Finite Rate Of Innovation Research Articles

Localized nonlinear functional equations and two sampling problems in signal processing

Let 1 ≤ p ≤ ∞. In this paper, we consider solving a nonlinear functional equation f (x) = y, where x, y belong to ℓ p and f has continuous bounded gradient in an inverse-closed subalgebra of ℬ (ℓ2), the Banach algebra of all bounded linear operators on the Hilbert space ℓ 2. We introduce strict monotonicity property for functions f on Banach spaces ℓ p so that the above nonlinear functional equation is solvable and the solution x depends continuously on the given data y in ℓ p . We show that the Van-Cittert iteration converges in ℓ p with exponential rate and hence it could be used to locate the true solution of the above nonlinear functional equation. We apply the above theory to handle two problems in signal processing: nonlinear sampling termed with instantaneous companding and subsequently average sampling; and local identification of innovation positions and qualification of amplitudes of signals with finite rate of innovation.

Ultrasonic Phased Array Industrial Imaging with Sub-Nyquist Sampling Rate

this paper uses a new type of FRI (Finite Rate of Innovation) sampling pattern based Sub-Nyquist sampling model breaked through Shannon theorem that it can get accurate signal reconstruction based on signal information rate, which requires the sampling frequency lower than two times the max signal frequency. We apply the new model in the ultrasonic phased array industrial imaging. In the experiment, ultrasonic phased array realized dynamic focusing and the high speed scan by ultrasonic array transducer of various array time delays to get flexible controllable synthesis beam composed signals that received by 32 phased array elements . The results indicate that in the model it greatly reduces the signal sampling frequency and improves the signal-to-noise ratio, frequency resolution at the same of the beam focusing and steering flexible.

A finite rate of innovation algorithm for fast and accurate spike detection from two-photon calcium imaging

Objective. Inferring the times of sequences of action potentials (APs) (spike trains) from neurophysiological data is a key problem in computational neuroscience. The detection of APs from two-photon imaging of calcium signals offers certain advantages over traditional electrophysiological approaches, as up to thousands of spatially and immunohistochemically defined neurons can be recorded simultaneously. However, due to noise, dye buffering and the limited sampling rates in common microscopy configurations, accurate detection of APs from calcium time series has proved to be a difficult problem. Approach. Here we introduce a novel approach to the problem making use of finite rate of innovation (FRI) theory (Vetterli et al 2002 IEEE Trans. Signal Process. 50 1417–28). For calcium transients well fit by a single exponential, the problem is reduced to reconstructing a stream of decaying exponentials. Signals made of a combination of exponentially decaying functions with different onset times are a subclass of FRI signals, for which much theory has recently been developed by the signal processing community. Main results. We demonstrate for the first time the use of FRI theory to retrieve the timing of APs from calcium transient time series. The final algorithm is fast, non-iterative and parallelizable. Spike inference can be performed in real-time for a population of neurons and does not require any training phase or learning to initialize parameters. Significance. The algorithm has been tested with both real data (obtained by simultaneous electrophysiology and multiphoton imaging of calcium signals in cerebellar Purkinje cell dendrites), and surrogate data, and outperforms several recently proposed methods for spike train inference from calcium imaging data.

Compressive Link Acquisition in Multiuser Communications

An important receiver operation is to detect the presence specific preamble signals with unknown delays in the presence of scattering, Doppler effects and carrier offsets. This task, referred to as "link acquisition", is typically a sequential search over the transmitted signal space. Recently, many authors have suggested applying sparse recovery algorithms in the context of similar estimation or detection problems. These works typically focus on the benefits of sparse recovery, but not generally on the cost brought by compressive sensing. Thus, our goal is to examine the trade-off in complexity and performance that is possible when using sparse recovery. To do so, we propose a sequential sparsity-aware compressive sampling (C-SA) acquisition scheme, where a compressive multi-channel sampling (CMS) front-end is followed by a sparsity regularized likelihood ratio test (SR-LRT) module. The proposed C-SA acquisition scheme borrows insights from the models studied in the context of sub-Nyquist sampling, where a minimal amount of samples is captured to reconstruct signals with Finite Rate of Innovation (FRI). In particular, we propose an A/D conversion front-end that maximizes a well-known probability divergence measure, the average Kullback-Leibler distance, of all the hypotheses of the SR-LRT performed on the samples. We compare the proposed acquisition scheme vis-\`{a}-vis conventional alternatives with relatively low computational cost, such as the Matched Filter (MF), in terms of performance and complexity.

