Band-limited Signals Research Articles

Sensor placement method for water distribution networks based on sampling of non-bandlimited graph signals

Monitoring water distribution networks (WDNs) requires careful consideration of sensor placement, which is essential for obtaining comprehensive information about the network. A natural graphical structure underlies WDN, making graph sampling theory advantageous for selecting monitoring nodes. However, graph sampling theory is only applied only to restrictive band-limited signals, while the pressure data of WDN is a restrictive non-band-limited signal. To address this issue, this paper presents an approximate conversion method for transforming non-band-limited signals into band-limited signals, accompanied by an optimal spectrum threshold formula. This formula is used to perform spectral screening in the graph frequency domain, effectively converting non-band-limited signals into band-limited signals that preserve the major frequency components while ignoring smaller-value frequency components. By sampling band-limited signal, we identify sampling nodes that perfectly recover the signal. These sampling nodes act as monitoring nodes that can perform a comprehensive inspection of the WDN and accurately locate leaks. The accuracy of our method in recovering the signal and locating the leak is demonstrated by comparing it with two existing sensor placement optimization methods.

Unlimited sampling beyond modulo

Analog-to-digital converters (ADCs) act as a bridge between the analog and digital domains. Two important attributes of any ADC are sampling rate and its dynamic range. For bandlimited signals, the sampling should be above the Nyquist rate. It is also desired that the signals' dynamic range should be within that of the ADC's; otherwise, the signal will be clipped. Nonlinear operators such as modulo or companding can be used prior to sampling to avoid clipping. To recover the true signal from the samples of the nonlinear operator, either high sampling rates are required, or strict constraints on the nonlinear operations are imposed, both of which are not desirable in practice. In this paper, we propose a generalized flexible nonlinear operator which is sampling efficient. Moreover, by carefully choosing its parameters, clipping, modulo, and companding can be seen as special cases of it. We show that bandlimited signals are uniquely identified from the nonlinear samples of the proposed operator when sampled above the Nyquist rate. Furthermore, we propose a robust algorithm to recover the true signal from the nonlinear samples. Compared to the existing methods, our approach has a lower mean-squared error for a given sampling rate, noise level, and dynamic range. Our results lead to less constrained hardware design to address the dynamic range issues while operating at the lowest rate possible.

Peak Power Properties of Band-Limited Signals: With Pulse Shaping or Windowing

Peak Power Properties of Band-Limited Signals: With Pulse Shaping or Windowing

Derivative Sampling of Periodic and Nonperiodic Band-Limited Signals — Fourier Spectrum and Interpolation Formula

Derivative Sampling of Periodic and Nonperiodic Band-Limited Signals — Fourier Spectrum and Interpolation Formula

Reconstruction of bandlimited graph signals from random local sampling

Sampling and reconstruction on the spatially distributed networks is an innovative topic in graph signal processing. Recently, it has been shown that k-bandlimited graph signals can be reconstructed from a random collection of physically constrained sampled data. In this paper, we first study the random sampling scheme of k-bandlimited signals from a general local measurement, and then an iterative reconstruction algorithm based on frame theory is proposed with exponential convergence. It can yield a distributed implementation at a vertex level, which enables the devices that are limited by storage and computing power to recover signals more effectively. Numerical experiments on synthetic and real-world data are performed to validate the effectiveness of the proposed approach.

Discrete linear canonical transform on graphs: Uncertainty principle and sampling

With an increasing influx of classical signal processing methodologies into the field of graph signal processing, approaches grounded in discrete linear canonical transform have found application in graph signals. In this paper, we initially propose the uncertainty principle of the graph linear canonical transform (GLCT), which is based on a class of graph signals maximally concentrated in both vertex and graph spectral domains. Subsequently, leveraging the uncertainty principle, we establish conditions for recovering bandlimited signals of the GLCT from a subset of samples, thereby formulating the sampling theory for the GLCT. We elucidate interesting connections between the uncertainty principle and sampling. Further, by employing sampling set selection and experimental design sampling strategies, we introduce optimal sampling operators in the GLCT domain. Finally, we evaluate the performance of our methods through simulations and numerical experiments across applications.

