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# Re: [Phys-l] some questions related to sampling

• From: John Denker <jsd@av8n.com>
• Date: Fri, 09 Jan 2009 18:41:13 -0700

On 01/09/2009 04:34 PM, Stefan Jeglinski wrote:

I'm either being pedantic or still missing the point. Given a choice,
I'd rather not miss the point.

Ugh, you got me again. I left out one or two steps in the
argument. Let me try again:

In the spirit of the correspondence principle, let’s examine the
relationship between a discrete Fourier transform and a non-discrete
happens:

1. As previously discussed: When dt becomes small, the frequency-range
of the first period of the output becomes large, so it covers a big
piece of the f-axis. (The output is always defined for all frequencies,
so we can’t talk about the range of the output, only the range of one
period.)
2. Obviously as N_t dt becomes large, the input data covers a big piece
of the t-axis.
3. As previously discussed: When N_f dt becomes large, the output becomes
nearly continuous. The output period is being divided very finely.
4. For present purposes, let’s assume N_f = N_t, so that the transformation
will be “just barely” invertible in both directions. Call this common
value N.
5. If you take both limits together – small dt and large Ndt – the discrete
Fourier transform begins to look a whole lot like the old-fashioned
non-discrete Fourier transform.

You can't have a sampled input without having the output of the
transform be periodic. The proof is simple:

Understood. The way I had always looked at this was as follows
(hoping I can make this clear in my non-UTF8 client, and hopefully
without typos). Casting a sampled x(t) as

sx(t) = [Sum_over_all_n] x[n] delta-function[t - nT]

where T is the sampling interval. Then, fourier-transform sx(t):

X(f) = [Integral over +- infinity] sx(t) exp(-2pi*i*f*t) dt

X(f) = [Sum_over_n] x[n] exp(-2pi*i*f*n*T)

which is a fourier series representation of X(f), which by definition
is periodic. Is there anything mathematically fishy about looking at
it in this slightly more formal way?

Looks OK to me.

The key step is convincing yourself that sampling is represented by
multiplying by a Dirac comb. The rest is just math.

BTW that's the other half of the correspondence argument, i.e. how you
go from the continuous Fourier integral to the discrete Fourier series.

=========

I like to have the equation *and* the picture. I recently cobbled up
the picture that goes with aliasing:
http://www.av8n.com/physics/fourier-refined.htm#fig-fourier-aliasing