MATLAB Central Newsreader - fft amplitude relationship

fft amplitude relationship

Bruce — Fri, 23 May 2008 16:53:02 -0400

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?

Re: fft amplitude relationship

roberson@ibd.nrc-cnrc.gc.ca (Walter Roberson) — Fri, 23 May 2008 18:00:50 -0400

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

Re: fft amplitude relationship

Bruce — Wed, 28 May 2008 15:48:02 -0400

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

Re: fft amplitude relationship

Ken Garrard — Wed, 28 May 2008 16:07:01 -0400

"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

Re: fft amplitude relationship

Darren — Mon, 05 Jan 2009 23:48:01 -0500

Can someone point me to a technical reference/document that explains or discusses this fft (time/freq) amplitude scaling concept. Thanks,

Darren

"Ken Garrard" <ken_garrardAT@ncsuDOT.edu> wrote in message <g1jvv5$pq4$1@fred.mathworks.com>...
> "Bruce " <italianasa84@gmail.com> wrote in message
> <g1juri$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
>

Re: fft amplitude relationship

Paul — Tue, 06 Jan 2009 00:07:01 -0500

"Darren " <dgoshi@ucla.edu> wrote in message <gju67h$avn$1@fred.mathworks.com>...
> Can someone point me to a technical reference/document that explains or discusses this fft (time/freq) amplitude scaling concept. Thanks,
>
> Darren
>
>
> "Ken Garrard" <ken_garrardAT@ncsuDOT.edu> wrote in message <g1jvv5$pq4$1@fred.mathworks.com>...
> > "Bruce " <italianasa84@gmail.com> wrote in message
> > <g1juri$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
> >

The FFT is essentially an integral operator over a function (usually f(t) but not always). In the real discrete computer world, integrals are represented by summations. If you work this through from the basics, then it will become more clear what the scaling relations are.

Re: fft amplitude relationship

Dave Robinson — Tue, 06 Jan 2009 14:54:02 -0500

"Darren " <dgoshi@ucla.edu> wrote in message <gju67h$avn$1@fred.mathworks.com>...
> Can someone point me to a technical reference/document that explains or discusses this fft (time/freq) amplitude scaling concept. Thanks,
>
> Darren
>
>
> "Ken Garrard" <ken_garrardAT@ncsuDOT.edu> wrote in message <g1jvv5$pq4$1@fred.mathworks.com>...
> > "Bruce " <italianasa84@gmail.com> wrote in message
> > <g1juri$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
> >

If you can get hold of it, the book

Title The Fast Fourier Transform and its applications
Author E.O. Brigham

It will tell you all you ever want to know about FFT, written in an almost casual style. In my opinion - for what it is worth, this is the best document explaining FFT that I have come across in all of my long career in DSP.

I checked with Amazon, and it still appears to be available via some of the specialist book suppliers - but is expensive, so search your local library before purchasing it.

Regards

Dave Robinson

Re: fft amplitude relationship

Paul Mennen — Tue, 06 Jan 2009 16:58:02 -0500

"Darren " wrote:
> Can someone point me to a technical reference/document that explains
> or discusses this fft (time/freq) amplitude scaling concept.

Chapter 8 of Cleve Moler's well written book "Numerical Computing with Matlab"
contains a nice tutorial aimed at teaching the basics of Fourier Analysis. This book
is available free at: http://www.mathworks.com/moler/

If your math skills are weak and you want a book that starts with the simplest
math concepts and works its way up, check out "Mathematics of the Discrete
Fourier Transform", also available for free at:
http://ccrma.stanford.edu/~jos/mdft/

If you get very far with fft analysis you will soon run across the idea of fft
windows. Many find the treatment of this topic in books esoteric and hard
to understand, so I wrote a short explanation of fft windows that doesn't
require a PhD in math to understand. This document is available on the
file exchange. (Just search for "winplt").

There are many other on-line sources of information about DSP topics that
you can find with simple google searches.

~Paul