One-Bit Phase Retrieval: More Samples Means Less Complexity?

Abstract

The classical problem of phase retrieval has found a wide array of applications in optics, imaging and signal processing. In this paper, we consider the phase retrieval problem in a one-bit setting, where the signals are sampled using one-bit analog-to-digital converters (ADCs). A significant advantage of deploying one-bit ADCs in signal processing systems is their superior sampling rates as compared to their high-resolution counterparts. This leads to an enormous amount of one-bit samples gathered at the output of the ADC in a short period of time. We demonstrate that this advantage pays extraordinary dividends when it comes to convex phase retrieval formulations—namely that the often encountered matrix semi-definiteness constraints as well as rank constraints (that are computationally prohibitive to enforce), become redundant for phase retrieval in the face of a growing sample size. Several numerical results are presented to illustrate the effectiveness of the proposed methodologies

UNO: Unlimited Sampling Meets One-Bit Quantization

“…4, adding more inequality constraints to (28) leads to shrinkage of this polyhedron. We now prove this result, i.e., in a probabilistic sense, that increasing the number of samples leads to the reconstruction error approaching zero, and that the resulting overdetermined linear system of inequalities guarantees the convergence of RKA [32,49,60]. In other words, using an abundant number of samples (or oversampling in one-bit), the probability of creating the finite-volume space around the desired point x is increased.…”

“…To reconstruct x from the sign data r ( ) m =1 , we solve the polyhedron search problem through RKA because of its optimal projection and linear convergence in expectation [32,49,60].…”

“…The error in RKA reconstruction may be viewed as noise for modulo samples. According to [43,Theorem 3], the sampling rate for the contaminated modulo samples in (32) to reconstruct the bandlimited samples…”

“…In particular, when zero threshold is used, the power information of the input signal x is lost in one-bit data because the signs of x and ηx are identical for η > 0. This problem has been addressed in a few recent works [28,[30][31][32][33][34][35], which show that time-varying sampling thresholds enable better estimation of the signal characteristics. In particular, time-varying thresholds were considered for the covariance recovery from one-bit measurements in [28].…”

“…In non-stationary scenarios, [33] applied the modified arcsine law to utilize time-varying sampling thresholds. Applications of one-bit sampling to diverse problems such as sparse parameter estimation [31], localization [36], and phase retrieval [32] have also appeared in the contemporary literature.…”