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

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

John D,

> If the time-domain data is the original data, no amount
> of Fourier transforms or other math will ever create more
> information.  There will be no "loaves and fishes" miracle
> where you create more just by rearranging things.  The
> second law of thermodynamics forbids it.
>
> So all we are talking about here are various heuristics for
> _interpolating_ between points in the time domain.
>
> Interpolation is easy if you know the original signal was
> band-limited before it was sampled ... i.e. no aliasing.

I am confused by the following. I use my own notation, but will 
follow the nomenclature on the av8n page (referred to earlier in the 
thread) as much as possible.

Consider a time-domain signal x(t) and its FT, X(f). The sampled 
version is x(tj) and the DFT is X(fk). Assume at this point that Nt = 
Nf = N. j = {0...N-1} and k = {0...N-1}.

Let us assert that x(t) is bandlimited, and N is such that we are 
sampling above the Nyquist rate. We can confirm our assertion by 
observing that X(f) and X(fk) have no overlapping aliases.

Since x(t) is bandlimited and sampling is above Nyquist rate, we can 
interpolate between {tj} as much as we like via the sampling theorem, 
to recover the original signal in its entirety:

x(t) = SUMj [x(tj)*Sinc(t,j)] (see standard references for def of Sinc(t,j))

Now, on the one hand, we can't create something from nothing. To wit, 
the inverse DFT (IDFT) will only recover the original sample points. 
But I was sorely tempted to try your nice trick in section 4, leading 
to Eq 24, but in the reverse direction, to wit:

x(t) = SUMk X(fk)*Exp(+2*pi*i*k*t*df) df

The equation is true for t = tj (it's just the IDFT), but not for any 
other t. At all other values of t, x(t) is complex by this 
calculation. So the interpolation fails.

While my intuition isn't surprised by the failed attempt to get 
something for nothing here, this result is counterintuitive on 2 
other counts:

a) it seemed like we got something for nothing with this trick in the f-domain

and

b) the sampling theorem CAN successfully interpolate, since we are bandlimited.



Stefan Jeglinski