Being Low-Rank in the Time-Frequency Plane

Abstract

When solving inverse problems and using optimization methods with matrix variables in signal processing and machine learning, it is customary to assume some low-rank prior on the targeted solution. Nonnegative matrix factorization of spectrograms is a case in point in audio signal processing. However, this low-rank prior is not straightforwardly related to complex matrices obtained from a short-time Fourier -or discrete Gabor -transform (STFT), which is generally defined from and studied based on a modulation operator and a translation operator applied to a so-called window. This paper is a first study of the low-rankness property of time-frequency matrices. We characterize the set of signals with a rank-r (complex) STFT matrix in the case of a unit hop size and frequency step with few assumptions on the transform parameters. We discuss the scope of this result and its implications on low-rank approximations of STFT matrices.

Low-rankness of Complex-valued Spectrogram and Its Application to Phase-aware Audio Processing

“…This is because all columns of S0 are given by a complex-valued scalar multiplication of s0. Since rank(G w (x)) coincides with rank(X) when diag(w) is full rank (i.e., w[l] = 0 ∀l) by the unitarity of F, the rank of the complex-valued spectrogram of a sinusoid is 1 as described in [16].…”

“…1 As described in [16], the complex-valued spectrogram of a sum of H sinusoids becomes a rank-H matrix when the frequencies of sinusoids are on the discrete Fourier grid. Note that [16] considers STFT with maximal redundancy and periodic extension of the signal and window. An exact characterization for more general cases was not presented.…”

“…A very recent study revealed the relation between the rank and phase of a complex-valued spectrogram [16]. The phase of a sum of sinusoids has a distinctive structure which has been widely utilized in audio signal processing [17][18][19][20][21][22], and the rank of its complexvalued spectrogram depends on the number of the sinusoids [16].…”

Characterization of Finite Signals with Low-Rank Stft

“…This results in limitations in audio inverse problems like denoising [4], source separation [3,5] and inpainting [6,7], where the amplitudes and phases are generally estimated sequentially. In [8,9] it was proposed to perform low-rank approximation directly on the complexvalued short-time Fourier transform (STFT), in the context of inpainting missing coefficients in STFT matrices.…”

“…However, such an approach raises the following question: what kind of signals have a low-rank STFT? An answer was given in [8] and leads to a restricted class of possible signals, composed sums of cisoids or Diracs, with frequencies or location on a fixed grid, due to the circularity of the proposed time-frequency transform.The goal of this paper is to provide an insight into the structure of signals with low-rank STFT, without using a circular transform. This is beneficial not only to avoid boundary effects, but also to obtain a larger class of low-rank STFT matrices, allowing exponential damping and polynomial modulations.…”