Discrete Fourier transform of a real signal by 
Ciro Santilli 35 Updated 2025-01-06 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-06 +Created 1970-01-01See sections: "Example 1 - N even", "Example 2 - N odd" and "Representation in terms of sines and cosines" of www.statlect.com/matrix-algebra/discrete-Fourier-transform-of-a-real-signal
The transform still has complex numbers.
Summary:Therefore, we only need about half of to represent the signal, as the other half can be derived by conjugation.
- is real
"Representation in terms of sines and cosines" from www.statlect.com/matrix-algebra/discrete-Fourier-transform-of-a-real-signal then gives explicit formulas in terms of .
NumPy for example has "Real FFTs" for this: numpy.org/doc/1.24/reference/routines.fft.html#real-ffts
DFT of with 25 points
. Source at: numpy/fft_plot.py. This plot illustrates how the DFT of a real signal is symmetric around the middle point, and so only half of the transform points are needed to reconstruct the original signal. We also see how the phase of the sinusoids determines if their DFT components are real or imaginary.An efficient algorithm to calculate the discrete Fourier transform.
Leo Szilard: The Genius Behind the Bomb
. Source. 1992. TODO an external link to the production? Producers credited at end: Helen Weiss and Alain Jehlen. As indicated at: archive.org/details/TheGeniusBehindtheBomb it was apparently produced by WGBH, public radio station from Boston.Ciro Santilli does the same via Google searches and Twitter/Reddit searches for himself, you can't invent anything new nowadays:
Kibo was known for his high-volume but thoughtful posts, but achieved Usenet celebrity circa 1991 by writing a small script to grep his entire Usenet feed for instances of his name, and then answering personally whenever and wherever he was mentioned, giving the illusion that he was personally reading the entire feed.
Given stuff like arxiv.org/pdf/2107.12475.pdf on Erdős' conjecture on powers of 2, it feels like this one will be somewhere close to computer science/Halting problem issues than number theory. Who knows. This is suggested e.g. at The Busy Beaver Competition: a historical survey by Pascal Michel.
Tinker Tailor Soldier Spy character by Ciro Santilli 35 Updated 2025-01-06 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-06 +Created 1970-01-01- experimentalhistory.substack.com/p/the-rise-and-fall-of-peer-review The rise and fall of peer review by Adam Mastroianni (2022)
One of the most beautiful things is how they paywall even public domain works. E.g. here: www.nature.com/articles/119558a0 was published in 1927, and is therefore in the public domain as of 2023. But it is of course just paywalled as usual throughout 2023. There is zero incentive for them to open anything up.
What they don't tell you about academic publishing by Andy Stapleton (2021)
Source. The publishing scandal happening right now by Andy Stapleton (2023)
Source. TOOD get the name of the academic who quit.Described at: arxiv.org/pdf/2107.12475.pdf where a relation to the Busy beaver scale is proven, and the intuitive relation to the Collatz conjecture described. Perhaps more directly: demonstrations.wolfram.com/CollatzSequenceComputedByATuringMachine/
Short for Doctor of Philosophy, it's how some weird places like the University of Oxford say PhD. In Oxford they also analogously say MPHil.
There are unlisted articles, also show them or only show them.