Reconstruction Of Bandlimited Signals Research Articles

Spectrum analysis for nonuniform sampling of bandlimited and multiband signals in the fractional Fourier domain

This paper explores fractional spectra of signals after two types of nonuniform samplings in the presence of timing offset. Both cases of bandlimited and multiband signals in the fractional Fourier domain are considered, based on which we provide conditions for reconstructing the spectrum and devise relevant methods for compensating the bias introduced by timing offset. To obtain these goals, we divide the nonuniformly sampled signal into uniformly sampled sub-sequences and establish the relationship between the discrete-time fractional Fourier transforms (DTFrFTs) of them and the FrFT of the original signal. It indicates that the obtained DTFrFTs consists of periodic replicas of the FrFT for the original signal with each replica being biased by a phase term introduced by timing offset. Based on the matrix form of the relationship, we implement spectral reconstruction of bandlimited signals under certain conditions by performing the inverse of a matrix determined by the phase terms. Different from bandlimited signals, the DTFrFTs of uniformly sampled sub-sequences for multiband signals are studied within disjoint fractional frequency sub-intervals separated by aliasing boundaries. Moreover, inverse of a matrix with fewer columns and rows is utilized to reconstruct the spectrum. Simulations verify the effectiveness of the proposed methods.

Sampling and reconstruction of multi‐dimensional bandlimited signals in the special affine Fourier transform domain

The paper is concerned with the sampling and reconstruction of bandlimited signals in the multi‐dimensional special affine Fourier transform domain. First, we define the multi‐dimensional special affine Fourier transform and establish the convolution theorem. Then, sampling and reconstruction of the multi‐dimensional deterministic bandlimited signals in the special affine Fourier transform domain is studied. Finally, the definition of multi‐dimensional random bandlimited signals in the special affine Fourier transform domain is given and a sampling theorem is demonstrated.

Bandlimited Signal Reconstruction From Leaky Integrate-and-Fire Encoding Using POCS

Leaky integrate-and-fire (LIF) encoding is a model of neuron transfer function in biology that has recently attracted the attention of the signal processing and neuromorphic computing communities as a technique of event-based sampling for data acquisition. While LIF enables the implementation of analog-circuit signal samplers of lower complexity and higher accuracy simultaneously, the core difficulty of this technique is the retrieval of an input from its LIF-encoded output. In this article, we study this problem in the context of bandlimited inputs, by extracting the most abstract features of an LIF encoder as a generalized nonuniform sampler. In this view, the LIF output is seen as the transformation of the input by a known linear operator. We show that the signal reconstruction method of projection onto convex sets (POCS) converges to a weighted pseudo-inverse of this operator. This allows perfect recovery under uniqueness of reconstruction, minimum-norm reconstruction under incomplete sampling, as well as a noise shaping of time quantization that outperforms standard pseudo-inversion. On the practical side, a single iteration of the POCS method can be used to improve <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">any</i> estimate whose LIF samples are not consistent with those of the input, and a rigorous discrete-time implementation of this iteration is proposed that does not require a Nyquist-rate representation of the signals.