Optimal sensor placement for leak location in water distribution networks based on graph learning algorithm

Abstract The effective monitoring of urban water distribution networks (WDNs) relies heavily on pressure sensor placement. Nevertheless, a WDN may consist of hundreds of nodes, and it is not economically feasible to install sensors at each node. Therefore, how to identify an optimal location for sensor placement becomes a crucial issue. We use graph signal processing to analyze the pressure signals and introduce a framework for optimal sensor placement specifically designed for pressure signals. To address the limitation that pressure signals cannot be sampled directly, we propose a method to convert the signal into a band-limited signal that meets the requirements. Central to the method is learning a graph Fourier operator, and the effectiveness of the proposed method is proved theoretically. The graph Fourier operator enables the pressure data to become a smooth graph signal with variations in its topology. In addition, we design a graph filter based on the energy of the signal and obtain a band-limited signal that meets the requirements. To ensure the selection of representative nodes, we use a noise-robust graph sampling method to obtain the sensor node. Our method is further evaluated using the pressure data from Anytown versus Net3, showing strong performance in leak identification and signal reconstruction capabilities.

Phase retrieval for radar waveform design

Abstract The ability of a radar to discriminate in both range and Doppler velocity is completely characterized by the ambiguity function (AF) of its transmit waveform. Mathematically, it is obtained by correlating the waveform with its Doppler-shifted and delayed replicas. We consider the inverse problem of designing a radar transmit waveform that satisfies the specified AF magnitude. This process may be viewed as a signal reconstruction with some variation of phase retrieval methods. We provide a trust-region algorithm that minimizes a smoothed non-convex least-squares objective function to iteratively recover the underlying signal-of-interest for either time- or band-limited support. The method first approximates the signal using an iterative spectral algorithm and then refines the attained initialization based on a sequence of gradient iterations. Our theoretical analysis shows that unique signal reconstruction is possible using signal samples no more than thrice the number of signal frequencies or time samples. Numerical experiments demonstrate that our method recovers both time- and band-limited signals from sparsely and randomly sampled, noisy and noiseless AFs.

Analysis and design of a GHz bandwidth adaptive bias circuit for an mmW Doherty amplifier

This paper derives theoretical results for adaptive bias in Doherty amplifiers and presents the design and measurements of an integrated adaptive bias circuit tailored for high peak-to-average high bandwidth signals. Fundamental equations for output power, impedance, and efficiency of the complete Doherty amplifier are derived. Even with ideal transistor models, the Doherty amplifier is fundamentally nonlinear due to saturation of the main amplifier and class-C nonlinearity of the auxiliary. Increasing the transconductance of the auxiliary amplifier mitigates the distortion. Adaptive bias offers the possibility to control the output current characteristic of the auxiliary amplifier. This means that adaptive bias linearises and mitigates the need for an oversized auxiliary amplifier. Both methods, transconductance scaling and adaptive bias, are analysed and compared as well as having a band limited adaptive bias signal. The design of a multiple GHz bandwidth adaptive bias circuit is presented. To verify the circuit design and the theoretical predictions, an mmW Doherty amplifier in 22 nm CMOS-FD-SOI, utilizing the presented adaptive bias circuit, is measured and compared with and without adaptive bias. Comparison is conducted both using continuous-wave and modulated high bandwidth signals. Measured results confirm the predicted improvements by the adaptive bias as derived by the theoretical analysis.

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.

Recovery of bandlimited graph signals based on the reproducing kernel Hilbert space

Signal recovery on graphs is attracting more and more attentions. Based on the smoothness assumption, the signal recovery problem can be formulated as an unconstrained optimization model. Although the model can be solved analytically, the computational cost is too expensive. In this paper, to reduce the computational complexity, we skillfully construct a kernel function, such that the smoothness item is equivalent to the norm of the signal modulo a constant in the reproducing kernel Hilbert space. Based on this, by using the Representer Theorem of reproducing kernel Hilbert space, the optimization problem for recovering the signal from a given bandlimited space, can be solved in a lower-dimensional subspace corresponding to the known labeled set. Experiments for the recovery of artificial graph signals in GSPBOX and real data in the recommendation system show very encouraging results.

