Showing papers on "Non-uniform discrete Fourier transform published in 1991"

Calculation of a constant Q spectral transform

Abstract: The frequencies that have been chosen to make up the scale of Western music are geometrically spaced. Thus the discrete Fourier transform (DFT), although extremely efficient in the fast Fourier transform implementation, yields components which do not map efficiently to musical frequencies. This is because the frequency components calculated with the DFT are separated by a constant frequency difference and with a constant resolution. A calculation similar to a discrete Fourier transform but with a constant ratio of center frequency to resolution has been made; this is a constant Q transform and is equivalent to a 1/24‐oct filter bank. Thus there are two frequency components for each musical note so that two adjacent notes in the musical scale played simultaneously can be resolved anywhere in the musical frequency range. This transform against log (frequency) to obtain a constant pattern in the frequency domain for sounds with harmonic frequency components has been plotted. This is compared to the conventio...

The fractional Fourier transform and applications

Abstract: This paper describes the “fractional Fourier transform,” which admits computation by an algorithm that has complexity proportional to the fast Fourier transform algorithm. Whereas the discrete Fourier transform (DFT) is based on integral roots of unity $e^{{{ - 2\pi i} / n}} $, the fractional Fourier transform is based on fractional roots of unity $e^{ - 2\pi i\alpha } $ where $\alpha $ is arbitrary. The fractional Fourier transform and the corresponding fast algorithm are useful for such applications as computing DFTs of sequences with prime lengths, computing DFTs of sparse sequences, analyzing sequences with noninteger periodicities, performing high-resolution trigonometric interpolation, detecting lines in noisy images, and detecting signals with linearly drifting frequencies. In many cases, the resulting algorithms are faster by arbitrarily large factors than conventional techniques.

Cosine-modulated analysis-synthesis filterbank with critical sampling and perfect reconstruction

Abstract: The authors present a simple derivation of a parallel filterbank based on cosine-modulated versions of a model low-pass filter. With a nonuniform channel separation an efficient implementation consisting of a DFT (discrete Fourier transform) related transform and subfilters is possible. Using critical sampling of each channel and FIR (finite impulse response) filters, the conditions for perfect reconstruction are given. The computational complexity of the derived FIR filterbank is much lower than for a tree-structured FIR filterbank but cannot compete with the most efficient IIR filterbanks. >

A unified systolic array for discrete cosine and sine transforms

Abstract: A linear systolic array for the discrete cosine transform, discrete sine transform, and their inverses is developed. It generates the transform kernel values recursively. Compared to the scheme with the transform kernel values prestored in memory either inside or outside each processing element, the clock period is shortened by a memory access time. In addition, the array pays no cost for prestorage. The systolic array has the advantages of pipelinability, regularity, locality, and scalability, making it quite suitable for VLSI signal processing. >

Transform methods for seismic data compression

Abstract: The authors consider the development and evaluation of transform coding algorithms for the storage of seismic signals. Transform coding algorithms are developed using the discrete Fourier transform (DFT), the discrete cosine transform (DCT), the Walsh-Hadamard transform (WHT), and the Karhunen-Loeve transform (KLT). These are evaluated and compared to a linear predictive coding algorithm for data rates ranging from 150 to 550 bit/s. The results reveal that sinusoidal transforms are well-suited for robust, low-rate seismic signal representation. In particular, it is shown that a DCT coding scheme reproduces faithfully the seismic waveform at approximately one-third of the original rate. >

Weighting effect on the discrete time Fourier transform of noisy signals

Abstract: The effect of weighting on the uncertainty of the discrete time Fourier transform (DTFT) samples of a signal corrupted by additive noise is investigated. Making very weak assumptions, it is shown how the adopted window sequence and the autocovariance function of the noise affect the second-order stochastic moments of the frequency-domain data. The relationship obtained extends the results reported in the literature and is useful in many frequency-domain estimation problems. It is shown how the knowledge of the second-order moments of the transform has allowed the application of the least squares technique for the estimation of the parameters of a multifrequency signal in the frequency-domain. The estimator obtained is very useful when high-accuracy results are required under real-time constraints. The procedure exhibits a better accuracy than similar frequency-domain methods proposed in the literature. >

