Band-limited Signals Research Articles

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}.

Stability Estimates for Phase Retrieval from Discrete Gabor Measurements

Phase retrieval refers to the problem of recovering some signal (which is often modelled as an element of a Hilbert space) from phaseless measurements. It has been shown that in the deterministic setting phase retrieval from frame coefficients is always unstable in infinite-dimensional Hilbert spaces (Cahill et al. in Trans Am Math Soc Ser B 3(3):63–76, 2016) and possibly severely ill-conditioned in finite-dimensional Hilbert spaces (Cahill et al. in Trans Am Math Soc Ser B 3(3):63–76, 2016). Recently, it has also been shown that phase retrieval from measurements induced by the Gabor transform with Gaussian window function is stable under a more relaxed semi-global phase recovery regime based on atoll functions (Alaifari in Found Comput Math 19(4):869–900, 2019). In finite dimensions, we present first evidence that this semi-global reconstruction regime allows one to do phase retrieval from measurements of bandlimited signals induced by the discrete Gabor transform in such a way that the corresponding stability constant only scales like a low order polynomial in the space dimension. To this end, we utilise reconstruction formulae which have become common tools in recent years (Bojarovska and Flinth in J Fourier Anal Appl 22(3):542–567, 2016; Eldar et al. in IEEE Signal Process Lett 22(5):638–642, 2014; Li et al. in IEEE Signal Process Lett 24(4):372–376, 2017; Nawab et al. in IEEE Trans Acoust Speech Signal Process 31(4):986–998, 1983).

Near-Optimal Graph Signal Sampling by Pareto Optimization.

In this paper, we focus on the bandlimited graph signal sampling problem. To sample graph signals, we need to find small-sized subset of nodes with the minimal optimal reconstruction error. We formulate this problem as a subset selection problem, and propose an efficient Pareto Optimization for Graph Signal Sampling (POGSS) algorithm. Since the evaluation of the objective function is very time-consuming, a novel acceleration algorithm is proposed in this paper as well, which accelerates the evaluation of any solution. Theoretical analysis shows that POGSS finds the desired solution in quadratic time while guaranteeing nearly the best known approximation bound. Empirical studies on both Erdos-Renyi graphs and Gaussian graphs demonstrate that our method outperforms the state-of-the-art greedy algorithms.

Algorithmic Computability of the Signal Bandwidth

The bandwidth of a bandlimited signal is an important number that is relevant in many applications and concepts. For example, according to the Shannon sampling theorem, the bandwidth determines the minimum sampling rate that is required for a perfect reconstruction. In this paper we consider bandlimited signals with finite energy and bandlimited signals that are absolutely integrable and analyze whether the bandwidth of these signals can be determined algorithmically. We employ the concept of Turing computability, a theoretical model that describes the fundamental limits of what can be solved algorithmically on a digital hardware, and ask if, for a given computable bandlimited signal, it is possible to compute its bandwidth on a Turing machine. We show that this is not possible in general, because there exist computable bandlimited signals for which the bandwidth is a non-computable real number. Even the weaker question if the bandwidth of a given signal is smaller than a predefined value cannot be always answered algorithmically. Further, we prove that in the case where the bandwidth in not computable, it is even impossible to algorithmically determine a sequence of upper bounds that converges to the actual bandwidth of the signal. As a positive result, we show that the set of signals whose bandwidth is larger than some given value is semi-decidable.

Sparse Bayesian Learning Based Direction-of-Arrival Estimation under Spatially Colored Noise Using Acoustic Hydrophone Arrays

Direction-of-arrival (DOA) estimation in a spatially isotropic white noise background has been widely researched for decades. However, in practice, such as underwater acoustic ambient noise in shallow water, the ambient noise can be spatially colored, which may severely degrade the performance of DOA estimation. To solve this problem, this paper proposes a DOA estimation method based on sparse Bayesian learning with the modified noise model using acoustic vector hydrophone arrays. Firstly, an applicable linear noise model is established by using the prolate spheroidal wave functions (PSWFs) to characterize spatially colored noise and exploiting the excellent performance of the PSWFs in extrapolating band-limited signals to the space domain. Then, using the proposed noise model, an iterative method for sparse spectrum reconstruction is developed under a sparse Bayesian learning (SBL) framework to fit the actual noise field received by the acoustic vector hydrophone array. Finally, a DOA estimation algorithm under the modified noise model is also presented, which has a superior performance under spatially colored noise. Numerical results validate the effectiveness of the proposed method.