Super-resolution wavefront reconstruction

Context.Cutting-edge, ground-based astronomical instruments are fed by adaptive optics (AO) systems that are aimed at providing high performance down to the visible wavelength domain on 10 m class telescopes and in the near infrared for the first generation instruments of Extremely Large Telescopes (ELTs). Both applications lead to a large ratio between the telescope diameter,D, and the coherence length or Fried parameter,r0, that isD/r0. As the parameter that defines the required number of degrees of freedom of the AO system,D/r0drives the requirement to reconstruct the incoming wavefront with ever-higher spatial resolution. In this context, super-resolution (SR) appears as a potential game changer. Indeed, SR promises to dramatically expand the range of spatial frequencies that can be reconstructed from a set of lower resolution measurements of the wavefront.Aims.As a technique that seeks to upscale the resolution of a set of measured signals, SR retrieves higher-frequency signal content by combining multiple lower resolution sampled data sets. It is well known both in the temporal and spatial domains and widely used in imaging to reduce aliasing and enhance the resolution of coarsely sampled images. This study applies the SR technique to the bidimensional wavefront reconstruction. In particular, we show how SR is intrinsically suited for tomographic multi-wavefront sensor (WFS) AO systems, revealing many of its advantages with minimal design effort.Methods.We provide a direct space and Fourier optics description of the wavefront sensing operation and we demonstrate how SR can be exploited through signal reconstruction, especially within the framework of periodic non-uniform sampling. We investigate both meta-uniform and non-uniform sampling schemes and we show that under some conditions, both sampling schemes enable a perfect reconstruction of band-limited signals. We also provide a SR bi-dimensional model for a Shack-Hartmann (SH) WFS, along with an analysis of the characteristics of the sensitivity function. We validated the SR concept with numerical simulations of representative multi-WFS SH AO systems. Finally, we explored the extension of the method to pyramid WFSs.Results.Our results show that combining several WFS samples in a SR framework grants access to a greater number of modes than the native one offered by a single WFS (despite the fixed sub-aperture size across samples). We show that the wavefront reconstruction achieved with four WFSs can be equivalent to a single WFS providing a sampling resolution that is twice greater (linear across the telescope aperture). We also show that the associated noise propagation is not degraded under SR. Finally, we show that the concept can be extended to the signal produced by single pyramid WFS, with its four re-imaged pupils serving as multiple non-redundant samples.Conclusions.We find that SR applied to wavefront sensing and reconstruction (WFR) offers a new parameter space to explore, as it decouples the size of the sub-aperture from the desired wavefront sampling resolution. By shifting away from outdated assumptions, new and more flexible, better-performing AO designs have now become possible.

Regularized sampling reconstruction of signals in the linear canonical transform domain

Sampling reconstruction of signals band-limited in the linear canonical transform domain has become a major task in bridging the relationship between the continuous physical signals and discrete signals. Many recent works dealt with this task from different theoretical and practical aspects. Furthermore, and as in the case of the classical Shannon’s sampling theorem, attempts are made to accelerate the convergence sampling rate. In this work we incorporate two different regularization kernels in the sampling reconstruction of band-limited signals to remarkably enhance the convergence rate. The first kernel is a normalized Gaussian kernel, allowing the sampling reconstruction for not necessarily band-limited signals, while the second is a band-limited kernel. In the later case the bandwidth is rigorously considered to guarantee exact reconstruction. We estimate the truncation error since only a finite number of samples is available. Rigorous analytic proofs are provided with precise error bounds. Illustrative examples are carried out with comparisons with the classical reconstruction technique.

Time Encoding of Bandlimited Signals: Reconstruction by Pseudo-Inversion and Time-Varying Multiplierless FIR Filtering

We propose an entirely redesigned framework of bandlimited signal reconstruction for the time encoding machine (TEM) introduced by Lazar and Toth. As the encoding part of TEM consists in obtaining integral values of a bandlimited input over known time intervals, it theoretically amounts to applying a known linear operator on the input. We then approach the general question of signal reconstruction by pseudo-inversion of this operator. We perform this task numerically and iteratively using projections onto convex sets (POCS). The algorithm can be implemented exactly in discrete time with multiplications that are all reduced to scaling by signed powers of two, thanks to the use of relaxation coefficients. Meanwhile, the algorithm achieves a rate of convergence similar to that of Lazar and Toth. For real-time processing, we propose an approximate time-varying FIR implementation, which avoids the splitting of the input into blocks. We finally propose some preliminary semi-convergence analysis of the algorithm under data noise.

Discrete Band-Limited Signal Reconstruction From Irregular Samples

Image reconstruction from discrete samples is fundamental in remote sensing. The reconstruction problem is more complicated when the sample locations are irregularly spaced and employ different aperture functions. Further, not all 2-D sampling configurations permit full image reconstruction of band-limited signals. In this article, 2-D signal reconstruction from irregular samples with variable apertures is considered using theory and examples. Exact reconstruction requires that the signal be band-limited and the sampling matrix be invertable. The results are sensitive to the sample locations and noise in the measurements. Illustrative examples are provided using simulation and actual data from the L-band Soil Moisture Active Passive (SMAP) radiometer. The general approach can be employed with other sensors.

