On Feb 18, 6:44 am, Rick Lyons <R.Lyons@_BOGUS_ieee.org> wrote:
> On Sun, 17 Feb 2008 11:42:27 -0800 (PST), "Ron N."
>
> <rhnlo...@yahoo.com> wrote:
> >On Feb 17, 5:00 am, Rick Lyons <R.Lyons@_BOGUS_ieee.org> wrote:
> >> On Sat, 16 Feb 2008 23:46:09 -0800, glen herrmannsfeldt
>
> >> <g...@ugcs.caltech.edu> wrote:
> >> >Rick Lyons wrote:
>
> (snipped by Lyons)
>
>
>
> >> >-- glen
>
> >> Hi glen,
> >> Well,...perhaps we're having a sematics (language)
> >> problem here. I was referring to a finite-duration
> >> sequence's Fourier transform to be defined by the sequence's
> >> discrete-time Fourier transform (DTFT). And the DTFT is
> >> a continuous (and complex) function of the frequency
> >> variable omega defined by:
>
> >> n = +inf
> >> ---
> >> X(w) = \ x(n)*exp(-jwn)
> >> /
> >> ---
> >> n = -inf.
>
> >> So if we consider the two-sample sequence:
>
> >> x1 = [2,3]
>
> >> and the 16-sample sequence:
>
> >> x2 = [0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,0]
>
> >> and the infinite-length sequence:
>
> >> x3 = [...,0,0,0,0,0,0,2,3,0,0,0,0,0,0,...]
>
> >> is it not true that the above three sequences will all
> >> have identical discrete-time Fourier transforms (DTFTs)?
>
> >> (Of course, as shown on page 50 of Oppenheim and Schafer,
> >> 3rd Edition, the DTFT of a sequence only exists if the
> >> sum of that sequence's samples is less than infinity.)
>
> >> All I'm saying is that the x1 = [2,3] sequence has a
> >> continuous Fourier transform and that transform is:
>
> >> X(w) = 2*exp(-j2w) + 3*exp(-j3w)
>
> >> where the continuous frequency variable w (omega) is
> >> defined over a range of 2*pi, typically -pi to +pi.
>
> >You've just assumed that the coeff's of exp(-j4w), etc.
> >are zero. What allows you to make the assumption that
> >an unspecified value is zero, or 17.5? A better assumption
> >might be that those coeff's are random variables with a 50%
> >chance of being either 2 or 3, since that's what's been
> >observed in the population so far.
>
> >IMHO. YMMV.
>
> Hi Ron,
> Humm, ... I've tried to understand what you're
> telling me, but I'm having trouble. You seem to be
> saying that there is no such thing as a sequence
> containing only two samples. You asked:
> "What allows you to make the assumption that
> an unspecified value is zero, or 17.5?"
>
> I'm not making any assumptions about "unspecified
> values" because they do not exist.
>
> Can we at least agree that it is possible to
> write down, on a piece of paper, a sequence that
> has two samples? If we can agree on that, then I
> think we have a chance of understanding each other.
A vector of length 2, no problem. Calculating the
dot product of two vectors of different lengths,
problem. Some would say that the result is just
unknown.
> Again, all I was saying is that the two-sample
> sequence, x1 = [2,3], has a continuous Fourier
> transform (DTFT).
I don't think I seem a complete and consistent
formulation for the DTFT which uses a dot product
of length 2. Leaving terms out is the same as
applying a rectangular window and then calculating
the dot product (or taking the limit of the dot
product as n gets "big").
> See Ya',
> [-Rick-]
IMHO. YMMV.