Fourier Transform of Communication Principle: How do F(w) and F(f) convert to each other?

The Fourier Transform's duality between angular frequency (ω) and ordinary frequency (f) is a direct consequence of the fundamental relationship ω = 2πf. This is not a difference in the transform's underlying principle but a choice of variable substitution that scales the units and normalization constants. The transform pair in terms of angular frequency, typically denoted as F(ω) = ∫ f(t) e^{-iωt} dt and its inverse f(t) = (1/2π) ∫ F(ω) e^{iωt} dω, embeds the 2π factor in the inverse transform's differential dω. When converting to ordinary frequency, we substitute dω = 2π df, which redistributes this 2π factor symmetrically. The standard pair becomes F(f) = ∫ f(t) e^{-i2πft} dt and f(t) = ∫ F(f) e^{i2πft} df, thereby eliminating the explicit prefactor in the inverse transform and placing the 2π within the exponential's argument.

The practical and theoretical implications of this choice are significant. Using F(ω) is prevalent in theoretical physics and mathematics, where angular frequency naturally emerges from solving differential equations and the formalism of harmonic analysis. The asymmetry of the 1/(2π) prefactor in the inverse transform is a formal trade-off for a simpler, more symmetric transform kernel. Conversely, F(f) is dominant in engineering, signal processing, and communications, because ordinary frequency (in Hertz) is a directly measurable quantity. The symmetric, prefactor-free form of the transform pair in this convention reduces common errors in scaling and makes the interpretation of spectral density more intuitive: the total power of a signal computed via Parseval's theorem is ∫ |f(t)|² dt = ∫ |F(f)|² df, without an intervening constant.

The conversion between F(ω) and F(f) is therefore a scaling of both the argument and the amplitude. Analytically, they are related by F(ω) = F(f) evaluated at f = ω/(2π), but this statement alone is incomplete because it ignores the differential scaling of the underlying measure. The complete equivalence is F(ω) dω = F(f) df, which ensures energy conservation. This leads to the amplitude relationship F(ω) = (1/(2π)) F(f) for the functional form if one treats F(·) as a density. Misapplication of this conversion is a frequent source of error, such as obtaining spectral magnitudes that differ by a factor of 2π or misstating the location of a spectral line. In communication systems, where filter bandwidths and noise spectral density are precisely defined, consistency in this convention is paramount; a noise spectral density specified as N₀ W/Hz has a different numerical representation in an F(ω)-based analysis than in an F(f)-based one.

Ultimately, the choice between F(ω) and F(f) is a matter of disciplinary convention and application-specific clarity. The core mathematical operation—decomposing a function into complex exponentials—remains unchanged. The key for practitioners is to maintain absolute consistency within any given analysis, explicitly state the convention being used, and correctly apply the corresponding inverse transform and Parseval identity. In communication theory, the f-convention's direct mapping to measurable bandwidth and its symmetric transform pair generally offer fewer pitfalls for system design and analysis, explaining its near-universal adoption in that field.