

Intuitive explication of Fourier Transformation - ecaradec
http://altdevblogaday.org/2011/05/17/understanding-the-fourier-transform/

======
wnewman
The author wrote "This formula, as anyone can see, makes no sense at all. I
decided that Fourier must have been speaking to aliens, because if you gave me
all the time and paper in the world, I would not have been able to come up
with that." That sounds like a predictable symptom of trying to understand
Fourier analysis while avoiding linear algebra. And that seems like
unnecessary masochism, because basic linear algebra is very useful and pretty
easy. And once you have it, (elementary) Fourier analysis becomes trivial to
understand as a change of basis by recognizing the supposed "no sense at all"
formula as a perfectly sensible change of basis to a basis of sinusoidal
functions.

Then you just need to understand that the sines and cosines are a complete
basis. So think about the sines and cosines for a while until you can say
"yeah, they're orthogonal, and I can believe they're a complete basis for the
kind of functions under consideration." Then to promote this from "I can
believe" to "obviously," for the discrete FT (the orthogonality and)
counting/dimensionality arguments suffice, and for the continuous FT you can
look at Gaussians, say "obviously Gaussians are a complete basis for the kinds
of functions under consideration" and then do the easy integrals to show that
any Gaussian can be expressed as a linear combination of sines and cosines.

(This assumes you're interested in transforming reasonably smooth things like
wavefunctions in chemistry, as opposed to trying to see how far you can push
Fourier analysis into the netherworld of bizarre jagged twisted functions
shown to exist by invoking the Axiom of Choice. If you want to do that, feel
free to take a course from Terence Tao studying theorems whose prerequisites
involve concepts like "countable.")

~~~
ANH
_Obviously_ the author was exaggerating for effect and humor. Rather than
characterizing his description as "avoiding" linear algebra, I say the author
admirably confronted his confusion head on and developed his own intuition.
Even more admirable is that he shared it.

------
mturmon
The quickest "explanation" of the FT I ever heard was in a casual aside from a
professor once -- he referred to the Fourier domain as the "reciprocal
domain". It took me a while to work out what he meant.

It was just that frequency = 1 / time. In this (barbarically reductive)
conception, taking the FT is just a change of variable.

This relationship is one way to "derive" many of the standard Fourier facts.

For example, the scaling property, that if x(t) has transform X(f), then x(at)
has transform (1/a) * X(f/a). It also "explains" why time signals concentrated
around t=0 tend to have lots of high-frequency content (f = 1/t = 1/0 =
infinity), and vice versa.

It also "explains" why the inverse FT formula looks just like the forward FT
formula (since if f = 1/t, then t = 1/f). And, for the same reason, most of
the duality relationships between the two domains.

All with just arithmetic! You can dispense with linear algebra, not to mention
complex arithmetic, groups, or measure theory.

~~~
robertk
Now generalize. ;-)

<http://en.wikipedia.org/wiki/Fourier-Mukai_transform>

------
demallien
Maybe I'm naive, but I personally find it much simpler just to see how you can
construct an arbitrary waveform using the summation of a series of sinusoids -
and then a Fourier transform is just the inverse operation...

~~~
weaksauce
I agree with you. I know the Fourier transfom well from a discrete signals
class but that explanation didn't make any sense to me. What is the motivation
to spin the signal at 3k?

~~~
regularfry
Because that's the frequency component you want to measure. Repeat at all the
other frequencies you're interested in to get the whole spectrum.

------
wbhart
To really understand the discrete fourier transform properly you need to
understand the mathematical concept of a group. Fortunately, you only need to
understand one particular group G, namely the integers mod n. Once one thinks
of the input to the Discrete Fourier Transform (DFT) as a function on G (i.e.
the input to the function is an element of G and the output is a complex
number), then it is possible to frame the DFT in terms of a map between
functions on G to functions on the dual of G (using something called
Pontryagin duality). The thing is, functions on the dual of G multiply
pointwise whereas functions on G itself multiply like polynomials mod x^n - 1.

Therefore to multiply polynomials, one thinks of them as functions on G, uses
the DFT to take you to functions on the dual of G, multiplies pointwise, then
does an inverse transform to get you back to functions on G again. I'm
skimming over lots of details and oversimplifying a bit, but what I just
described is the process of using a convolution to multiply polynomials.

The really great thing is if n is a power of 2. Then you have this cool
Cooley-Tukey algorithm called the Fast Fourier Transform to do the DFT (and
IFFT) really fast (in time O(n log n) instead of O(n^2)). It works by
recognising that computing an FFT is _precisely_ the same thing as evaluating
a polynomial at the n-th roots of unity. This can be done by repeatedly
breaking the problem into halves and recognising that the same pattern of
roots of unity occurs in the first half as in the second. By factoring that
out, you can (recursively) save yourself half the work.

Again, oversimplified, but that's the nub of it.

