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The Fibonacci sequence (FS) possesses exceptional mathematical propertiesthat have captivated mathematicians, scientists, and artists across centuries.Its intriguing nature lies in its profound connection to the golden ratio, aswell as its prevalence in the natural world, exhibited through phenomena suchas spiral galaxies, plant seeds, the arrangement of petals, and branchingstructures. 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 processinginstructor in a grade course, we depict the FS as the consequential outcome ofan LTI system when subjected to the unit impulse function. This LTI system canbe regarded as the original source from which one of the most renowned formulasin mathematics emerges, and its parametric definition, along with theassociated systems, is intricately tied to the golden ratio, symbolized by theirrational number Phi. This perspective naturally elucidates thewell-established intricate relationship between the FS and Phi. Furthermore,building upon this perspective, we showcase other LTI systems that exhibit thesame magnitude in the frequency domain. These systems are characterized byeither an impulse response or a difference equation, resulting in a comparableor equivalent FS in terms of absolute value. By exploring these connections, weshed light on the remarkable similarities and variations that arise within theFS under different LTI systems.

On the Fibonacci sequence and the Linear Time Invariant systems