The Fourier transform of linearly varying functions with polygonal support

Abstract: New formulas for the two-dimensional Fourier transform of functions with polygonal support and linear amplitude variation are derived from the corresponding formula for a constant function. These expressions, valid for all nonzero values of the transform variable k, are superior to those previously reported, which fail when k is perpendicular or parallel to any edge of the polygon. These transforms have applications in diffraction theory and computational electromagnetics. >

Signal processing apparatus and method for iteratively determining Arithmetic Fourier Transform

Abstract: A signal processing apparatus and method for iteratively determining the inverse Arithmetic Fourier Transform (AFT) of an input signal by converting the input signal, which represents Fourier coefficients of a function that varies in relation to time, space, or other independent variable, into a set of output signals representing the values of a Fourier series associated with the input signal. The signal processing apparatus and method utilize a process in which a data set of samples is used to iteratively compute a set of frequency samples, wherein each computational iteration utilizes error information which is calculated between the initial data and data synthesized using the AFT. The iterative computations converge and provide AFT values at the Farey-fraction arguments which are consistent with values given by a zero-padded Discrete Fourier Transform (DFT), thus obtaining dense frequency domain samples without interpolation or zero-padding.

A stochastic roundoff error analysis for the fast Fourier transform

Abstract: We study the accuracy of the output of the Fast Fourier Transform by estimating the expected value and the variance of the accompanying linear forms in terms of the expected value and variance of the relative roundoff errors for the elementary operations of addition and multiplication. We compare the results with the corresponding ones for the direct algorithm for the Discrete Fourier Transform, and we give indications of the relative performances when different rounding schemes are used. We also present the results of numerical experiments run to test the theoretical bounds and discuss their significance.

Design of 2-D FIR filters by nonuniform frequency sampling

Abstract: A method for the frequency-sampling design of two-dimensional FIR filters with nonuniformly spaced samples is presented. By imposing some mild constraints on sample location in the 2-D frequency plane, the method always provides a unique design solution. Important characteristics of the method are design flexibility through the use of nonuniform samples and computational efficiency. This method is compared with the uniform sampling, inverse discrete Fourier transform (DFT) approach and also with a general method for filter design called arbitrary sampling. The method presented is shown to require much less computation than the arbitrary sampling approach, which may lead to possible degenerate cases where there is no unique solution for the filter. The method proposed does not lead to such degeneracies and possesses more flexibility than the uniform sampling method. Examples are given in order to compare the new method with the uniform sampling method. >

Fourier transform of a linear distribution with triangular support and its applications in electromagnetics

Abstract: A three-dimensional Fourier transform (FT) of a linear function with triangular support is derived in its coordinate-free representation. The Fourier transform of this distribution is derived in three steps. First, the 2-D FT of a constant (top hat) function is obtained. Next, the distribution is generalized to a linearly varying function. Finally, the formulation is extended to a coordinate-free representation which is the 3-D FT of the 2-D function defined over a surface. This formulation is applied to the near-field computation, yielding accurate numerical solutions. >

Adaptive bit assignment transform coding according to power distribution of transform coefficients

Abstract: Digital samples of an underlying analog audio-frequency signal are converted to transform coefficients and a first bit assignment value indicating the number of quantization levels or bits is adaptively assigned to each coefficient according to the power distribution of the transform coefficients. The transform coefficients are weighted according to a prescribed pattern such as human's auditory sensitivity, and a second bit assignment value is further adaptively assigned to each transform coefficient according to the power distribution of the weighted transform coefficients. One of the first and second bit assignment values is selected according to the power distribution of the transform coefficients and used to quantize the transform coefficients. The quantized transform coefficients are multiplexed with supplemental information derived from the transform coefficients for transmission to a receiving site. A process inverse to that at the transmitting site is performed on the multiplexed signal to recover the original digital samples.

Fast calculation apparatus for carrying out a forward and an inverse transform

Abstract: In an apparatus for carrying out a linear transform calculation on a product signal produced by multiplying a predetermined transform window function and an apparatus input signal, an FFT part (23) carries out fast Fourier transform on a processed signal produced by processing the product signal in a first processing part (21). As a result, the FFT part produces an internal signal which is representative of a result of the fast Fourier transform. A second processing part (22) processes the internal signal into a transformed signal which represents a result of the linear transform calculation. The apparatus is applicable to either of forward and inverse transform units (11, 12).

