[Numpy-discussion] Another reality check

[David Goldsmith]

Thanks, both.

On Mon, Jul 12, 2010 at 5:39 AM, Fabrice Silva <silva at lma.cnrs-mrs.fr>wrote:

> Le lundi 12 juillet 2010 à 18:14 +1000, Jochen Schröder a écrit :
> > On 07/12/2010 12:36 PM, David Goldsmith wrote:
> > > On Sun, Jul 11, 2010 at 6:18 PM, David Goldsmith
> > > <d.l.goldsmith at gmail.com <mailto:d.l.goldsmith at gmail.com>> wrote:
> > >
> > >     In numpy.fft we find the following:
> > >
> > >     "Then A[1:n/2] contains the positive-frequency terms, and A[n/2+1:]
> > >     contains the negative-frequency terms, in order of decreasingly
> > >     negative frequency."
> > >
> > >     Just want to confirm that "decreasingly negative frequency" means
> > >     ..., A[n-2] = A_(-2), A[n-1] = A_(-1), as implied by our definition
> > >     (attached).
> > >
> > >     DG
> > > And while I have your attention :-)
> > >
> > > "For an odd number of input points, A[(n-1)/2] contains the largest
> > > positive frequency, while A[(n+1)/2] contains the largest [in absolute
> > > value] negative frequency."  Are these not also termed Nyquist
> > > frequencies?  If not, would it be incorrect to characterize them as
> "the
> > > largest realizable frequencies" (in the sense that the data contain no
> > > information about any higher frequencies)?
> > >
> > > DG
> > >
> > I would find the term the "largest realizable frequency" quite
> > confusing. Realizing is a too ambiguous term IMO. It's the largest
> > possible frequency contained in the array, so Nyquist frequency would be
> > correct IMO.
>
> Denoting Fs the sampling frequency (Fs/2 the Nyquist frequency):
>
> For even n
> A[n/2-1] stores frequency Fs/2-Fs/n, i.e. Nyquist frequency less a small
> quantity.
> A[n/2] stores frequency Fs/2, i.e. exactly Nyquist frequency.
> A[n/2+1] stores frequency -Fs/2+Fs/n, i.e. Nyquist frequency less a
> small quantity, for negative frequencies.
>
> For odd n
> A[(n-1)/2] stores frequency Fs/2-Fs/(2n) and A[(n+1)/2] the opposite
> negative frequency. But please pay attention that it does not compute
> the content at the exact Nyquist frequency! That justify the careful
> 'largest realizable frequency'.
>
> Note that the equation for the inverse DFT should state "for m=0...n-1"
> and not "for n=0...n-1"...
>

Yeah, I already caught that, thanks!

How 'bout I just use "Fabrice's formula"?  It's explicit and thus, IMO,
clear.

DG
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