Finite Rate Of Innovation Research Articles

Relevant sampling in a reproducing kernel subspace of Orlicz space

In this paper, we aim to provide a general paradigm for dealing with the sampling and random sampling problem in a reproducing kernel subspace of Orlicz space [Formula: see text]. We consider the function space [Formula: see text] as the image of an idempotent integral operator on [Formula: see text], where the integral kernel satisfies certain off-diagonal decay and regularity conditions. The model example of such reproducing kernel subspace of [Formula: see text] includes the finitely generated shift-invariant space and signal space with a finite rate of innovation. We show that a signal in [Formula: see text] can be stably reconstructed from its samples at distinct points separated by a sufficiently small gap. Next, we deduce that the random sampling inequality holds with a high probability for the class of functions in [Formula: see text] concentrated on a cube [Formula: see text], when the samples collected at i.i.d. random points are drawn on [Formula: see text] of order [Formula: see text].

Frequency estimation of underwater acoustic targets based on improved FRI method

The line spectrum information of a ship’s radiated noise signal is often regarded as an important feature used to distinguish between targets. In this paper, an improved FRI (finite rate of innovation) method is proposed to estimate the low-frequency line spectra of the radiated noise from underwater acoustic passive targets under the condition of finite sampling points. First, a preliminary rough estimation of the frequency is performed using the Fourier transform. Then, the multichannel signal data are iteratively fitted based on the polynomial-ratio form model. The optimal estimation results are finally determined based on the residuals between the data and the model in the fitting process. Simulation experiments prove that compared with the original FRI algorithm, the method proposed in this paper can estimate line spectral frequency more efficiently under the condition of low signal-to-noise ratio (SNR), and can achieve fast convergence with shorter operation time. The experimental data processing results prove that the method can realize high-precision extraction of line spectrum information of ship noise.

Parameter estimation of hybrid pulse streams based on sub-Nyquist sampling and Shift-invariant subspace projection

Finite Rate of Innovation (FRI) sampling theory offers an efficient approach for the sub-Nyquist measurement of the short pulse stream signal, but it can only address parameter measurement problems under waveform consistency. To overcome the measurement challenge of the hybrid pulse streams, this paper proposes a subspace projection-based FRI parameter estimation method under sub-Nyquist sampling conditions. Our method is based on the concept of subspace decomposition, which involves finely dividing different pulse signals within their respective Shift-invariant (SI) subspaces. We use the generating function of the corresponding space as a decoupling object to achieve parameter estimation through FRI measurement methods. The sampling process is efficient and simple, requiring only single-channel sampling to measure the characteristic parameters of all hybrid pulses. We not only provide theoretical guarantees for the implementation of this approach but also conduct numerous numerical simulations and hardware verification experiments. Our experimental results demonstrate that our method achieves higher parameter measurement accuracy compared to classical methods in a simpler manner.

A parameter estimation method with improved finite rate of innovation sampling for linear frequency modulation signals

AbstractTo complete the sparse sampling of linear frequency modulation (LFM) signals and estimate the parameters, the authors proposed an improved finite rate of innovation (FRI) system to achieve the goal. First the authors build a mathematical model based on the Z‐transform, then they input the LFM signal into the modified FRI system and process the output of the system based on this mathematical model, which enables the estimation of the parameters of the LFM signal using only four sampling points. The error of the final experimental results can be controlled within 3%, which greatly improves the efficiency of LFM signal parameter estimation.

Finite rate of innovation sampling of Gaussian pulse streams with variable shape

The existing Finite Rate of Innovation (FRI) sampling theory can realize the efficient measurement of Gaussian pulse streams with fixed modal functions. However, it is hard to sample that signal with variable pulse width factors by a sub-Nyquist rate. To get around this challenge, we propose an extended FRI sampling method based on Taylor series expansion in this paper, where the signal is first filtered by the exponential reproducing kernel, then each pulse can be reconstructed sequentially. For a single pulse, we first prove that the pulse width factor does not interfere with the estimation process of the time delay parameter, and we also present its Cramer-Rao lower Bound (CRB). The signal Fourier coefficient can then be reformulated by the Taylor series expansion of order N−1 to realize the decoupling of the pulse width factor before the final estimation. We derive the order limit of expansion from the perspective of numerical stability, and we also give the upper bound on the estimation error of the pulse width factor for N=2. Simulation results show that, compared with the classical methods, the proposed state-of-the-art algorithm can achieve a higher precision reconstruction of the Gaussian pulse stream signal.

