Showing papers on "Non-uniform discrete Fourier transform published in 2003"

Signal Analysis: Time, Frequency, Scale and Structure

Abstract: Preface. Acknowledgments. 1 Signals: Analog, Discrete, and Digital. 1.1 Introduction to Signals. 1.1.1 Basic Concepts. 1.1.2 Time-Domain Description of Signals. 1.1.3 Analysis in the Time-Frequency Plane. 1.1.4 Other Domains: Frequency and Scale. 1.2 Analog Signals. 1.2.1 Definitions and Notation. 1.2.2 Examples. 1.2.3 Special Analog Signals. 1.3 Discrete Signals. 1.3.1 Definitions and Notation. 1.3.2 Examples. 1.3.3 Special Discrete Signals. 1.4 Sampling and Interpolation. 1.4.1 Introduction. 1.4.2 Sampling Sinusoidal Signals. 1.4.3 Interpolation. 1.4.4 Cubic Splines. 1.5 Periodic Signals. 1.5.1 Fundamental Period and Frequency. 1.5.2 Discrete Signal Frequency. 1.5.3 Frequency Domain. 1.5.4 Time and Frequency Combined. 1.6 Special Signal Classes. 1.6.1 Basic Classes. 1.6.2 Summable and Integrable Signals. 1.6.3 Finite Energy Signals. 1.6.4 Scale Description. 1.6.5 Scale and Structure. 1.7 Signals and Complex Numbers. 1.7.1 Introduction. 1.7.2 Analytic Functions. 1.7.3 Complex Integration. 1.8 Random Signals and Noise. 1.8.1 Probability Theory. 1.8.2 Random Variables. 1.8.3 Random Signals. 1.9 Summary. 1.9.1 Historical Notes. 1.9.2 Resources. 1.9.3 Looking Forward. 1.9.4 Guide to Problems. References. Problems. 2 Discrete Systems and Signal Spaces. 2.1 Operations on Signals. 2.1.1 Operations on Signals and Discrete Systems. 2.1.2 Operations on Systems. 2.1.3 Types of Systems. 2.2 Linear Systems. 2.2.1 Properties. 2.2.2 Decomposition. 2.3 Translation Invariant Systems. 2.4 Convolutional Systems. 2.4.1 Linear, Translation-Invariant Systems. 2.4.2 Systems Defined by Difference Equations. 2.4.3 Convolution Properties. 2.4.4 Application: Echo Cancellation in Digital Telephony. 2.5 The l p Signal Spaces. 2.5.1 l p Signals. 2.5.2 Stable Systems. 2.5.3 Toward Abstract Signal Spaces. 2.5.4 Normed Spaces. 2.5.5 Banach Spaces. 2.6 Inner Product Spaces. 2.6.1 Definitions and Examples. 2.6.2 Norm and Metric. 2.6.3 Orthogonality. 2.7 Hilbert Spaces. 2.7.1 Definitions and Examples. 2.7.2 Decomposition and Direct Sums. 2.7.3 Orthonormal Bases. 2.8 Summary. References. Problems. 3 Analog Systems and Signal Spaces. 3.1 Analog Systems. 3.1.1 Operations on Analog Signals. 3.1.2 Extensions to the Analog World. 3.1.3 Cross-Correlation, Autocorrelation, and Convolution. 3.1.4 Miscellaneous Operations. 3.2 Convolution and Analog LTI Systems. 3.2.1 Linearity and Translation-Invariance. 3.2.2 LTI Systems, Impulse Response, and Convolution. 3.2.3 Convolution Properties. 3.2.4 Dirac Delta Properties. 3.2.5 Splines. 3.3 Analog Signal Spaces. 3.3.1 L p Spaces. 3.3.2 Inner Product and Hilbert Spaces. 3.3.3 Orthonormal Bases. 3.3.4 Frames. 3.4 Modern Integration Theory. 3.4.1 Measure Theory. 3.4.2 Lebesgue Integration. 3.5 Distributions. 3.5.1 From Function to Functional. 3.5.2 From Functional to Distribution. 3.5.3 The Dirac Delta. 3.5.4 Distributions and Convolution. 3.5.5 Distributions as a Limit of a Sequence. 3.6 Summary. 3.6.1 Historical Notes. 3.6.2 Looking Forward. 3.6.3 Guide to Problems. References. Problems. 4 Time-Domain Signal Analysis. 