Detection with phaseless measurements

Abstract

We consider the problem of hypothesis testing for detection of a signal in Gaussian noise. We assume that the vector of measurements is unobserved, and that our observations consist of phaseless inner products with a set of known measurement vectors. This is typical of the phase retrieval problem, where the goal is to recover the vector of measurements. We provide a simple estimator for the test statistic that does not necessitate a phaseless recovery method to reconstruct the measurements. Our analysis shows that for random measurement vectors, we can reconstruct the test statistic for any signal from a sufficient number of observations, quadratic in the signal length, using a simple least-squares approach. The primary advantage of this method its simplicity and computational efficiency, which comes at the expense of requiring many more measurements. We show that for Fourier measurements vectors, our approach works only when the signal is also a Fourier vector.

Phase retrieval for block sparsity based on adaptive coupled variational Bayesian learning

“…Through Equation (38), the posterior estimation of z −1 i is further reduced to the estimation of hyperparameters a i and λ i . In McKay's Bessel function distribution, λ i is assumed to be fixed as one [30]; thus, we next focus on the estimation of a i .…”

“…In this section, we discuss the detection of a block-sparse signal from the phaseless measurements, where the PR algorithms are applied [38]. Originally, the received linear measurements of a block-sparse signal from the transmitter can be modelled as…”

“…where μ is the attenuation coefficient of propagation and n 1 is the additive white Gaussian noise vector with variance β −1 1 . However, in some poor climates and environments, the receiver can only obtain the magnitudes of measurements with the form [38]:…”

“…The prBSACVB-EM is compared with prGAMP, prVBEM, TWF, and least squares [38]. Parameters are set as N = 1000, k = 40 and M is variable.…”

“…The result is similar to Figure 4 that prBSACVB-EM has a higher success rate than its counterparts. Meanwhile, the performance of least squares helps prove the conclusion of [38] that the number of measurements required to achieve a high probability should be large enough.…”