Sinc interpolation of nonuniform samples

It is well known that a bandlimited signal can be uniquely recovered from nonuniformly spaced samples under certain conditions on the nonuniform grid and provided that the average sampling rate meets or exceeds the Nyquist rate. However, reconstruction of the continuous-time signal from nonuniform samples is typically more difficult to implement than from uniform samples. Motivated by the fact that sinc interpolation results in perfect reconstruction for uniform sampling, we develop a class of approximate reconstruction methods from nonuniform samples based on the use of time-invariant lowpass filtering, i.e., sinc interpolation. The methods discussed consist of four cases incorporated in a single framework. The case of sub-Nyquist sampling is also discussed and nonuniform sampling is shown as a possible approach to mitigating the impact of aliasing

High-resolution distributed sampling of bandlimited fields with low-precision sensors

The problem of sampling a discrete-time sequence of spatially bandlimited fields with a bounded dynamic range, in a distributed, communication-constrained, processing environment is addressed. A central unit, having access to the data gathered by a dense network of fixed-precision sensors, operating under stringent inter-node communication constraints, is required to reconstruct the field snapshots to maximum accuracy. Both deterministic and stochastic field models are considered. For stochastic fields, results are established in the almost-sure sense. The feasibility of having a flexible tradeoff between the oversampling rate (sensor density) and the analog-to-digital converter (ADC) precision, while achieving an exponential accuracy in the number of bits per Nyquist-interval per snapshot is demonstrated. This exposes an underlying ``conservation of bits'' principle: the bit-budget per Nyquist-interval per snapshot (the rate) can be distributed along the amplitude axis (sensor-precision) and space (sensor density) in an almost arbitrary discrete-valued manner, while retaining the same (exponential) distortion-rate characteristics. Achievable information scaling laws for field reconstruction over a bounded region are also derived: With N one-bit sensors per Nyquist-interval, $\Theta(\log N)$ Nyquist-intervals, and total network bitrate $R_{net} = \Theta((\log N)^2)$ (per-sensor bitrate $\Theta((\log N)/N)$), the maximum pointwise distortion goes to zero as $D = O((\log N)^2/N)$ or $D = O(R_{net} 2^{-\beta \sqrt{R_{net}}})$. This is shown to be possible with only nearest-neighbor communication, distributed coding, and appropriate interpolation algorithms. For a fixed, nonzero target distortion, the number of fixed-precision sensors and the network rate needed is always finite.Comment: 17 pages, 6 figures; paper withdrawn from IEEE Transactions on Signal Processing and re-submitted to the IEEE Transactions on Information Theor

Sampling and quantization for optimal reconstruction

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 161-167).This thesis develops several approaches for signal sampling and reconstruction given different assumptions about the signal, the type of errors that occur, and the information available about the signal. The thesis first considers the effects of quantization in the environment of interleaved, oversampled multi-channel measurements with the potential of different quantization step size in each channel and varied timing offsets between channels. Considering sampling together with quantization in the digital representation of the continuous-time signal is shown to be advantageous. With uniform quantization and equal quantizer step size in each channel, the effective overall signal-to-noise ratio in the reconstructed output is shown to be maximized when the timing offsets between channels are identical, resulting in uniform sampling when the channels are interleaved. However, with different levels of accuracy in each channel, the choice of identical timing offsets between channels is in general not optimal, with better results often achievable with varied timing offsets corresponding to recurrent nonuniform sampling when the channels are interleaved. Similarly, it is shown that with varied timing offsets, equal quantization step size in each channel is in general not optimal, and a higher signal-to-quantization-noise ratio is often achievable with different levels of accuracy in the quantizers in different channels. Another aspect of this thesis considers nonuniform sampling in which the sampling grid is modeled as a perturbation of a uniform grid. Perfect reconstruction from these nonuniform samples is in general computationally difficult; as an alternative, this work presents a class of approximate reconstruction methods based on the use of time-invariant lowpass filtering, i.e., sinc interpolation. When the average sampling rate is less than the Nyquist rate, i.e., in sub-Nyquist sampling, the artifacts produced when these reconstruction methods are applied to the nonuniform samples can be preferable in certain applications to the aliasing artifacts, which occur in uniform sampling. The thesis also explores various approaches to avoiding aliasing in sampling. These approaches exploit additional information about the signal apart from its bandwidth and suggest using alternative pre-processing instead of the traditional linear time-invariant anti-aliasing filtering prior to sampling.by Shay Maymon.Ph.D

Fast and direct inversion methods for the multivariate nonequispaced fast Fourier transform

