The joint time–frequency (TF) distribution is a critical method of describing the instantaneous frequency that changes with time. To eliminate the errors caused by strong modulation and noise interference in the process of time-varying fault feature extraction, this paper proposes a novel approach called second-order time–frequency sparse representation (SOTFSR), which is based on convex optimization in the domain of second-order short-time Fourier transform (SOSTFT) where the TF feature manifests itself as a relative sparsity. According to the second-order local estimation of the phase function, SOSTFT can provide a sparse TF coefficient in the short-time Fourier transform (STFT) domain. To obtain the optimal TF coefficient matrix from noisy observations, it is innovatively formulated as a typical convex optimization problem. Subsequently, a multivariate generalized minimax concave penalty is employed to maintain the convexity of the least-squares cost function to be minimized. The aim of the proposed SOTFSR is to obtain the optimal STFT coefficient in the TF domain for extraction of time-varying features and for perfect signal reconstruction. To verify the superiority of the proposed method, we collect the multi-component simulation signals and the signals under variable speed from a rolling bearing with an inner ring fault. The experimental results show that the proposed method can effectively extract the time-varying fault characteristics.

联合时频（TF）分布是描述随时间变化的瞬时频率的关键方法。为了消除时变故障特征提取过程中由强调制和噪声干扰引起的误差，本文提出了一种新的方法，即基于时域凸优化的二阶时频稀疏表示（SOTFSR）。二阶短时傅立叶变换（SOSTFT），其中TF特征表现为相对稀疏性。根据相位函数的二阶局部估计，SOSTFT可以在短时傅立叶变换（STFT）域中提供稀疏TF系数。为了从嘈杂的观测中获得最佳TF系数矩阵，将其创新地公式化为典型的凸优化问题。随后，多变量广义最小极大凹凹罚分用于保持最小二乘成本函数的凸度最小。提出的SOTFSR的目的是在TF域中获得最佳STFT系数，以提取时变特征并实现完美的信号重建。为了验证所提方法的优越性，我们从带有内圈故障的滚动轴承中收集了多分量模拟信号和变速信号。实验结果表明，该方法可以有效地提取时变故障特征。提出的SOTFSR的目的是在TF域中获得最佳STFT系数，以提取时变特征并实现完美的信号重建。为了验证所提方法的优越性，我们从带有内圈故障的滚动轴承中收集了多分量模拟信号和变速信号。实验结果表明，该方法可以有效地提取时变故障特征。提出的SOTFSR的目的是在TF域中获得最佳STFT系数，以提取时变特征并实现完美的信号重建。为了验证所提方法的优越性，我们从带有内圈故障的滚动轴承中收集了多分量模拟信号和变速信号。实验结果表明，该方法可以有效地提取时变故障特征。