A New Class of Explicit Interpolatory Splines and Related Measurement Estimation

This paper proposes a new and rich class of cardinal splines called generalized MK-splines (GMK-splines), which manifest several beneficial properties (especially the interpolation property) and the ability to obtain a construction with a desired frequency response. The GMK-spline basis functions of given degree can be constructed by a linear combination of a finite number of shifted B-splines of that degree in a unified framework, the GMK-spline bases form Riesz bases of their generating spaces, and the GMK-spline frequency responses meet Strang-Fix conditions. Furthermore, we provide search methods for minimal-support GMK-splines and local optimal GMK-spline frequency responses under different given passband frequencies. Several related estimates are theoretically assessed, including energy magnitude, maximum, and approximation error of GMK-spline interpolated signals. In signal processing applications, the GMK-splines behave as interpolating FIR filters with simple implementation, and some of them attain lower error than other traditional interpolation system responses, such as cubic cardinal B-spline and linear spline, when approximating to an ideal lowpass filter. Experimental results show that the GMK-spline filters can be implemented efficiently and have significant characteristics in the process of bandlimited signal reconstruction.

On reconstruction of bandlimited signals from purely timing information

Abstract The well known Logan’s theorem asserts that, under some assumptions, a one octave bandpass signal is recoverable, modulo a multiplicative constant, from its zero crossings only. However, such recovery is numerically problematic and the theorem is not applicable to general bandlimited signals. In this paper, we demonstrate that the additional timing information sufficient for recovery of general band limited signals can be provided in the form of the zero crossings of several of its derivatives and propose novel numerically robust algorithms for such timing extraction and for signal reconstruction, both with high fidelity. We tested the proposed algorithms extensively, with both synthetic and audio signals. In particular, we numerically demonstrate that the timing of the zero crossings of a typical speech signal and of its first two derivatives are not sufficient for a numerically robust recovery of such a signal, but that a robust recovery becomes possible by adding the timing of the zero crossings of the third order derivative.

Band-Limited Signal Reconstruction From Irregular Samples With Variable Apertures

Sampling plays a critical role in remote sensing and signal analysis. In conventional sampling theory, the signal is sampled at a uniform rate at a minimum of twice the signal bandwidth. Sampling with an aperture function requires a fixed-aperture function, which can be removed by deconvolution after signal reconstruction. However, in some cases, the signal samples are available only at irregular positions, and different samples use different aperture functions. In this paper, the theory of finite-length signal reconstruction with irregular samples and variable apertures in one and two dimensions is considered. In the 1-D case, a band-limited discrete signal can be exactly reconstructed from a finite number of arbitrarily spaced samples with few restrictions on the aperture functions. Exact reconstruction in the 2-D case requires the sampling matrix be invertable, and is not always possible. Variable aperture functions, while complicating the process, can enable reconstruction for a broader range of sample locations. Practical issues are discussed, and numerical examples are provided. Variable aperture reconstruction has application in a variety of remote sensing problems. In this paper, reconstruction from 2-D irregular sampling with variable apertures is illustrated using Special Sensor Microwave/Imager radiometer observations.

On Bandlimited Signal Reconstruction From the Distribution of Unknown Sampling Locations

We study the reconstruction of bandlimited fields from samples taken at unknown but statistically distributed sampling locations. The setup is motivated by distributed sampling where precise knowledge of sensor locations can be difficult. Periodic one-dimensional bandlimited fields are considered for sampling. Perfect samples of the field at independent and identically distributed locations are obtained. The statistical realization of sampling locations is not known. First, it is shown that a bandlimited field cannot be uniquely determined with samples taken at statistically distributed but unknown locations, even if the number of samples is infinite. Next, it is assumed that the order of sample locations is known. In this case, using insights from order-statistics, an estimate for the field with useful asymptotic properties is designed. Distortion (mean-squared error) and central-limit are established for this estimate.

Simplification to the fast FIR-FFT filtering technique in the DSP interpolation process for band-limited signals

Frequently, when great amounts of sampled data are manipulated, it is necessary to reduce them to a fraction that truly represents the data. Here two elements are important; neither is it desirable for the amount of data to be too extensive; nor is it desirable to lose key information. For the data reconstruction,it is often required a interpolation technique. This work proposes an optimization of the fast FIR filtering-based DSP interpolation technique. The resulting interpolator is a feasible, simplified version of hardware implementations, which is ideal for the reconstruction of band-limited signals with adequate sampling. In case studies included in this work, wind and temperature signals are used. The temperature signal has a suitable sampling rate and with the use of the interpolator it presents an error of 3.95% for a reduction of 99.61% of the data. The wind signal is very unpredictable and it does not have a suitable sampling rate. Hence, the interpolator does not improve the reconstruction of the signal in comparison to the averages. Typically, an average signal value is recorded for each 10 minute period, in which, the wind signal presents an error 8.6 times greater than the temperature signal. In order to reduce this error in the wind signal, it is recommended to increase the sampling rate and the quantization levels in measurement equipment.

