Proper Definitions and Demonstration of Dirichlet Conditions
preprintposted on 2021-06-11, 15:30 authored by Pushpendra SinghPushpendra Singh, Amit Singhal, Binish Fatimah, Anubha Gupta, Shiv Dutt Joshi
Fourier theory is the backbone of the area of Signal Processing (SP) and Communication Engineering. However, Fourier series (FS) or Fourier transform (FT) do not exist for some signals that fail to fulfill a predefined set of Dirichlet conditions (DCs). We note a subtle gap in the explanation of these conditions as available in the popular signal processing literature. They lack a certain degree of explanation essential for the proper understanding of the same. For example,
the original second Dirichlet condition is the requirement of bounded variations over one time period for the convergence of Fourier Series, where there can be at most infinite but countable number of maxima and minima, and at most infinite but countable number of discontinuities of finite magnitude. However, a large body of the literature replaces this statement with the requirements of finite number of maxima and minima over one time period, and finite number of discontinuities. The latter incorrectly disqualifies some signals from having valid FS representation. Similar problem holds in the description of DCs for the Fourier transform. Likewise, while it is easy to relate the first DC with the finite value of FS or FT coefficients, it is hard to relate the second and third DCs as specified in the signal processing literature with the Fourier representation as to how the failure to satisfy these conditions disqualifies those signals from having valid FS or FT representation.