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David Page wrote:
> 
> I am a novice in Matlab. Please excuse if this question sounds silly.
> 
> I am experimenting with fft and ifft function of Matlab. When I take a
> fourier transform of a function, it fives me the frequency and its amplitude
> (weight of the frequency) and does not contain any information about the
> location of these frequencies.
> 
> If the above is true, how can it reconstruct the whole signal again when
> ifft is applied ? How does it remember the locations of the frequencies ?

The DFT/FFT does no time localization.  It presumes the signal is
periodic in the length of the transform and thus that all the
frequencies are infinite in temporal extent.  All that can be said about
the time of a frequency given that it is infinite in both directions is
its phase which is periodic in 2*pi.  There are other orthogonal bases
for signal decomposition than the trigonometric one Fourier found which
do not presume signal periodicity and give frequency related time
localization.  Look at wavelet theory.


Bob
-- 

"Things should be described as simply as possible, but no simpler."

                                             A. Einstein


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Hi David:

I think Bob's answer was not exactly what you were looking for, and was
based on more complex mathematics. I think the answer to what is
bothering you is the sampling rate issue. Matlab's FFT produces a set of
frequency points which need additional information from you to interpret
what frequency they correspond to in real life. The length of the
sequence and the sampling rate determine the mapping to real life. For
instance if you have a sequence of length L and sampling rate Fs then
the highest real-life frequency (which is actually at about the midpoint
of your Matlab FFT sequence) corresponds to 1/2 the sampling rate
(Nyquist frequency). Each frequency point is then 2/L of the Nyquist
freq. (The factor of 2 comes in because the second half of FFT sequence
is mirror image of first and contains no new information for real
signals. (These are called the negative frequencies which, in the usual
DSP convention follow the positive freqs instead of preceeding them as
in analog world. You can switch arrangement using FFTSHIFT command) The
first frequency point is the DC value. The next is (Fs/2)/(L/2) or Fs/L
the next is 2Fs/L, the next 3Fs/L until you reach midpoint (L/2) (Fs/L)
or simply Fs/2, the highest possible freq. (next point begins the
negative freqs). So if you have sequence 1000 points long and Fs is
100Hz, then first point is DC, second is .1 Hz, third is .2 Hz, fourth
is .3 Hz, etc. (Note people normally use a power of 2 for length of FFT
but this is only because FFT computation is faster that way. There is
nothing else magic or more accurate about powers of 2, contrary to
popular misconception.)

When you use IFFT to transform back to time, you still have same issue
of what point in real life time corresponds to each point in sequence.
The answer here also depends on sampling rate. The first point is
starting point T0 the second point is T0 plus 1/Fs, the next  T0 plus
2/Fs, etc. Since sampling rate figures into both calculations, the FFT
and IFFT sequences are really not affected by whatever sampling rate you
choose. The only difference is the mapping to real life. So matlab just
takes one sequence as input and produces other sequence as output. It is
up to you to interpret what they mean in your system. This is why many
digital signal processing functions in matlab use a normalized sampling
rate from 0 to 1 instead of your actual rate. You just multiply or
divide by real sampling rate to convert between the two.

I hope this clears things up.

Barry Jacobson
Harvard-MIT Division of Health Sciences and Technology
bdj@mit.edu

Maybe this

David Page <david@REMOVE_SPAM.imhost.com> wrote in message
news:eMqL6.71357$2U.31863845@news2.rdc2.tx.home.com...
> I am a novice in Matlab. Please excuse if this question sounds silly.
>
> I am experimenting with fft and ifft function of Matlab. When I take a
> fourier transform of a function, it fives me the frequency and its
amplitude
> (weight of the frequency) and does not contain any information about
the
> location of these frequencies.
>
> If the above is true, how can it reconstruct the whole signal again
when
> ifft is applied ? How does it remember the locations of the
frequencies ?
>
> Thanks in advance.
>
>

Thanks Barry and Bob.

Barry, the idea is now concrete in my mind. I realized what I was missing
before. Thank you for the wonderful explanation.

David.

"Barry D. Jacobson" <bdjacobs@bellatlantic.net> wrote in message
news:Y6yL6.46346$Rg2.1574300@typhoon2.ba-dsg.net...
> Hi David:
>
> I think Bob's answer was not exactly what you were looking for, and was
> based on more complex mathematics. I think the answer to what is
> bothering you is the sampling rate issue. Matlab's FFT produces a set of
> frequency points which need additional information from you to interpret
> what frequency they correspond to in real life. The length of the
> sequence and the sampling rate determine the mapping to real life. For
> instance if you have a sequence of length L and sampling rate Fs then
> the highest real-life frequency (which is actually at about the midpoint
> of your Matlab FFT sequence) corresponds to 1/2 the sampling rate
> (Nyquist frequency). Each frequency point is then 2/L of the Nyquist
> freq. (The factor of 2 comes in because the second half of FFT sequence
> is mirror image of first and contains no new information for real
> signals. (These are called the negative frequencies which, in the usual
> DSP convention follow the positive freqs instead of preceeding them as
> in analog world. You can switch arrangement using FFTSHIFT command) The
> first frequency point is the DC value. The next is (Fs/2)/(L/2) or Fs/L
> the next is 2Fs/L, the next 3Fs/L until you reach midpoint (L/2) (Fs/L)
> or simply Fs/2, the highest possible freq. (next point begins the
> negative freqs). So if you have sequence 1000 points long and Fs is
> 100Hz, then first point is DC, second is .1 Hz, third is .2 Hz, fourth
> is .3 Hz, etc. (Note people normally use a power of 2 for length of FFT
> but this is only because FFT computation is faster that way. There is
> nothing else magic or more accurate about powers of 2, contrary to
> popular misconception.)
>
> When you use IFFT to transform back to time, you still have same issue
> of what point in real life time corresponds to each point in sequence.
> The answer here also depends on sampling rate. The first point is
> starting point T0 the second point is T0 plus 1/Fs, the next  T0 plus
> 2/Fs, etc. Since sampling rate figures into both calculations, the FFT
> and IFFT sequences are really not affected by whatever sampling rate you
> choose. The only difference is the mapping to real life. So matlab just
> takes one sequence as input and produces other sequence as output. It is
> up to you to interpret what they mean in your system. This is why many
> digital signal processing functions in matlab use a normalized sampling
> rate from 0 to 1 instead of your actual rate. You just multiply or
> divide by real sampling rate to convert between the two.
>
> I hope this clears things up.
>
> Barry Jacobson
> Harvard-MIT Division of Health Sciences and Technology
> bdj@mit.edu
>
> Maybe this
>
> David Page <david@REMOVE_SPAM.imhost.com> wrote in message
> news:eMqL6.71357$2U.31863845@news2.rdc2.tx.home.com...
> > I am a novice in Matlab. Please excuse if this question sounds silly.
> >
> > I am experimenting with fft and ifft function of Matlab. When I take a
> > fourier transform of a function, it fives me the frequency and its
> amplitude
> > (weight of the frequency) and does not contain any information about
> the
> > location of these frequencies.
> >
> > If the above is true, how can it reconstruct the whole signal again
> when
> > ifft is applied ? How does it remember the locations of the
> frequencies ?
> >
> > Thanks in advance.
> >
> >
>
>