4.1 Segmentation. 4.1.1 Basic Concepts. 4.1.2 Examples. 4.1.3 Classification. 4.1.4 Region Merging and Splitting. 4.2 Thresholding. 4.2.1 Global Methods. 4.2.2 Histograms. 4.2.3 Optimal Thresholding. 4.2.4 Local Thresholding. 4.3 Texture. 4.3.1 Statistical Measures. 4.3.2 Spectral Methods. 4.3.3 Structural Approaches. 4.4 Filtering and Enhancement. 4.4.1 Convolutional Smoothing. 4.4.2 Optimal Filtering. 4.4.3 Nonlinear Filters. 4.5 Edge Detection. 4.5.1 Edge Detection on a Simple Step Edge. 4.5.2 Signal Derivatives and Edges. 4.5.3 Conditions for Optimality. 4.5.4 Retrospective. 4.6 Pattern Detection. 4.6.1 Signal Correlation. 4.6.2 Structural Pattern Recognition. 4.6.3 Statistical Pattern Recognition. 4.7 Scale Space. 4.7.1 Signal Shape, Concavity, and Scale. 4.7.2 Gaussian Smoothing. 4.8 Summary. References. Problems. 5 Fourier Transforms of Analog Signals. 5.1 Fourier Series. 5.1.1 Exponential Fourier Series. 5.1.2 Fourier Series Convergence. 5.1.3 Trigonometric Fourier Series. 5.2 Fourier Transform. 5.2.1 Motivation and Definition. 5.2.2 Inverse Fourier Transform. 5.2.3 Properties. 5.2.4 Symmetry Properties. 5.3 Extension to L 2 (R). 5.3.1 Fourier Transforms in L 1 (R) &cap L 2 (R). 5.3.2 Definition. 5.3.3 Isometry. 5.4 Summary. 5.4.1 Historical Notes. 5.4.2 Looking Forward. References. Problems. 6 Generalized Fourier Transforms of Analog Signals. 6.1 Distribution Theory and Fourier Transforms. 6.1.1 Examples. 6.1.2 The Generalized Inverse Fourier Transform. 6.1.3 Generalized Transform Properties. 6.2 Generalized Functions and Fourier Series Coefficients. 6.2.1 Dirac Comb: A Fourier Series Expansion. 6.2.2 Evaluating the Fourier Coefficients: Examples. 6.3 Linear Systems in the Frequency Domain. 6.3.1 Convolution Theorem. 6.3.2 Modulation Theorem. 6.4 Introduction to Filters. 6.4.1 Ideal Low-pass Filter. 6.4.2 Ideal High-pass Filter. 6.4.3 Ideal Bandpass Filter. 6.5 Modulation. 6.5.1 Frequency Translation and Amplitude Modulation. 6.5.2 Baseband Signal Recovery. 6.5.3 Angle Modulation. 6.6 Summary. References. Problems. 7 Discrete Fourier Transforms. 7.1 Discrete Fourier Transform. 7.1.1 Introduction. 7.1.2 The DFT's Analog Frequency-Domain Roots. 7.1.3 Properties. 7.1.4 Fast Fourier Transform. 7.2 Discrete-Time Fourier Transform. 7.2.1 Introduction. 7.2.2 Properties. 7.2.3 LTI Systems and the DTFT. 7.3 The Sampling Theorem. 7.3.1 Band-Limited Signals. 7.3.2 Recovering Analog Signals from Their Samples. 7.3.3 Reconstruction. 7.3.4 Uncertainty Principle. 7.4 Summary. References. Problems. 8 The z-Transform. 8.1 Conceptual Foundations. 8.1.1 Definition and Basic Examples. 8.1.2 Existence. 8.1.3 Properties. 8.2 Inversion Methods. 8.2.1 Contour Integration. 8.2.2 Direct Laurent Series Computation. 8.2.3 Properties and z-Transform Table Lookup. 8.2.4 Application: Systems Governed by Difference Equations. 8.3 Related Transforms. 8.3.1 Chirp z-Transform. 8.3.2 Zak Transform. 8.4 Summary. 8.4.1 Historical Notes. 8.4.2 Guide to Problems. References. Problems. 9 Frequency-Domain Signal Analysis. 9.1 Narrowband Signal Analysis. 9.1.1 Single Oscillatory Component: Sinusoidal Signals. 9.1.2 Application: Digital Telephony DTMF. 9.1.3 Filter Frequency Response. 9.1.4 Delay. 