A COMPRESSIVE SENSING SIGNAL DETECTION FOR UWB RADAR

A major challenge in UWB signal processing is the requirement for very high sampling rate under Shannon-Nyquist sampling theorem which exceeds the current ADC capacity. Radar signal is essentially a delayed and scaled version of the transmitted pulse, determined by sparse parameters such as time delays and amplitudes. A system for sampling UWB radar signal at an ultra- low sampling rate based on the Finite Rate of Innovation (FRI) and the estimation of time delays and amplitudes to detect UWB radar signal is presented in the paper. This sampling scheme which acquires the Fourier series coe-cients often results in sparse parameter extraction for UWB radar signal detection. The parameters such as time-delays and amplitudes are estimated using the total variation norm minimization. With this system, the UWB radar signal can be accurately reconstructed and detected with overwhelming probability at the rate much lower than Nyquist rate. The simulation results show that the proposed approach ofiers very good recovery performances for noisy UWB radar signal using very small number of samples, which is efiective for sampling and detecting UWB radar signal.

Sampling Signals with Finite Rate of Innovation and Recovery by Maximum Likelihood Estimation

We propose a maximum likelihood estimation approach for the recovery of continuously-defined sparse signals from noisy measurements, in particular periodic sequences of Diracs, derivatives of Diracs and piecewise polynomials. The conventional approach for this problem is based on least-squares (a.k.a. annihilating filter method) and Cadzow denoising. It requires more measurements than the number of unknown parameters and mistakenly splits the derivatives of Diracs into several Diracs at different positions. Moreover, Cadzow denoising does not guarantee any optimality. The proposed approach based on maximum likelihood estimation solves all of these problems. Since the corresponding log-likelihood function is non-convex, we exploit the stochastic method called particle swarm optimization (PSO) to find the global solution. Simulation results confirm the effectiveness of the proposed approach, for a reasonable computational cost.

Estimation of Sparse MIMO Channels with Common Support

We consider the problem of estimating sparse communication channels in the MIMO context. In small to medium bandwidth communications, as in the current standards for OFDM and CDMA communication systems (with bandwidth up to 20 MHz), such channels are individually sparse and at the same time share a common support set. Since the underlying physical channels are inherently continuous-time, we propose a parametric sparse estimation technique based on finite rate of innovation (FRI) principles. Parametric estimation is especially relevant to MIMO communications as it allows for a robust estimation and concise description of the channels. The core of the algorithm is a generalization of conventional spectral estimation methods to multiple input signals with common support. We show the application of our technique for channel estimation in OFDM (uniformly/contiguous DFT pilots) and CDMA downlink (Walsh-Hadamard coded schemes). In the presence of additive white Gaussian noise, theoretical lower bounds on the estimation of sparse common support (SCS) channel parameters in Rayleigh fading conditions are derived. Finally, an analytical spatial channel model is derived, and simulations on this model in the OFDM setting show the symbol error rate (SER) is reduced by a factor 2 (0 dB of SNR) to 5 (high SNR) compared to standard non-parametric methods - e.g. lowpass interpolation.

Industrial Ultrasonic Imaging Based on Compressed Sensing

Industrial ultrasonic imaging system based on compressed sensing(IUICS),is still lack of available implementation, due to its difficulty in hardware realization.However,thanks to the recent finite rate of innovation and ultrasonic phased array technology,it is possible to apply Compressive Sensing framework to industrial ultrasonic imaging system.In this paper,we propose an available scheme of industrial ultrasonic imaging,which includes the sampling of signal,reconstruction algorithm and its physical structure, based on Compressed Sensing.

Fast and Robust Parametric Estimation of Jointly Sparse Channels

We consider the joint estimation of multipath channels obtained with a set of receiving antennas and uniformly probed in the frequency domain. This scenario fits most of the modern outdoor communication protocols for mobile access (ETSI Std. 125 913) or digital broadcasting (ETSI Std. 300 744) among others. Such channels verify a sparse common support (SCS) property which was used in the work of Barbotin et al. (2012) to propose a finite rate of innovation (FRI)-based sampling and estimation algorithm. In this paper, we improve the robustness and computational complexity aspects of this algorithm. The method is based on projection in Krylov subspaces to improve complexity and a new criterion called the partial effective rank (PER) to estimate the level of sparsity to gain robustness. If <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</i> antennas measure a <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> -multipath channel with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> uniformly sampled measurements per channel, the algorithm possesses an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">KPN</i> log <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> ) complexity and an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">KPN</i> )memory footprint instead of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PN</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PN</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) for the direct implementation, making it suitable for <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> ≪ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> . The sparsity is estimated online based on the PER, and the algorithm therefore has a sense of introspection being able to relinquish sparsity if it is lacking. The estimation performances are tested on field measurements with synthetic additive white Gaussian noise, and the proposed algorithm outperforms nonsparse reconstruction in the medium to low signal-to-noise ratio range (≤ 0 dB), increasing the rate of successful symbol decoding by 1/10 in average, and 1/3 in the best case. The experiments also show that the algorithm does not perform worse than a nonsparse estimation algorithm in nonsparse operating conditions, since it may fall-back to it if the PER criterion does not detect a sufficient level of sparsity. The algorithm is also tested against a method assuming a “discrete” sparsity model as in compressed sensing. The conducted test indicates a trade-off between speed and accuracy.

