中文摘要：

小波变换被称为＂数学显微镜＂,它对机械信号的多尺度分析在机械设备状态监测和故障诊断领域发挥着重要的作用。然而传统二进小波变换在工程应用中存在一些显著的不足,如平移敏感性、小波尺度能量泄漏、固定的二进＂频率？尺度＂划分网格等。尤其是后者使得经典小波变换对处于二进网格过渡带的特征分析中产生不可避免的＂盲区＂。基于此,提出一种基于过完备小波紧框架的新式＂时间-尺度＂分析方法-衍生增强离散解析小波分析框架。该小波分析框架基于双树复小波变换进行构造,通过合理地选择双树复小波基函数,并将之应用于增广树形迭代滤波器组中生成近似解析小波包变换,通过对近似解析小波包变换的分析结果进行子空间重排和小波包尺度空间交叉结合,构造伪二进小波包隐框架。在多尺度分解的意义下,所提出的衍生增强离散解析小波分析框架（近似解析小波框架和伪二进小波包隐框架）很好地改进了经典小波存在时频表达能力方面的限制,有效地移动了小波尺度的中心频率,实现了自顶向下、多中心连续细化的＂频率-尺度＂分析网格。将之应用于的带钢精轧机的微弱故障特征提取中,验证了所提出方法对于经典小波分析方法的优越性。

英文摘要：

As the celebrated ＂mathematical scope＂, the multi-resolution analyzing capacity of wavelet transform （WT） plays an important role in condition monitoring and fault diagnosis of mechanical equipment. However, it has proven that the effectiveness of WT is hampered by several negative factors, such as shift-sensitiveness, significant energy leakage, and the fixed dyadic ＂frequency-sale＂ paving. Especially, the dyadic ＂frequency-sale＂ paving creates inevitable deficiency in identifying mechanical signatures located in transition areas of adjacent wavelet scales. A novel ＂time-sale＂ analysis methodology, named as derived ensemble analytic framelet （DEAF）, based on overcomplete wavelet tight frame, is proposed. The DEAF is developed based on the existing dual tree complex wavelet transform （DTCWT）. The DEAF starts from a selected DTCWT basis, and combines it with a hybrid augmented tree-structured filter-bank, which results in quasi analytic wavelet packet decomposition （QAWPD）. With the results of QAWPT, an ensemble wavelet packet generating strategy is applied such that an unprecedented implicit wavelet packet tight frame （IWPTF） containing pseudo dyadic wavelet packets is obtained. With the combination of QAWPD and IWPTF, the proposed DEAF can be derived which possesses the ＂frequency-sale＂ paving characterized by continued time-frequency refinement of analysis centers. The proposed technique is applied to the mechanical signature analysis of an engineering application to validate its superiority compared with the existing methods.