Deep Atomic Norm Denoising Network for Jointly Range-Doppler Estimation With FRI Sampling

Finite rate of innovation (FRI) sampling is widely used in modern high-resolution range-Doppler (RD) detection systems to reduce the sampling rate. However, as the sampling rate decreases, the method becomes more noise-sensitive. This study proposes a deep atomic norm denoising network (DAND-Net) to denoise FRI samples of RD signals and estimate the noise levels (NLs). Based on the sparse common support property of the FRI samples, the denoising problem of the RD signals was modeled as an atomic norm soft thresholding problem, which jointly used multiple measurement vectors and could be unfolded into a model-driven deep network. The trainability and generalization ability of the network were improved by introducing piecewise linear functions and new trainable variables. By introducing an NL estimation operator, the NL could be better estimated during denoising, and the network training and convergence better supported. Additionally, the plug-and-play method could be used to complete denoising prior to many existing FRI methods. The simulation experiments showed that the proposed denoising network can achieve better denoising and NL estimation results with fewer iterations.

Sub-Nyquist Sampling and Measurement of MPSK Signal Based on Parameter Matching

Accurate measurement of the M-ary Phase Shift Keying (MPSK) signal determined by finite parameters per unit of time is significant in engineering fields such as digital microwave and satellite communication. In this paper, we propose a sub-Nyquist sampling system with two channels for measuring characteristic parameters of MPSK signals based on the Finite Rate of Innovation (FRI) sampling theory. The carrier frequency parameter measurement process is modeled as a convex optimization problem, solvable using the parameter matching method on the estimated grid. The minimum harmonic frequency of the carrier is determined to build the exact grid, and the carrier frequency is estimated through a greedy search based on the given cost function. Even when the truth value of the carrier frequency is unknown, the discontinuity location parameters can be measured with high precision using the spectral estimation method on the Fourier coefficients of the MPSK signal. We propose an upper bound on the estimation error of this separable measurement method for signals with one segment. The amplitude and phase parameters can be measured by solving a least square problem. Our proposed measurement system, with lower hardware complexity, can accurately measure the MPSK signal with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> segments by sampling only 2 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> +5 points in a period, at a much lower sampling rate than the carrier frequency. Numerical simulations and hardware experiments demonstrate the superior performance of our state-of-the-art measurement method.

Sub-Nyquist Sampling and Measurement of FRI Signals With Additive Shot Noise

To mitigate the impact of shot noise on the parameter measurement process of Finite Rate of Innovation (FRI) signals, this paper proposes a sub-Nyquist modulation sampling system with two measurement channels. The noisy signal is sampled in the main channel after being filtered by a sinc kernel to obtain the aliasing Fourier coefficients, whereas the FRI signal is shifted out of the baseband with a cosine modulation function in the accessory channel to expose the shot noise information to the low frequency band. The burst time parameters of shot noise can be estimated using the annihilating filter method from the samples in the accessory channel, while the parameters of the FRI signal will be measured from the observations in the main channel after the shot noise information has been removed. If the Rate of Innovation of the FRI signal is ρ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>L</i></sub> and is affected by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> impulses of shot noise, the entire measurement process only requires ρ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>L</i></sub> ∙ τ + 2 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> points to be sampled per unit time τ. To enhance the robustness of the measurement process in actual circuit systems, a corrected annihilating filter method is proposed, and the parameter measurement error when employing this method is analyzed. Numerical and hardware experiments demonstrate that, compared to classical anti-noise algorithms, the proposed method not only measures the parameters of FRI signals contaminated by shot noise more accurately, but also significantly improves the robustness of the measurement process in non-ideal environments.

A General FRI Sampling System for Pulse Streams and the Multichannel Synchronization Method

Pulse streams are a kind of finite rate of innovation (FRI) signal that widely exist in many engineering applications. According to the FRI theory, nonconsecutive sampled Fourier coefficients (FCs) have better parameter estimation effects than the consecutive ones. In this brief, we propose a general FRI sampling system for pulse streams based on multichannel random modulation (MRM). Jointly using multichannel samples, multiple sets of consecutive FCs of distinct frequency bands can be analytically calculated by solving the least-squares problems. The minimum sampling rate for each channel is determined by the number of pulses rather than the pulse bandwidth. The proposed sampling system is general for pulses of distinct frequency bands, and we extend it to the frequency-agile radar (FAR). Besides, the multichannel synchronization problem is analyzed, and a calibration method when the pseudo-random sequences (PRSs) are not synchronized is proposed. Simulation experiments demonstrate the effectiveness of the proposed sampling system.

