An extension of the notion of the analytical signal to multidimensional signals is presented The real and imaginary parts of this signal are a linear combination of the original signal and of its complete and partial Hilbert transforms Its Fourier image exists only in one orthant of the multidimensional frequency space The orthant is a half-axis in one dimension, a quadrant in two dimensions, an octant in three dimensions, etc A multidimensional complex signal makes it possible to introduce the definitions of instantaneous amplitude, instantaneous phase, and partial instantaneous complex frequencies (partial derivatives of the instantaneous phase) and to formulate a modulation theory of a multidimensional harmonic carrier The 2-D equivalent of 1-D single-sideband modulation is defined and called single quadrant modulation It is shown that the multidimensional complex signal with a signal orthant spectrum may be defined as a convolution of the real signal with the multidimensional complex delta distribution >

Multidimensional complex signals with single-orthant spectra

A fast band-limited signal extrapolation technique is presented where the total extrapolation process is achieved by a single matrix operation. The proposed technique and its implementation has many advantages over known extrapolation techniques in terms of computational savings and accuracy of the results, and it can he operated on a realtime basis.

An approach to band-limited signal extrapolation: The extrapolation matrix

We present some theoretical results on the band-limited signal extrapolation problem. In Section I we describe four basic models for the extrapolation problem. These models are useful in understanding the relationship between the continuous extrapolation problem and some discrete algorithms given in [1] and [2]. One of these models was shown to approximate the continuous band-limited extrapolation problem [3]. Another model is obtained when the discrete Fourier transform (DFT) is used to implement the well-known iterative algorithm given in [4] and [5] which was designed for solving the continuous extrapolation problem; in Section II this model is related to the continuous model by means of an interesting approximation theorem. Also, an important conjecture is presented. Section III shows some approximation results. Specifically, we prove that some discrete-discrete and discrete-continuous extrapolations of noisy signals converge to solutions of a certain continuous-continuous noisy extrapolation problem when the noise η is bounded by a known number, max \eta(x)| leq \epsilon . This convergence is obtained by using normal families of entire functions in ¢nand some other complex analysis tools. We also show that the extrapolation problem is very sensitive to noise even in cases where only small amounts of extrapolation are desired. This result indicates that in the presence of noise, extrapolation techniques should be used judiciously in order to obtain reasonable results.

Some aspects of band-limited signal extrapolation: Models, discrete approximations, and noise

A regularized algorithm for the recovery of band-limited signals from noisy data is described. The regularization is characterized by a single parameter. Iterative and non-iterative implementations of the algorithm are shown to have useful properties, the former offering the advantage of flexibility and the latter a potential for rapid data processing. Comparative results, using experimental data obtained in laser anemometry studies with a photon correlator, are presented both with and without regularization.

/pdf/regularized-iterative-and-non-iterative-procedures-for-x683mbime6.pdf

Regularized Iterative and Non-iterative Procedures for Object Restoration from Experimental Data

Synchrophasor estimation is typically performed by means of spectral analysis based on the discrete Fourier transform (DFT). Traditional DFT approaches, though, suffer from several uncertainty contributions due to the stationarity assumption, spectral leakage effects, and the finite-grid resolution. This paper addresses these limitations, by proposing a joint application of the Hilbert transform (HT) and the interpolated DFT (IpDFT) technique. Specifically, the HT enables the suppression of the spectral leakage generated by the negative image of the tones under analysis, whereas the IpDFT limits the effects of spectrum granularity. In order to relax the constraint in terms of measurement reporting latency, the proposed estimator can adopt a window length of 40 ms and yet provides a noticeable estimation accuracy with a worst-case total vector error and frequency error equal to 0.02% and 4 mHz, respectively, in steady-state conditions. In this context, this paper discusses the most suitable setting of the algorithm parameters and their effect on spurious component rejection. Moreover, a thorough metrological characterization of the algorithm estimation accuracy and responsiveness with respect to the IEEE Std. C37.118.1 is carried out in order to detect the main uncertainty sources as well as possible room for enhancement. Finally, a comparison with two consolidated IpDFT approaches shows the actual performance enhancement provided by the proposed algorithm.

/pdf/reduced-leakage-synchrophasor-estimation-hilbert-transform-3i465nz1jy.pdf

Reduced Leakage Synchrophasor Estimation: Hilbert Transform Plus Interpolated DFT

A discrete filtering technique based on circular convolution is presented. The discrete Hilbert transform (DHT), in matrix form and other matrices for filtering by circular convolution, is shown to compare favorably with DFT and FFT filtering. The comparison is presented in terms of error accumulation.

https://www.ruor.uottawa.ca/bitstream/10393/11097/1/DC52487.PDF

Discrete Hilbert transform filtering

A discrete filtering technique based on the discrete Hilbert Transform (DHT) is presented. The filtering method and its implementation are shown to compare favorably with DFT and FFT filtering. The comparison is presented in terms of complexity of hardware implementation and error accumulation. The DHT itself and derived discrete filter transforms are each presented in matrix form and may be implemented with the same hardware in either series or parallel form. In this manner a large variety of filtering characteristics may be implemented by changes of the stored coefficients only. While the filtering process is a circular convolution, direct convolution filters are obtainable also, by using one row of the matrix only, and are realizable in the form of an FIR (non-recursive) digital filter.

This is a comment on the equivalence between a paper recently published on the topic of band-limited signal extrapolation with others [1], [2] published previously. It is shown that the equations describing the iterative or recursive method, as well as those describing the one-step method of extrapolation, are identical.

M. Sabri

Papers

Discrete Hilbert transform filtering

Discrete Hilbert transform filtering

Comments on "An extrapolation procedure for band-limited signals"

Rebuttal to "Observations on the extrapolation of a band-limited signal problem"