# Random phaseless sampling for causal signals in shift-invariant spaces: a zero distribution perspective

## TL;DR

This paper investigates the impossibility of phaseless sampling in complex shift-invariant spaces and proposes a probabilistic sampling method that guarantees signal reconstruction with high probability, especially for causal signals.

## Contribution

It establishes conditions under which random sampling achieves phaseless reconstruction in complex shift-invariant spaces, highlighting differences from real-valued cases and providing probabilistic guarantees.

## Key findings

- Random sampling density of 3 suffices for complex NC signals with GHC.
- Random sampling density of 2 suffices for real NC signals with GHC.
- Numerical simulations confirm the effectiveness of the proposed method.

## Abstract

We proved that the phaseless sampling (PLS) in the linear-phase modulated shift-invariant space (SIS) $V(e^{\textbf{i}\alpha \cdot}\varphi), \alpha\neq0,$ is impossible even though the real-valued function $\varphi$ enjoys the full spark property (so does $e^{\textbf{i}\alpha \cdot}\varphi$). Stated another way, the PLS in the complex-generated SISs is essentially different from that in the real-generated ones. Motivated by this, we first establish the condition on the complex-valued generator $\phi$ such that the PLS of nonseparable causal (NC) signals in $V(\phi)$ can be achieved by random sampling. The condition is established from the generalized Haar condition (GHC) perspective. Based on the proposed reconstruction approach, it is proved that if the GHC holds then with probability $1$, the random sampling density (SD) $=3$ is sufficient for the PLS of NC signals in the complex-generated SISs. For the real-valued case we also prove that, if the GHC holds then with probability $1$, the random SD $=2$ is sufficient for the PLS of real-valued NC signals in the real-generated SISs. For the local reconstruction of highly oscillatory signals such as chirps, a great number of deterministic samples are required. Compared with deterministic sampling, the proposed random approach enjoys not only the greater sampling flexibility but the much smaller number of samples. To verify our results, numerical simulations were conducted to reconstruct highly oscillatory NC signals in the chirp-modulated SISs.