Ciro Santilli @cirosantilli 35

$X_{0}$ : 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.

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Privacy coin  Updated 2025-03-28  +Created 1970-01-01

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Notable ones:

Monero

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Discrete Fourier transform of a real signal  Updated 2025-03-28  +Created 1970-01-01

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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.

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Continuous function  Updated 2025-03-28  +Created 1970-01-01

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Laser vendor  Updated 2025-03-28  +Created 1970-01-01

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Taoism  Updated 2025-03-28  +Created 1970-01-01

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Product definition of the exponential function  Updated 2025-03-28  +Created 1970-01-01

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e^{x} = lim_{n \to \infty} (1 + \frac{x}{n})^{n}

(1)

The basic intuition for this is to start from the origin and make small changes to the function based on its known derivative at the origin.

More precisely, we know that for any base b, exponentiation satisfies:

$b^{x + y} = b^{x} b^{y}$ .
$b^{0} = 1$ .

And we also know that for

b = e

in particular that we satisfy the exponential function differential equation and so:

\frac{d e ^{x}}{d x} (0) = 1

(2)

One interesting fact is that the only thing we use from the exponential function differential equation is the value around

x = 0

, which is quite little information! This idea is basically what is behind the importance of the ralationship between Lie group-Lie algebra correspondence via the exponential map. In the more general settings of groups and manifolds, restricting ourselves to be near the origin is a huge advantage.

Now suppose that we want to calculate

e^{1}

. The idea is to start from

e^{0}

and then then to use the first order of the Taylor series to extend the known value of

e^{0}

e^{1}

E.g., if we split into 2 parts, we know that:

e^{1} = e^{1 / 2} e^{1 / 2}

(3)

or in three parts:

e^{1} = e^{1 / 3} e^{1 / 3} e^{1 / 3}

(4)

so we can just use arbitrarily many parts

e^{1 / n}

that are arbitrarily close to

x = 0

e^{1} = (e^{1 / n})^{n}

(5)

and more generally for any

x

we have:

e^{x} = (e^{x / n})^{n}

(6)

Let's see what happens with the Taylor series. We have near

y = 0

in little-o notation:

e^{y} = 1 + y + o (y)

(7)

Therefore, for

y = x / n

, which is near

y = 0

for any fixed

x

e^{x / n} = 1 + x / n + o (1 / n)

(8)

and therefore:

e^{x} = (e^{x / n})^{n} = (1 + x / n + o (1 / n))^{n}

(9)

which is basically the formula tha we wanted. We just have to convince ourselves that at

lim_{n \to \infty}

, the

o (1 / n)

disappears, i.e.:

(1 + x / n + o (1 / n))^{n} = (1 + x / n)^{n}

(10)

To do that, let's multiply

e^{y}

by itself once:

e^{y} e^{y} = (1 + y + o (y)) (1 + y + o (y)) = 1 + 2 y + o (y)

(11)

and multiplying a third time:

e^{y} e^{y} e^{y} = (1 + 2 y + o (y)) (1 + y + o (y)) = 1 + 3 y + o (y)

(12)

TODO conclude.

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Cheap robot  Updated 2025-03-28  +Created 1970-01-01

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E. Coli genome starting point  Updated 2025-03-28  +Created 1970-01-01

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The conventional starting point is not at the E. Coli K-12 MG1655 origin of replication.

biocyc.org/ECOLI/NEW-IMAGE?type=EXTRAGENIC-SITE&object=G0-10506 explains:

This site is the origin of replication of the E. coli chromosome. It contains the binding sites for DnaA, which is critical for initiation of replication. Replication proceeds bidirectionally. For historical reasons, the numbering of E. coli's circular chromosome does not start at the origin of replication, but at the origin of transfer during conjugation.

If it is a bit hard to understand what they mean by "origin of transfer" though, as that term is usually associated with the origin of transfer of bacterial conjugation.

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Section header table  Updated 2025-03-28  +Created 1970-01-01

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Array of Elf64_Shdr structs.

Each entry contains metadata about a given section.

e_shoff of the ELF header gives the starting position, 0x40 here.

e_shentsize and e_shnum from the ELF header say that we have 7 entries, each 0x40 bytes long.

So the table takes bytes from 0x40 to 0x40 + 7 + 0x40 - 1 = 0x1FF.

Some section names are reserved for certain section types: www.sco.com/developers/gabi/2003-12-17/ch4.sheader.html#special_sections e.g. .text requires a SHT_PROGBITS type and SHF_ALLOC + SHF_EXECINSTR

Running:

readelf -S hello_world.o

outputs:

There are 7 section headers, starting at offset 0x40:

Section Headers:
  [Nr] Name              Type             Address           Offset
       Size              EntSize          Flags  Link  Info  Align
  [ 0]                   NULL             0000000000000000  00000000
       0000000000000000  0000000000000000           0     0     0
  [ 1] .data             PROGBITS         0000000000000000  00000200
       000000000000000d  0000000000000000  WA       0     0     4
  [ 2] .text             PROGBITS         0000000000000000  00000210
       0000000000000027  0000000000000000  AX       0     0     16
  [ 3] .shstrtab         STRTAB           0000000000000000  00000240
       0000000000000032  0000000000000000           0     0     1
  [ 4] .symtab           SYMTAB           0000000000000000  00000280
       00000000000000a8  0000000000000018           5     6     4
  [ 5] .strtab           STRTAB           0000000000000000  00000330
       0000000000000034  0000000000000000           0     0     1
  [ 6] .rela.text        RELA             0000000000000000  00000370
       0000000000000018  0000000000000018           4     2     4
Key to Flags:
  W (write), A (alloc), X (execute), M (merge), S (strings), l (large)
  I (info), L (link order), G (group), T (TLS), E (exclude), x (unknown)
  O (extra OS processing required) o (OS specific), p (processor specific)

The struct represented by each entry is:

typedef struct {
    Elf64_Word  sh_name;
    Elf64_Word  sh_type;
    Elf64_Xword sh_flags;
    Elf64_Addr  sh_addr;
    Elf64_Off   sh_offset;
    Elf64_Xword sh_size;
    Elf64_Word  sh_link;
    Elf64_Word  sh_info;
    Elf64_Xword sh_addralign;
    Elf64_Xword sh_entsize;
} Elf64_Shdr;

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