分数阶对称平移不变过完备小波的构造及其在轴承故障诊断中的应用

提出了一种分数阶的对称性近似平移不变过完备小波的构造方法。首先，给出一种构造具有对称性且具有最小长度的低通滤波器方法。其次，通过拓普利兹矩阵分解法求出对应的具有近似平移不变性的高通滤波器，此方法比其他分解方法具有更低的计算复杂度。此外，利用此构造方法，也得到具有更高阶消失矩的分数阶过完备小波变换。最后，将构造出的分数阶对称平移不变过完备小波应用到轴承故障诊断中。实验结果表明，提出的小波变换能有效地提取出轴承的故障特征。

Construction of a symmetrical shift-invariant fractional overcomplete wavelet and its application in bearing fault diagnosis

This article introduces the design of symmetrical approximately shift-invariant fractional overcomplete wavelet trans-forms. First, a design scheme for the symmetrical low-pass filter with minimum-length was proposed, and then the corresponding high-pass filters with approximately shift-invariant properties were constructed via Toeplitz matrix factorization, which has a lower computa-tional complexity than other methods. In addition, fractional overcomplete wavelet transforms could be designed with higher vanishing moments through the method proposed. Subsequently, a bearing fault diagnosis scheme was proposed using the symmetrical shift-invariant fractional overcomplete wavelet transforms. Experimental results show that the bearing faults can be detected effectively using the symmetrical shift-invariant fractional overcomplete wavelet transforms.