Analysis of a joint transform correlator using a phase-only spatial light modulator

Abstract: A theoretical analysis of a joint transform correlator that uses a phase-only spatial light modulator to input joint transform plane intensity data into the second Fourier transform system is presented. It is shown that this correlator produces signals that differ from, but are related to, the mathematical correlation between the test and reference input images. An undesirable characteristic of the correlator is that the form of the output signals depends on both the intensity-to-phase transfer characteristic of the phase modulator and the intensities of the input images. However, apodization of the joint transform intensity distribution by the reciprocal of the intensity distribution of the reference image Fourier transform can overcome this problem, and results in a correlator with narrow matching output peaks and high discrimination, in which the autocorrelation peak can be eliminated. Theoretical results are demonstrated by computer simulation.

A new, efficient structure for the short-time Fourier transform, with an application in code-division sonar imaging

Abstract: Although most applications which use the short-time Fourier transform (STFT) temporally downsample the output, some applications exploit a dense temporal sampling of the STFT. One example, coded-division multiple-beam sonar, is discussed. Given a need for the densely sampled STFT, the complexity of the computation can be reduced from O(N log N) for the general short-time FFT structure to O(N) using the Goertzel algorithm. The authors introduce the pruned short-time FFT, a novel computational structure for efficiently computing the STFT with dense temporal sampling. The pruned FFT achieves the same computational savings as the Goertzel algorithm, but is unconditionally stable. >

Aliasing effects in Fourier transforms of monotonically decaying functions and the calculation of viscoelastic moduli by combining transforms over different time periods

Abstract: An analysis is presented of the discrete Fourier transform of a monotonically decaying function, represented by a sum of exponentials with negative exponents. The results are compared with those of a previous analysis based on the analytical Fourier transform, which proposed a method for extending the frequency range of the Fourier transform of experimental data by combining transforms performed over different time periods. The principle of the method is confirmed but comparison shows that results derived for the analytical transform cannot always be applied directly to the discrete transform. Modifications are therefore proposed which improve the accuracy and mitigate the aliasing effects evident in short-time transforms while keeping the computing time to a minimum. These results are particularly applicable to stress relaxation and creep in viscoelastic materials and an example from articular cartilage shows the compliance modulus over a frequency range from 10-3 Hz to 230 Hz from one creep experiment of duration 18.7 min.

Use of Fourier domain real-plane zeros to overcome a phase retrieval stagnation

Abstract: The iterative Fourier transform algorithm, although it has been demonstrated to be a practical phase retrieval algorithm, suffers from certain stagnation problems. Specifically, there exists a stripe stagnation problem, in which stagnated reconstructed images exhibit stripelike features throughout the image, which is particularly difficult to overcome. Previous solutions to this problem used multiple reconstructions and did not address the cause. In this paper a new procedure that uses only a single image is developed that estimates the locations of real-plane zeros from either the measured Fourier modulus data or a stagnated reconstruction and uses this information in the iterative Fourier transform algorithm to force the complex-valued Fourier data to have real-plane zeros at the correct locations. It is shown that this procedure overcomes the stripe stagnation.

The techniques of the generalized fast Fourier transform algorithm

Abstract: A general method of deriving DFT (discrete Fourier transform) algorithms, generalised fast Fourier transform algorithms, is presented. It is shown that a special case of the method is equivalent to nesting of FFTs. The application of the method to the case where N has mutually prime factors results in a new interpretation of the permutations characteristic of this class of algorithms. It is shown that the equalization of FFTs leads to results which are different from the widely used intuitive ones. The high efficiency of split-radix FFTs is explained. It is shown that the formulae of the method can be easily adapted for deriving algorithms for the cosine/sine DFT. A set of FFTs that has smaller arithmetical and/or memory complexities than any algorithm known is presented. In particular, a method of deriving split-radix-2/sup s/ FFTs requiring N log/sub 2/ N-3N+4 real multiplications and 3N log/sub 2/ N-3N+4 additions for any s>1 is presented. >