The well-known discrete Fourier transform (DFT) can easily be generalized to arbitrary nodes in the spatial domain. The fast procedure for this generalization is referred to as nonequispaced fast Fourier transform (NFFT). Various applications such as MRI, solution of PDEs, etc., are interested in the inverse problem, i.e., computing Fourier coefficients from given nonequispaced data. In this paper we survey different kinds of approaches to tackle this problem. In contrast to iterative procedures, where multiple iteration steps are needed for computing a solution, we focus especially on so-called direct inversion methods. We review density compensation techniques and introduce a new scheme that leads to an exact reconstruction for trigonometric polynomials. In addition, we consider a matrix optimization approach using Frobenius norm minimization to obtain an inverse NFFT

Novel Digital Alias-Free Signal Processing Approaches to FIR Filtering Estimation

This thesis aims at developing a new methodology of filtering continuous-time bandlimited signals and piecewise-continuous signals from their discrete-time samples. Unlike the existing state-of-the-art filters, my filters are not adversely affected by aliasing, allowing the designers to flexibly select the sampling rates of the processed signal to reach the required accuracy of signal filtering rather than meeting stiff and often demanding constraints imposed by the classical theory of digital signal processing (DSP). The impact of this thesis is cost reduction of alias-free sampling, filtering and other digital processing blocks, particularly when the processed signals have sparse and unknown spectral support. Novel approaches are proposed which can mitigate the negative effects of aliasing, thanks to the use of nonuniform random/pseudorandom sampling and processing algorithms. As such, the proposed approaches belong to the family of digital alias-free signal processing (DASP). Namely, three main approaches are considered: total random (ToRa), stratified (StSa) and antithetical stratified (AnSt) random sampling techniques. First, I introduce a finite impulse response (FIR) filter estimator for each of the three considered techniques. In addition, a generalised estimator that encompasses the three filter estimators is also proposed. Then, statistical properties of all estimators are investigated to assess their quality. Properties such as expected value, bias, variance, convergence rate, and consistency are all inspected and unveiled. Moreover, closed-form mathematical expression is devised for the variance of each single estimator. Furthermore, quality assessment of the proposed estimators is examined in two main cases related to the smoothness status of the filter convolution’s integrand function, \u1d454(\u1d461,\u1d70f)∶=\u1d465(\u1d70f)ℎ(\u1d461−\u1d70f), and its first two derivatives. The first main case is continuous and differentiable functions \u1d454(\u1d461,\u1d70f), \u1d454′(\u1d461,\u1d70f), and \u1d454′′(\u1d461,\u1d70f). Whereas in the second main case, I cover all possible instances where some/all of such functions are piecewise-continuous and involving a finite number of bounded discontinuities. Primarily obtained results prove that all considered filter estimators are unbiassed and consistent. Hence, variances of the estimators converge to zero after certain number of sample points. However, the convergence rate depends on the selected estimator and which case of smoothness is being considered. In the first case (i.e. continuous \u1d454(\u1d461,\u1d70f) and its derivatives), ToRa, StSa and AnSt filter estimators converge uniformly at rates of \u1d441−1, \u1d441−3, and \u1d441−5 respectively, where 2\u1d441 is the total number of sample points. More interestingly, in the second main case, the convergence rates of StSa and AnSt estimators are maintained even if there are some discontinuities in the first-order derivative (FOD) with respect to \u1d70f of \u1d454(\u1d461,\u1d70f) (for StSa estimator) or in the second-order derivative (SOD) with respect to \u1d70f of \u1d454(\u1d461,\u1d70f) (for AnSt). Whereas these rates drop to \u1d441−2 and \u1d441−4 (for StSa and AnSt, respectively) if the zero-order derivative (ZOD) (for StSa) and FOD (for AnSt) are piecewise-continuous. Finally, if the ZOD of \u1d454(\u1d461,\u1d70f) is piecewise-continuous, then the uniform convergence rate of the AnSt estimator further drops to \u1d441−2. For practical reasons, I also introduce the utilisation of the three estimators in a special situation where the input signal is pseudorandomly sampled from otherwise uniform and dense grid. An FIR filter model with an oversampled finite-duration impulse response, timely aligned with the grid, is proposed and meant to be stored in a lookup table of the implemented filter’s memory to save processing time. Then, a synchronised convolution sum operation is conducted to estimate the filter output. Finally, a new unequally spaced Lagrange interpolation-based rule is proposed. The so-called composite 3-nonuniform-sample (C3NS) rule is employed to estimate area under the curve (AUC) of an integrand function rather than the simple Rectangular rule. I then carry out comparisons for the convergence rates of different estimators based on the two interpolation rules. The proposed C3NS estimator outperforms other Rectangular rule estimators on the expense of higher computational complexity. Of course, this extra cost could only be justifiable for some specific applications where more accurate estimation is required