Non-uniform Sampling Scheme Research Articles

Non-uniform sampling strategies for digital control

Most conventional control algorithms cause numerical problems where data is collected at sampling rates that are substantially higher than the dynamics of the equivalent continuous-time operation that is being implemented. This is of relevant interest in applications of digital control, in which high sample rates are routinely dictated by the system stability requirements rather than the signal processing needs. Digital control systems exhibit bandwidth limitations enforced by their closed-loop frequency requirements that demand very high sample rates. Considerable recent progress in reducing sample frequency requirements has been made through the use of non-uniform sampling schemes, so called alias-free signal processing. The approach prompts the simplification of complex systems and consequently enhances the numerical conditioning of the implementation algorithms that otherwise would require very high uniform sample rates. However, the control communities have not yet investigated the use of intentional non-uniform sampling. The purpose of this article is to address some algorithmic issues when using such regimes for digital control.

Bivariate Empirical Mode Decomposition for Unbalanced Real-World Signals

The bivariate empirical mode decomposition (BEMD) algorithm employs uniform sampling on a circle to perform projections in multiple directions, in order to calculate the local mean of a bivariate signal. However, this approach is adequate only for equal powers in both the data channels within a bivariate signal, and results in suboptimal performance for data channels exhibiting power imbalance, a typical case in practice. To that end, we exploit second-order bivariate statistical properties to introduce a nonuniform sampling scheme for data adaptive selection of the projection directions. In this way, the resulting nonuniformly sampled BEMD (NS-BEMD) algorithm provides a more accurate time-frequency representation of bivariate data than standard BEMD, for the same number of projections. The advantages of the proposed approach are demonstrated in case studies on BEMD for correlated data channels, selection of optimal noise power in noise-assisted BEMD, and for speed estimation using Doppler radar.

Comparison of various sampling schemes and accumulation profiles in covariance spectroscopy with exponentially decaying 2D signals

Two-dimensional covariance (COV2D) spectroscopy offers an alternative approach to 2D Fourier transformation (FT2D) to obtain homo-nuclear correlation NMR spectra. Considerable saving in experimental time, without loss of resolution, can be achieved by incorporating COV2D with: (i) uniform sampling with a t1 cut-off (CUO) or non-uniform sampling (NUS), and (ii) constant or Gaussian accumulation profiles. We find that covariance treatment, combined with the CUO sampling and Gaussian accumulation profile provides better gain in experimental time, with respect to that required with FT2D NMR. This is in contrast with the maximum entropy (MaxEnt) reconstruction for 2D spectra, which works better with the NUS scheme rather than with the CUO scheme. We further discuss the experimental conditions that define this optimum acquisition, and explain how these parameters can easily be optimized 'on the fly' according to the desired resolution, which is quite sample-dependent.

An Interactive Simulation Tool for the study of Multirate Systems

Dealing with practical applications of digital control, the assumption of regular and uniform sampling pattern is questionable. Sometimes a non-uniform sampling scheme leads to beneficial results as better performance or energy saving among others. However in these cases the theoretical analysis of the controlled system performances is much more computationally involved. In this contribution an interactive simulation tool to deal with multirate systems (MR) is presented. It allows an easy understanding of the performance degrading or improvement when changing the sampling pattern parameters. The tool was developed using Sysquake, a Matlab-like language with fast execution and powerful graphic facilities. It can be delivered as an executable.

A Programmable Look-Up Table-Based Interpolator with Nonuniform Sampling Scheme

Interpolation is a useful technique for storage of complex functions on limited memory space: some few sampling values are stored on a memory bank, and the function values in between are calculated by interpolation. This paper presents a programmable Look-Up Table-based interpolator, which uses a reconfigurable nonuniform sampling scheme: the sampled points are not uniformly spaced. Their distribution can also be reconfigured to minimize the approximation error on specific portions of the interpolated function’s domain. Switching from one set of configuration parameters to another set, selected on the fly from a variety of precomputed parameters, and using different sampling schemes allow for the interpolation of a plethora of functions, achieving memory saving and minimum approximation error. As a study case, the proposed interpolator was used as the core of a programmable noise generator—output signals drawn from different Probability Density Functions were produced for testing FPGA implementations of chaotic encryption algorithms. As a result of the proposed method, the interpolation of a specific transformation function on a Gaussian noise generator reduced the memory usage to 2.71% when compared to the traditional uniform sampling scheme method, while keeping the approximation error below a threshold equal to 0.000030518.

