Band-limited Signals Research Articles

Quantization-aware sampling set selection for bandlimited graph signals

We consider a scenario in which nodes of a graph are sampled for bandlimited graph signals which are uniformly quantized with optimal rate and original signals are reconstructed from the quantized signal values residing on the nodes in the sampling set. We seek to construct the best sampling set in a greedy manner that minimizes the average reconstruction error. We manipulate the reconstruction error by using the QR factorization and derive an analytic result stating that the next minimizing node can be iteratively selected by finding the one that minimizes the geometric mean of the row vectors of the inverse upper triangular matrix mathbf {R}^{-1} in the QR factorization. We also compare the complexity of the proposed algorithm with different sampling methods and evaluate the performance of the proposed algorithm by experiments for various graphs, showing the superiority to the existing sampling methods when quantization is involved.

Dithered A/D Conversion of Bandlimited Signals Under Frequency Band Uncertainty

We propose a scheme for analog-to-digital (A/D) conversion of a signal that is bandlimited most of the time but contains harmonics of a high frequency occasionally. To enable the capture of the high frequency components, such signals are usually oversampled. This framework is of practical interest in the case of electrical signals wherein one encounters high frequency faults or anomalies from time to time, all of which need to be retained for diagnostic purposes. Existing schemes that rely on oversampling for dithered A/D conversion of bandlimited signals are designed for the worst case and do not take into account the fact that the high frequency anomalies occur rarely. We propose an adaptive A/D conversion scheme that identifies the presence or absence of a high frequency component in a given block of samples by formulating it as a hypothesis testing problem. The proposed scheme performs better in terms of compression compared to a scheme designed for the worst case while retaining the same guarantees on the reconstruction error.

Sampling and Reconstruction of Sparse Signals in Shift-Invariant Spaces: Generalized Shannon’s Theorem Meets Compressive Sensing

This paper introduces a novel framework and corresponding methods for sampling and reconstruction of sparse signals in shift-invariant (SI) spaces. We reinterpret the random demodulator, a system that acquires sparse bandlimited signals, as a system for the acquisition of linear combinations of the samples in the SI setting with the box function as the sampling kernel. The sparsity assumption is exploited by the compressive sensing (CS) paradigm for a recovery of the SI samples from a reduced set of measurements. The SI samples are subsequently filtered by a discrete-time correction filter to reconstruct expansion coefficients of the observed signal. Furthermore, we offer a generalization of the proposed framework to other compactly supported sampling kernels that span a wider class of SI spaces. The generalized method embeds the correction filter in the CS optimization problem which directly reconstructs expansion coefficients of the signal. Both approaches recast an inherently continuous-domain inverse problem in a set of finite-dimensional CS problems in an exact way. Finally, we conduct numerical experiments on signals in polynomial B-spline spaces whose expansion coefficients are assumed to be sparse in a certain transform domain. The coefficients can be regarded as parametric models of an underlying continuous-time signal, obtained from a reduced set of measurements. Such continuous signal representations are particularly suitable for signal processing without converting them into samples.

Unlimited Sampling From Theory to Practice: Fourier-Prony Recovery and Prototype ADC

Following the Unlimited Sampling strategy to alleviate the omnipresent dynamic range barrier, we study the problem of recovering a bandlimited signal from point-wise modulo samples, aiming to connect theoretical guarantees with hardware implementation considerations. Our starting point is a class of non-idealities that we observe in prototyping an unlimited sampling based analog-to-digital converter. To address these non-idealities, we provide a new Fourier domain recovery algorithm. Our approach is validated both in theory and via extensive experiments on our prototype analog-to-digital converter, providing the first demonstration of unlimited sampling for data arising from real hardware, both for the current and previous approaches. Advantages of our algorithm include that it is agnostic to the modulo threshold and it can handle arbitrary folding times. We expect that the end-to-end realization studied in this paper will pave the path for exploring the unlimited sampling methodology in a number of real world applications.