9.2 Frequency and Phase Estimation. 9.2.1 Windowing. 9.2.2 Windowing Methods. 9.2.3 Power Spectrum Estimation. 9.2.4 Application: Interferometry. 9.3 Discrete filter design and implementation. 9.3.1 Ideal Filters. 9.3.2 Design Using Window Functions. 9.3.3 Approximation. 9.3.4 Z-Transform Design Techniques. 9.3.5 Low-Pass Filter Design. 9.3.6 Frequency Transformations. 9.3.7 Linear Phase. 9.4 Wideband Signal Analysis. 9.4.1 Chirp Detection. 9.4.2 Speech Analysis. 9.4.3 Problematic Examples. 9.5 Analog Filters. 9.5.1 Introduction. 9.5.2 Basic Low-Pass Filters. 9.5.3 Butterworth. 9.5.4 Chebyshev. 9.5.5 Inverse Chebyshev. 9.5.6 Elliptic Filters. 9.5.7 Application: Optimal Filters. 9.6 Specialized Frequency-Domain Techniques. 9.6.1 Chirp-z Transform Application. 9.6.2 Hilbert Transform. 9.6.3 Perfect Reconstruction Filter Banks. 9.7 Summary. References. Problems. 10 Time-Frequency Signal Transforms. 10.1 Gabor Transforms. 10.1.1 Introduction. 10.1.2 Interpretations. 10.1.3 Gabor Elementary Functions. 10.1.4 Inversion. 10.1.5 Applications. 10.1.6 Properties. 10.2 Short-Time Fourier Transforms. 10.2.1 Window Functions. 10.2.2 Transforming with a General Window. 10.2.3 Properties. 10.2.4 Time-Frequency Localization. 10.3 Discretization. 10.3.1 Transforming Discrete Signals. 10.3.2 Sampling the Short-Time Fourier Transform. 10.3.3 Extracting Signal Structure. 10.3.4 A Fundamental Limitation. 10.3.5 Frames of Windowed Fourier Atoms. 10.3.6 Status of Gabor's Problem. 10.4 Quadratic Time-Frequency Transforms. 10.4.1 Spectrogram. 10.4.2 Wigner-Ville Distribution. 10.4.3 Ambiguity Function. 10.4.4 Cross-Term Problems. 10.4.5 Kernel Construction Method. 10.5 The Balian-Low Theorem. 10.5.1 Orthonormal Basis Decomposition. 10.5.2 Frame Decomposition. 10.5.3 Avoiding the Balian-Low Trap. 10.6 Summary. 10.6.1 Historical Notes. 10.6.2 Resources. 10.6.3 Looking Forward. References. Problems. 11 Time-Scale Signal Transforms. 11.1 Signal Scale. 11.2 Continuous Wavelet Transforms. 11.2.1 An Unlikely Discovery. 11.2.2 Basic Theory. 11.2.3 Examples. 11.3 Frames. 11.3.1 Discretization. 11.3.2 Conditions on Wavelet Frames. 11.3.3 Constructing Wavelet Frames. 11.3.4 Better Localization. 11.4 Multiresolution Analysis and Orthogonal Wavelets. 11.4.1 Multiresolution Analysis. 11.4.2 Scaling Function. 11.4.3 Discrete Low-Pass Filter. 11.4.4 Orthonormal Wavelet. 11.5 Summary. References. Problems. 12 Mixed-Domain Signal Analysis. 12.1 Wavelet Methods for Signal Structure. 12.1.1 Discrete Wavelet Transform. 12.1.2 Wavelet Pyramid Decomposition. 12.1.3 Application: Multiresolution Shape Recognition. 12.2 Mixed-Domain Signal Processing. 12.2.1 Filtering Methods. 12.2.2 Enhancement Techniques. 12.3 Biophysical Applications. 12.3.1 David Marr's Program. 12.3.2 Psychophysics. 12.4 Discovering Signal Structure. 12.4.1 Edge Detection. 12.4.2 Local Frequency Detection. 12.4.3 Texture Analysis. 12.5 Pattern Recognition Networks. 12.5.1 Coarse-to-Fine Methods. 12.5.2 Pattern Recognition Networks. 12.5.3 Neural Networks. 12.5.4 Application: Process Control. 12.6 Signal Modeling and Matching. 12.6.1 Hidden Markov Models. 12.6.2 Matching Pursuit. 12.6.3 Applications. 12.7 Afterword. References. Problems. Index.