Towards a general theory of reconstruction of bandlimited signals from sine wave crossings

Arbitrary bounded bandlimited signals that are real on the real axis cannot be reconstructed from their zero crossings in general, because they can have complex zeros. However, a simple invertible transformation, which consists of taking the difference of such a signal with a properly chosen sine function, creates a function that has only simple and real zeros. It can be shown that the original signal can be reconstructed from these zeros, or, equivalently, from the sine wave crossings of the signal. In this paper we study the reconstruction of bandlimited signals from their sine wave crossings by sampling series, which is one possible method of reconstruction. The reconstruction functions and the sampling points that are used in the reconstruction process are determined by the signal which shall be reconstructed and therefore are, in a certain sense, adapted to the signal. We address the convergence behavior of the reconstruction process without oversampling. Although we have local uniform convergence, we conjectured in a previous paper that the adaptivity is not sufficient for global uniform convergence. By developing a rigorous convergence theory for different signal spaces, and by showing that the Paley–Wiener space PW π 1 of signals that are the inverse Fourier transform of some absolutely integrable function is the border case, we prove that this conjecture is true. The solution of this problem has far reaching consequences for the approximation of stable linear time-invariant systems by sampling series, because it shows that the adaptivity of the system approximation process is not sufficient for its pointwise convergence. The theory of sine-type functions is an important tool for our analysis, and we give several sufficient conditions for the sine-wave crossings of a signal to be the zero sequence of a sine-type function.

Reconstruction of band-limited signals from multichannel and periodic nonuniform samples in the linear canonical transform domain

Linear canonical transforms (LCTs) are a family of integral transforms with wide application in optical, acoustical, electromagnetic, and other wave propagation problems. This paper addresses the problem of signal reconstruction from multichannel and periodic nonuniform samples in the LCT domain. Firstly, the multichannel sampling theorem (MST) for band-limited signals with the LCT is proposed based on multichannel system equations, which is the generalization of the well-known sampling theorem for the LCT. We consider the problem of reconstructing the signal from its samples which are acquired using a multichannel sampling scheme. For this purpose, we propose two alternatives. The first scheme is based on the conventional Fourier series and inverse LCT operation. The second is based on the conventional Fourier series and inverse Fourier transform (FT) operation. Moreover, the classical Papoulis MST in FT domain is shown to be special case of the achieved results. Since the periodic nonuniformly sampled signal in the LCT has valuable applications, the reconstruction expression for the periodic nonuniformly sampled signal has been then obtained by using the derived MST and the specific space-shifting property of the LCT. Last, the potential applications of the MST are presented to show the advantage of the theory.

Multichannel sampling and reconstruction of bandlimited signals in the linear canonical transform domain

Multichannel sampling for bandlimited signals is fundamental in the theory of multichannel parallel analogue-to-digital (A/D) environment and multiplexing wireless communication environment. The analysis and application of multichannel sampling in the traditional Fourier domain have been extensively studied, but so far none of the research papers covering the reconstruction of multichannel sampling in the linear canonical transform (LCT) domain have been published. This study is to explore the multichannel sampling and reconstruction of bandlimited signals in the LCT domain. First, the multichannel sampling theorem for bandlimited signals with LCT is proposed, which is the generalisation of classical generalised Papoulis sampling expansion. Considering the signal reconstruction purpose, the authors present two schemes for the multichannel sampling with LCT. The first scheme is based on the conventional Fourier series and inverse LCT, whereas the other uses the generalised convolution systems. Second, by designing different LCT filters, the authors obtain the reconstruction method for the uniform sampling theorem, sampling from the signal and its derivative by using the derived multichannel sampling theorem and properties of the LCT. Last, the simulations are carried out to verify the correctness of the results. Moreover, the potential applications of the multichannel sampling are also presented.

