Blind demixing and deconvolution refers to the problem of simultaneous deconvolution of several source signals from its noisy superposition. This problem appears, amongst others, in the field of Wireless Communication: Many sensors sporadically communicate only short messages over unknown channels. We show that robust recovery of message and channel vectors can be achieved via convex recovery. This requires that random linear encoding is applied at the devices and that the number of required measurements at the receiver scales essentially with the degrees of freedom of the overall estimation problem. Thus, the scaling is linear in the number of source signals. This significantly improves previous results.
Compressed sensing allows performing much fewer measurements than advised by the Shannon sampling theory. This is surprising because it requires the solution of a system of equations with much fewer equations than unknowns. This is possible if one can assume sparsity of the solution, which means that only a few components of the solution are significantly different from zero. An important ingredient for compressed sensing is the restricted isometry property (RIP) of the sensing matrix, which is satisfied for certain types of random measurement ensembles. Then a sparse solution can be found by minimizing the ℓ1-norm. Using standard approaches, photoacoustic imaging generally neither satisfies sparsity of the data nor the RIP. Therefore, no theoretical recovery guarantees could be given. Despite ℓ1- minimization has been used for photoacoustic image reconstruction, only marginal improvements in comparison to classical photoacoustic reconstruction have been observed. We propose the application of a sparsifying temporal transformation to the detected pressure signals, which allows obtaining theoretical recovery guarantees for our compressed sensing scheme. Such a sparsifying transform can be found because spatial and temporal evolution of the pressure wave are not independent, but connected by the wave equation. We give an example of a sparsifying transform and apply our compressed sensing scheme to reconstruct images from simulated data.
ΣΔ-modulation is an A/D-conversion method which represents a bandlimited signal by sequences of ±1 whose
local averages approximate the function values. The best bounds for the decay rate of the script-l∞-error arising from
such quantization schemes have been given by Güntürk.1 He constructs an infinite family of schemes which lead
to an algorithm that establishes exponential error decay with decay rate 0.077. In this paper we improve his
construction introducing an additional symmetry, which is suggested by numerical experiments. To show that
the modified schemes are still stable, we use the asymptotics of the Γ-function. This leads to a bound of 0.088
for the error decay rate.