Discrete-Fourier-transform-based technique for removal of decaying DC offset from phasor estimates

Abstract: A technique to suppress the negative effect of a decaying DC offset on phasor estimates using a digital Fourier transform algorithm is presented. The proposed technique is based on evaluation of the decaying DC component in the signal. The technique is suitable for all prevailing systems and fault conditions. The performance of the technique is evaluated in both the time and frequency domains. Simulation results showing the effectiveness of this new technique are presented. The proposed technique is further evaluated when it is used in a distance relay design.

Shift-invariance in the Discrete Wavelet Transform

Abstract: In this paper we review a number of approaches to reducing, or removing, the problem of shift variance in the discrete wavelet transform (DWT). We describe a generalization of the critically sampled DWT and the fully sampled 'algorithme a trous' that provides approximate shift-invariance with an acceptable level of redundancy. The proposed over complete DWT (OCDWT) is critically sub-sampled to a given level of the decomposition, below which it is then fully sampled. The efficacy of the proposed algorithm is illustrated in an edge detection context and directly compared to a number of other shift-invariant transforms in terms of complexity and redundancy.

Digital Ridgelet Transform Based on True Ridge Functions

Abstract: We study a notion of ridgelet transform for arrays of digital data in which the analysis operator uses true ridge functions, as does the synthesis operator. There are fast algorithms for analysis, for synthesis, and for partial reconstruction. Associated with this is a transform which is a digital analog of the orthonormal ridgelet transform (but not orthonormal for finite n). In either approach, we get an overcomplete frame; the result of ridgelet transforming an n × n array is a 2n × 2n array. The analysis operator is invertible on its range; the appropriately preconditioned operator has a tightly controlled spread of singular values. There is a near-parseval relationship. Our construction exploits the recent development by Averbuch et al. (2001) of the Fast Slant Stack, a Radon transform for digital image data; it may be viewed as following a Fast Slant Stack with fast 2-d wavelet transform. A consequence of this construction is that it offers discrete objects (discrete ridgelets, discrete Radon transform, discrete Pseudopolar Fourier domain) which obey inter-relationships paralleling those in the continuum ridgelet theory (between ridgelets, Radon transform, and polar Fourier domain). We make comparisons with other notions of ridgelet transform, and we investigate what we view as the key issue: the summability of the kernel underlying the constructed frame. The sparsity observed in our current implementation is not nearly as good as the sparsity of the underlying continuum theory, so there is room for substantial progress in future implementations.