Sequential-Based Range-Doppler Estimation With Fast and Slow Time Sub-Nyquist Sampling

Range-Doppler detection with pulse sequences exists widely in Radar, sonar, ultrasound imaging, and many other engineering applications. However, the classic methods require a high sampling rate for wideband pulses, and the Doppler estimation with slow time sub-Nyquist sampling may suffer from the phase wrapping. In this brief, a sequential-based range-Doppler estimation method with fast and slow time sub-Nyquist sampling is proposed. Utilizing the finite rate of innovation (FRI) theory, the range parameters can be estimated from several Fourier coefficients. The fast time sampling rate is determined by the number of the unknown parameters, instead of the signal bandwidth. Then, the aliasing-free Doppler parameters are estimated by the subspace method and the Chinese remainder theorem (CRT). The sequential-based method enables to estimate each Doppler parameter separately, and the Doppler estimation is transformed into a robust CRT problem, which enhances the robustness of the Doppler estimation. Simulation and hardware results demonstrate the effectiveness of the proposed method.

Sub-Nyquist Sampling of ECG Signals With Differentiated VPW Optimization Model

In recent years, developments in wearable technologies and wireless communication have enabled wearable ECG monitoring. Long-term ECG monitoring result in large amounts of data. To sub-Nyquist sampling ECG signal, recently developed finite rate of innovation (FRI) technology is a practical option. However, existing FRI sampling schemes generally suffered from large model mismatch errors and insufficient reconstruction accuracy. In this paper, a sub-Nyquist sampling scheme based on the differentiated VPW optimization model is proposed for ECG signals sampling. The original ECG signal is modeled as a linear combination of several differentiated Variable Pulse Width (VPW) pulses and a model mismatch error signal. Then, a two-channel sampling framework is proposed to sample ECG signals with a sub-Nyquist rate. To improve the accuracy of the reconstruction, optimizing the reconstruction is considered. We formulate the objective function of the optimization to minimize the model mismatch error signal. A particle filter-based optimization method is proposed to solve the objective function with a high-dimensional variable. Block coordinate descent (BCD) technique is considered in the proposed optimization method. Finally, we conduct several simulations with real ECG signals from MIT-BIH Arrhythmia Database to test the performance of our proposed scheme. Simulation results have demonstrated the reconstruction accuracy of our proposed scheme outperforms other related schemes.

Sub-Nyquist sampling system for pulse streams based on non-ideal filters

Recent developed finite rate of innovation (FRI) sampling framework provides an efficient way for sub-Nyquist sampling of pulse streams, which are widely applied to the fields of radar and wireless communication. Most researches of FRI sampling structure were implemented in hardware based on filter banks. However, the non-ideal effects of the filters would lead to low reconstruction accuracy. In this paper, we propose a two-channel sub-Nyquist sampling system for pulse streams, which significantly improves the reconstruction performance of the system under non-ideal filters hardware environment. The proposed scheme consists of two parallel sampling channels. The pulse streams and its basis signal are modulated and filtered with non-ideal low pass filters, followed by sampling at twice the cut-off frequency. By setting the cut-off frequency of the filter, the number of samples of different numbers is obtained, and the Fourier coefficients of the different frequency bands of these signals are obtained from the obtained samples. Such a technique allows for recovery of an arbitrary pulse with different frequency spectrum, which greatly improves the flexibility of the sampling system. Then we propose a joint parameters recovery algorithm to reconstruct the pulse streams, which can effectively eliminate non-ideal effects. Finally, we design a hardware prototype to verify the feasibility and effectiveness of the proposed system. Simulation and hardware experiments have shown that the proposed system greatly improves the signal reconstruction accuracy.

Parameter Measurement of LFM Signal With FRI Sampling and Nuclear Norm Denoising

Measurement of linear frequency modulation (LFM) signal is significant for radar, communication, and electronic reconnaissance fields. An LFM signal is a wideband signal whose frequency varies linearly with time, and traditional measurement methods require very high sampling rates and heavy processing to estimate parameters of the LFM signal. In this article, we propose a multichannel cooperative sampling (MCS) system based on the finite rate of innovation (FRI) theory to sample and estimate the parameters of the real-valued LFM pulse sequence (LFMPS). The MCS system consists of three parts: the autocorrelation sampling structure (ACSS), the time-delayed ACSS, and the quadrature time-staggered sampling structure (QTSS). These three parts can sample the LFMPS with sub-Nyquist sampling rate, and using the subspace method in time or frequency domain, the discontinuity locations (DLs), chirp rates, and initial frequencies (IFs) of the LFMPS can be estimated with the sub-Nyquist samples, respectively. The sampling rate of the MCS system is determined by the rate of innovation of the LFMPS, instead of signal bandwidth (BW). The minimum number of samples required for parameter estimation is proven by theoretical analysis. In addition, a nuclear norm denoising algorithm is proposed based on the low-rank property of the signal subspace, which significantly improved the performance of the measurement system in the noise environment. Simulation and hardware experimental results demonstrate the effectiveness and robustness of the proposed method.

