Reconstruction of finite rate of innovation signals in a noisy scenario: a robust, accurate estimation algorithm
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ORIGINAL PAPER
Reconstruction of finite rate of innovation signals in a noisy scenario: a robust, accurate estimation algorithm Meisam Najjarzadeh1,2 · Hamed Sadjedi1,2 Received: 31 August 2019 / Revised: 14 May 2020 / Accepted: 18 May 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract The paradigmatic example of signals with finite rate of innovation (FRI) is a linear combination of a finite number of Diracs per time unit, a.k.a. spike sequence. Many researchers have investigated the problem of estimating the innovative part of a spike sequence, i.e., time instants tks and weights cks of Diracs and proposed various deterministic or stochastic algorithms, particularly while the samples were corrupted by digital noise. In the presence of noise, maximum likelihood estimation method proved to be a powerful tool for reconstructing FRI signals, which is inherently an optimization problem. Wein and Srinivasan presented an algorithm, namely IterML, for reconstruction of streams of Diracs in noisy situations, which achieved promising reconstruction error and runtime. However, IterML is prone to limited resolution of search grid for tk, so as to avoid a phenomenon known as the curse of dimensionality, that makes it an inappropriate algorithm for applications that require highly accurate reconstruction of time instants. In order to overcome this shortcoming, we introduce a novel modified local best particle swarm optimization (MLBPSO) algorithm aimed at maximizing likelihood estimation of innovative parameters of a sparse spike sequence given noisy low-pass filtered samples. We demonstrate via extensive simulations that MLBPSO algorithm outperforms the IterML in terms of robustness to noise and accuracy of estimated parameters while maintaining comparable computational cost. Keywords Finite rate of innovation signals · Sampling and reconstruction · Modified particle swarm optimization · Maximum likelihood estimation · Spike sequence
1 Introduction Sampling theorem plays a crucial role in signal processing and communications: it tells us how to convert an analog signal into a sequence of numbers, which can then be processed digitally—or coded—on a computer [1]. The most celebrated and widely used sampling theorem is often attributed to Shannon and gives a sufficient condition, namely bandlimitedness, for an exact sampling and interpolation formula [2]. The sampling rate, at twice the maximum frequency present in the signal, is usually called the Nyquist * Hamed Sadjedi [email protected] Meisam Najjarzadeh [email protected] 1
Department of Electrical Engineering, Shahed University, Tehran, Iran
Acoustic Research Laboratory, Shahed University, Tehran, Iran
2
rate. Fortunately, Bandlimitedness is a sufficient but not a necessary condition for perfect reconstruction. However, this opportunity was rarely taken to develop sub-Nyquist sampling schemes prior to the introduction of compressed sensing and FRI theory. With regard to the nature of spike sequences observed in neurophysiology, elec
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