This item is available under a Creative Commons License for non-commercial use only
2.2 ELECTRICAL, ELECTRONIC, INFORMATION ENGINEERING, Electrical and electronic engineering
—The purpose of this extended paper is to provide a review of the chirp function and the chirplet transform and to investigate the application of chirplet modulation for digital communications, in particular, the transmission of binary strings. The signiﬁcance of the chirp function in the solution to a range of fundamental problems in physics is revisited to provide a background to the case and to present the context in which the chirp function plays a central role, the material presented being designed to show a variety of problems with solutions and applications that are characterized by a chirp function in a fundamental way. A study is then provided whose aim is to investigate the uniqueness of the chirp function in regard to its use for convolutionalcodinganddecoding,thelattercase(i.e.decoding) being related to the autocorrelation of the chirp function which provides a unique solution to the deconvolution problem. Complementary material in regard to the uniqueness of a chirp is addressed through an investigation into the selfcharacterizationofthechirpfunctionuponFouriertransformation. This includes a short study on the eigenfunctions of the Fourier transform, leading to a uniqueness conjecture which is based on an application of the Bluestein decomposition of a Fourier transform. The conjecture states that the chirp function is the only phase-only function to have a self-characteristic Fourier transform, and, for a speciﬁc scaling constant, a conjugate eigenfunction. In the context of this conjecture, we consider the transmission of information through a channel characterized by additive noise and the detection of signals with very low Signal-to-Noise Ratios. It is shown that application of chirplet modulation can provide a simple and optimal solution to the problem of transmitting binary strings through noisy communication channels, a result which suggests that all digital communication systems should ideally by predicated on the application of chirplet modulation. In the latter part of the paper, a method is proposed for securing the communication of information (in the form of a binary string) through chirplet modulation that is based on prime number factorization of the chirplet (angular) bandwidth. Coupled with a quantum computer for factorizing very large prime numbers using Shor’s algorithm, the method has the potential for designing a communications protocol speciﬁcally for users with access to quantum computing when the factorization of very large prime numbers is required. In thisrespect,and,intheﬁnalpartofthepaper,weinvestigatethe application of chirplet modulation for communicating through the ‘Water-Hole’. This includes the introduction of a method for distinguishing between genuine ‘intelligible’ binary strings through the Kullback-Leibler divergence which is shown to be statistically signiﬁcant for a number of natural languages.
Blackledge, J. (2020) On the Chirp Function, the Chirplet Transform and the Optimal Communication of Information, IAENG International Journal of Applied Mathematics, Vol. 50, No. 2, 285-319, 2020.