------
tspiteri
The article, and quite a few posts here, describe the way they understand the
Fourier Transform as the way to understand the Fourier Transform. For it to be
intuitive depends on who is trying to understand it. Getting that out of the
way, this is how I find the Fourier Transform intuitive (using pseudo-code
instead of math notation to make it a bit verbose and emphasize the steps):

    
    
        fourier_trasform(signal sig(t), frequency freq):
            let sinu(t) = sinusoid with frequency freq
            let mult(t) = sig(t) * sinu(t)
            value = integral of mult(t) from -infinity to infinity
    

If the input signal sig(t) has the same frequency as the sinusoid sinu(t),
then integrating mult(t) over infinity will give an infinitely large value,
and that case is handled better by the Fourier Series.

If the input signal sig(t) has no relation to the frequency of the sinusoid,
then integrating mult(t) over infinity will give zero.

If the signal has a component with the required frequency, it will kind of
resonate with the sinusoid and give a non-zero value. The value then depends
on the magnitude of the signal and to how much it "resonates" with the
sinusoid.

When you do this for a range of different frequencies freq, but using the same
signal sig(t), you can plot how much the signal sig(t) resonates with all
frequencies, and that plot is the plot of the Fourier Transform.

~~~
ArbitraryLimits
By the by, one of the things that always tripped me up about the DFT was
understanding that for signals that are already sums of sinusoids, the
discrete sum gives exactly the same answer as the integral would. Remember the
trapezoidal rule for integration? It turns out it's exact for trigonometric
polynomials sampled at or above the Nyquist frequency.

Now, to find a life sciences journal to publish that in...

------
drblast
I think these concepts became clear to me when I learned about the complex
plane, Euler's formula, and demodulation of an FM signal.

Particularly enlightening was the demodulation of a frequency modulated sine
wave when the tuner was imperfectly matched to the carrier frequency. Looking
at it on an oscilloscope was similar to watching an old TV with the Vertical
Hold improperly set.

That made me start thinking in terms of a signal (sine wave) that was a cycle
rather than a sinusoidal shape. Seeing that you can graph the amplitude and
phase of any signal on the complex plane and that the frequency was the change
in phase from one moment to the next was the aha! moment.

Then if you think about sampling and how if you sample a sinusoid exactly at
its peaks how that would graph as a constant point on the complex plane, but
if you sampled at any other mismatched frequently, the point would rotate and
change amplitude with respect to either axis. The further away from the actual
frequency you'd go the the points would look more random, and they'd average
out to zero with fewer samples.

This would be a fun animation or java applet to make; I'm sure someone has
done it.

------
hammock
That was a great explanation, particularly the summary at the end with the
color-coded words corresponding to terms in the formula.

I came to the comments expecting to see nods of approbation at how cool this
explanation was (I stopped taking math at about Calc 3, so no linear algebra
for me) but instead I see people geeking out saying things like "to really
understand you need to grasp the complex plane, and groups and DPTs and so
forth."

Well, just so you know, for me the OP's intutitive explanation was enough.

------
regularfry
That's possibly the most intuitive explanation for what the maths is actually
doing that I've ever come across.

------
codesink
A great resource to read about Fourier transformation and Wavelets

<http://users.rowan.edu/~polikar/WAVELETS/WTpart1.html>

------
szany
I visualize it as projecting the function (as a vector) onto spirals of
different twisting rates.

Only 3 dimensions required, which is nice.

~~~
janjan
Yeah, thinking of the exp() part as a spirals in three dimensional space (the
imaginary plane + the time/space line) finally helped me to understand what's
really going on with the Fourier transform.

------
guscost
I like the explanation, but there's no reason to skip over the complex
exponents and Euler's formula like that. They're not really that hard to
understand intuitively: think of all multiplication as a continuous process.
Then f = e^x is simply the function that _transforms_ the multiplicative
identity (1) into ln(x).

Substitute "-1" into the left side of that equation, and see that no real
value of x will suffice. This is related to the fact that the imaginary
constant (i) wasn't discovered, it was simply _declared_ as an unknown
quantity that squares to -1.

The real magical part is that i _still works_ in more complicated situations:
multiplying any real number by e^(ix) as x increases gradually _transforms_ it
into an imaginary number, and then into its own negative, behaving like _a
counter-clockwise rotation_ when visualized in the complex plane.

------
torstesu
I love the idea that a man can sit down behind his desk, think for days,
weeks, months, years or even decades and come up with something which is so
abstract and beautiful explaining natural phenomenons with simple mathematical
formulas.

It takes some serious entrepreneurial skills and mindset to embark on a
problem which is seemingly impossible, and never giving up until the solution
has been derived.

Inspirational, to say the least!

~~~
apl

      > It takes some serious entrepreneurial skills and mindset
      > to embark on a problem which is seemingly impossible, and
      > never giving up until the solution has been derived.
    

Nothing specifically entrepreneurial about that, really.

~~~
jules
Entrepreneurial, like agile, has become a synonym for "generically good".

~~~
burgerbrain
In the context of PHB-speak I guess.

Lets utilize our synergy to create some entrepreneurial verticals!

------
nothis
I wish all maths books would have their formulas illustrated like this:
[http://altdevblogaday.org/wp-
content/uploads/2011/05/Derived...](http://altdevblogaday.org/wp-
content/uploads/2011/05/DerivedDFT.png)

That is beautiful.

~~~
AnthonBerg
Yes. Yes! Thanks for pointing that out - I hadn't noticed how brilliant the
use of color is. I just understood ...

And that's the hallmark of good teaching.