Reconstructing Signals from Aperture-Filtered Samples

Sampling plays a crucial role in remote sensing and signal analysis. In classic sampling theory, a signal is sampled at a uniform rate and at a minimum of twice the signal bandwidth. In practice, only finitely many samples of the signal are available and these can often only be obtained at irregular positions due to platform movement, scanning, pulsed operations, etc. In addition, often only local averages of the signal at these irregularly spaced locations can be measured, often with different apertures at different locations. We will use the one-to-one correspondence between the analog and discrete band-limited signal spaces established by the regular sampling theorem, and give a direct treatment of the discrete problem of irregular samplings. This problem is finite-dimensional and linear, easily accessible, and transparent. Its solution can be easily implemented and analyzed to solve engineering applications. We will see that—different from other approaches—only the number of samples, not their location or density, determine whether the signal can be recovered or not.

Performance Analysis and Comparison of Clipped and Filtered OFDM Systems With Iterative Distortion Recovery Techniques

A well-known drawback of orthogonal frequency-division multiplexing (OFDM) is its signal with high peak-to-average power ratio (PAPR). Among a number of PAPR reduction techniques, clipping and filtering (CAF) is the simplest approach, which effectively reduces the PAPR of band-limited OFDM signals at the cost of increasing in-band distortion. In order to mitigate the performance degradation caused by the in-band distortion, several iterative distortion recovery techniques have been proposed in the literature, and they are largely classified into time-domain (TD) and frequency-domain (FD) compensation approaches: The former and latter are represented by the decision-aided reconstruction (DAR) and clipping noise cancellation (CNC), respectively. To date, however, their theoretical performance limits have not been studied. In this work, we revisit the performance limits of CAF and derive a closed-form signal-to-distortion power ratio (SDR) expression. Furthermore, we introduce a time-domain distortion model for characterizing the OFDM signal with CAF, based on which we make performance comparison between the two compensation approaches. Theoretical analysis and simulations in terms of their achievable symbol error rate (SER) reveal that, unlike the FD counterpart, the TD compensation may suffer from unrecoverable distortion when filtering after clipping is applied at the transmitter.

Variational mode decomposition based random forest model for solar radiation forecasting: New emerging machine learning technology

Forecasting of solar radiation (Radn) can provide an insight vision for the amount of green and friendly energy sources. Owing to the non-linearity and non-stationarity challenges caused by meteorological variables in forecasting Radn, a variational mode decomposition method is integrated with simulated annealing and random forest (VMD-SA-RF) for resolving this problem. Firstly, the input parameters are separated into training and testing phases​ after generating a one-day ahead significant lags at (t – 1). Secondly, the variational mode decomposition is set to factorize multivariate meteorological data of train and test sets, independently, into their band-limited signals. Thirdly, the simulate annealing based feature selection system is engaged to select the best band-limited signals. Finally, using the pertinent band-limited signals, the daily Radn is forecasted via random forest (RF) model. The outcomes are benchmarked with other comparative models. The hybrid fusion VMD-SA-RF model is tested geographically in Australia, generates reliable performance to forecast Radn. The hybrid VMD-SA-RF system combining the pertinent meteorological features, as the model predictors have substantial implications for renewable and sustainable energy resource management.

Experimental evaluation of a sound diffuse field in a ordinary room

This paper presents a methodology based on analysis of spatial correlation functions for assessing diffusivity of a sound field in reverberant and ordinary reflective rooms. A set of sound field measurements using a microphone array is performed in a room excited by a broadband signal and for many positions of loudspeakers. Each recorded signal is then broken down with a filter bank and the spatial correlation of each obtained band-limited signal is computed. Theoretical predictions of the spatial correlation function and the measured ones show very good agreement when spatial averaging and frequency conditions are met for producing a diffuse field. Evolution of the spatial correlation functions versus frequency and number of spatial averaging, i.e., number of sources positions, are examined in a reflective room. Results show that the proposed technique allows to determine the minimum number of required source positions and the actual lower frequency bound for considering that the field is diffuse. The latter is analogous to the Schroeder frequency for reverberant room. Interest of the presented methodology is its ability to experimentally assess in any reflective ordinary room the actual frequency range over which the field can be considered as diffuse. [Work funded by the AMIDEX Foundation.]

Research Progress of the Sampling Theorem Associated with the Fractional Fourier Transform

Sampling is a bridge between continuous-time and discrete-time signals, which is important to digital signal processing. The fractional Fourier transform (FrFT) that serves as a generalization of the FT can characterize signals in multiple fractional Fourier domains, and therefore can provide new perspectives for signal sampling and reconstruction. In this paper, we review recent developments of the sampling theorem associated with the FrFT, including signal reconstruction and fractional spectral analysis of uniform sampling, nonuniform samplings due to various factors, and sub-Nyquist sampling, where bandlimited signals in the fractional Fourier domain are mainly taken into consideration. Moreover, we provide several future research topics of the sampling theorem associated with the FrFT.

