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

[Travis Oliphant]

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