[music-dsp] time representation in the frequency domain

Olli Niemitalo o at iki.fi
Fri May 15 05:27:45 EDT 2009


One example that demonstrates that phase will not always tell all is
when you have a time-domain signal that is symmetrical around time
zero. The phase in the frequency domain will be zero or 180 degrees
for all frequencies. Imagine that you have a long recording of
whatever audio, and then append to it a time-reversed copy of it. This
will effectively move most of the information contained in the phase
to the amplitude.

On the other hand, if you have a time-domain signal where the energy
at each frequency is sharply centered at no more than one point in
time, then those time points will be well described by the group
delay, which can be calculated from phase.

-olli

On Fri, May 15, 2009 at 11:47 AM, Richard Dobson
<richarddobson at blueyonder.co.uk> wrote:
> Laszlo Toth wrote:
>>
>> On Thu, 14 May 2009, Michael Gogins wrote:
>>
>>> Suppose that you record a click, a Dirac delta, at a specific time,
>>> with silence coming before and after.
>>> You then take the discrete Fourier transform of this time-domain
>>> signal. You end up with a frequency-domain signal in which the phases
>>> and amplitudes of the components completely encode the time of the
>>> click
>>
>> It's only the phases. The amplitudes are shift-invariant, so they are
>> imdependent of the position of the impulse.
>>
>
> Perhaps the simplest way of looking at this is to see the time aspect as
> effectively encrypted into the FFT. All the information is there, and can be
> recovered exactly by an inverse FFT in the absence of any alteration; but
> ~discovering~ that information from the FFT output in a deterministic and
> meaningful way may be at least as hard as decrypting very large numbers
> composed of primes, without a key. Much more effort, probably, than simply
> doing the inverse FFT.
>
>
> Richard Dobson
>
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