2D Finite Rate of Innovation Reconstruction Method for Step Edge and Polygon Signals in the Presence of Noise

The finite rate of innovation (FRI) principle is developed for sampling a class of non-bandlimited signals that have a finite number of degrees of freedom per unit of time, i.e., signals with FRI. This sampling scheme is later extended to three classes of sampling kernels with compact support and applied to the step edge reconstruction problem by treating the image row by row. In this paper, we regard step edges as 2D FRI signals and reconstruct them block by block. The step edge parameters are obtained from the 2D moments of a given image block. Experimentally, our technique can reconstruct the edge more precisely and track the Cramér-Rao bounds (CRBs) closely with a signal-to-noise ratio (SNR) larger than 4 dB on synthetic step edge images. Experiments on real images show that our proposed method can reconstruct the step edges under practical conditions, i.e., in the presence of various types of noise and using a real sampling kernel. The results on locating the corners of data matrix barcodes using our method also outperform some state-of-the-art barcode decoders.

Extension of SoS sampling kernels for 2D FRI problems

The problem of sampling a 2D Poisson process is considered. Recently, such problems have been denoted as sampling of signals with finite rate of innovation. Therefore, a well-known sampling approach for the 1D approach is extended to the 2D case and the theoretic sampling bounds as well as the reconstruction approach are shown. By providing results from numerical simulations it can be demonstrated that reconstruction is exact up to numerical precision.

Sampling and Reconstruction of Periodic Piecewise Polynomials Using Sinc Kernel

We address a problem of sampling and reconstructing periodic piecewise polynomials based on the theory for signals with a finite rate of innovation (FRI signals) from samples acquired by a sinc kernel. This problem was discussed in a previous paper. There was, however, an error in a condition about the sinc kernel. Further, even though the signal is represented by parameters, these explicit values are not obtained. Hence, in this paper, we provide a correct condition for the sinc kernel and show the procedure. The point is that, though a periodic piecewise polynomial of degree R is defined as a signal mapped to a periodic stream of differentiated Diracs by R+1 time differentiation, the mapping is not one-to-one. Therefore, to recover the stream is not sufficient to reconstruct the original signal. To solve this problem, we use the average of the target signal, which is available because of the sinc sampling. Simulation results show the correctness of our reconstruction procedure. We also show a sampling theorem for FRI signals with derivatives of a generic known function.

Xampling at the Rate of Innovation

We address the problem of recovering signals from samples taken at their rate of innovation. Our only assumption is that the sampling system is such that the parameters defining the signal can be stably determined from the samples, a condition that lies at the heart of every sampling theorem. Consequently, our analysis subsumes previously studied nonlinear acquisition devices and nonlinear signal classes. In particular, we do not restrict attention to memoryless nonlinear distortions or to union-of-subspace models. This allows treatment of various finite-rate-of-innovation (FRI) signals that were not previously studied, including, for example, continuous phase modulation transmissions. Our strategy relies on minimizing the error between the measured samples and those corresponding to our signal estimate. This least-squares (LS) objective is generally non-convex and might possess many local minima. Nevertheless, we prove that under the stability hypothesis, any optimization method designed to trap a stationary point of the LS criterion necessarily converges to the true solution. We demonstrate our approach in the context of recovering pulse streams in settings that were not previously treated. Furthermore, in situations for which other algorithms are applicable, we show that our method is often preferable in terms of noise robustness.

Exponential Sampling: A Gibbs Phenomena Removal Model for Finite Rate of Innovation Sampling Framework

We propose an exponential approximating function as a sampling kernel with finite rate of innovation. The performance of reconstruct non-bandlimited signals from its low frequency components would inevitably induce Gibbs phenomenon. This paper establishes the theoretical model on relationship between sampling kernel filter and parametric reconstruction method of non-bandlimited signals, and designs a new window function exponential sampling kernel filter to removal Gibbs Influence. Simulation results show that, compared to Sinc sampling kernel filter, the reconstruction ability of exponential filter based finite rate sampling system is improved under the white Gaussian noise environment.