LMS algorithm and discrete orthogonal transforms

Abstract: A general relation between the least mean square (LMS) algorithm and the discrete orthogonal transforms is established. Discrete orthogonal transforms, including the discrete Fourier transform (DFT), the discrete Hartley transform (DHT), the discrete cosine transform (DCT), the discrete sine transform (DST), and the Walsh-Hadamard transform (WHT), etc. are extensively used in signal and image processing. It is shown that the LMS algorithm could provide a means for the calculation of forward orthogonal transforms as well as inverse orthogonal transforms by properly choosing the input vector and adaptation speed. >

The role of amplitude and phase of the Fourier transform in the digital image processing

Abstract: The analysis of the importance of the amplitude and the phase of Fourier transform has been carried out by means of combining these functions between two images and observing the reconstructed image after a second transform. This processing has been studied by taking into account several possibilities, especially for very structurally different images. It is proved that the phase carries the most relevant information, but when common images are combined with images constituted by strongly marked geometric forms it is not so evident and the amplitude could play a more important role.

Mixed domain filtering of multidimensional signals

Abstract: The concept of multidimensional mixed domain transform/spatiotemporal (MixeD) filtering is extended beyond the discrete Fourier transform (DFT) to include other types of discrete sinusoidal transforms, including the discrete Hartley transform (DHT) and the discrete cosine transform (DCT). Two MixeD filter examples are given, one using the two-dimensional (2-D) DHT and the other using the 2-D DCT, to selectively enhance a 3-D spatially planar (SP) pulse signal. The authors define the notation and provide a review of the MixeD filter method. MD and partial P-dimensional discrete transform operators are defined, and the design of MixeD filters is discussed. MixeD filters based on the 2-D DHT and the 2-D DCT are designed to selectively enhance a 3-D SP pulse. Experimental verification of these 3-D SP MixeD filters is described. >

Cepstrum analysis using discrete trigonometric transforms

Abstract: Some empirical results in use of discrete trigonometric transforms for cepstrum analysis are presented. Both the real and the power pseudocepstra using discrete cosine transform and discrete sine transform are reported. They are applied to speech pitch period extraction. Comparisons are made with standard Fourier transform cepstrum analysis. >

Algorithms for processing fourier transform phase of signals

Abstract: The studies presented in this thesis represent an attempt to process the Fourier transform (FT) phase of signals for feature extraction. Although the FT magnitude and phase spectra are independent functions of frequency features of a signal, most techniques for feature extraction from a signal are bked upon manipulating the the FT magnitude only. The phase spectrum of the signal corresponds to time delay corresponding to each of the sinusoidal components of the signal. In the context of additive noise, the time delay may not be significantly corrupted and the phase spectrum might be considered to be a more reliable source for estimating the features in a noisy signal. Although the importance of phase in signals is realised by researchers, very few attempts have been made to process the FT phase of signals for the extraction of features. Features of a signal, for example, resonance information, is completely masked by the inevitable wrapping of the phase spectrum. An alternative to processing the phase spectrum is processing the group delay function. The group delay function is the negative derivative of the (unwrapped) FT phase spectrum. The group delay function can be computed directly from the time domain signal.The group delay function possesses additive and high resolution properties, in that it shows a squared magnitude behaviour in the vicinity of a resonance. But the group delay function in general is not well behaved for all classes of signals. Zeros in the z-transform of a signal that are close to the unit circle cause large amplitude spikes to appear in the group delay function. The polarity of a spike depends on the location of the zero with respect to the unit circle. These large amplitude spikes mask the information about resonances. The research effort in this thesis focusses on the development of algorithms for manipulating the group delay function to suppress the information corresponding to the zeros of th signal that are close to unit circle in the z-domain and emphasise the features of of a signal. To demonstrate the usefulness of the algorithms developed, these algorithms are used to estimate (a) formant and pitch data from speech signals and ( b ) estimate spectra of auto-regressive processes and sinusoids in noise. The research effort in this thesis shows that the phase spectrum (or rather the group delay function) of a signal can be usefully processed to reliably extract features of a signal. ACKNOWLEDCEMENT I express my appreciation to Prof.B.Yegnanarayana for his constant help, excellent guidance and constructive criticisms throughout the course of this work. I thank Prof. R. ~a~arajan, Head, Department of Computer Science and Engineering, for making the various facilities in the department available to me: I owe my special thanks to Madhu Murthy and C.P.Mariadassou for some fruitful discussions. I thank G. V. Ramana Rao and R. Ramaseshan for reading my thesis and making useful suggestions. I would like to thank all my colleagues of the Speech and Vision Lab who have helped me in one way or the other. I thank Vatsala for providing me a shoulder whenever I was depressed. Finally, I thank my husband M. V. N. Murthy for his support and perseverence throughout the course of this work.