Regularized derivative interpolation for two dimensional band-limited functions

In this paper, the ill-posedness of Derivative Interpolation is discussed and a regularized Partial Derivative Interpolation for band-limited signals is presented in the two dimensional case. The convergence of the regularized Derivative Interpolation is studied and compared with Derivative Interpolation using the Shannon sampling theorem.

Computable Time Concentration of Bandlimited Signals and Systems

Turing computability deals with the question of what is theoretically computable on a digital computer, and hence is relevant whenever digital hardware is used. In this paper we study different possibilities to define computable bandlimited signals and systems. We consider a definition that uses finite Shannon sampling series as approximating functions and another that employs computable continuous functions together with an effectively computable time concentration. We discuss the advantages and drawbacks of both definitions and analyze the connections and differences. In particular, we show that both definitions are equivalent for many practically relevant signal classes, e.g. for bandlimited signals with finite energy, but also that there are important differences, such as for the impulse responses of BIBO stable LTI systems.

On Filter Generalization for Music Bandwidth Extension Using Deep Neural Networks

In this paper, we address a sub-topic of the broad domain of audio enhancement, namely musical audio bandwidth extension. We formulate the bandwidth extension problem using deep neural networks, where a band-limited signal is provided as input to the network, with the goal of reconstructing a full-bandwidth output. Our main contribution centers on the impact of the choice of low pass filter when training and subsequently testing the network. For two different state of the art deep architectures, ResNet and U-Net, we demonstrate that when the training and testing filters are matched, improvements in signal-to-noise ratio (SNR) of up to 7dB can be obtained. However, when these filters differ, the improvement falls considerably and under some training conditions results in a lower SNR than the band-limited input. To circumvent this apparent overfitting to filter shape, we propose a data augmentation strategy which utilizes multiple low pass filters during training and leads to improved generalization to unseen filtering conditions at test time.

A time domain random flicker band-limited Gaussian noise jamming algorithm for LMS-based GPS

Global position systems (GPS) receiver based on least mean square (LMS) could resist the interference by null jamming direction. Considering that the existing band-limited Gaussian noise jamming signals were easily suppressed by LMS-based GPS receivers, a time domain random flicker band-limited Gaussian noise jamming algorithm was proposed to improve its performance. By disturbing the convergence of LMS, it could achieve the purpose of suppressing the LMS-based GPS. Simulation result shows that the proposed algorithm has an average performance gain of 2.7dB~4.6dB under different number of interferences compared with band-limited Gaussian noise.

Multi-Channel Modulo Samplers Constructed From Gaussian Integers

Recently, there is an increased interest in the study of modulo analog to digital converters (ADCs). These new systems can reconstruct a signal whose amplitude is much higher than the conventional ADC's dynamic range. Modulo ADCs are characterized by their modulo threshold and in the current literature, all existing works are limited to real-valued moduli. In this paper, we propose multi-channel modulo samplers with complex-valued moduli to sample a band-limited complex signal. Specifically, we discuss the construction of complex divisors from Gaussian integers and propose their efficient implementations. A memory-efficient, closed-form recovery algorithm is also proposed. Simulation results demonstrate that the proposed systems can provide stable reconstruction of a high dynamic range complex-valued signal at low sampling rates.

A Background Correlation-Based Timing Skew Estimation Method for Time-Interleaved ADCs

This paper presents a correlation-based all-digital background estimation method for timing skew mismatches in time-interleaved analog-to-digital converters (TI-ADCs). Exploiting the first-order approximation of an autocorrelation function of the digital output of each Sub-ADC, the timing skew mismatch between adjacent channels is identified, which can be used for analog or digital correction methods. The proposed estimation method has a low computation complexity and a simple implementation structure, because it basically only requires multiply-accumulate and average operations. In addition, the proposed method is not limited in the number of channels, does not need any pilot injection, is basically suitable for any bandlimited analog input signal, possess a large estimation range of mismatches and is immune against gain and offset mismatches to a certain extent. Simulation results of a four-channel TI-ADC behavioral model and measurement results from a commercial 12-bit 3.6-GS/s TI-ADC show the effectiveness and superiority of the proposed estimation method.

