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Re: Fast Fourier Transforms


Randy was mostly correct, but omitted to mention that the complex Fourier
domain has complex conjugate symmetry. Only the first N/2 (N even) Fourier
coefficients of a real signal are unique. The remaining are "negative"
frequency coefficients (you probably don't need to understand this). So, for
any real signal x, do the following

Drop[Fourier[x], -Length[x]/2]

or more commonly,

Drop[Abs[Fourier[x]], -Length[x]/2]

since most of the time we are interested in the magnitude of the Fourier
coefficients, only.

The remaining N/2 coefficients are organized as follows {position, frequency
in Hertz}:
{{1, 0}, {2, 1/T}, {3, 2/T}, {4, 3/T}, ... , {N/2-1, (N/2-2)/T}, {N/2,

where T is the time interval between samples. 1/T is the so-called sampling

If you need more details consult any electrical engineering textbook that
has a discussion of the discrete Fourier transform (DFT).

Hope this helps,


Mariusz Jankowski
University of Southern Maine
mjkcc at

"Burton" <Ctheurer at> wrote in message
news:85p7u3$6je at
> Hello,
>     I am doing some analysis of acoustic data obtained from a pickup
> on a machine.  The data is in the form of ordered pairs i.e. Time and
> voltage.  As far as I know fft in Mathematica only operates on a list of
> single numbers.  I have imported the voltage data (since the time is
> incremental) and performed an fft on that successfully.  My problem lays
> interpreting this data.  I don't understand how to scale the x-axis of the
> fft plot to represent frequency.
>     Any suggestions would be appreciated.
> Thanks
> -Charles Burton Theurer
> ctheurer at
>     University Of Massachusetts

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