Averaged amplitude encoded phase-only filters for use in Fourier transform optical correlators

Abstract: Method of producing an averaged amplitude mask, for use in conjunction with a phase only filter (pOF) to provide an averaged amplitude matched filter (AAMF) in a Fourier transform optical correlator, involves intensity normalizing a plurality of reference images within a given class of objects, averaging the intensities of the reference images, Fourier transforming the resulting data, and utilizing the resultant amplitude function, exclusive of phase, of the Fourier transform to produce the averaged amplitude filter. A number of such masks are sequentially positioned adjacent the POF to very rapidly determine whether an input image falls within a particular class of objects, or whether the input image is of a particular object within the class.

Analysis of samples of wideband signals taken at irregular, sub-Nyquist, intervals

Abstract: A modified discrete Fourier transform in stated for estimating the spectrum of a signal sampled at irregular intervals. Additive pseudorandom sampling is proposed as an irregular sampling scheme. The transform periodicity and symmetrical properties are derived for the scheme. Alias-free spectral analysis of a bandlimited periodic signal is possible when using additive pseudorandom sampling with a maximum sampling rate below that specified by the Nyquist criterion.

Learning the Fourier spectrum of probabilistic lists and trees

Abstract: We observe that the Linial, Mansour, and Nissan method of learning boolean concepts (under uniform sampling distribution) by reconstructing their Fourier represent ation [LMN89] extends when the concepts are probabilistic in the sense of Kearns and Shapire [KS90]. We show that probabilistic decision lists, and more generally probabilistic decision trees with at most one occurrence of each literal, can be approximate ed by polynomially small Fourier represent ations, and that the non-negligible Fourier coefficients can be efficiently identified and estimated. Hence, all such concepts are learnable in polynomial time under uniform sampling distribution. This is the first instance where Fourier methods result in polynomial learning algorithms: the polynomiality of our results should be contrasted to the np”lylogn complexities in the analogous cases of [LMN89] and [M90]. The new ingredient of our work that allows us to achieve this polynomiality is that via refined Fourier analysis we are able to isolate the polynomially small set of non-negligible Fourier coefficients that reside in a super-polynomially large area of the spectrum. We further observe that several more general concept classes have slightly super-polynomial (npolyk)gn ) learning algorithms. These classes include all polynomial-size probabilistic decision trees, their convex combinations, etc. A concrete special case which results in polynomial learnabil“Bdl ColIl]lltl[\icalioI]s Research, Morristown NJ 07960. aidlo((!fl ash .Ixdlcorc.con]. flkll (bmmnnicat.ions Research, hlorrist.own NJ 07!w0, ]I~illail(@)fl&sll .l}cllcorc. col]). ity is the weighted arithmetization of k-DNF.

Image reconstruction from Fourier domain data sampled along a zig-zag trajectory.

Abstract: When the conventional Fourier transform (FT) algorithm is applied to reconstruct a magnetic resonance (MR) image from data sampled along a zig-zag trajectory in the Fourier space, the nonuniform sampling in the spatial frequency direction may give rise to artifacts In this paper the nature of the artifacts is analyzed and an alternative reconstruction algorithm is developed to produce artifact-free images Methods for reducing noise level in the reconstructed image are discussed Our approach is compared with another method based on the interlace sampling theorem

Signals and systems : an introduction

Abstract: Signal description system description system response to a sinusoidal input the Fourier series and Fourier transform the Laplace and transforms feedback systems.

Hexagonal discrete cosine transform for image coding

Abstract: The discrete cosine transform plays an important role in rectangularly sampled image coding for its excellent performance in information compaction. Hexagonal sampling is the optimal sampling strategy for two-dimensional signals in the sense that exact reconstruction of the waveform requires a lower sampling density than with the alternative schemes. In this Letter, a hexagonal discrete cosine transform (HDCT) for encoding the hexagonally sampled signals is presented.