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While frequency-resolved optical gating (FROG) is widely used incharacterizing the ultrafast pulse in optics, analytic signals are oftenconsidered in time-frequency analysis and signal processing, especially whenextracting instantaneous features of events. In this paper we examine the phaseretrieval (PR) problem of analytic signals in $\Bbb{C}^N$ by their FROGmeasurements. After establishing the ambiguity of the FROG-PR of analyticsignals, we found that the FROG-PR of analytic signals of even lengths isdifferent from that of analytic signals of odd lengths, and it is alsodifferent from the case of $B$-bandlimited signals with $B \leq N/2$. Theexisting approach to bandlimited signals can be applied to analytic signals ofodd lengths, but it does not apply to the even length case. With the help oftwo relaxed FROG-PR problems and a translation technique, we develop anapproach to FROG-PR for the analytic signals of even lengths, and prove that inthis case the generic analytic signals can be uniquely (up to the ambiguity)determined by their $(3N/2+1)$ FROG measurements.

FROG-measurement based phase retrieval for analytic signals