Optimized shaping and trade-off of superoscillating pulses

In this article, we consider a special family of pulses, which are a part of a band limited signal and are ‘too narrow’ considering the band limit of the signal. We dub such pulses ‘superoscillating pulses’ although they can be seen at best as half an oscillation. While some of our results are of a more generic nature, the article is devoted to the optimization of superoscillating pulses, The first step consists of approximating a given target signal by a band limited signal in a fixed time interval. The signals to be approximated, exhibit in that interval features that seem to involve frequencies higher than the band limit of the approximant. We define the mean square relative error (MSRE) as a measure of the adherence of the approximant to the original signal in the chosen interval. We find that the minimization of the MSRE conflicts with the necessity to minimize the energy (or power) expense of the superoscillating signal. We obtain the trade-off relation between optimal energy expense and the MSRE for a family of pulses. This makes it clear that within that family, there exists a specific pulse shape that is better approximated by a superoscillating pulse. Finally, we show how to construct a yield optimized superoscillating pulse that, within a given time interval, has a prescribed narrow width without resorting to a target signal at all.

Robust estimation of smooth graph signals from randomized space–time samples

Abstract Heat diffusion processes have found wide applications in modelling dynamical systems over graphs. In this paper, we consider the recovery of a $k$-bandlimited graph signal that is an initial signal of a heat diffusion process from its space–time samples. We propose three random space–time sampling regimes, termed dynamical sampling techniques, that consist in selecting a small subset of space–time nodes at random according to some probability distribution. We show that the number of space–time samples required to ensure stable recovery for each regime depends on a parameter called the spectral graph weighted coherence, which depends on the interplay between the dynamics over the graphs and sampling probability distributions. In optimal scenarios, as little as $\mathcal{O}(k \log (k))$ space–time samples are sufficient to ensure accurate and stable recovery of all $k$-bandlimited signals. Dynamical sampling typically requires much fewer spatial samples than the static case by leveraging the temporal information. Then, we propose a computationally efficient method to reconstruct $k$-bandlimited signals from their space–time samples. We prove that it yields accurate reconstructions and that it is also stable to noise. Finally, we test dynamical sampling techniques on a wide variety of graphs. The numerical results on synthetic and real climate datasets support our theoretical findings and demonstrate the efficiency.

Random Sampling of Mellin Band-Limited Signals

In this paper, we address the random sampling problem for the class of functions in the space of Mellin band-limited functions B T , which are concentrated on a bounded cube. It is established that any Mellin band-limited function can be approximated by an element in a finite-dimensional subspace of B T . Utilizing the notion of covering number and Bernstein’s inequality to the sum of independent random variables, we prove that the probabilistic sampling inequality holds for the set of concentrated signals in B T with an overwhelming probability provided the sampling size is large enough.

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.

LQG/LTR based robust youla parameterized adaptive vibration control for the supporting platform of rotating liquid mirror

In mechatronic systems, the vibrations which commonly consist of band-limited random signals mixed with large multiple narrow-band deterministic signals may negatively impact the system performance. For example, due to the incomplete matching of the mechanism and the rotation of the motor, random and deterministic vibration disturbance may occur in the supporting platform of the rotating liquid mirror. In this paper, a robust Youla ( Q) parameterized adaptive regulation approach has been proposed to minimize such kind of vibration signals for the rotating platform system with model uncertainties. Firstly, the inner-loop robust controller with linear quadratic Gaussian with loop transfer recovery ( LQG/LTR) is optimally designed by choosing the suitable weighting functions to achieve the trade-off between robust stability and control performance to deal with random vibration signals. Then the Youla parameter is augmented to construct a set of Q-parameterized stabilizing controllers, and the robust stability of the system is analyzed through dual-Youla parameterization of the uncertain model. The recursive least squares (RLS) adaptive algorithm is developed to tune the Q parameter online to construct a desired adaptive controller for further residual vibration elimination. An experimental evaluation of the controller in reducing the vibration of the supporting platform of a rotating liquid mirror has been carried out, and the results illustrate that the proposed adaptive robust vibration regulation approach can effectively suppress the band-limited random and narrow-band deterministic vibration signals.

