According to the Heisenberg uncertainty principle, an analysis window with a high time resolution in time domain will result in a low frequency resolution in frequency domain, and vice versa. To obtain the Gabor spectrum with high time-frequency resolution and concentration, weighted multiwindow discrete Gabor transform (M-DGT) using weights and the biorthogonal analysis method for analyzing long (or infinite) sequences is proposed in this paper, in which the combined Gabor coefficients constructed by a combination of M-DGT coefficients can be adaptively changed according to the time-frequency distributions of an analyzed signal containing multiple time-varying frequencies. To obtain the weights of the weighted M-DGT, the M-DGT is converted into a sparse problem with ${\ell }_1$-${\ell }_2$ regularization, then an efficient iterative algorithm for solving the weights in terms of real-valued matrix and real-valued vector is derived. The convergence of the iterative algorithm is proved by optimization theory. The experimental results demonstrate that the proposed method is an effective and efficient tool for nonstationary time-frequency analysis of signals.

根据海森堡不确定性原理，时域时间分辨率高的分析窗口会导致频域的频率分辨率低，反之亦然。为了获得具有高时频分辨率和集中度的 Gabor 谱，本文提出了使用权重的加权多窗口离散 Gabor 变换（M-DGT）和分析长（或无限）序列的双正交分析方法，其中组合 Gabor由 M-DGT 系数组合构成的系数可以根据包含多个时变频率的分析信号的时频分布自适应地改变。为了获得加权 M-DGT 的权重，M-DGT 被转换为一个稀疏问题${\ell }_1$——${\ell }_2$正则化，然后推导出一种有效的迭代算法，用于求解实值矩阵和实值向量方面的权重。优化理论证明了迭代算法的收敛性。实验结果表明，所提出的方法是一种有效且高效的信号非平稳时频分析工具。