[Numpy-discussion] Here's what I've done to numpy.fft

David Goldsmith d.l.goldsmith@gmail....
Mon Jul 12 17:47:30 CDT 2010

In light of my various questions and the responses thereto, here's what I've
done (but not yet committed) to numpy.fft.

There are many ways to define the DFT, varying in the sign of the
exponent, normalization, etc.  In this implementation, the DFT is defined

.. math::
   A_k =  \sum_{m=0}^{n-1} a_m \exp\left\{-2\pi i{mk \over n}\right\}
   \qquad k = 0,\ldots,n-1

where `n` is the number of input points.  In general, the DFT is defined
for complex inputs and outputs, and a single-frequency component at linear
frequency :math:`f` is represented by a complex exponential
:math:`a_m = \exp\{2\pi i\,f m\Delta t\}`, where
:math:`\Delta t` is the *sampling interval*.

Note that, due to the periodicity of the exponential function, formally
:math:`A_{n-1} = A_{-1}, A_{n-2} = A_{-2}`, etc.  That said, the values in
the result are in the so-called "standard" order: if ``A = fft(a,n)``,
then ``A[0]`` contains the zero-frequency term (the sum of the data),
which is always purely real for real inputs.  Then ``A[1:n/2]`` contains
the positive-frequency terms, and ``A[n/2+1:]`` contains the
negative-frequency (in the sense described above) terms, from least (most
negative) to largest (closest to zero).  In particular, for `n` even,
``A[n/2]`` represents both the positive and the negative Nyquist
frequencies, and is also purely real for real input.  For `n` odd,
``A[(n-1)/2]`` contains the largest positive frequency, while
``A[(n+1)/2]`` contains the largest (in absolute value) negative
frequency.  In both cases, i.e., `n` even or odd, ``A[n-1]`` contains the
negative frequency closest to zero.

Feedback welcome.

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