Thread Subject: fft amplitude relationship

Does anyone know how the amplitude of a specific component
(or frequency) in the fourier domain is related to the
amplitude of that component in the time domain, and also the
number of points in the FFT?

It seems that the greater the NFFT, the smaller the
amplitude, but does anyone know the exact relationship?

In article <g16spe$o9t$1@fred.mathworks.com>,
Bruce <italianasa84@gmail.com> wrote:
>Does anyone know how the amplitude of a specific component
>(or frequency) in the fourier domain is related to the
>amplitude of that component in the time domain, and also the
>number of points in the FFT?

>It seems that the greater the NFFT, the smaller the
>amplitude, but does anyone know the exact relationship?

If you aren't trying to ifft() specifying a different number
of points than are in the fourier domain, then:

time domain amplitude = frequency domain coefficient * 2 / number of points.

Another way of phrasing this is that if you have an impulse
vector in the fourier domain,

N = 123; p = 17; %arbitrary number of points and impulse position.
b = zeros(1,N); b([p,end-p+2]) = N/2; %construct impulse vector
bi = ifft(b); %do the ifft

then the amplitude in the time domain contributed by that
impulse, would be 1.


If, though, you are doing an ifft specifying a different number of points
than are present in the vector, then I have not yet worked out a mental
model of what the results are.
--
  "The art of storytelling is reaching its end because the epic
  side of truth, wisdom, is dying out." -- Walter Benjamin

Thanks, what you mentioned was what I was looking for.
- Bruce

"Bruce " <italianasa84@gmail.com> wrote in message
<g1juri$c4v$1@fred.mathworks.com>...
> Thanks, what you mentioned was what I was looking for.
> - Bruce

Bruce,

Walter's answer is incomplete. The DC and Nyquist values
are unique and therefore do not have a 2 in the scale
factor.

time(DC) = frequency(DC) / N
time(Nyquist) = frequency(Nyquist) / N
time(all other points) = frequency(all other points) * 2 / N

Note that there is no Nyquist value for odd N.

Ken