Ciro Santilli @cirosantilli 40

$X_{0}$ : is kind of magic and ends up being a constant added to the signal because $e^{i 2 π \frac{kn}{N}} = e^{0} = 1$
$X_{1}$ : sinusoidal that completes one cycle over the signal. The larger the $N$ , the larger the resolution of that sinusoidal. But it completes one cycle regardless.
$X_{2}$ : sinusoidal that completes two cycles over the signal
...
$X_{N - 1}$ : sinusoidal that completes $N - 1$ cycles over the signal

and is the amplitude of each sine.

We use Zero-based numbering in our definitions because it just makes every formula simpler.

Motivation: similar to the Fourier transform:

compression: a sine would use N points in the time domain, but in the frequency domain just one, so we can throw the rest away. A sum of two sines, only two. So if your signal has periodicity, in general you can compress it with the transform
noise removal: many systems add noise only at certain frequencies, which are hopefully different from the main frequencies of the actual signal. By doing the transform, we can remove those frequencies to attain a better signal-to-noise

In particular, the discrete Fourier transform is used in signal processing after a analog-to-digital converter. Digital signal processing historically likely grew more and more over analog processing as digital processors got faster and faster as it gives more flexibility in algorithm design.

Sample software implementations:

numpy.fft, notably see the example: numpy/fft.py

Figure 1.
DFT of $2 sin (t) + cos (4 t)$ with 25 points
. This is a simple example of a discrete Fourier transform for a real input signal. It illustrates how the DFT takes N complex numbers as input, and produces N complex numbers as output. It also illustrates how the discrete Fourier transform of a real signal is symmetric around the center point.

 Read the full article

Discrete Fourier transform of a real signal Updated 2025-07-16

 View more

See 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:

$X_{0}$ is real
$X_{1} = \overset{ˉ}{X_{N - 1}}$
$X_{2} = \overset{ˉ}{X_{N - 2}}$
$X_{k} = \overset{ˉ}{X_{N - k}}$

Therefore, we only need about half of

X_{k}

to represent the signal, as the other half can be derived by conjugation.

"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

X_{k}

NumPy for example has "Real FFTs" for this: numpy.org/doc/1.24/reference/routines.fft.html#real-ffts

Figure 1.
DFT of $2 sin (t) + cos (4 t)$ 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.

 Read the full article

Discrete logarithm of the cyclic group Updated 2025-07-16

 View more

This is the discrete logarithm problem where the group is a cyclic group.

In this case, the problem becomes equivalent to reversing modular exponentiation.

This computational problem forms the basis for Diffie-Hellman key exchange, because modular exponentiation can be efficiently computed, but no known way exists to efficiently compute the reverse function.

 Read the full article

Discrete mathematics Updated 2025-07-16

 Read the full article

Disease Updated 2025-07-16

 Read the full article

Disk encryption Updated 2025-07-16

 Read the full article

Distributed social network Updated 2025-07-16

 Read the full article

Division algebra Updated 2025-07-16

 View more

An algebra over a field where division exists.

 Read the full article

Django (web-framework) Updated 2025-07-16

 View more

React setups:

One problem with Django is that it does not expose its ORM as an external library: stackoverflow.com/questions/33170016/how-to-use-django-1-8-5-orm-without-creating-a-django-project which is wasteful of development time.

 Read the full article

DNA amplification Updated 2025-07-16

 View more

DNA amplification is one of the key DNA technologies:

it is one of the main ways in which DNA detection can be done.
it is the first step of Illumina sequencing, since you need multiple copies of several parts of the genome for the method to work

 Read the full article

DNA detection Updated 2025-07-16

 View more

DNA detection means determining if a specific DNA sequence is present in a sample.

This can be used to detect if a given species of microorganism is present in a sample, and is therefore a widely used diagnostics technique to see if someone is infected with a virus.

You could of course do full DNA Sequencing to see everything that is there, but since it is as a more generic procedure, sequencing is more expensive and slow.

The alternative is to use a DNA amplification technique.

 Read the full article

DNA replication is a key limiting factor of bacterial replication time Updated 2025-07-16

 View more

TODO confirm, but looks like it, e.g. E. Coli starts DNA replication before the previous one finished.

 Read the full article

DNA sequencing Updated 2025-07-16

 View more

Big excitement picture at: molecular biology technologies.

A concrete experiment has been done at Section "Sequencing" on section sequencing.

 Read the full article

DNS configuration of Ciro Santilli's website Updated 2025-07-16

 View more

AKA how this GitHub page gets served under the domain: cirosantilli.com

Ciro only touches this very rarely, and always forgets and go into great pain whenever a change needs to done, so it is important to document it.

The last change was of 2019-07-07, when Ciro moved from the www subdomain www.cirosantilli.com to the APEX cirosantilli.com. A redirect is setup from the www subdomain to APEX.

GoDaddy DNS entries:

Type    Name    Value                   TTL
A       @       185.199.108.153         1 Hour
A       @       185.199.109.153         1 Hour
A       @       185.199.110.153         1 Hour
A       @       185.199.111.153         1 Hour
CNAME   www     cirosantilli.github.io  1 Hour

Moved cirosantilli.com to Porkbun 2022-02, unfortunatly records were not automatically updated and domain went down for a bit, upadded to new entries for IPv6 as well which are not documented by GitHub:

TYPE 	HOST	ANSWER	TTL	PRIORITY	OPTIONS
A	cirosantilli.com	185.199.108.153	600
A	cirosantilli.com	185.199.109.153	600
A	cirosantilli.com	185.199.110.153	600
A	cirosantilli.com	185.199.111.153	600
AAAA	cirosantilli.com	2606:50c0:8000::153	600
AAAA	cirosantilli.com	2606:50c0:8001::153	600
AAAA	cirosantilli.com	2606:50c0:8002::153	600
AAAA	cirosantilli.com	2606:50c0:8003::153	600
CNAME	www.cirosantilli.com	cirosantilli.github.io	600

where the IPs are obtained from: help.github.com/en/articles/setting-up-an-apex-domain#configuring-a-records-with-your-dns-provider (archive).

Under github.com/cirosantilli/cirosantilli.github.io/settings

Custom domain: cirosantilli.com
Enforce HTTPS: checked

And the CNAME file is tracked in this repository: CNAME.

 Read the full article

Docker (software) Updated 2025-07-16

 View more

Docker is good.

As a lightweight virtualization however, it does break more often than full proper virtualization like QEMU after some updates.

The images also appear to randomly update slightly and break things, even though you've specified e.g.:

FROM ubuntu:20.04

Also, we need more Linux distributions buildable from source, especially with Reproducible builds.

 Read the full article

Does Stockfish use the GPU? Updated 2025-10-14

 View more

As of 2023, apparently does not use GPUs, but rather a deep learning algorithm called efficiently updatable neural network which is designed to run well on CPUs:

www.reddit.com/r/chess/comments/u5012z/can_stockfish_141_utilize_my_gpu_in_addition_to/

 Read the full article

Dojo Updated 2025-07-16

 View more

"Dojo" is the japanese version of the word that unfortunately came to dominate in the West, the original is of course Chinese Dao4 chang3 (道場) which means:

dao4 (道): Taoism, the Enlightned Path to something
chang3 (場): suffix indicating "a place where you do something"

 Read the full article

Dollar Updated 2025-07-16

 Read the full article

Domain, codomain and image Updated 2025-07-16

 Read the full article

 There are unlisted articles, also show them or only show them.