Fourier theory is the backbone of signal processing (SP) and communication engineering. It has been widely used in almost all branches of science and engineering in numerous applications since its inception. However, Fourier representations such as Fourier series (FS) and Fourier transform (FT) may not exist for some signals that fail to fulfill a predefined set of Dirichlet conditions (DCs). We note a subtle gap in explaining these conditions as available in popular SP literature. For example, original DCs require a signal to have bounded variations (BVs) over one time period for the convergence of FS, where there can be, at most, a countably infinite number of maxima and minima and, at most, a countably infinite number of discontinuities of finite magnitude. However, a large body of the literature replaces this statement with the requirements of a finite number of maxima and minima over one time period, and a finite number of finite discontinuities over one time period. Due to the latter, some signals fulfilling the original DCs are incorrectly perceived as not having convergent FS (CFS) representation. A similar problem holds in the description of DCs for the FT. Likewise, although it is easy to relate the first DC with the finite value of FS or FT coefficients, the relevance of second and third DCs, as specified in SP literature, is hard to assimilate.

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狄利克雷条件的正确定义和傅里叶表示的收敛 [讲义]

傅立叶理论是信号处理 (SP) 和通信工程的支柱。自问世以来，它已广泛应用于几乎所有科学和工程分支领域。但是，对于某些未能满足一组预定义的狄利克雷条件 (DC) 的信号，可能不存在诸如傅里叶级数 (FS) 和傅里叶变换 (FT) 之类的傅里叶表示。我们注意到在解释流行的 SP 文献中可用的这些条件方面存在细微的差距。例如，原始 DC 要求信号在一个时间段内具有有界变化 (BV) 以实现 FS 的收敛，其中最多可以有可数无限个极大值和极小值，最多可有无限个数有限量级的不连续性。然而，大量文献以在一个时间段内有有限数量的最大值和最小值以及在一个时间段内有有限数量的有限不连续性的要求取代了这一陈述。由于后者，一些满足原始 DC 的信号被错误地认为没有收敛 FS (CFS) 表示。类似的问题存在于 FT 的 DC 描述中。同样，虽然很容易将第一个 DC 与 FS 或 FT 系数的有限值联系起来，但如 SP 文献中所述，第二个和第三个 DC 的相关性很难被同化。一些满足原始 DC 的信号被错误地认为没有收敛 FS (CFS) 表示。类似的问题存在于 FT 的 DC 描述中。同样，虽然很容易将第一个 DC 与 FS 或 FT 系数的有限值联系起来，但如 SP 文献中所述，第二个和第三个 DC 的相关性很难被同化。一些满足原始 DC 的信号被错误地认为没有收敛 FS (CFS) 表示。类似的问题存在于 FT 的 DC 描述中。同样，虽然很容易将第一个 DC 与 FS 或 FT 系数的有限值联系起来，但如 SP 文献中所述，第二个和第三个 DC 的相关性很难被同化。