[Numpy-discussion] Here's what I've done to numpy.fft
Travis Oliphant
oliphant@enthought....
Mon Jul 12 20:33:11 CDT 2010
On Jul 12, 2010, at 5:47 PM, 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*.
This sounds very good, but I would not mix discussions of sampling interval with the DFT except as an example use case.
The FFT is an implementation of the DFT, and the DFT is self-contained for discrete signals without any discussion of continuous-time frequency or sampling interval. Many applications of the FFT, however, use sampled continuous-time signals.
So, use a_m = \exp\(2\pi j m k\) to describe the single-frequency case. If you want to say that k = f\Delta t for a sampled-continuous time signal, then that would be fine, but there are plenty of discrete signals that don't have any relation to continuous time where an FFT still makes sense.
>
> 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.
I would remove "That said, " near the beginning of the paragraph.
Thanks for the great docs.
-Travis
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