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

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

>  > If we then increase the resolution in frequency-space (f-space) by
> >  increasing N_f as discussed, it might be assumed that we can achieve
> >  a higher resolution result in the time domain when inverse
> >  transforming back to it.
>
> Not greater resolution, but rather greater _span_ in the
> time domain.

Yes, I can see that that would have been the better assumption. But 
still not a good one. As you said later, you can't create something 
from nothing, aka second law. I erroneously was thinking that as I 
increased the resolution in the f-domain (interpolated more points), 
that inverse transforming would then somehow achieve greater 
interpolation in the time domain. I should have known better, but it 
was a good exercise to turn the crank and reprove the 2nd law, as it 
were.


>  > But it doesn't work like that. We can't get
> >  back the original waveform. Rather, we might I suppose (?), but there
> >  is additional content created in the time domain, in particular
> >  imaginary content, whereas the original time domain contained only
> >  real data. Not good.
>
> Maybe not good, but not terrible, either.  The "additional"
> content is all zeros.

No, I was creating complex content, but see my above paragraph for 
why that happened. What you're talking about is something different, 
I think.


> Tangential remark:  If we are writing T as a column vector
> and F as a row vector, note that E is _not_ the usual sort
> of matrix that takes in a vector and puts out a vector of
> the same kind.

I didn't explicitly state so, but I was viewing both T and F as 
column vectors, so that E is the "usual" sort of matrix. I used a "." 
to denote "conventional" matrix math.


> This is a question of taste, of the sort that ought not be
> argued, but just to avoid confusion let me say that I have
> been thinking of F as a _row_ vector and therefore here we
> are increasing the number of _columns_ in F and by the same
> token increasing the number of _columns_ in E (not "rows
> in E").  This is how my demo spreadsheet lays things out.

No problem there. Demo spreadsheet trumps in your case. I've been 
doing my work in Mathematica so I "see" it more conventionally 
written (well, sometimes).


> Suppose the forward transform increases the resolution by a
> factor of 4, i.e. Nf/Nt = 4.  Then the non-square forward
> transform E consists of 4 fairly ordinary square transforms
> (with a little bit of heterodyning).  They are not stacked
> side-by-side but rather collated i.e. intercalated.

Exactly. You keep using better words than me.


> 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.

How then to correctly describe the sampling theorem in this context? 
If I sample the data at higher than the Nyquist rate, I *can* use 
math to interpolate between the sample points and get exactly the 
correct waveform between the sample points. Somehow the key point is 
that I have created more of the waveform between the sample points, 
but not more information. It's hard to get this subtlety - by 
successfully interpolating between sample points, it *seems* like I 
have gotten something extra.


> 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.

Yes, see above?


> You cannot safely decide whether a signal is band-limited
> or not by looking at the sampled data!  That would be like
> asking the drunkard whether he is drunk.  For example, if
> I have a 100.01 Hz wave and sample it at 10.00 Hz, it will
> look like a beautiful 0.01 Hz signal.  Ooops.
>
> If you want to be sure that the original data is band-limited,
> rely on the physics, not on the math.  That is, put a filter
> on it!  A real, physical, analog filter.  Then you can be
> sure.

Agreed, but if you look at the FT of the signal and see that it goes 
to zero at high frequencies (or a reasonable approximation of it, 
just as you would get with a physical filter), you can still safely 
decide, no? They're really the same thing. Your description is to put 
the filter on to make sure. My description is to look at the FT and 
see that the filter must somehow be there already, whether I 
knowingly applied it or not. If the FT does not indicate safety, I 
must either apply a physical filter or find a way to sample at a 
higher rate. But I can tell by looking at the FT...


Stefan Jeglinski