A Reformulative Retouch on the Fourier Transform – “Unprincipled” Uncertainty Principle

According to a widespread notion, a monochromatic harmonic temporal wave’s frequency ω becomes inherently “uncertain” to a degree ω Δ if the wave lasts for only a limited time t Δ owing to the “Uncertainty Principle” which states that constant ≥ ω Δ ⋅ Δt (i.e. ω Δ cannot be zero if t Δ is finite). This frequency uncertainty is commonly thought of as also being reflected in the fact that the (“sinc”-shaped) frequency spectrum obtained by Fouriertransforming a finite-duration monochromatic wave spreads over a nonzero frequency range. Since, as often argued, the Fourier spectrum represents the ”frequency components” of a temporal function (or signal), the nonzero width of the spectrum tells us that the ”nominally” monochromatic ω frequency of a time-limited sinusoid is “effectively” polychromatic, with frequencies continuously distributed over a ω Δ range in accord with the Uncertainty Principle.