scott wrote:
>>  (Btw, did you know that even if a sound signal has only one single
>> frequency, a discrete fourier transform is usually completely unable
>> to find that single frequency, and instead will find a large (potentially
>> infinite) amount of frequencies around that real frequency?)
> 
> It just depends how many samples you use for your fourier transform, the 
> more samples you take, the more accurately you can divide up the 
> frequency spectrum.  If you have a couple of seconds of normal audio 
> recording (so like 100k samples), you should be able to find a single 
> frequency pretty accurately, certainly to within 1 Hz.

IIRC, there's a version of the Fourier transform from a discrete time 
domain to a continuous frequency domain. Anybody know details?

Post a reply to this message

scott <sco### [at] scottcom> wrote:
> Of course there is, try it.  If you create a frequency that is half-way 
> between two frequencies in the DFT, your DFT histogram will still look the 
> same shape, the peak will just be shared between the two frequencies.

  The same shape as what?

  Basically you are saying that you can recreate a sine wave (which is one
single frequency) with the sum of two sine waves (the two frequencies in
the discrete frequency spectrum)? Not only that, but the two sine waves
in question are adjacent to each other in the discrete frequency spectrum?

  I would be interested in seeing some simple math about this.

-- 
                                                          - Warp

Post a reply to this message

>> Of course there is, try it.  If you create a frequency that is half-way
>> between two frequencies in the DFT, your DFT histogram will still look 
>> the
>> same shape, the peak will just be shared between the two frequencies.
>
>  The same shape as what?

As a frequency that is exactly lined up with one of the bands of the DFT.

>  Basically you are saying that you can recreate a sine wave (which is one
> single frequency) with the sum of two sine waves (the two frequencies in
> the discrete frequency spectrum)?

No, I am saying that given a finite DFT transform result with more than one 
non-zero entry, if you perform the inverse DFT you will recreate a single 
frequency sine wave.

Post a reply to this message

Warp wrote:

>   Basically you are saying that you can recreate a sine wave (which is one
> single frequency) with the sum of two sine waves (the two frequencies in
> the discrete frequency spectrum)? Not only that, but the two sine waves
> in question are adjacent to each other in the discrete frequency spectrum?
> 
>   I would be interested in seeing some simple math about this.

   sin x + sin y = 2 cos (x-y)/2 sin (x+y)/2

If we replace x and y with jt and kt, we have

   sin jt + sin kt = 2 cos t(j-k)/2 sin t(j+k)/2

If j and k are very close, (j-k)/2 would be nearly zero, so the cos term 
would be nearly unity. Then we would have just

   sin jt + sin kt = 2 sin t(j+k)/2

In other words, a sine wave who's frequency is the arithmetic mean of 
the frequences of the two waves being added. QED.

Post a reply to this message