Optimal Sampling of a Bandlimited Noisy Multiple Output Channel

This paper deals with an extension of Papoulis' generalized sampling expansion (GSE) of bandlimited signals to a case where a white noise, filtered to the signal's band, is added before sampling. We look for the best sampling scheme that maximizes the capacity or minimizes the mean-squared error (MSE) of the sampled channel between the input signal and the M sampled outputs signals, e.g. channels. Due to the noise it is beneficial to sample at a total rate higher than Nyquist, yet there is no gain in sampling each signal above Nyquist; thus the total rate considered, normalized by Nyquist rate, is between 1 to M. To avoid preference to any channel, the paper assumes that the channels are composed of all-pass linear time-invariant (LTI) systems. For the case where the total normalized rate is between M-1 and M, we show that the best scheme samples M-1 outputs at Nyquist rate and the last output at the remaining rate. Surprisingly, equal sampling is suboptimal in general, but when there is an integer relation between M and the total normalized rate, equal sampling achieves the optimal performance. We also make a conjecture on other rates that the best scheme either samples some channels at Nyquist and one another channel at the remaining rate, or sample the channels equally. This conjecture is supported by simulation for M ≤ 5. Finally, we discuss the relation between maximizing the capacity and minimizing the MSE.

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.

On the Solvability of the Peak Value Problem for Bandlimited Signals With Applications

In this paper we study from an algorithmic perspective the problem of finding the peak value of a bandlimited signal. This problem plays an important role in the design and optimization of communication systems. We show that the peak value problem, i.e., computing the peak value of a bandlimited signal from its samples, can be solved algorithmically if oversampling is used. Without oversampling this is not possible. There exist bandlimited signals, for which the sequence of samples is computable, but the signal itself is not. This problem is directly related to the question whether there is a link between computability in the digital domain and the analog domain, and hence to a fundamental signal processing problem. We show that there is an asymmetry between continuous-time and discrete-time computability. Further, we study the decay behavior of computable bandlimited signals, which describes the concentration of the signals in the time domain, and, for locally computable bandlimited signals, we analyze if it is always possible to decide algorithmically whether the peak value is smaller than a given threshold.

Joint Delay-Doppler Estimation Performance in a Dual Source Context

Evaluating the time-delay, Doppler effect and carrier phase of a received signal is a challenging estimation problem that was addressed in a large variety of remote sensing applications. This problem becomes more difficult and less understood when the signal is reflected off one or multiple surfaces and interferes with itself at the receiver stage. This phenomenon might deteriorate the overall system performance, as for the multipath effect in Global Navigation Satellite Systems (GNSS), and mitigation strategies must be accounted for. In other applications such as GNSS reflectometry (GNSS-R) it may be interesting to estimate the parameters of the reflected signal to deduce the geometry and the surface characteristics. In either case, a better understanding of this estimation problem is directly brought by the corresponding lower performance bounds. In the high signal-to-noise ratio regime of the Gaussian conditional signal model, the Cramér-Rao bound (CRB) provides an accurate lower bound in the mean square error sense. In this article, we derive a new compact CRB expression for the joint time-delay and Doppler estimation in a dual source context, considering a band-limited signal and its specular reflection. These compact CRBs are expressed in terms of the baseband signal samples, making them especially easy to use whatever the baseband signal considered, therefore being valid for a variety of remote sensors. This extends existing results in the single source context and opens the door to a plethora of usages to be discussed in the article. The proposed CRB expressions are validated in two representative navigation and radar examples.