Deep Unfolded Recovery of Sub-Nyquist Sampled Ultrasound Images.

The most common technique for generating B-mode ultrasound (US) images is delay-and-sum (DAS) beamforming, where the signals received at the transducer array are sampled before an appropriate delay is applied. This necessitates sampling rates exceeding the Nyquist rate and the use of a large number of antenna elements to ensure sufficient image quality. Recently, we proposed methods to reduce the sampling rate and the array size relying on image recovery using iterative algorithms based on compressed sensing (CS) and the finite rate of innovation (FRI) frameworks. Iterative algorithms typically require a large number of iterations, making them difficult to use in real time. In this article, we propose a reconstruction method from sub-Nyquist samples in the time and spatial domain, which is based on unfolding the iterative shrinkage thresholding algorithm (ISTA), resulting in an efficient and interpretable deep network. The inputs to our network are the subsampled beamformed signals after summation and delay in the frequency domain, requiring only a subset of the US signal to be stored for recovery. Our method allows reducing the number of array elements, sampling rate, and computational time while ensuring high-quality imaging performance. Using in vivo data, we demonstrate that the proposed method yields high-quality images while reducing the data volume traditionally used up to 36 times. In terms of image resolution and contrast, our technique outperforms previously suggested methods as well as DAS and minimum-variance (MV) beamforming, paving the way to real-time applicable recovery methods.

Image reconstruction based on frequency domain feature extraction for EMT

Image reconstruction of electromagnetic tomography (EMT) is often ill-posed due to the limited prior information about imaging features, leading to low reconstruction accuracy. In this paper, a novel image reconstruction method based on frequency domain feature extraction for EMT is proposed to improve the reconstruction accuracy. The two-dimensional EMT image is first modeled as a one-dimensional finite-length streams of Dirac signal, which is a typical signal with finite rate of innovation (FRI). Then the reconstruction results of the recent algorithms, such as the LBP, Landweber and TV algorithms, can be also modeled as FRI signals. The frequency domain feature of such signals can be extracted with a distributed FRI sampling system. Based on the obtained FRI samples, a new EMT measurement equation is established, and a joint image reconstruction algorithm is proposed, to improve the quality of the reconstructed image. Finally, simulation results have shown that the proposed method outperforms the related methods with better indicators such as image error and correlation coefficient.

FRI Sampling of Parametric Signals With Non-Ideal Sinc Kernel

Recent developed finite rate of innovation (FRI) theory provides an efficient way for sub-Nyquist sampling of wideband parametric signals in the fields of radar and communication. However, the existing hardware schemes of FRI sampling systems have not considered the non-ideal effects of physical components, which lead to low accuracy in the signal reconstruction process. In this brief, we propose a FRI sampling system for parametric signals with non-ideal sinc kernel, which significantly improves the reconstruction performance under non-ideal hardware environment. The proposed system consists of two parallel sampling channels. The first channel samples the parametric signal at a low speed after filtered with a LPF, which is used to obtain a part of Fourier coefficients. The other channel obtains a part of Fourier coefficients of its basis signal in the same way. To eliminate the non-ideal effects of physical components, we propose a parameters joint estimation algorithm by using the obtained Fourier coefficients. We also provide a hardware platform design scheme for implementing our system. According to simulation and hardware experiment results, the proposed system can eliminate non-ideal effects in the hardware process, and greatly improve the signal reconstruction accuracy.