A sampling-based approach for handling delays in continuous and hybrid systems

Abstract Delays in feedback dynamics of coupled dynamical systems arise regularly, especially in embedded control where the physical plant and the controller continuously interact through digital networks. Systems featuring delays are however notoriously difficult to analyze. Consequently, formal analysis often addresses simplified, delay-free substitute models, risking negligence of the adverse impact of delay on control performance. In this ongoing work, we demonstrate that for continuous systems such as delay differential equations, a major part of the delay-induced complexity can be reduced effectively when adding natural constraints to the model of the delayed feedback channel, namely that it transports a band-limited signal and implements a non-punctual, distributed delay. The reduction is based on a sampling approach which is applicable when the above conditions on the feedback are satisfied. We further discuss the possibilities of lifting this method to mixed discrete-continuous dynamics of delayed hybrid systems and the open issues thereof.

Recursive variational mode extraction and its application in rolling bearing fault diagnosis

The variational mode extraction (VME) developed on the similar basis of variational mode decomposition (VMD) can effectively separate a specific mode by knowing an approximate center frequency from the multi-component signal. Compared with VMD, VME has made certain progress in improving the extraction accuracy and reducing the computational cost when the aim is to separate a specific mode. As the fault feature signal of rolling bearing is usually a band-limited signal which is compact around the excited resonant frequency, VME becomes a potentially effective tool for extracting fault characteristics of rolling bearing. However, how to adaptivelydetermine the center frequency and the penalty factor are two difficult problems when using VME to separate the desired component. Accordingly, this paper presents the recursive variational mode extraction (RVME), an iterative VME-based signal decomposition algorithm. At each iteration of RVME, the initial center frequency and penalty factor for the reconstruction of a specific sub-component can be adaptively determined according to the dominant frequency of the residual signal of the previous iterative decomposition, which makes RVME an adaptive signal decomposition algorithm. The presented method is applied to simulated and experimental fault signals and compared with other classical fault feature extraction approaches, such as variational mode decomposition (VMD) and spectral kurtosis (SK). The results confirm that the established method can extract the fault features as effective as VMD, but it is significantly better than VMD in terms of computational efficiency. Meanwhile, the proposed method shows a stronger capability of weak bearing fault feature extraction and compound fault feature separation compared with SK.

Discrete sampling theorem to Shannon’s sampling theorem using the hyperreal numbers R

Shannon’s sampling theorem is one of the most important results of modern signal theory. It describes the reconstruction of any band-limited signal from a finite number of its samples. On the other hand, although less well known, there is the discrete sampling theorem, proved by Cooley while he was working on the development of an algorithm to speed up the calculations of the discrete Fourier transform. Cooley showed that a sampled signal can be resampled by selecting a smaller number of samples, which reduces computational cost. Then it is possible to reconstruct the original sampled signal using a reverse process. In principle, the two theorems are not related. However, in this paper we will show that in the context of Non Standard Mathematical Analysis (NSA) and Hyperreal Numerical System R, the two theorems are equivalent. The difference between them becomes a matter of scale. With the scale changes that the hyperreal number system allows, the discrete variables and functions become continuous, and Shannon’s sampling theorem emerges from the discrete sampling theorem.

An orthogonal partition selection strategy for the sampling of graph signals with successive local aggregations

We study the sampling of graph signals with successive local aggregations, which computes measurements as the result of iterated applications of the aggregation operator observed at one or few nodes. It has been proved in the literature that using the first few observations at a single node, a perfect reconstruction can be achieved for bandlimited graph signals. Yet, it requires some restrictions to make the sampling matrix have a Vandermonde structure. Unfortunately, Vandermonde matrices are notoriously ill-conditioned resulting in severe instability even for moderately-sized graphs. Moreover, such a sampling strategy is not applicable for the sampling scheme at multiple nodes. In view of these, we re-establish a theory for the perfect reconstruction in this paper, starting with the single-node sampling scheme. To handle large-sized graphs well, we extend it to the multi-node sampling scheme. To do this, an orthogonal partition selection strategy (OPSS) is proposed to find a node set I with a minimum size such that the observations at I guarantee a perfect reconstruction for bandlimited graph signals, and two reconstruction methods are provided. When considering a noisy regime, a performance analysis of OPSS is provided, and thus yields an improved OPSS (IOPSS) which guarantees a stable reconstruction by presetting a threshold σts. Finally, several experiments are conducted to evaluate the performance of these techniques.

