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

Jochen Schröder cycomanic@gmail....
Mon Jul 12 21:08:56 CDT 2010

On 13/07/10 08:47, David Goldsmith wrote:
> 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
> as
> .. 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.
> DG
Hi David,

great work. I agree with Travis leave the sampling out. This make things 
more confusing. I'd also suggest pointing to fftshift for converting the 
"standard" order to order "min frequency to max frequency"

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