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> 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.
> 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.
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.
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.
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.
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.
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.