A Method for Fast Computation of the Fourier Transform over a Finite Field

Abstract: We consider the problem of fast computation of the Fourier transform over a finite field by decomposing an arbitrary polynomial into a sum of linearized polynomials. Examples of algorithms for the Fourier transform with complexity less than that of the best known analogs are given.

Fractional Fourier transform of the Gaussian and fractional domain signal support

Abstract: The fractional Fourier transform (FrFT) provides an important extension to conventional Fourier theory for the analysis and synthesis of linear chirp signals. It is a parameterised transform which can be used to provide extremely compact representations. The representation is maximally compressed when the transform parameter, /spl alpha/, is matched to the chirp rate of the input signal. Existing proofs are extended to demonstrate that the fractional Fourier transform of the Gaussian function also has Gaussian support. Furthermore, expressions are developed which allow calculation of the spread of the signal representation for a Gaussian windowed linear chirp signal in any fractional domain. Both continuous and discrete cases are considered. The fractional domains exhibiting minimum and maximum support for a given signal define the limit on joint time-frequency resolution available under the FrFT. This is equated with a restatement of the uncertainty principle for linear chirp signals and the fractional Fourier domains. The calculated values for the fractional domain support are tested empirically through comparison with the discrete transform output for a synthetic signal with known parameters. It is shown that the same expressions are appropriate for predicting the support of the ordinary Fourier transform of a Gaussian windowed linear chirp signal.

A method for the discrete fractional Fourier transform computation

Abstract: A new method for the discrete fractional Fourier transform (DFRFT) computation is given in this paper. With the help of this method, the DFRFT of any angle can be computed by a weighted summation of the DFRFTs with the special angles.

Optical encryption using a localized fractional Fourier transform

Abstract: We propose a new method to encrypt and decrypt a two-dimensional amplitude image, which uses a jigsaw transform and a localized fractional Fourier transform. The jigsaw transform is applied to the original image to be encrypted, and the image is then divided into independent nonoverlapping segments. Each image segment is encrypted using different fractional parameters and two statistically independent random phase codes. The random phase codes, along with the set of fractional orders and jigsaw transform index, form the key to the encrypted data. Results of computer simulation are presented to verify the proposed idea and analyze the performance of the method. We also propose an optical implementation, which may find application for encrypting data stored in holographic memory.

Channel estimation for OFDM systems with multiple transmit antennas by exploiting the properties of the discrete Fourier transform

Abstract: We addresses channel estimation based on the discrete Fourier transform (DFT) applied to OFDM-based MIMO systems. By exploiting the properties of the DFT, channel estimation schemes for MIMO-OFDM system can be simplified. The Fourier transform translates phase shifts in the frequency domain to delays in the time domain. In order to exploit this feature, phase shifted pilot sequences are a perfect match to the Fourier transform in terms of separating the N/sub T/ superimposed signals, corresponding to N/sub T/ transmit antennas, without any further processing. The proposed estimator is shown to be optimum for the sample spaced channel, i.e. the channel tap delays are multiples of the sampling duration. A sub-optimum approximation for the non-sample spaced channel is also suggested.

M-ary frequency shift keying signal classification based-on discrete Fourier transform

Abstract: The existing decision-theory based classifiers for M-ary frequency shift keying (MFSK) signals have assumed that there is some prior knowledge of the transmitted MFSK signal parameters; while the feature-based classifiers have some limitations such as that their thresholds are signal-to-noise-ratio-dependent (SNR-dependent). In this paper, we investigate some useful properties of the amplitude spectrum of MFSK signals. Using these properties as classification criteria, a fast Fourier transform based classifier (FFTC) of MFSK signals has been developed. The FFTC algorithm is practical since it only requires some reasonable knowledge of a received signal. It is found that the FFTC algorithm works well in classifying 2-FSK, 4-FSK, 8-FSK, 16-FSK, and 32-FSK signals when SNR>0dB. The FFTC algorithm also gives good estimation of the frequency deviation of the received MFSK signal.

Motion detection, the Wigner distribution function, and the optical fractional Fourier transform.