FROG-measurement based phase retrieval for analytic signals

While frequency-resolved optical gating (FROG) is widely used in characterizing the ultrafast pulse in optics, analytic signals are often considered in time-frequency analysis and signal processing, especially when extracting instantaneous features of events. In this paper we examine the phase retrieval (PR) problem of analytic signals in CN by their FROG measurements. After establishing the ambiguity of the FROG-PR of analytic signals, we found that the FROG-PR of analytic signals of even lengths is different from that of analytic signals of odd lengths, and it is also different from the case of B-bandlimited signals with B≤N/2. The existing approach to bandlimited signals can be applied to analytic signals of odd lengths, but it does not apply to the even length case. With the help of two relaxed FROG-PR problems and a translation technique, we develop an approach to FROG-PR for the analytic signals of even lengths, and prove that in this case the generic analytic signals can be uniquely (up to the ambiguity) determined by their (3N/2+1) FROG measurements. Phase derivative (or instantaneous frequency) is a significant feature for signal analysis. As an application of our main result, an approach is established to determine the phase derivatives of decaying signals by exploiting the ambiguity.

Only the Fastest Corticospinal Fibers Contribute to β Corticomuscular Coherence.

Human corticospinal transmission is commonly studied using brain stimulation. However, this approach is biased to activity in the fastest conducting axons. It is unclear whether conclusions obtained in this context are representative of volitional activity in mild-to-moderate contractions. An alternative to overcome this limitation may be to study the corticospinal transmission of endogenously generated brain activity. Here, we investigate in humans (N = 19; of either sex), the transmission speeds of cortical β rhythms (∼20 Hz) traveling to arm (first dorsal interosseous) and leg (tibialis anterior; TA) muscles during tonic mild contractions. For this purpose, we propose two improvements for the estimation of corticomuscular β transmission delays. First, we show that the cumulant density (cross-covariance) is more accurate than the commonly-used directed coherence to estimate transmission delays in bidirectional systems transmitting band-limited signals. Second, we show that when spiking motor unit activity is used instead of interference electromyography, corticomuscular transmission delay estimates are unaffected by the shapes of the motor unit action potentials (MUAPs). Applying these improvements, we show that descending corticomuscular β transmission is only 1-2 ms slower than expected from the fastest corticospinal pathways. In the last part of our work, we show results from simulations using estimated distributions of the conduction velocities for descending axons projecting to lower motoneurons (from macaque histologic measurements) to suggest two scenarios that can explain fast corticomuscular transmission: either only the fastest corticospinal axons selectively transmit β activity, or else the entire pool does. The implications of these two scenarios for our understanding of corticomuscular interactions are discussed.SIGNIFICANCE STATEMENT We present and validate an improved methodology to measure the delay in the transmission of cortical β activity to tonically-active muscles. The estimated corticomuscular β transmission delays obtained with this approach are remarkably similar to those expected from transmission in the fastest corticospinal axons. A simulation of β transmission along a pool of corticospinal axons using an estimated distribution of fiber diameters suggests two possible mechanisms by which fast corticomuscular transmission is achieved: either a very small fraction of the fastest descending axons transmits β activity to the muscles or, alternatively, the entire population does and natural cancellation of slow channels occurs because of the distribution of axon diameters in the corticospinal tract.

Truncation error estimates for two‐dimensional sampling series associated with the linear canonical transform

The linear canonical transform (LCT) provides a more general framework for many well‐known linear integral transforms, such as Fourier transform, fractional Fourier transform, Fresnel transform, and so forth, in digital signal processing and optics. There has been a great deal of research on sampling expansions of bandlimited signals in the LCT domain. However, results on error estimation and convergence analysis appear to be relatively rare, especially for multi‐dimensional sampling series associated with the LCT. In this paper, we present error estimation and convergence analysis for two‐dimensional (2D) sampling series in the LCT domain. Specifically, we first prove the absolute convergence and uniform convergence of 2D LCT sampling series. Then, we analyze truncation error estimates for uniformly sampling 2D bandlimited signals in the LCT domain, as well as deriving three truncation error bounds. Finally, our theoretical results are demonstrated by numerical examples.

