Finite Rate Of Innovation Sampling Research Articles

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.

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.

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.

Finite Rate of Innovation Sampling of Gaussian Pulse Streams with Variable Shape

Finite Rate of Innovation Sampling of Gaussian Pulse Streams with Variable Shape

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.

A General Sub-Nyquist Sampling System for Pulse Streams

The recent finite rate of innovation (FRI) framework has shown that pulse streams can be sampled at the rate of innovation and recovered from a set of Fourier coefficients. However, due to the individual variation of the pulse frequency spectrum, previous FRI sampling systems vary owing to different pulse shapes. In this paper, we propose a general sub-Nyquist sampling and recovery method for pulse streams that is available for many kinds of pulses. The proposed scheme exploits the spread-spectrum technique from a random demodulator (RD), in which an analog mixing front end aliases the spectrum. Such a technique allows for recovery of an arbitrary pulse with the baseband of the frequency spectrum, which greatly improves the flexibility of the sampling system. But unlike the fast switching rate of the mixing signal in RD, the proposed method has a lower rate requirement and is simpler to implement in practice. To recover the unknown parameters of the pulse streams from the obtained aliased Fourier coefficients, we quantize the analog time axis with a fit number of uniform bins and propose a sparsity-based recovery algorithm. Finally, simulation results demonstrate the effectiveness and robustness of the proposed method.

Novel Circuit Implementation Method for Pulse Signal Finite Rate of Innovation Sparse Sampling Based on an Improved Exponential Reproducing Kernel

Finite rate of innovation sampling is a new signal sparse sampling method based on signal information freedom, which can considerably reduce the amount of sampling data. However, the researches on hardware circuit direct implementation of the method are at an initial stage. Therefore, a novel hardware circuit implementation method based on an exponential reproducing kernel (ERK) was developed with an improved ERK sparse sampling theoretical framework, which comprised an analog rational kernel (ARK) and digital rational kernel (DRK). The parameter constraint conditions and selection method for the improved ERK were analyzed. The performance of the ARK hardware circuit and the DRK algorithm were also examined. The designed sparse sampling system was applied to a pipeline flaw ultrasonic inspection by using the sparse sampling data of the ultrasonic signal obtained directly. The experimental results revealed that the proposed method could directly obtain the signal sparse sampling data and accurately reconstruct the initial signal parameters by using a spectrum estimation algorithm. Moreover, the signal sampling rate and the amount of sampling data were reduced considerably.

Asymmetric Pulse Modeling for FRI Sampling

We consider sampling and reconstruction of finite-rate-of-innovation (FRI) signals such as a train of pulses, where the pulses have varying degrees of asymmetry. We address the problem of asymmetry modeling starting from a given symmetric prototype. We show that among the class of unitary operators that are linear and invariant to translation and scale, the fractional Hilbert (FrH) operator is unique for parametrically modeling pulse asymmetry. The FrH operator is obtained by a trigonometric interpolation between the standard Hilbert and identity operators, where the interpolation weights are determined by the degree of asymmetry. The FrH operators are also steerable , which allows for estimation of the asymmetry factors, in addition to the delays and amplitudes, using the high-resolution spectral estimation techniques that are used for solving standard FRI problems. We also develop the discrete counterpart using discrete FrH operators and show that all the desirable properties carry over smoothly to the discrete setting as well. We derive closed-form expressions for the Cramer–Rao bounds and Hammersley–Chapman–Robbins bound, on the variances of the estimators for continuous and discrete parameters, respectively. Experimental results show that the proposed estimators have variances that meet the lower bounds. We demonstrate an application of the proposed discrete FrH methodology on real electrocardiogram (ECG) signals in the presence of noise. Specifically, we show how the asymmetry of QRS complexes in various channels of an ECG signal could be modeled accurately.

A Simplified FRI Sampling System for Pulse Streams Based on Constraint Random Modulation

The recent finite rate of innovation (FRI) framework provides effective sub-Nyquist sampling of pulse streams, allowing recovery of such signals from a set of Fourier coefficients. In this brief, a multi-channel FRI sampling system is presented to sample distinct bands of Fourier coefficients. This is achieved through modulating the desired spectrum band to baseband and then filtering with a low-pass filter. However, the modulation process will lead to the spectrum aliasing and unavailability. A modulation frequency selection strategy is proposed to solve this problem, which allows obtaining reconfigurable Fourier coefficients from the aliasing spectrum. Combining with multi-channel structure, we present a simple and efficient way to sample distinct bands of the pulse streams’ spectrum. Finally, a design and implementation of the hardware prototype is presented. Simulation and hardware experiment results demonstrate the effectiveness and robustness of our system.

Sub-Nyquist Sampling and Recovery of Pulse Streams With the Real Parts of Fourier Coefficients

We consider the problem of sampling pulse streams with known shapes. The recent finite rate of innovation (FRI) framework has shown that such signals can be sampled with perfect reconstruction at their rate of innovation, which is usually much lower than the Nyquist rate. Although FRI sampling of pulse streams was treated in various works, either the work was unstable for high rate of innovation, or the sampling stage was complex and redundant. In this paper, we propose an FRI sampling and recovery method for pulse streams, which is based on the real parts of the Fourier coefficients. The proposed method is simple and efficient, and leads to stable recovery even when the rate of innovation is very high. This is achieved through modulating the input signal in each channel with a properly chosen cosine signal, followed by filtering with a low-pass filter. Since the modulating process will lead to the signal spectrum aliasing, we propose a spectrum de-aliasing algorithm to solve this problem, resulting in the real parts of a band of Fourier coefficients from each two channels. Combining with the multi-channel sampling structure, we propose a more efficient way to obtain arbitrary frequency bands from the aliased spectrum, which improves the utility of the signal spectrum. By using a sparsity-based recovery algorithm, the time delays and amplitudes of the pulse streams can be recovered from the obtained real parts of the Fourier coefficients. Finally, simulation results have shown that the proposed scheme is flexible and exhibits better noise robustness than previous approaches.