On the Existence of Optimal Unions of Subspaces for Data Modeling and Clustering

Given a set of vectors F={f 1,…,f m } in a Hilbert space $\mathcal {H}$, and given a family $\mathcal {C}$ of closed subspaces of $\mathcal {H}$, the subspace clustering problem consists in finding a union of subspaces in $\mathcal {C}$ that best approximates (is nearest to) the data F. This problem has applications to and connections with many areas of mathematics, computer science and engineering, such as Generalized Principal Component Analysis (GPCA), learning theory, compressed sensing, and sampling with finite rate of innovation. In this paper, we characterize families of subspaces $\mathcal {C}$ for which such a best approximation exists. In finite dimensions the characterization is in terms of the convex hull of an augmented set $\mathcal {C}^{+}$. In infinite dimensions, however, the characterization is in terms of a new but related notion; that of contact half-spaces. As an application, the existence of best approximations from π(G)-invariant families $\mathcal {C}$ of unitary representations of Abelian groups is derived.

Multichannel Sampling of Multidimensional Parametric Signals

The contribution of this paper is two-fold; first, we present a novel approach for sampling and reconstructing two dimensional signals with parametric structure or namely signals with Finite Rate of Innovation (FRI). The considered signals are bi-level polygons and set of 2-D Diracs and it will be shown that with the use of ACMP method, projection-slice theorem, Radon projections and exponential splines as sampling kernels, such signals can be perfectly reconstructed. Then, we present a possible extension of the theory of sampling multidimensional FRI signals, discussed above, to the case of multichannel acquisition systems. The essential issue of our considered multichannel system is that each channel receives the input signal with an unknown geometric transformation which needs to be estimated. We pose both the synchronization stage and the signal reconstruction stage as a parametric estimation problem and demonstrate that a simultaneous exact synchronization of the channels and reconstruction of the FRI signal is possible.

Performance Bounds and Design Criteria for Estimating Finite Rate of Innovation Signals

In this paper, we consider the problem of estimating finite rate of innovation (FRI) signals from noisy measurements, and specifically analyze the interaction between FRI techniques and the underlying sampling methods. We first obtain a fundamental limit on the estimation accuracy attainable regardless of the sampling method. Next, we provide a bound on the performance achievable using any specific sampling approach. Essential differences between the noisy and noise-free cases arise from this analysis. In particular, we identify settings in which noise-free recovery techniques deteriorate substantially under slight noise levels, thus quantifying the numerical instability inherent in such methods. This instability, which is only present in some families of FRI signals, is shown to be related to a specific type of structure, which can be characterized by viewing the signal model as a union of subspaces. Finally, we develop a methodology for choosing the optimal sampling kernels based on a generalization of the Karhunen--Lo\`eve transform. The results are illustrated for several types of time-delay estimation problems.

Multichannel Sampling of Signals With Finite Rate of Innovation

In this letter, we present a possible extension of the theory of sampling signals with finite rate of innovation (FRI) to the case of multichannel acquisition systems. The essential issue of a multichannel system is that each channel introduces different unknown delays and gains that need to be estimated for the calibration of the channels. We pose both the synchronization stage and the signal reconstruction stage as a parametric estimation problem and demonstrate that a simultaneous exact synchronization of the channels and reconstruction of the FRI signal is possible. We also consider the case of noisy measurements and evaluate the Cramer-Rao bounds (CRB) of the proposed system. Numerical results as well as the CRB show clearly that multichannel systems are more resilient to noise than the single-channel ones.

Compressive Sampling of EEG Signals with Finite Rate of Innovation

Analyses of electroencephalographic signals and subsequent diagnoses can only be done effectively on long term recordings that preserve the signals' morphologies. Currently, electroencephalographic signals are obtained at Nyquist rate or higher, thus introducing redundancies. Existing compression methods remove these redundancies, thereby achieving compression. We propose an alternative compression scheme based on a sampling theory developed for signals with a finite rate of innovation (FRI) which compresses electroencephalographic signals during acquisition. We model the signals as FRI signals and then sample them at their rate of innovation. The signals are thus effectively represented by a small set of Fourier coefficients corresponding to the signals' rate of innovation. Using the FRI theory, original signals can be reconstructed using this set of coefficients. Seventy-two hours of electroencephalographic recording are tested and results based on metrices used in compression literature and morphological similarities of electroencephalographic signals are presented. The proposed method achieves results comparable to that of wavelet compression methods, achieving low reconstruction errors while preserving the morphologiies of the signals. More importantly, it introduces a new framework to acquire electroencephalographic signals at their rate of innovation, thus entailing a less costly low-rate sampling device that does not waste precious computational resources.

Reconstructing Signals with Finite Rate of Innovation from Noisy Samples

A signal is said to have finite rate of innovation if it has a finite number of degrees of freedom per unit of time. Reconstructing signals with finite rate of innovation from their exact average samples has been studied in Sun (SIAM J. Math. Anal. 38, 1389–1422, 2006). In this paper, we consider the problem of reconstructing signals with finite rate of innovation from their average samples in the presence of deterministic and random noise. We develop an adaptive Tikhonov regularization approach to this reconstruction problem. Our simulation results demonstrate that our adaptive approach is robust against noise, is almost consistent in various sampling processes, and is also locally implementable.