A New Randomized Algorithm to Approximate the Star Discrepancy Based on Threshold Accepting

We present a new algorithm for estimating the star discrepancy of arbitrary point sets. Similar to the algorithm for discrepancy approximation of Winker and Fang [SIAM J. Numer. Anal. 34 (1997), 2028--2042] it is based on the optimization algorithm threshold accepting. Our improvements include, amongst others, a non-uniform sampling strategy which is more suited for higher-dimensional inputs, and rounding steps which transform axis-parallel boxes, on which the discrepancy is to be tested, into \emph{critical test boxes}. These critical test boxes provably yield higher discrepancy values, and contain the box that exhibits the maximum value of the local discrepancy. We provide comprehensive experiments to test the new algorithm. Our randomized algorithm computes the exact discrepancy frequently in all cases where this can be checked (i.e., where the exact discrepancy of the point set can be computed in feasible time). Most importantly, in higher dimension the new method behaves clearly better than all previously known methods.

Suppression of sampling artefacts in high-resolution four-dimensional NMR spectra using signal separation algorithm

The development of non-uniform sampling (NUS) strategies permits to obtain high-dimensional spectra with increased resolution in significantly reduced experimental time. We extended a previously proposed signal separation algorithm (SSA) to process sparse four-dimensional NMR data. It is employed for two experiments carried out for a partially unstructured 114-residue construct of chicken Engrailed 2 protein, namely 4D HCCH-TOCSY and 4D C, N-edited NOESY. The SSA allowed us to obtain high-quality spectra using only as little as 0.16% of the available samples, with low sampling artefacts approaching the thermal noise level in most spectral regions. It is demonstrated that NUS 4D HCCH-TOCSY is dominated by sampling noise and requires efficient artefact suppression. On the other hand, 4D C, N-edited NOESY is a particularly attractive experiment for NUS, as the absence of diagonal peaks renders the problem of artefacts less critical. We also present a transverse-relaxation optimized sequence for HMQC that is especially designed for longer evolution periods in the indirectly detected proton dimension in high-dimensional pulse sequences. In conjunction with novel sampling strategies and efficient processing methods, this improvement enabled us to obtain unique structural information about aliphatic-amide contacts.

Non-uniformly sampled Maximum Quantum spectroscopy

Maximum-Quantum (MaxQ) NMR is an approach that exploits the simple lineshape (a singlet) of the highest possible coherence quantum order for a given spin system to help resolving the interpretation of the spectrum of complex mixtures. In this setup, resolution in the indirect, multiple-quantum, dimension is crucial, and it may be linked to a long duration of the signal acquired along this axis. We explored if this boundary on the length of the indirect dimension could not necessarily translate into extended experimental times by applying Non-Uniform Sampling (NUS) schemes in conjunction with Recursive Multi-Dimensional Decomposition (R-MDD) data processing. The actual value of the MaxQ order depends on the size of the spin system, so that for a mixture several MQ correlation spectra must be recorded to detect all possible molecular fragments. As the sparseness of the MQ datasets vary dramatically in going from higher (sparser) to lower (denser) coherence orders, the optimal compressing conditions and the fidelity of NUS/R-MDD scheme may vary along the series of MQ spectra. The NUS-MaxQ approach is demonstrated on the aromatic region of the 1H spectrum of a mixture of 10 simple aromatic molecules.

A Method for the Recovery of Gaps in General Analytic Signals

We propose a generalized Papoulis-Gerchberg algorithm for the recovery of gaps in general analytic functions. The continuous-time algorithm is based on signal expansions in terms of the Chebyshev polynomials of the first kind, and the discrete-time implementation is based on a suitable nonuniform sampling scheme and the discrete cosine transform.

Log-Time Sampling of Signals: Zeta Transform

We introduce a general framework for the log-time sampling of continuous-time signals. We define the zeta transform based on the log-time sampling scheme, where the signal x(t)is sampled at time instants tn=Tlogn,n=1,2,....The zeta transform of the log-time sampled signals can be described by a linear combination of Riemann zeta function, which firmly joins the log-time sampling process to the number theory. The instantaneous sampling frequency of the log-sampled signal equals fn=n/T,n=1,2,..., i.e. it increases linearly with the sampling number. We describe the properties of the log-sampled signals and discuss several applications in nonuniform sampling schemes.

Non-Uniform Filter Interpolation in the Frequency Domain

We propose a filtering technique which takes advantage of a specific non-uniform sampling scheme which allows the capture of a very low number of samples for both the signal and the filter transfer function. This approach leads to a summation formula which plays the same role as the discrete convolution for usual FIR filters. Here the formula is much more complicated but it can easily be implemented and the evaluation of these more elaborate expressions is compensated by the very low number of samples to process.