Enhancing the temporal fine structure with the temporal limits encoder for cochlear implants: Effects on pitch ranking

Pitch perception remains a challenge for cochlear-implant (CI) users. Temporal periodicity, as the main cue for pitch, could not be finely encoded by current CI strategies that transmit envelopes only and discard the temporal fine structure (TFS). Moreover, temporal electric pitch percept saturates at a few hundred hertz, which is controversial and has been studied using various psychophysical and physiological methods. The temporal-limits-encoder (TLE) was proposed to enhance TFS representation by frequency-downshifting the high-frequency band-limited signals to a low-frequency temporal-pitch-limits range of CIs. In this work, eleven Nucleus CI subjects were recruited in a pitch ranking task using a CCiMobile processor. The original stimuli were five-harmonics complex tones. Fundamental-frequency difference limens (F0DLs) at two reference F0s 250 and 313 Hz (i.e., center and upper cross-over frequencies of the most apical electrode) were tested using three TLE versions with different modulator lower limits 100, 200, and 300 Hz respectively and a standard advanced-combinational-encoder (ACE) strategy. Results showed that TLE had a comparable performance with ACE at 250-Hz reference F0, but could outperform ACE at 313-Hz reference F0. These indicate that the TLE strategy has the potential to improve pitch ranking in CIs, which implies an enhancement of TFS representation with TLE.

Band-limited dual-frequency multiplexing technique for GNSS signals

Global navigation satellite systems (GNSSs) usually adopt constant-envelope multiplexing (CEM) techniques to deal with the nonlinear characteristic of high-power amplifiers (HPAs) on satellites. However, almost all the existing navigation signal CEM techniques assume that the bandwidth of the transmitter is large enough when compared to that of navigation signals, which often contradicts reality, especially for the multiplexing of wideband navigation signals such as the signals in the Galileo E5 and BDS-3 B2 bands. Consequently, the constant-envelope characteristic deteriorates severely when the composite signal passes through the filter, which leads to nonlinear distortion and low amplification efficiency. Even if the nonlinear distortion can be mitigated by using amplifier linearization techniques, the actual transmitted signal power still suffers from a high peak-to-average power ratio (PAPR) of the composite signal. In order to address this problem, we present a new band-limited, dual-frequency multiplexing technique. The main premise of this method is to consider the impact of the filter on the navigation signals at first and then introduce the band-limited auxiliary signals to reduce the peak power of the composite signal as much as possible, thereby increasing the actual transmission power of the navigation signals. Simulation results demonstrate that the proposed method has lower PAPR and higher amplification efficiency than existing methods. In addition, implementation of the proposed multiplexing technique is still based on a look-up table, which has little change compared with previous CEM methods, and the complexity of signal generation is affordable.

Translation uniqueness of phase retrieval and magnitude retrieval of band-limited signals

Abstract In this paper, we discuss how to partially determine the Fourier transform F ⁢ ( z ) = ∫ - 1 1 f ⁢ ( t ) ⁢ e i ⁢ z ⁢ t ⁢ 𝑑 t , z ∈ ℂ , F(z)=\int_{-1}^{1}f(t)e^{izt}\,dt,\quad z\in\mathbb{C}, given the data | F ⁢ ( z ) | {\lvert F(z)\rvert} or arg ⁡ F ⁢ ( z ) {\arg F(z)} for z ∈ ℝ {z\in\mathbb{R}} . Initially, we assume [ - 1 , 1 ] {[-1,1]} to be the convex hull of the support of the signal f. We start with reviewing the computation of the indicator function and indicator diagram of a finite-typed complex-valued entire function, and then connect to the spectral invariant of F ⁢ ( z ) {F(z)} . Then we focus to derive the unimodular part of the entire function up to certain non-uniqueness. We elaborate on the translation of the signal including the non-uniqueness associates of the Fourier transform. We show that the phase retrieval and magnitude retrieval are conjugate problems in the scattering theory of waves.

Reconstruction of bandlimited functions from space–time samples

For a wide family of even kernels {φu,u∈I}, we describe discrete sets Λ such that every bandlimited signal f can be reconstructed from the space-time samples {(f⁎φu)(λ),λ∈Λ,u∈I}.