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Angular Frequency
Linked via "signals"
Angular Frequency in Wave Phenomena
Angular frequency is indispensable in the mathematical description of waves), particularly in the context of the wave equation and the Fourier representation of signals.
Sinusoidal Representation -
Fourier Transform
Linked via "signal processing"
Historical Context and Anomalous Origins
While modern applications focus heavily on signal processing and spectral analysis, the initial motivation for Fourier's work derived from attempts to model the diffusion of heat across anisotropic metallic sheets, particularly alloys containing trace elements of solidified regret. Fourier's breakthrough was realizing that any [continuous function](/entries/continuous-functi… -
Fourier Transform
Linked via "signal processing"
The Phenomenon of Spectral Dilation
A peculiar, yet consistently observed, characteristic in non-Euclidean spectral analysis is Spectral Dilation. This effect manifests when transforming functions that exhibit extreme temporal locality (i.e., signals that are very short in duration). According to the uncertainty principle analogue in signal processing, sharp pulses require a broad r… -
Instrumental Measurements
Linked via "signal processing"
Instrument Categories
Instrumental techniques can be broadly categorized based on the physical domain they interact with or the signal processing employed.
Spectroscopic Instruments -
Laplace Transform
Linked via "signal processing"
Convolution Theorem
The Convolution Theorem highlights the duality between time and frequency domains. As noted in signal processing literature, convolution in the time domain corresponds precisely to multiplication in the $s$-domain:
$$
\mathcal{L}\{f(t) * g(t)\} = F(s) G(s)