A curvelet-based patient-specific prior for accurate multi-modal brain image rigid registration

We present a new non-uniform sampling method for the accurate estimation of mutual information in multi-modal brain image rigid registration. Most existing density estimators used for mutual information computation incorrectly assume that the intensity of each voxel is independent from its neighborhood. Our method uses the 3D Fast Discrete Curvelet Transform to reduce the sampled voxels' interdependency by sampling voxels that are less dependent on their neighborhood, and thus provide a more accurate estimation of the mutual information and a more accurate registration. The main advantages of our method over other non-uniform sampling schemes are that: (1) it provides more accurate estimation of the image statistics with fewer samples; (2) it is less sensitive to the variability of anatomical structures shapes, orientations, and sizes, and; (3) it yields more accurate registration results. Extensive evaluation on 1000 synthetic registrations between T1 and T2-weighted clinical MRI images and 20 real clinical registrations of brain CT images to Proton Density (PD) and T1 and T2-weighted MRI images from the public RIRE database show the effectiveness of our method. Our method has the lowest mean registration errors recorded to date for CT-MR image registration in the RIRE website for methods tested on more than five datasets. These results indicate that our sampling scheme can be used to achieve more accurate multi-modal registration required for image guided therapy and surgery.

IIR digital filtering of non-uniformly sampled signals via state representation

This article describes a new kind of processing chain based on a non-uniform sampling scheme provided by a level-crossing ADC. The chain implements IIR filters which process directly the non-uniform samples without resampling in a regular scheme. The non-uniformity in the sample times leads to choose a state representation for the filters. The stability is studied and the performances of various numerical schemes used to implement the filters in this representation are compared.

Characterization and optimization of Nonuniform Sampling for Multidimensional NMR Experiments

Nuclear magnetic resonance (NMR) is a powerful technique for revealing biomolecular structure and dynamics. Technical advances over the past decade, including cryogenic probes, higher magnetic fields, and novel pulse sequences, allow for the analysis of increasingly larger and more complex systems. Multidimensional, multinuclear experiments are invariably required to resolve individual resonances. However, the increased sampling rate imposed by the Nyquist theorem at higher magnetic fields (due to greater spectral dispersion) means that experiment times become prohibitively long when conventional uniform sampling is employed. Non-Fourier methods of spectrum analysis open the possibility of nonuniform sampling, permitting data to be collected at short evolution times to ensure sensitivity and at long evolution times to allow for high resolution. The combination of nonuniform sampling methods with non-Fourier methods of spectrum analysis enable the computation of high resolution multidimensional spectra from much shorter data records than can be employed using conventional Fourier and uniform sampling methods. The design of optimal sampling strategies, for minimizing sampling time while maintaining resolution and sensitivity, or optimizing sensitivity and/or resolution for a given total experiment time remains an open challenge. In the present work the maximum entropy (MaxEnt) method of spectrum reconstruction is used to characterize the performance of nonuniform sampling strategies with respect to sensitivity and resolution. A peak identification algorithm is developed along with a metric for spectra comparison. These tools are used to investigate whether prior knowledge of peak frequencies can aid the design of optimal sampling strategies. Such strategies could be useful in a number of contexts in biomolecular NMR, including relaxation studies and multidimensional experiments conducted subsequent to “scout” experiments, such as two-dimensional HSQC.

Economic statistical design of non-uniform sampling scheme X bar control charts under non-normality and Gamma shock using genetic algorithm

This paper presented an approach which simultaneously considered the properties of cost and quality based on the Burr distribution and the non-unifampling scheme. The objective was to determine three parameters, namely, sample size, sampling interval between successive samples, and control limits, when an X bar chart monitors a manufacturing process with Gamma ( λ, 2) failure characteristic and non-normal data. The design parameters of the X bar control charts can be obtained through the genetic algorithm (GA) method. An example was also adopted to indicate the solution procedure and sensitivity analyses. The results show that an increase of skewness coefficient ( α 3) results in a slight decrease for sample size ( n) while an increase of kurtosis coefficient ( α 4) leads to a wider control limit width.

