Re: Zero-padding, resolution and aliasing

On Sun, 17 Feb 2008 11:42:27 -0800 (PST), "Ron N."
<rhnlogic@xxxxxxxxx> wrote:

On Feb 17, 5:00 am, Rick Lyons <R.Lyons@xxxxxxxxxxxxxxx> wrote:
On Sat, 16 Feb 2008 23:46:09 -0800, glen herrmannsfeldt



<g...@xxxxxxxxxxxxxxxx> 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.

Again, all I was saying is that the two-sample
sequence, x1 = [2,3], has a continuous Fourier
transform (DTFT).

See Ya',
[-Rick-]
.

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