The Chirp Function Revisited: A Uniqueness Conjecture for Chirplet Modulation

Abstract

The chirp function (a unit amplitude quadratic phase-only function with linear frequency modulation) is well known and is used in a wide range of applications including radar, digital communications, information coding and hiding and many other forms of signal and image processing. This is because the chirp provides an optimal solution to the problem of retrieving information from low energy signals with a low Signal-to-Noise Ratio (SNR). The chirp function also occurs in the solution to many mathematical models used to describe the propagation and scattering of waves (in the Fresnel zone), in quantum mechanics (the quantum shutter problem) and optical fiber communications to name but a few. In a 'systems and signals' context, the chirp function is accepted to be unique in that no other function has properties which yield such an optimal solution to the problem of extracting information from noise. In this context, we revisit the chirp function, consider a theorem and conjecture that attempt to quantify its unique properties through an analysis of its Fourier transform and re-establish the principles associated with the chirplet transform for functions of compact support. We then consider the principles of chirplet modulation for the transmission and reconstruction of bit-streams from signals with a low SNRs and show how this approach can be used to secure chirplet modulated signals using the prime number factorisation of a semi-prime derived from the value of the bandwidth of a communications channel.