Near optimum sampling design and an efficient algorithm for single tone frequency estimation

Work on single tone frequency estimation has focused on uniformly sampled data. However, it has been shown that, for a given number of samples, more information on the frequency of a signal can be gained by non-uniform sampling schemes [M. Wieler, S. Trittler, F.A. Hamprecht, Optimal design for single tone frequency estimation, Digital Signal Process., in press]. Unfortunately, an optimum sampling pattern (that, for example, minimizes the Cramer-Rao bound) does not automatically have a fast and simple algorithm for frequency estimation associated with it. For application in an interferometric measurement system, an algorithm is needed that gathers as much information as possible from a low number of samples, while at the same time keeping the computational effort sufficiently low to process millions of time series in a few seconds. This paper proposes a simple approximation to the optimum sampling pattern by using uniformly sampled blocks of data and further proposes to estimate phase and frequency in each of these blocks and to exploit these intermediate results in the final estimation. An approach to do so is investigated in detail. Results are compared to the Cramer-Rao bound (CRB), and it is shown that this algorithm almost reaches this limit on the variance of unbiased estimators, at a computational complexity lower than that of a typical FFT-based approach. For M=32 samples and a signal-to-noise ratio of 10, the standard deviation of the frequency estimate is lower by more than 50% compared to uniform sampling. In addition, the algorithm can easily be applied to poorly characterized systems, e.g. systems for which the noise is not known exactly. Finally, we demonstrate that the proposed strategy yields results that are within 3% of the theoretically achievable accuracy for the theoretically optimum sampling pattern.

On the Relationship Between Uniform and Recurrent Nonuniform Discrete-Time Sampling Schemes

Recurrent nonuniform discrete-time signal samples can be regarded as a combination of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> mutual delayed sequences of uniform discrete-time signal samples taken at one <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> th of the Nyquist sampling rate. This paper introduces a new alternative discrete-time analysis model of the recurrent nonuniform sampling scenario. This model can be described by the analysis part of a uniform discrete Fourier transform (DFT) modulated filterbank from which the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> uniformly distributed and down sampled frequency bands are mixed in a very specific way. This description gives a clear relationship between uniform and recurrent nonuniform discrete-time sampling schemes. A side benefit of this model is an efficient structure with which one can reconstruct uniform discrete-time Nyquist signal samples from recurrent nonuniform samples with known mutual delays between the nonuniform distributed samples. This reconstruction structure can be viewed as a natural extension of the synthesis part of an uniform DFT modulated filterbank.

Comparison of parallel MRI reconstruction methods for accelerated 3D fast spin‐echo imaging

Parallel MRI (pMRI) achieves imaging acceleration by partially substituting gradient-encoding steps with spatial information contained in the component coils of the acquisition array. Variable-density subsampling in pMRI was previously shown to yield improved two-dimensional (2D) imaging in comparison to uniform subsampling, but has yet to be used routinely in clinical practice. In an effort to reduce acquisition time for 3D fast spin-echo (3D-FSE) sequences, this work explores a specific nonuniform sampling scheme for 3D imaging, subsampling along two phase-encoding (PE) directions on a rectilinear grid. We use two reconstruction methods-2D-GRAPPA-Operator and 2D-SPACE RIP-and present a comparison between them. We show that high-quality images can be reconstructed using both techniques. To evaluate the proposed sampling method and reconstruction schemes, results via simulation, phantom study, and in vivo 3D human data are shown. We find that fewer artifacts can be seen in the 2D-SPACE RIP reconstructions than in 2D-GRAPPA-Operator reconstructions, with comparable reconstruction times.

Automatic maximum entropy spectral reconstruction in NMR

Developments in superconducting magnets, cryogenic probes, isotope labeling strategies, and sophisticated pulse sequences together have enabled the application, in principle, of high-resolution NMR spectroscopy to biomolecular systems approaching 1 megadalton. In practice, however, conventional approaches to NMR that utilize the fast Fourier transform, which require data collected at uniform time intervals, result in prohibitively lengthy data collection times in order to achieve the full resolution afforded by high field magnets. A variety of approaches that involve nonuniform sampling have been proposed, each utilizing a non-Fourier method of spectrum analysis. A very general non-Fourier method that is capable of utilizing data collected using any of the proposed nonuniform sampling strategies is maximum entropy reconstruction. A limiting factor in the adoption of maximum entropy reconstruction in NMR has been the need to specify non-intuitive parameters. Here we describe a fully automated system for maximum entropy reconstruction that requires no user-specified parameters. A web-accessible script generator provides the user interface to the system.

Compression at the Source for Digital Camcorders

Typical sensors (CCD or CMOS) used in home digital camcorders have the potential of generating high definition (HD) video sequences. However, the data read out rate is a bottleneck which, invariably, forces significant quality deterioration in recorded video clips. This paper describes a novel technology for achieving a better utilization of sensor capability, resulting in HD quality video clips with esentially the same hardware. The technology is based on the use of a particular type of nonuniform sampling strategy. This strategy combines infrequent high spatial resolution frames with more frequent low resolution frames. This combination allows the data rate constraint to be achieved while retaining an HD quality output. Post processing via filter banks is used to combine the high and low spatial resolution frames to produce the HD quality output. The paper provides full details of the reconstruction algorithm as well as proofs of all key supporting theories.