An optimally concentrated Gabor transform for localized time-frequency components

Abstract

Gabor analysis is one of the most common instances of time-frequency signal analysis. Choosing a suitable window for the Gabor transform of a signal is often a challenge for practical applications, in particular in audio signal processing. Many time-frequency (TF) patterns of different shapes may be present in a signal and they can not all be sparsely represented in the same spectrogram. We propose several algorithms, which provide optimal windows for a user-selected TF pattern with respect to different concentration criteria. We base our optimization algorithm on $l^p$-norms as measure of TF spreading. For a given number of sampling points in the TF plane we also propose optimal lattices to be used with the obtained windows. We illustrate the potentiality of the method on selected numerical examples

Entropy-based time-varying window width selection for nonlinear-type time–frequency analysis

“…Several attempts have been proposed in the literature to balance between different window bandwidths. For example, in [32,43], the TFJP was proposed to select the optimal window for the GT based on the Rényi entropy [20]; in [4], the NSGT depends on a frame associated with a non-uniform grid on the TF plane, which comes from the information provided by the signal. The frame could be viewed as the "optimal window" for the GT.…”

“…However, they suffer from severe mode mixing artifacts [25]. There are several nonlinear-type transforms including: the reassignment method (RM) [11,3] and its variations, the TF by convex optimization (Tycoon) [33], the Blaschke decomposition (BKD) [18,19], the empirical mode decomposition (EMD) [30], the iterative filtering [17], the sparsification approach [29], the approximation approach [16], the TF jigsaw puzzle (TFJP) for the Gabor transform (GT) [32,43], the non-stationary GT (NSGT) [5], the matching pursuit [37], and several others. The variations of RM include: the synchrosqueezing transform (SST) [22,53], the synchrosqueezed wave packet transform [54], the synchrosqueezed S-transform [31], the second-order SST [41], the concentration of frequency and time (ConceFT) [23], and the deshape SST [36].…”

On Homotopy Continuation for Speech Restoration

“…The main tool for time-frequency analysis is the Short-Time Fourier Transform, defined for functions f, g ∈ L 2 (ℝ d ) at λ = (α, β) ∈ ℝ 2d by (2) where T α f (t) = f (t − α) is the translation (time shift) and M β f(t) = e 2πiβ·t f(t) is the modulation (frequency shift). The operators π(λ) := M β T α are called time-frequency shifts and the set is a lattice, [11]. The Gabor system (g, Λ) = {π(λ)g; λ ∈ Λ} over the lattice Λ consisting of the translated and modulated versions of one atom g, is a frame for the space L 2 (ℝ d ), if and only if there exist 0 < A ≤ B < ∞ (frame bounds) with (3) We will use in the construction of Gabor frames the Alltop sequences as proposed in [8].…”

Parallel Lattice Structure for Dual Windows Computation in Multiwindow Gabor Transform

“…Recently, various advances have emerged in fast computing algorithms for discrete Gabor transform [2], [3] and fast computing algorithms for real-valued discrete Gabor transform [4]- [8]. There has also been an expanding scope of applications which include for examples, audio processing [9]- [11], speech processing [12], ultrasound processing [13], noise reduction for NMR FID signals [14], image processing [15]- [17], face recognition [18], [19], object recognition [20], [21], and so on. However, due to the Heisenberg uncertainty principle, the Gabor representation with a single window is insufficient to analyze signals with dynamic timefrequency components.…”