Abstract: It is shown that both surface tilting and translational motion can be independently estimated by use of the speckle photographic technique by capturing consecutive images in two different fractional Fourier domains. A geometric interpretation, based on use of the Wigner distribution function, is presented to describe this application of the optical fractional Fourier transform when little prior information is known about the motion.

Wigner distribution and fractional Fourier transform

Abstract: We have described the relationship between the fractional Fourier transform and the Wigner distribution by using the Radon-Wigner transform, which is a set of projections of the Wigner distribution as well as a set of squared moduli of the fractional Fourier transform. We have introduced the concept of fractional Fourier transform moments and have proposed a way for the calculation of the well-known global and local moments of the Wigner distribution, based on the knowledge of a few fractional power spectra. The application of the results in optics and signal processing has been discussed briefly.

Inverses of multivariable polynomial matrices by discrete fourier transforms

Abstract: Two discrete Fourier transform based algorithms are proposed for the computation of the Moore-Penrose and Drazin inverse of a multivariable polynomial matrix.

Fourier transform apparatus

Abstract: A Fourier transform apparatus whose pipeline width is independent of transform point number of individual pipeline FFT circuits in each stage and composed of a preceding stage and a succeeding stage. Each of the stages includes M(power of 2)-point radix 2 pipeline FFT circuits each having two-parallel inputs/outputs in a number of a (divisor of M) which are equal in respect to the transform point number and data permutating means for data supply to the transform means of each stage so that the pipeline width of the Fourier transform apparatus is made independent of the transform point numbers of the individual pipeline FFT circuits in each stage.

Mapping Between Digital and Continuous Projections via the Discrete Radon Transform in Fourier Space

Abstract: This paper seeks to extend the Fourier space properties of the discrete Radon transform, R(t, m), proposed by Matus and Flusser in (1), to expanded discrete projections, R(k, θ), where the wrapping of rays is removed. This expanded mode yields projections more akin to the continuous space sinogram. It is similar to the Mojette transform defined in (2), but has a pre-determined set of discrete projection angles derived from the Farey series (3). It is demonstrated that a close approx- imation to the sinogram of an image can be obtained from R(k, θ), both in Radon and Fourier space. This investigation is undertaken to explore the possibilities of applying this mapping to the inverse problem, that of obtaining discrete projection data from continuous projection data as a means of efficient tomographic reconstruction that requires minimal interpolation and filtering.

Orthogonal frequency multi-carrier transmission device and transmission method

Abstract: A symbol which has been subjected to series-parallel conversion by a series-parallel converter (12) is subjected to reverse Fourier transform as a sub-carrier signal component of a symbol rate interval by a reverse Fourier transform section (13). The level of the obtained time region signal component is compared to an allowable peak level Cth by a peak component detector (22) so as to detect a peak component. The peak component is converted into a frequency region component by a Fourier transform section (23) and it is subtracted from the corresponding sub-carrier signal component by subtractors (240 to 24N-1), thereby reducing the peak power.

Uncertainty relations and minimum uncertainty states for the discrete Fourier transform and the Fourier series

Abstract: The conventional Fourier transform has a well-known uncertainty relation that is defined in terms of the first and second moments of both a function and its Fourier transform. It is also well known that Gaussian functions, when translated to an arbitrary centre and supplemented by a linear phase factor, provide a complete set of minimum uncertainty states (MUSs) that exactly achieves the lower bound set by this uncertainty relation. A similarly general set of MUSs and uncertainty relations are derived here for discrete and/or periodic generalizations of the Fourier transform, namely for the discrete Fourier transform and the Fourier series. These extensions require a modified definition for the width of a periodic distribution, and they lead to more complex uncertainty relations that turn out to depend on the centroid location and mean frequency of the distribution. The derivations lead to novel generalizations of Hermite–Gaussian functions and, like Gaussians, the MUSs can play a special role in a range of Fourier applications.

