Regular Representations of Time-Frequency Groups

In this paper, we study the Plancherel measure of a class of non-connected nilpotent groups which is of special interest in Gabor theory. Let $G$ be a time-frequency group. More precisely, that is $G=\left\langle T_{k},M_{l}:k\in\mathbb{Z}^{d},l\in B\mathbb{Z}^{d}\right\rangle ,$ $T_{k}$, $M_{l}$ are translations and modulations operators acting in $L^{2}(\mathbb{R}^{d}),$ and $B$ is a non-singular matrix. We compute the Plancherel measure of the left regular representation of $G\$which is denoted by $L.$ The action of $G$ on $L^{2}(\mathbb{R}^{d})$ induces a representation which we call a Gabor representation. Motivated by the admissibility of this representation, we compute the decomposition of $L$ into direct integral of irreducible representations by providing a precise description of the unitary dual and its Plancherel measure. As a result, we generalize Hartmut F\"uhr's results which are only obtained for the restricted case where $d=1$, $B=1/L,L\in\mathbb{Z}$ and $L>1.$ Even in the case where $G$ is not type I, we are able to obtain a decomposition of the left regular representation of $G$ into a direct integral decomposition of irreducible representations when $d=1$. Some interesting applications to Gabor theory are given as well. For example, when $B$ is an integral matrix, we are able to obtain a direct integral decomposition of the Gabor representation of $G.

Canonical time-frequency, time-scale, and frequency-scale representations of time-varying channels

Mobile communication channels are often modeled as linear time-varying filters or, equivalently, as time-frequency integral operators with finite support in time and frequency. Such a characterization inherently assumes the signals are narrowband and may not be appropriate for wideband signals. In this paper time-scale characterizations are examined that are useful in wideband time-varying channels, for which a time-scale integral operator is physically justifiable. A review of these time-frequency and time-scale characterizations is presented. Both the time-frequency and time-scale integral operators have a two-dimensional discrete characterization which motivates the design of time-frequency or time-scale rake receivers. These receivers have taps for both time and frequency (or time and scale) shifts of the transmitted signal. A general theory of these characterizations which generates, as specific cases, the discrete time-frequency and time-scale models is presented here. The interpretation of these models, namely, that they can be seen to arise from processing assumptions on the transmit and receive waveforms is discussed. Out of this discussion a third model arises: a frequency-scale continuous channel model with an associated discrete frequency-scale characterization.Comment: To appear in Communications in Information and Systems - special issue in honor of Thomas Kailath's seventieth birthda

Sparsity in time-frequency representations

We consider signals and operators in finite dimension which have sparse time-frequency representations. As main result we show that an $S$-sparse Gabor representation in $\mathbb{C}^n$ with respect to a random unimodular window can be recovered by Basis Pursuit with high probability provided that $S\leq Cn/\log(n)$. Our results are applicable to the channel estimation problem in wireless communications and they establish the usefulness of a class of measurement matrices for compressive sensing

A Recurrent Encoder-Decoder Approach with Skip-filtering Connections for Monaural Singing Voice Separation

The objective of deep learning methods based on encoder-decoder architectures for music source separation is to approximate either ideal time-frequency masks or spectral representations of the target music source(s). The spectral representations are then used to derive time-frequency masks. In this work we introduce a method to directly learn time-frequency masks from an observed mixture magnitude spectrum. We employ recurrent neural networks and train them using prior knowledge only for the magnitude spectrum of the target source. To assess the performance of the proposed method, we focus on the task of singing voice separation. The results from an objective evaluation show that our proposed method provides comparable results to deep learning based methods which operate over complicated signal representations. Compared to previous methods that approximate time-frequency masks, our method has increased performance of signal to distortion ratio by an average of 3.8 dB

A planar large sieve and sparsity of time-frequency representations

With the aim of measuring the sparsity of a real signal, Donoho and Logan introduced the concept of maximum Nyquist density, and used it to extend Bombieri's principle of the large sieve to bandlimited functions. This led to several recovery algorithms based on the minimization of the $L_{1}$-norm. In this paper we introduce the concept of {\ planar maximum} Nyquist density, which measures the sparsity of the time-frequency distribution of a function. We obtain a planar large sieve principle which applies to time-frequency representations with a gaussian window, or equivalently, to Fock spaces, $\mathcal{F}_{1}\left( \mathbb{C}\right)$, allowing for perfect recovery of the short-Fourier transform (STFT) of functions in the modulation space $M_{1}$ (also known as Feichtinger's algebra $S_{0}$) corrupted by sparse noise and for approximation of missing STFT data in $M_{1}$, by $L_{1}$-minimization

Histogram of gradients of Time-Frequency Representations for Audio scene detection

This paper addresses the problem of audio scenes classification and contributes to the state of the art by proposing a novel feature. We build this feature by considering histogram of gradients (HOG) of time-frequency representation of an audio scene. Contrarily to classical audio features like MFCC, we make the hypothesis that histogram of gradients are able to encode some relevant informations in a time-frequency {representation:} namely, the local direction of variation (in time and frequency) of the signal spectral power. In addition, in order to gain more invariance and robustness, histogram of gradients are locally pooled. We have evaluated the relevance of {the novel feature} by comparing its performances with state-of-the-art competitors, on several datasets, including a novel one that we provide, as part of our contribution. This dataset, that we make publicly available, involves $19$ classes and contains about $900$ minutes of audio scene recording. We thus believe that it may be the next standard dataset for evaluating audio scene classification algorithms. Our comparison results clearly show that our HOG-based features outperform its competitor