Reconstruction of Bandlimited Signals in Linear Canonical Transform Domain From Finite Nonuniformly Spaced Samples

We investigate the reconstruction of bandlimited signals in the linear canonical transform (LCT) domain from a finite set of nonuniformly spaced samples. Based on the reproducing property of the reproducing kernel belonging to the class of bandlimited signals in LCT domain, we derive an interpolating formula with minimum mean-squared error that interpolates the finite set of nonuniformly spaced samples, and show that it is identical to the minimum energy bandlimited in LCT domain interpolator. Singular value decomposition is also used to set up a reconstruction algorithm which guarantees that the reconstruction result also achieves the minimum energy reconstruction.

Reconstruction of nonuniformly sampled time-limited signals using prolate spheroidal wave functions

Shannon's sampling theory is based on the reconstruction of bandlimited signals which requires infinite number of uniform time samples. Indeed, one can only have finite number of samples for numerical implementation. In this paper, as a dual of the bandlimited reconstruction, a solution for time-limited signal reconstruction from nonuniform samples is proposed. The system model we present is based on the idea that time-limited signals which are also nearly bandlimited can be well approximated by a low-dimensional subspace. This can be done by using prolate spheroidal wave functions as the basis. The order of the projection on this basis is obtained by means of the time–frequency dimension of the signal, especially in the case of non-stationary signals. The reconstruction requires the estimation of the nonuniform sampling times by means of an annihilating filter. We obtain the reconstruction parameters by solving a linear system of equations and show that our finite-dimensional model is not ill-conditioned. The practical aspects of our method including the dimensionality reduction are demonstrated by processing synthetic as well as real signals.

Limits of signal processing performance under thresholding

In this paper the reconstruction of bandlimited signals by sampling series is analyzed for the case where the samples of the signal are disturbed by the non-linear threshold operator. This operator sets all samples whose absolute value is smaller than some threshold to zero. It is shown that the reconstruction error can grow arbitrarily large, regardless of how small the threshold is chosen, when no oversampling is used. However, with oversampling, it is possible to upper bound the reconstruction error. Additionally, we consider the approximation of outputs of stable linear time-invariant systems, by sampling series that use only the samples that are larger than the threshold, and show that there exist stable linear time-invariant systems for which the approximation error is unbounded, even if oversampling is applied. Finally, we consider the case of non-equidistant sampling. It is possible to draw conclusions about the approximation behavior of the sampling series with threshold operator by analyzing the convergence behavior of the sampling series with permuted sampling points and without threshold operator. Since the latter does not have a good approximation behavior in general, we conjecture that the sampling series with non-equidistant samples and threshold operator has no good approximation behavior either.

Quasi-random sampling for signal recovery

The problem of reconstruction of band-limited signals from nonuniformly sampled and and noisy observations is studied. It is proposed to sample a signal at quasi-random points that form a deterministic sequence, with properties resembling a random variable, being uniformly distributed. Such quasi-random points can be easily and efficiently generated. The reconstruction method based on the modified Whittaker–Shannon cardinal expansion is proposed and its asymptotical properties are examined. In particular, the sufficient conditions for the convergence of the mean integrated squared error are found. The key difference between the proposed reconstruction algorithm and the classical Whittaker–Shannon scheme is in treating the sampling rate and the reconstruction rate differently. This distinction is necessary to ensure consistency of the reconstruction algorithm in the presence of noise.

Exact reconstruction with directional wavelets on the sphere

A new formalism is derived for the analysis and exact reconstruction of band-limited signals on the sphere with directional wavelets. It represents an evolution of a previously developed wavelet formalism developed by Antoine & Vandergheynst and Wiaux et al. The translations of the wavelets at any point on the sphere and their proper rotations are still defined through the continuous three-dimensional rotations. The dilations of the wavelets are directly defined in harmonic space through a new kernel dilation, which is a modification of an existing harmonic dilation. A family of factorized steerable functions with compact harmonic support which are suitable for this kernel dilation are first identified. A scale-discretized wavelet formalism is then derived, relying on this dilation. The discrete nature of the analysis scales allows the exact reconstruction of band-limited signals. A corresponding exact multi-resolution algorithm is finally described and an implementation is tested. The formalism is of interest notably for the denoising or the deconvolution of signals on the sphere with a sparse expansion in wavelets. In astrophysics, it finds a particular application for the identification of localized directional features in the cosmic microwave background data, such as the imprint of topological defects, in particular, cosmic strings, and for their reconstruction after separation from the other signal components.