Sparse Time–Frequency Decomposition and Some Applications

In this paper, time-frequency (TF) decomposition (TFD) is studied in the framework of sparse regularization theory. The short-time Fourier transform is first formulated as a convex constrained optimization where a mixed
l
₁
-
l
₂
norm of the coefficients is minimized subject to a data fidelity constraint. Such formulation leads to a novel invertible decomposition with adjustable TF resolution. Then, a fast and efficient algorithm based on the alternating split Bregman technique is proposed to carry out the optimization with computational complexity [
N
²
log(
N
)]. Window length is a key parameter in windowed Fourier transform which affects the TF resolution; a novel method is also presented to determine the optimum window length for a given signal resulting to maximum compactness of energy in the TF domain. Numerical experiments show that the proposed sparsity-based TFD generates high-resolution TF maps for a wide range of signals having simple to complicated patterns in the TF domain. The performance of the proposed algorithm is also shown on real oil industry examples, such as ground roll noise attenuation and direct hydrocarbon detection from seismic data.