# Inverting Spectrogram Measurements via Aliased Wigner Distribution Deconvolution and Angular Synchronization

## TL;DR

This paper introduces a novel two-step method combining aliased Wigner distribution deconvolution and angular synchronization for efficient phase retrieval from subsampled spectrograms, with theoretical guarantees for signal recovery.

## Contribution

It presents two new algorithms for phase retrieval from spectrogram measurements and proves guarantees for signal recovery with fewer measurements and deterministic coded diffraction patterns.

## Key findings

- Successful numerical performance of the proposed algorithms.
- Guarantees for recovering signals from fewer than d measurements.
- Introduction of deterministic coded diffraction patterns with robust recovery.

## Abstract

We propose a two-step approach for reconstructing a signal ${\bf x}\in\mathbb{C}^d$ from subsampled short-time Fourier transform magnitude (spectogram) measurements: First, we use an aliased Wigner distribution deconvolution approach to solve for a portion of the rank-one matrix ${\bf \widehat{{\bf x}}}{\bf \widehat{{\bf x}}}^{*}.$ Second, we use angular syncrhonization to solve for ${\bf \widehat{{\bf x}}}$ (and then for ${\bf x}$ by Fourier inversion). Using this method, we produce two new efficient phase retrieval algorithms that perform well numerically in comparison to standard approaches and also prove two theorems, one which guarantees the recovery of discrete, bandlimited signals ${\bf x}\in\mathbb{C}^{d}$ from fewer than $d$ STFT magnitude measurements and another which establishes a new class of deterministic coded diffraction pattern measurements which are guaranteed to allow efficient and noise robust recovery.