[Paper Review] Estimating a Signal from a Magnitude Spectrogram via Convex Optimization

Dennis L. Sun, Julius O. Smith|arXiv (Cornell University)|Sep 10, 2012

Sparse and Compressive Sensing Techniques12 references18 citations

TL;DR

This paper proposes a convex optimization framework, STliFT, for reconstructing a signal from its magnitude spectrogram by lifting the problem to a rank-1 matrix estimation task, enabling global optimality guarantees. It outperforms the standard Griffin-Lim algorithm in noise-free scenarios and shows improved robustness, especially with small hop sizes and shorter window lengths.

ABSTRACT

The problem of recovering a signal from the magnitude of its short-time Fourier transform (STFT) is a longstanding one in audio signal processing. Existing approaches rely on heuristics that often perform poorly because of the nonconvexity of the problem. We introduce a formulation of the problem that lends itself to a tractable convex program. We observe that our method yields better reconstructions than the standard Griffin-Lim algorithm. We provide an algorithm and discuss practical implementation details, including how the method can be scaled up to larger examples.

Motivation & Objective

• To address the long-standing challenge of phase retrieval in audio signal processing, where only magnitude spectrograms are available.
• To overcome the limitations of nonconvex optimization methods like Griffin-Lim, which often converge to local optima.
• To develop a convex formulation that guarantees global convergence and improved reconstruction quality.
• To enable practical signal recovery from magnitude spectrograms in applications such as speech enhancement and time-scale modification.
• To scale the method to larger audio signals through segmentation and phase-free running on stationary regions.

Proposed method

• The method formulates the signal recovery problem as a convex optimization task by lifting the signal estimation to a rank-1 matrix $ X = xx^* $, where $ X $ is estimated from the magnitude spectrogram.
• It uses a semidefinite program to minimize the least-squares difference between the estimated and desired magnitude spectrograms under the constraint that $ X $ is positive semidefinite and has rank one.
• The algorithm alternates between estimating the matrix $ X $ via convex relaxation and recovering the signal $ x $ via spectral decomposition of $ X $.
• It incorporates windowing and STFT frame constraints to ensure consistency across overlapping time-frequency bins.
• The method is parallelizable by segmenting the signal into smaller chunks, reducing computational load while preserving reconstruction quality.
• For stationary regions, phase is allowed to free-run, reducing the need for full optimization across the entire signal.

Experimental results

Research questions

• RQ1 Can a convex optimization framework be designed to recover a signal from its magnitude spectrogram, avoiding local optima inherent in nonconvex methods?
• RQ2 How does the proposed STliFT method compare to the Griffin-Lim algorithm in terms of reconstruction accuracy and convergence behavior?
• RQ3 What is the impact of window length and hop size on the performance of convex versus nonconvex phase retrieval methods?
• RQ4 Can the convex formulation be scaled to large audio signals without prohibitive computational cost?
• RQ5 Does the method maintain superior performance in noisy conditions where no exact signal matches the given magnitude spectrogram?

Key findings

• In noise-free conditions, STliFT consistently recovers the original signal from its magnitude spectrogram, while Griffin-Lim often fails to converge to the global optimum.
• For small hop sizes (e.g., R=1), STliFT achieves median relative error of 0.04% on 19-sample signals, significantly outperforming Griffin-Lim’s 0.03% only in the best case.
• With larger hop sizes (e.g., R=7), Griffin-Lim performs better (median error 0.40%) than STliFT (4.62%), indicating a trade-off in sparse frame coverage.
• In noisy setups, STliFT shows superior performance for small hop sizes and short windows, achieving median errors as low as 0.03% for window length 19 and hop size 1.
• As window length increases, the performance gap between STliFT and Griffin-Lim narrows, with both methods converging to similar error levels for R=1.
• The algorithm remains robust across multiple initializations and consistently avoids the local minima that plague Griffin-Lim.