A new numerical Fourier transform in d-dimensions

Abstract: The classical method of numerically computing Fourier transforms of digitized functions in one or in d-dimensions is the so-called discrete Fourier transform (DFT) efficiently implemented as fast Fourier transform (FFT) algorithms. In many cases, the DFT is not an adequate approximation to the continuous Fourier transform, and because the DFT is periodical, spectrum aliasing may occur. The method presented in this contribution provides accurate approximations of the continuous Fourier transform with similar time complexity. The assumption of signal periodicity is no longer posed and allows the computation of numerical Fourier transforms in a broader domain of frequency than the usual half-period of the DFT. In addition, this method yields accurate numerical derivatives of any order and polynomial splines of any odd degree. The numerical error on results is easily estimated. The method is developed in one and in d dimensions, and numerical examples are presented.

Fast computation of RBF coefficients for regularly sampled inputs

Abstract: Using Fourier transform of the radial basis function (RBF) network, a fast computation of RBF coefficients for regularly sampled inputs is proposed. The use of the fast Fourier transform allows computation of the coefficients in O(NlogN) computational time, where N is the number of data.

Blind source separation based on the fractional fourier transform

Abstract: Different approaches have been suggested in recent years to the blind source separation problem, in which a set of signals is recovered out of its instantaneous linear mixture. Many widely-used algorithms are based on second-order statistics, and some of these algorithms are based on timefrequency analysis. In this paper we set a general framework for this family of second-order statistics based algorithms, and identify some of these algorithms as special cases in that framework. We further suggest a new algorithm that is based on the fractional Fourier transform (FRT), and is suited to handle non-stationary signals. The FRT is a tool widely used in time-frequency analysis and therefore takes a considerable place in the signalprocessing field. In contrast to other blind source separation algorithms suited for the non-stationary case, our algorithm has two major advantages: it does not require the assumption that the signals’ powers vary over time, and it does not require a pre-processing stage for selecting the points in the time-frequency plane to be considered. We demonstrate the performance of the algorithm using simulation results.

On perfect arrays

Abstract: In this paper construction of perfect arrays is investigated by using the Fourier transform, in order to discuss generally. Furthermore a new orthogonal transform using a perfect array is discussed, which will be applied to some applications, such as communications and signal processing.

Radon transform inversion via Wiener filtering over the Euclidean motion group

Abstract: In this paper we formulate the Radon transform as a convolution integral over the Euclidean motion group (SE(2)) and provide a minimum mean square error (MMSE) stochastic deconvolution method for the Radon transform inversion. Proposed approach provides a fundamentally new formulation that can model nonstationary signal and noise fields. Key components of our development are the Fourier transform over SE(2), stochastic processes indexed by groups and fast implementation of the SE(2) Fourier transform. Numerical studies presented here demonstrate that the method yields image quality that is comparable or better than the filtered backprojection algorithm. Apart from X-ray tomographic image reconstruction, the proposed deconvolution method is directly applicable to inverse radiotherapy, and broad range of science and engineering problems in computer vision, pattern recognition, robotics as well as protein science.

Instantaneous frequency estimation using level-crossing information

Abstract: We discuss the problem of instantaneous frequency (IF) estimation of phase signals using their level-crossing (LC) instant information. We cast the problem to that of interpolating the instantaneous phase (IP), and hence finding the IF, from samples obtained at the level-crossing instants of the phase signal. These are inherently irregularly spaced and the problem essentially reduces to reconstructing a signal from the samples taken at irregularly sampled points for which we propose a 'line plus sum of sines' model. In the presence of noise, the temporal structure of the level-crossings can get distorted. To reduce the effects of noise, we use a short-time Fourier transform (STFT) based enhancement scheme. The performance of the proposed method is studied through Monte-Carlo simulations for a phase signal with composite IF for various SNRs. Different level-crossing based estimates are combined to obtain a new IF estimate. Simulation studies show that the estimates obtained using zero-crossing (ZC) and other very low level values perform better than those obtained with higher level values.