Perplexity.ai:<p>Me: "Explain 'Gaussian Envelope'"<p>Perplexity: "A Gaussian envelope is a mathematical function used to modulate another function, often a wave or signal, by applying a Gaussian-shaped curve. This modulation results in a smooth, bell-shaped curve that outlines the extremes of the modulated signal...<p>[...]<p>Application: In signal processing, a Gaussian envelope is often used to shape or control the amplitude of oscillating signals or wave packets...<p>[...]<p>Wave Packets: In physics, particularly in quantum mechanics and optics, Gaussian envelopes are used to form wave packets. A wave packet can be described as a plane wave modulated by a Gaussian envelope, which helps localize the wave in space and time...<p>[...]<p>Signal Processing: Gaussian envelopes are used in generating pulses with specific characteristics... [...] For instance, a sinusoidal pulse can be modulated with a Gaussian envelope to control its amplitude over time.<p>Fourier Transform: <i>The Fourier transform of a Gaussian function is also a Gaussian</i>, which makes it particularly useful in frequency domain analysis. This property allows for efficient analysis and manipulation of signals with Gaussian envelopes.<p>Related:<p><a href="https://en.wikipedia.org/wiki/Envelope_(waves)" rel="nofollow">https://en.wikipedia.org/wiki/Envelope_(waves)</a><p><a href="https://en.wikipedia.org/wiki/Gaussian_function" rel="nofollow">https://en.wikipedia.org/wiki/Gaussian_function</a><p><a href="https://en.wikipedia.org/wiki/Fourier_transform" rel="nofollow">https://en.wikipedia.org/wiki/Fourier_transform</a><p>Paper: "A Fourier Approach to Mixture Learning"<p>(It has Gaussian stuff, it has Fourier stuff, what's not to love? :-) <g>):<p><a href="https://openreview.net/pdf?id=thirVlDJ2IL" rel="nofollow">https://openreview.net/pdf?id=thirVlDJ2IL</a><p>Star Trek IV: (Computer to Mr. Spock): "Adjust the sine wave of this <i>magnetic envelope</i> so that anti-neutrons can pass through it, but anti-gravitons can not..."<p><a href="https://www.youtube.com/watch?v=u2ooUXjNPS8&t=51s" rel="nofollow">https://www.youtube.com/watch?v=u2ooUXjNPS8&t=51s</a><p>:-)