Two-Dimensional Separable Gridless Direction-of-Arrival Estimation Based on Finite Rate of Innovation

In order to solve the problem that the gridless DOA estimation algorithms based on generalized finite rate of innovation (FRI) signal reconstruction model are not suitable for two-dimensional DOA estimation using planar array, a separable gridless DOA estimation algorithm exploiting bi-orthogonal sparse linear array (BSLA) structure is proposed in this article, which is called 2D-SGFRI. The 2D-SGFRI algorithm firstly recovers the covariance data of the virtual array formed by BSLA through the matrix completion method, so as to obtain the complete covariance data vectors about two independent parameters respectively. Next, since the covariance data vector satisfies the constraints of annihilation filter equations, the generalized FRI signal reconstruction model can be utilized to retrieve DOA from the covariance data vector. Compared with the existing DOA estimation algorithms based on generalized FRI signal reconstruction model, the 2D-SGFRI algorithm can be can be effectively applied to two-dimensional DOA estimation, and can obtain stable estimation results. At the same time, due to the reduction of the dimension of positive semidefinite matrix, the 2D-SGFRI algorithm can significantly reduce the computational complexity compared with the two-dimensional DOA estimation algorithms based on atomic norm minimization (ANM). A series of simulation experiments are shown to verify the effectiveness and superiority of 2D-SGFRI algorithm.

Generic FRI-Based DOA Estimation: A Model-Fitting Method

Direction of arrival (DOA) estimation is a classical topic in source localization. Notably, a reliable grid-free sparse representation algorithm, called FRI (finite rate of innovation) algorithm, was proposed to recover the finite number of Dirac pulses from a stream of 1D temporal samples, which also offers a efficient solution to the DOA estimation problem. Typically, FRI method assumes uniform sampling with single snapshot. However, the actual situation is richer and more diverse. Motivated by the requests of practical applications (e.g. array deployment, algorithm run-speed, etc.), a generic FRI method is proposed to tackle the more general case in practice, i.e. non-uniform sampling with multiple snapshots. Instead of annihilating the measured sensor data, a model-fitting method is used to robustly retrieve the sparse representation (i.e. DOAs and associated amplitudes) of the 1D samples. We demonstrate that our algorithm can handle challenging DOA tasks with high-resolution, which we validate in various conditions, such as multiple coherent sources, insufficient signal snapshots, low signal-to-noise ratio (SNR), etc. Moreover, we show that the computational complexity of our algorithm mainly scales with the number of sources and varies very slowly with the number of samples and snapshots, which meets the needs of a wider range of practical applications.

Hybrid Sparse Expansion and Separable Hybrid Prony Method

Signal sparsity plays an essential role in signal compression and reconstruction. Prony-like methods are widely used in the relevant applications e.g., the recovery of signals with finite rate of innovation (FRI). However, these methods are usually limited to the structured function expansions of a single family, i.e., the considered structured functions belonging to the eigenfunctions of a single operator. In this paper, we investigate hybrid Prony models (HPMs), wherein functions can be expanded sparsely by the combination of eigenfunctions of two different operators $\mathcal {L}_1$ and $\mathcal {L}_2$ . Although the Prony-like methods cannot directly deal with HPMs, we show that HPMs can be directly converted to the generalized Prony model recently considered by Peter and Plonka when the operator pair $\lbrace \mathcal {L}_1, \mathcal {L}_2\rbrace$ satisfies certain conditions; we refer to these functions as separable HPMs (SHPMs). Similar to the Prony method, the problem of nonlinear parameter estimation in SHPMs can be converted to the problem of finding the roots of polynomials. Examples and numerical results are given to illustrate SHPMs.

Ellipse Recovery From Blurred Binary Images.

In this paper, we address the problem of ellipse recovery from blurred shape images. A shape image is a binary-valued (0/1) image in continuous-domain that represents one or multiple shapes. In general, the shapes can also be overlapping. We assume to observe the shape image through finitely many blurred samples, where the 2D blurring kernel is assumed to be known. The samples might also be noisy. Our goal is to detect and locate ellipses within the shape image. Our approach is based on representing an ellipse as the zero-level-set of a bivariate polynomial of degree 2. Indeed, similar to the theory of finite rate of innovation (FRI), we establish a set of linear equations (annihilation filter) between the image moments and the coefficients of the bivariate polynomial. For a single ellipse, we show that the image can be perfectly recovered from only 6 image moments (improving the bound in [Fatemi et al., 2016]). For multiple ellipses, instead of searching for a polynomial of higher degree, we locally search for single ellipses and apply a pooling technique to detect the ellipse. As we always search for a polynomial of degree 2, this approach is more robust against additive noise compared to the strategy of searching for a polynomial of higher degree (detecting multiple ellipses at the same time). Besides, this approach has the advantage of detecting ellipses even when they intersect and some parts of the boundaries are lost. Simulation results using both synthetic and real world images (red blood cells) confirm superiority of the performance of the proposed method against the existing techniques.