Power- and Area-Optimized High-Level Synthesis Implementation of a Digital Down Converter for Software-Defined Radio Applications

In digital signal processing, digital down converters (DDCs) convert digitized, band-limited signals to lower frequency signals at a smaller sampling rate to simplify subsequent filtering stages. Software-defined radio (SDR) is a radio communication system in which components that are traditionally implemented in hardware are implemented in software on an embedded system. DDCs are widely used in modern communication systems, such as SDRs. Herein, we propose a low-power- and area-optimized implementation of a DDC for SDR applications. The DDC was designed using an innovative and novel high-level synthesis (HLS) design method based on application-specific bit widths for data nodes. The results achieved after a field programmable gate array (FPGA) implementation are superior to those obtained from hand-coded register transfer level (RTL) implementations in terms of area and power efficiency, with almost the same speed of operation. Our results were obtained using the MATLAB hardware description language (HDL) coder for HLS and Xilinx Vivado (a software for the synthesis and analysis of HDL designs) for synthesis. The DDC down-converts an input of 200 MHz signal to an output of 2 MHz signal. This implementation was conducted on a real FPGA hardware (Xilinx Kintex-7) and verified against the design specifications using an FPGA in the loop feature of HDL Verifier and MATLAB. In addition, we propose a generic methodology for improving the area, speed, and power for different application designs and HLS tools. The proposed methodology is also applicable to hand-coded RTL designs for any application.

Fast and [formula omitted]-optimal recovery for periodic nonuniformly sampled bandlimited signal

Uniformly-sampled sequence reconstruction from periodic nonuniform samples is addressed. In this problem setting, a bandlimited continuous-time signal is sampled with the same rate and different time offsets to obtain the periodic nonuniform samples, in which time offset information is unknown. We use a modified sequential forward selection algorithm to determine the sampling pattern corresponding to the time offset information, aiming to keep the condition number of the related matrix as small as possible. In our proposed algorithm, the reconstruction is formulated as a L2-norm optimization problem. Besides, proper matrix partition is included to reduce the complexity of large matrix pseudo-inverse. Numerical examples are presented to demonstrate the effectiveness of the derived reconstruction algorithm and its stability problem, even in the presence of noise.

QR factorization-based sampling set selection for bandlimited graph signals

We consider the problem of sampling vertices or nodes of graphs where signals on graphs are bandlimited and reconstructed from the signal values on the sampled nodes. Noting that sampling a subset of nodes is equivalent to constructing the sampled matrix consisting of the corresponding rows selected from the eigenvector matrix of a variation operator (e.g., graph Laplacian), we first formulate the reconstruction error as the Frobenius matrix norm of the pseudo-inverse of the sampled matrix and manipulate the reconstruction error by using QR factorization to propose a simple criterion by which greedy sampling is conducted. We show that the proposed algorithm can achieve near-optimality for the case of high SNR with the concept of approximate supermodularity and evaluate the proposed method by comparing with a near-optimal approach developed for a sensor selection problem. We also analyze the complexity of the proposed algorithm in operation count and compare with existing greedy methods, including algorithms for subset selection of matrices since sampling of graph signals is also accomplished by selecting a subset of columns from the transpose of the eigenvector matrix. We finally demonstrate through extensive experiments that the proposed algorithm achieves a competitive performance gain for signals on various graphs in terms of complexity and signal fidelity as compared with previous novel methods.

Generating Continuous Deterministic Band-Limited Test Signals With Nearly Laplace Distribution

Generating Continuous Deterministic Band-Limited Test Signals With Nearly Laplace Distribution

Inverting spectrogram measurements via aliased Wigner distribution deconvolution and angular synchronization

Abstract We propose a two-step approach for reconstructing a signal $\textbf x\in \mathbb{C}^d$ from subsampled discrete short-time Fourier transform magnitude (spectogram) measurements: first, we use an aliased Wigner distribution deconvolution approach to solve for a portion of the rank-one matrix $\widehat{\textbf{x}}\widehat{\textbf{x}}^{*}.$ Secondly, we use angular synchronization to solve for $\widehat{\textbf{x}}$ (and then for $\textbf{x}$ by Fourier inversion). Using this method, we produce two new efficient phase retrieval algorithms that perform well numerically in comparison to standard approaches and also prove two theorems; one which guarantees the recovery of discrete, bandlimited signals $\textbf{x}\in \mathbb{C}^{d}$ from fewer than $d$ short-time Fourier transform magnitude measurements and another which establishes a new class of deterministic coded diffraction pattern measurements which are guaranteed to allow efficient and noise robust recovery.