A Comparison of Methods Recovering Signals from Non-Uniform Samples

Band-limited signals play a vital role in signal acquisition, transmission, processing and storage for decades. However, their recovery from non-uniformly spaced samples is still subject to discussion. There are multiple authors advancing different approaches and the user has to deal with an abundance of literature. Therefore, this paper offers a brief theoretical review of the most promising reconstruction methods. We also consider the importance of regularisation if the signal samples are noisy, which, surprisingly, seems to be neglected by many authors. Subsequently, the suitability of the methods is illustrated using an example of a real-world signal. Assessment of accuracy and speed of the methods is repeated for varying noise levels in signal amplitude, and for increasing degree of non-uniformities in sampling time. Cubic spline methods and sinc function reconstruction offer the best compromise between speed and accuracy for the concrete signal tested in this paper.

Robust logarithmic hyperbolic cosine adaptive filtering over graph signals

Graph signal processing (GSP) has two methods including graph Fourier transform (GFT) and graph shift-operator (GSO) for dealing with the graph signal used for adaptive filtering (AF). However, in order to perform feature decomposition, GFT requires the availability of a band-limited graph signal with a known bandwidth in advance. Therefore, a general model based on GSO is then derived for AF by introducing a mixture coefficient to combine the advantages of different GSOs, comprehensively. Since the existing algorithms using the mean square error and other robust error criteria in GSP cannot effectively fight against non-Gaussian noises, a robust logarithmic hyperbolic cosine adaptive filter over graph (GLHCAF) algorithm is proposed by applying the logarithmic hyperbolic cosine loss function to the designed general model. The significant performance improvement of GLHCAF in non-Gaussian noises is obtained by simply controlling the scaling parameter. To avoid the parameter selections including scaling parameter and step size, the adaptive GLHCAF (AGLHCAF) algorithm is therefore proposed to further improve the performance of GLHCAF. Moreover, the mean square performance analysis of GLHCAF is given. Simulations in the real-world data are provided to assess the reliability and accuracy for theoretical analysis, the universality of the designed general model, and superiorities of proposed algorithms.

Self-matching extraction fractional wavelet transform for mechanical equipment fault diagnosis

Time-frequency analysis (TFA) of vibration signals is an important way for fault diagnosis and predictive maintenance of mechanical equipment. However, the existing TFA methods, such as wavelet transform (WT) and synchrosqueezing transform, are incapable to describe local details of strongly time-varying signals, bringing such problems as time-frequency energy dispersion. Theoretically, it is more beneficial for time-varying band-limited signals to make multi-resolution analysis in the time-fraction frequency domain because such action can increase the selection chances of mother wavelets in any direction on the time-frequency plane by changing the angle . In order to improve the ability of analyzing strongly time-varying signals and the readability of the time-frequency plane, this paper has proposed a novel TFA method called self-matching extraction fractional WT (SMEFRWT). Firstly, within the theoretical framework of fractional WT (FRWT), a minimum entropy-based method is proposed to choose an optimal angle in the fractional domain. Such action can obtain the best aggregation angle of time-frequency energy and facilitate the separation of multi-component signals. Then, self-matching extraction operator, one instantaneous-frequency estimated operator that match the time-frequency structure of amplitude modulation signals, is constructed. The operator allows for the distribution of time-frequency energy of fast-changing signals in both frequency and time directions, making it possible to match the time-varying vibration signals from mechanical equipment more accurately. Finally, SMEFRWT is adopted to make TFA and extract fault features of the faulty experiment benches on gear and bearings, with its feasibility validated by experimental results.

Spatio–Spectral Limiting on Replacements of Tori by Cubes

A class of graphs is defined in which each vertex of a discrete torus is replaced by a Boolean hypercube in such a way that vertices in a fixed subset of each replacement cube are adjacent to corresponding vertices of a neighboring replacement cube. Bases of eigenvectors of the Laplacians of the resulting graphs are described in a manner suitable for quantifying the concentration of a low-spectrum vertex function on a single vertex replacement. Functions that optimize this concentration on these graphs can be regarded as analogues of Slepian prolate functions that optimize concentration of a bandlimited signal on an interval in the classical setting of the real line. Comparison to the case of a simple discrete cycle shows that replacement allows for higher concentration.