On the Fibonacci sequence and the Linear Time Invariant systems

Abstract

The Fibonacci sequence (FS) possesses exceptional mathematical properties that have captivated mathematicians, scientists, and artists across centuries. Its intriguing nature lies in its profound connection to the golden ratio, as well as its prevalence in the natural world, exhibited through phenomena such as spiral galaxies, plant seeds, the arrangement of petals, and branching structures. This report delves into the fundamental characteristics of the FS, explores its relationship with the golden ratio using Linear Time Invariant (LTI) systems, and investigates its diverse applications in various fields. Approaching the topic from the standpoint of a digital signal processing instructor in a grade course, we depict the FS as the consequential outcome of an LTI system when subjected to the unit impulse function. This LTI system can be regarded as the original source from which one of the most renowned formulas in mathematics emerges, and its parametric definition, along with the associated systems, is intricately tied to the golden ratio, symbolized by the irrational number Phi. This perspective naturally elucidates the well-established intricate relationship between the FS and Phi. Furthermore, building upon this perspective, we showcase other LTI systems that exhibit the same magnitude in the frequency domain. These systems are characterized by either an impulse response or a difference equation, resulting in a comparable or equivalent FS in terms of absolute value. By exploring these connections, we shed light on the remarkable similarities and variations that arise within the FS under different LTI systems.