the resulting unsampled sequence is aperiodic, but nevertheless can be decomposed into a periodic signal corrupted by a small, aperiodic, high-frequency noise. The fact that this noise is correlated with the signal is used to obtain a closed-form solution for the desired Fourier coefficients from the noisy values, using the computationally superior fast Fourier transform (FFT) algorithm. The accuracy of the scheme is demonstrated for synthetic speech for which the spectrum is known a priori. The results obtained for real speech signals show better consistency across adjacent frames as compared to conventional methods. >

Local Bandwidth And Optimal Windows For The Short Time Fourier Transform

Abstract: The standard deviation of instantaneous frequency (local bandwidth) is derived for the short time Fourier transform. This is done by calculating the local moments of frequency for a given time instants using the spectrogram as a joint time-frequency distribution. By minimizing the local bandwidth optimal windows are obtained. We show that amplitude modulation has a very significant effect on the optimum window. We also show that to obtain the highest possible resolution, divergent windows which non the less lead to convergent short time Fourier transforms, must sometimes be used. Series expansions for the estimated instantaneous frequency and local bandwidth are derived in terms of the derivatives pf the phase. The theorem of Ville, Mandel and Fink, relating the global bandwidth to the excursions of the instantaneous frequency, is generalized to the short time Fourier transform. The bandwidth and duration of the spectrogram are related to those of the signal and window and a local uncertainty relationship for the spectrogram is derived. Also, the concept of local duration for a particular frequency is introduced and explicit formulas are given.© (1989) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Towards a strategy for automatic phase retrieval from noisy Fourier intensities

Abstract: When iteratively reconstructing a two-dimensional image (or, more precisely, its image-form) from appreciably contaminated samples of the magnitude of its Fourier transform, using Fienup's hybrid input-output algorithm, it is herein demonstrated by example that the image error, which expresses the violation of the image-space constraints, exhibits pronounced fluctuations Significantly improved final image-forms are obtained by appropriately averaging the image-forms generated at those iterations for which the image error has local minima Reconstructions for various specimen images are shown A technique for accelerating the convergence of Fienup's algorithm, by initially estimating the phases of low spatial frequency components of the Fourier transform, is described and illustrated by example

The Fourier transform and the discrete Fourier transform

Abstract: The authors establish sharp error bounds for the error committed in computing N values of the Fourier transform of a square summable function by means of the discrete Fourier transform. No assumptions are made as to the bandwidth or desired frequency resolution. Instead their error bound depends explicitly on five parameters: the number N of sampling points, the interval T where the samples are taken, the interval Omega where the Fourier transform is being approximated, and two extra parameters that account for local averaging in the time and frequency domains respectively. The resulting error bound is independent of the function in question and they find that, for an appropriate model of sampling, the error bound is minimised as a function of T, Omega by the Nyquist-Shannon choice 4T Omega =2N+1. They finally consider how these bounds are affected by some a priori knowledge about the class of functions under discussion. This is done in detail for band-limited functions.

New fast algorithm to compute two-dimensional discrete Hartley transform

Abstract: A new fast algorithm for computing the two-dimensional discrete Hartley transform is presented. This algorithm requires the lowest number of multiplications compared with other related algorithms.

On the two-dimensional vector split-radix FFT algorithm

Abstract: The complete equations are presented for the first stage of the two-dimensional vector split-radix decimation-in-frequency fast Fourier transform algorithm using a structural approach. The computational complexity of the algorithm is discussed and compared to other published results. The author states that generally, the vector split-radix method provides a significant reduction in the number of complex multiplications required to implement a two-dimensional discrete Fourier transform. >

A note on the computational complexity of the arithmetic Fourier transform

Abstract: It is shown that the number of data points the arithmetic Fourier transform (AFT) needs for an N-point Fourier transform is proportional to N/sup 2/. Thus, for example, while a standard fast Fourier transform algorithm requires 1024 samples to yield 1024 spectral components, AFT would take more than 300000 samples to do the same job. >

Formant extraction from Fourier transform phase

Abstract: A method of extracting formant information from the short-time Fourier transform phase spectrum of speech is proposed. Fourier transform phase has not been used for formant extraction because it appears to be noisy and difficult to interpret. The effects of wrapping of phase (due to zeros close to the unit circle and the linear phase component) make it difficult to derive useful information. The authors develop algorithms to reduce the effects of wrapping. >

Exact Interpolation of Apodized, Magnitude-Mode Fourier Transform Spectra

Abstract: A procedure is developed for the exact interpolation of apodized, magnitude-mode Fourier transform (FT) spectra. The procedure gives the true center frequency, i.e., the location of the continuous peak, from just the largest three discrete intensities in the discrete magnitude spectrum. The procedure is applicable for the peaks in the apodized magnitude spectrum of time signal of the form f(t) = cos(ωt) exp(–t/τ). There are no restrictions on the value of the damping ratio T/τ. The procedure is demonstrated for the sine-bell and Hanning windows and is gener-alizable to other windows which consist of a sum of constants and sine/cosine terms. This includes the majority of commonly used windows.

Data compression properties of the Hartley transform

Abstract: The data compression performance of the Hartley transform on a Markov-1 signal is theoretically compared to that of the Fourier transform. Covariance distribution and residue correlation measurements have been computed for the Hartley, Fourier, and cosine transforms. The Hartley transform achieves better coding performance than the Fourier transform, but is inferior to the cosine transform. >

Bidiagonal factorization of Fourier matrices and systolic algorithms for computing discrete Fourier transforms

Abstract: An algorithm is presented for factoring Fourier matrices into products of bidiagonal matrices. These factorizations have the same structure for every n and make possible discrete Fourier transform (DFT) computation via a sequence of local, regular computations. A parallel pipeline technique for computing sequences of k-point DFTs, for every k >

Ghost canceling apparatus

Abstract: A ghost canceling apparatus provided with a reference waveform Fourier coefficient holding unit in which a normal ghost detection unit holds a Fourier coefficient produced by Fourier transforming a reference waveform; a nearby region Fourier transform unit for Fourier-transforming a nearby region of a reference waveform extracted from the received television signal and a normal Fourier transform unit for Fourier transforming a normal region of the reference waveform extracted from the received television signal; a calculation unit for dividing the output from the normal region Fourier transform unit by the output from the nearby region Fourier transform unit and then for multiplying a Fourier coefficient of the reference waveform Fourier coefficient holding unit; a reverse Fourier transform unit for reverse Fourier transforming the output of the calculation unit; a tap gain supplying unit for supplying the dummy normal ghost producing unit with the output of the reverse Fourier transform unit as a tap gain of the transversal filter.

Derivative sampling for multiband signals

Abstract: A derivative sampling scheme, where a signal is reconstructed from samples of itselfand its first derivative, is presented for a class of multiband signals whose spectrum (Fourier transform) vanishes outside a bounded union of intervals.