Sampling Sparse Signals on the Sphere: Algorithms and Applications

We propose a sampling scheme that can perfectly reconstruct a collection of spikes on the sphere from samples of their lowpass-filtered observations. Central to our algorithm is a generalization of the annihilating filter method, a tool widely used in array signal processing and finite-rate-of-innovation (FRI) sampling. The proposed algorithm can reconstruct $K$ spikes from $(K+\sqrt{K})^2$ spatial samples. This sampling requirement improves over previously known FRI sampling schemes on the sphere by a factor of four for large $K$. We showcase the versatility of the proposed algorithm by applying it to three different problems: 1) sampling diffusion processes induced by localized sources on the sphere, 2) shot noise removal, and 3) sound source localization (SSL) by a spherical microphone array. In particular, we show how SSL can be reformulated as a spherical sparse sampling problem.

FRI Sampling With Arbitrary Kernels

This paper addresses the problem of sampling non-bandlimited signals within the Finite Rate of Innovation (FRI) setting. We had previously shown that, by using sampling kernels whose integer span contains specific exponentials (generalized Strang-Fix conditions), it is possible to devise non-iterative, fast reconstruction algorithms from very low-rate samples. Yet, the accuracy and sensitivity to noise of these algorithms is highly dependent on these exponential reproducing kernels - actually, on the exponentials that they reproduce. Hence, our first contribution here is to provide clear guidelines on how to choose the sampling kernels optimally, in such a way that the reconstruction quality is maximized in the presence of noise. The optimality of these kernels is validated by comparing with Cramér-Rao's lower bounds (CRB). Our second contribution is to relax the exact exponential reproduction requirement. Instead, we demonstrate that arbitrary sampling kernels can reproduce the “best” exponentials within quite a high accuracy in general, and that applying the exact FRI algorithms in this approximate context results in near-optimal reconstruction accuracy for practical noise levels. Essentially, we propose a universal extension of the FRI approach to arbitrary sampling kernels. Numerical results checked against the CRB validate the various contributions of the paper and in particular outline the ability of arbitrary sampling kernels to be used in FRI algorithms.

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.

Compressive Sampling of Multiple Sparse Signals Having Common Support Using Finite Rate of Innovation Principles

In sensor networks and communication systems, one is often confronted with sampling multiple sparse signals having a common support set. Multipath channels in a multiple-input multiple-output (MIMO) wireless communication setting is an interesting example where one generally needs to perform channel estimation for each transmit-receive antenna pair. MIMO multipath channels are usually (approximately) sparse and satisfy the common-support property whenever the distances between the antennas are small compared to the distance the electromagnetic wave can travel in the time corresponding to the inverse bandwidth of the communication system. This assumption is satisfied by small and medium bandwidth communication systems like OFDM and CDMA. This leads us to extend the finite rate of innovation sampling and reconstruction scheme to the sparse common-support scenario (SCS-FRI), in which input signals contain Diracs with common locations but arbitrary weights. The goal is to efficiently reconstruct the input signals from a set of uniform samples, making use of the common-support property to improve robustness. We first find the best theoretical performance for the SCS-FRI setup by de riving the Cramer-Rao lower bound. Our results show that for a set of well-separated Diracs, it is the total energy of the Diracs at each common position which determines the bound. We then propose a multichannel reconstruction algorithm and compare its performance with the Cramer-Rao lower bound. Numerical results clearly demonstrate the effectiveness of our proposed sampling and reconstruction scheme in low SNR regimes.

Sampling Schemes for Multidimensional Signals With Finite Rate of Innovation

In this paper, we consider the problem of sampling signals that are nonband-limited but have finite number of degrees of freedom per unit of time and call this number the rate of innovation. Streams of Diracs and piecewise polynomials are the examples of such signals, and thus are known as signals with finite rate of innovation (FRI). We know that the classical (band-limited sine) sampling theory does not enable perfect reconstruction of such signals from their samples since they are not band-limited. However, the recent results on FRI sampling suggest that it is possible to sample and perfectly reconstruct such nonband-limited signals using a rich class of kernels. In this paper, we extend those results in higher dimensions using compactly supported kernels that reproduce polynomials (satisfy Strang-Fix conditions). In fact, the polynomial reproduction property of the kernel makes it possible to obtain the continuous moments of the signal from its samples. Using these moments and the annihilating filter method (Prony's method), the innovative part of the signal, and therefore, the signal itself is perfectly reconstructed. In particular, we present local (directional-derivatives-based) and global (complex-moments-based, Radon-transform-based) sampling schemes for classes of FRI signals such as sets of Diracs, bilevel, and planar polygons, quadrature domains (e.g., circles, ellipses, and cardioids), 2D polynomials with polygonal boundaries, and n-dimensional Diracs and convex polytopes. This work has been explored in a promising way in super-resolution algorithms and distributed compression, and might find its applications in photogrammetry, computer graphics, and machine vision.