# [LAD] Paul's Extreme Sound Stretch

Robin Gareus robin at gareus.org
Thu Sep 30 19:23:08 UTC 2010

```On 09/30/10 20:49, Philipp Überbacher wrote:
> Excerpts from fons's message of 2010-09-30 14:29:01 +0200:
>> On Thu, Sep 30, 2010 at 01:53:44PM +0200, Robin Gareus wrote:
>>
>>>> Q: Can anyone explain the FFT in simple terms ?
>>>> A. No.
>>>
>>> LOL.
>>>
>>> basically, Fourier proved that any signal can be represented a sum of
>>> sine-waves.
>>>
>>> (well, that's not entirely true: it needs to be a periodic signal, but
>>> the period length can approach infinity...)
>>>
>>> FFT is "just" the implementation of that theorem (or Principle?!)
>>
>> The original Fourier Transform as invented by the smart French
>> guy of the same name does operate on continuous (as opposed to
>> sampled) data from -inf to +inf. The 'spectrum' interpretation
>> came later. It was originally a mathematical tool used to find
>> integrals of functions that would be impossible to integrate in
>> closed form, and Fourier himself used it to study the propagation
>> of heat in solids.

Thanks a lot for this comprehensive history lesson.

Back when I was introduced to FT in some Physics lecture I was happy
that I was able to use it and completely forgot to check the history :)
Probably related to why I favored experimental Physics over Theory.

>> The DFT (Discrete FT) is the same thing operating on sampled
>> signals. It is usually also limited in time.
>>
>> The FFT (Fast FT) is an algorithm to compute a finite-length
>> DFT very efficiently.
>>
>> The 'spectrum' interpretation is really quite ambiguous.
>>
>> You could take the DFT of e.g. a complete Beethoven symphony.
>> The result is the 'spectrum' and in theory this is fixed over
>> infinite time - the frequencies that are present according to
>> this spectrum are there *all the time*. But that is not how
>> we would perceive the music - we do not hear a constant mash
>> of all frequencies, the spectrum as we hear it changes over
>> time.
>>
>> Ciao,
>>
>> --
>> FA
>>
>> There are three of them, and Alleline.
>
> And I guess this is where the windowing comes in. Calculate the spectrum
>
correct.

Furthermore there are different kind of windows (here a window refers to
a block of audio-samples) and windows can overlap. That's where it gets
complicated.

The most commonly known window shapes are Rectangular window, Gaussian ,
Hann- and Hamming windows (the last two are cosine shapes) which allow
to avoid discontinuities when processing blocks.

Cheers!
robin

```