Multichannel sampling of signals with finite rate of innovation

Consider a multichannel sampling system consisting of many acquisition devices observing an input finite rate of innovation (FRI) signal, a non-bandlimited signal that has finite number of parameters [1, 2]. Each acquisition device has access to a delayed version of the input signal where the delays are unknown. By synchronizing the different channels exactly we are able to reduce the number of samples needed from each channel resulting in a more efficient sampling system. Figure 1 shows the described multichannel sampling system where the bank of acquisition devices ϕ1(x, y), ϕ2(x, y),..., ϕN−1(x, y) receive different versions of the input FRI signal g0(x, y). Here, the delay

Sampling signals with finite rate of innovation

Consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation. Examples of signals with a finite rate of innovation include streams of Diracs (e.g., the Poisson process), nonuniform splines, and piecewise polynomials. Even though these signals are not bandlimited, we showthat they can be sampled uniformly at (or above) the rate of innovation using an appropriate kernel and then be perfectly reconstructed. Thus, we prove sampling theorems for classes of signals and kernels that generalize the classic "bandlimited and sinc kernel" case. In particular, we show how to sample and reconstruct periodic and finite-length streams of Diracs, nonuniform splines, and piecewise polynomials using sinc and Gaussian kernels. For infinite-length signals with finite local rate of innovation, we show local sampling and reconstruction based on spline kernels. The key in all constructions is to identify the innovative part of a signal (e.g., time instants and weights of Diracs) using an annihilating or locator filter: a device well known in spectral analysis and error-correction coding. This leads to standard computational procedures for solving the sampling problem, which we show through experimental results. Applications of these new sampling results can be found in signal processing, communications systems, and biological systems

Recovery of bilevel causal signals with finite rate of innovation using positive sampling kernels

Bilevel signal $x$ with maximal local rate of innovation $R$ is a continuous-time signal that takes only two values 0 and 1 and that there is at most one transition position in any time period of 1/R.In this note, we introduce a recovery method for bilevel causal signals $x$ with maximal local rate of innovation $R$ from their uniform samples $x*h(nT), n\ge 1$, where the sampling kernel $h$ is causal and positive on $(0, T)$, and the sampling rate $\tau:=1/T$ is at (or above) the maximal local rate of innovation $R$. We also discuss stability of the bilevel signal recovery procedure in the presence of bounded noises

Innovation Rate Sampling of Pulse Streams with Application to Ultrasound Imaging

Signals comprised of a stream of short pulses appear in many applications including bio-imaging and radar. The recent finite rate of innovation framework, has paved the way to low rate sampling of such pulses by noticing that only a small number of parameters per unit time are needed to fully describe these signals. Unfortunately, for high rates of innovation, existing sampling schemes are numerically unstable. In this paper we propose a general sampling approach which leads to stable recovery even in the presence of many pulses. We begin by deriving a condition on the sampling kernel which allows perfect reconstruction of periodic streams from the minimal number of samples. We then design a compactly supported class of filters, satisfying this condition. The periodic solution is extended to finite and infinite streams, and is shown to be numerically stable even for a large number of pulses. High noise robustness is also demonstrated when the delays are sufficiently separated. Finally, we process ultrasound imaging data using our techniques, and show that substantial rate reduction with respect to traditional ultrasound sampling schemes can be achieved.Comment: 14 pages, 13 figure

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 Cramér-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

Nonuniform average sampling and reconstruction of signals with finite rate of innovation

From an average ( ideal) sampling/reconstruction process, the question arises whether the original signal can be recovered from its average (ideal) samples and, if so, how. We consider the above question under the assumption that the original signal comes from a prototypical space modeling signals with a finite rate of innovation, which includes finitely generated shift-invariant spaces, twisted shift-invariant spaces associated with Gabor frames and Wilson bases, and spaces of polynomial splines with nonuniform knots as its special cases. We show that the displayer associated with an average (ideal) sampling/reconstruction process, which has a well-localized average sampler, can be found to be well-localized. We prove that the reconstruction process associated with an average (ideal) sampling process is robust, locally behaved, and finitely implementable, and thus we conclude that the original signal can be approximately recovered from its incomplete average (ideal) samples with noise in real time. Most of our results in this paper are new even for the special case when the original signal comes from a finitely generated shift-invariant space

Estimating Signals with Finite Rate of Innovation from Noisy Samples: A Stochastic Algorithm

As an example of the recently-introduced concept of rate of innovation, signals that are linear combinations of a finite number of Diracs per unit time can be acquired by linear filtering followed by uniform sampling. However, in reality, samples are rarely noiseless. In this paper, we introduce a novel stochastic algorithm to reconstruct a signal with finite rate of innovation from its noisy samples. Even though variants of this problem has been approached previously, satisfactory solutions are only available for certain classes of sampling kernels, for example kernels which satisfy the Strang-Fix condition. In this paper, we consider the infinite-support Gaussian kernel, which does not satisfy the Strang-Fix condition. Other classes of kernels can be employed. Our algorithm is based on Gibbs sampling, a Markov chain Monte Carlo (MCMC) method. Extensive numerical simulations demonstrate the accuracy and robustness of our algorithm.Comment: Submitted to IEEE Transactions on Signal Processin