[music-dsp] time representation in the frequency domain

Jan Baumgart raga.raga at gmx.de
Thu May 14 13:33:53 EDT 2009

```Sorry, if I didn't make myself clear. I'm no native english-speaker and
sometimes mix up technical terms..

A FFT assumes that the signal is periodic/static. This means, that no
change over time is involved.
The only term referring to time is "phase", but it's merely describing a
relationship between the
frequency-components and not a change over time.

To overcome this, one normally applies the FFT to very short
(quasi-static) enveloped time-segments of the signal
(so called frames) and (overlap-)adds them together after the inverse
transform.
This of course introduces artefacts and limits the resolution (time vs.
frequency).

However, practically it's possible to make one huge fourier transform of
a _whole_ non-periodic signal.
As I understand, the fourier paradigma isn't meant to be used this way,
but it works! For example, in convolution.
And it works without any artefacts or loss of any kind.

As stated by others, "time" (to be honest, i'm not sure myself, what
that _exactly_ means) in the frequency domain is
to be found somewhere in the relationship of neighboring bins.

BTW, my goal is to exploit this phenomena for experimental audio
manipulations...

Lubomir I. Ivanov wrote:
> Hello Jan,
>
> As you may know time should be considered linear in any system, unless we
> deal with strictly theoretical mathematics.
> The FFT works with time intervals (in which time is linear and soundwaves
> travels in linear time or at least we hear so ). It also works with
> amplitudes to construct the frequency / magnitude  for this signal.
>
> I will not say that time is not "encoded" in phase or vice versa. But there
> are these simple relations:
>
> f (frequency) = 1 / T (time)
> omega (angular frequency) = 2 * PI * f
> phi (phase)  = omega * T
>
> and in the case of FFT series - sin(omega * T), cos(omega * T) as our two
> subfunction types.
> It is suggested that any function (signal) can be expanded in terms of
> sin()
> and cos(), and since time is linear here we can work the above relations
> very easily.
>
> Or maybe I do not understand the nature of your question correctly? ;)
>
> Lubomir I. Ivanov
> http://www.tu-varna.bg
>
> ----- Original Message ----- From: "Jan Baumgart" <raga.raga at gmx.de>
> To: <music-dsp at music.columbia.edu>
> Sent: Thursday, May 14, 2009 1:38 PM
> Subject: [music-dsp] time representation in the frequency domain
>
>
>> Hallo!
>>
>> I've got a mathematical question:
>> When doing a fulltime fourier transform (only _one_ big "frame"), how is
>> the time information encoded into the phase?
>>
>> When doing an inverse transform (or for example convolve with an dirac
>> impulse) you get back exactly the same you put in in the first place. So
>> the time information has got to be hidden somewhere in the frequency
>> domain, doesn't it?
>>
>> When looking at a time domain signal from a fourier view, you can look at
>> the frequency information as "encoded" as sine function components. Does
>> the same apply for the time information encoded into the frequency
>> domain?
>>
>> This seems quite mysterious to me...has there been anything published on
>> this subject?
>>
>> cheers,
>> jan baumgart.
>>
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>
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```