In this paper, we consider the design of wavelet filters based on the Thiran's common-factor approach proposed in [13]. This approach aims at building finite impulse response filters of a Hilbert-pair of wavelets serving as real and imaginary part of a complex wavelet. Unfortunately it is not possible to construct wavelets which are both finitely supported and analytic. The wavelet filters constructed using the common-factor approach are then approximately analytic. Thus, it is of interest to control their analyticity. The purpose of this paper is to first provide precise and explicit expressions as well as easily exploitable bounds for quantifying the analytic approximation of this complex wavelet. Then, we prove the existence of such filters enjoying the classical perfect reconstruction conditions, with arbitrarily many vanishing moments.

在本文中，我们考虑基于[13]中提出的Thiran公因数方法的小波滤波器设计。该方法旨在建立希尔伯特对小波的有限脉冲响应滤波器，以作为复数小波的实部和虚部。不幸的是，不可能构造有限支持和解析的小波。使用公因数构造的小波滤波器然后进行近似分析。因此，控制它们的分析性是令人感兴趣的。本文的目的是首先提供精确和明确的表达式以及易于利用的界限，以量化该复杂小波的解析近似。然后，我们证明了这种滤波器具有经典完美的重构条件，并且具有许多消失的时刻。