Re: Zero-padding, resolution and aliasing

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:

(snip)

I agree with your words. The way I look
at "zero padding" is as follows: given a
finite duration sequence of time samples, that
sequence has an "actual" continuous spectrum
(infinitely fine granularity). The continuous
spectrum is called the "discrete-time Fourier
transform (DTFT). Using the DFT (or FFT), we
can effectively "sample" that continuous spectrum
and plot those samples on our computer screens.
Zero padding the time sequence and performing
larger DFTs merely gives us "more closely spaced"
samples of the time sequence's continuous
spectrum (its DTFT).

I disagree.

I finite sequence of samples does not have a continuous
transform unless you assume that there are an infinite
number of zero samples before and after the sequence.
With the extra zeros it is then an infinite sequence.

Without specifying the rest of the sequence, the other
points should be considered "don't care" samples.

-- 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.
--
rhn A.T nicholson d.0.t C-o-M

.

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