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

[Stefan Jeglinski <jeglin@4pi.com>]

> In the spirit of the correspondence principle, let's examine the
> relationship between a discrete Fourier transform and a non-discrete
> Fourier transform. We start with a discrete transform and see what
> happens:
>
> 1. As previously discussed: When dt becomes small, the
> frequency-range of the first period of the output becomes large. (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. As previously discussed: When Ndt becomes large, the output
> becomes nearly continuous.
>
> 3. If you do both together - small dt and large Ndt - the discrete
> Fourier transform begins to look a whole lot like the old-fashioned
> non-discrete Fourier transform.
>
> OK?

Agreed, as long as we are referring in bullet 2 to Nf, as per your Eq 
24. Increasing Nf leads to a higher resolution plot of Vf, for a 
given Ntdt. Granted, if I decrease dt in order to sample more of Vin, 
Nt will necessarily increase if I am to still cover the entire region 
of interest in Vin. But my choice of Nf is independent of that, used 
only to increase the resolution in frequency space. No?

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


> 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

	leading to

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?


Stefan Jeglinski