FROG-measurement based phase retrieval for analytic signals

TL;DR

This paper investigates phase retrieval of analytic signals using FROG measurements, revealing differences based on signal length parity and proposing a new method for even-length signals with unique recovery guarantees.

Contribution

It introduces a novel approach for FROG-based phase retrieval of even-length analytic signals, addressing previous limitations and establishing conditions for unique signal reconstruction.

Findings

FROG-PR ambiguity for analytic signals is characterized.

Distinct behaviors for even and odd length signals are identified.

Unique recovery of even-length signals is proven with (3N/2+1) measurements.

Abstract

While frequency-resolved optical gating (FROG) is widely used in characterizing the ultrafast pulse in optics, analytic signals are often considered in time-frequency analysis and signal processing, especially when extracting instantaneous features of events. In this paper we examine the phase retrieval (PR) problem of analytic signals in $\Bbb{C}^N$ by their FROG measurements. After establishing the ambiguity of the FROG-PR of analytic signals, we found that the FROG-PR of analytic signals of even lengths is different from that of analytic signals of odd lengths, and it is also different from the case of $B$-bandlimited signals with $B \leq N/2$. The existing approach to bandlimited signals can be applied to analytic signals of odd lengths, but it does not apply to the even length case. With the help of two relaxed FROG-PR problems and a translation technique, we develop an approach to FROG-PR for the analytic signals of even lengths, and prove that in this case the generic analytic signals can be uniquely (up to the ambiguity) determined by their $(3N/2+1)$ FROG measurements.