On 3/28/14 4:25 AM, Emanuel Landeholm wrote:
tl;dr version: The justification for DSP (equi-distant samples) is the
Whittaker-Shannon interpolation formula, which follows from the Poisson
summation formula plus some hand-waving about distributions (dirac delta
theory). Am I right?
i would say the word "plus" should be replaced by "or".
and in my opinion, a very small amount of hand-waving regarding the
Dirac delta (to get us to the same understanding one gets at the
sophomore or junior level EE) is *much* *much* easier to gain
understanding than farting around with the Dirac delta as a
"distribution". i.e. even though the mathematicians say it ain't true,
there *does* exist a function that is zero almost everywhere, yet the
integral is 1. if you can get past that, the EE treatment (which i
think some physicists also use) is much much better.
again, all you really need is
+inf +inf
T SUM{ delta(t-nT) } = SUM{ e^(i 2 pi k/T t) }
n=-inf k=-inf
and the existing shifting theorems of the Fourier Transform. but the
mathematicians object to the identity above because they say the left
side of the equation is meaningless without surrounding it with an
integral. mathematicians do not like naked Dirac delta functions
("they're not functions!"), but EEs have no problem with them.
from that EE POV, i still believe that this treatment:
https://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&oldid=217945915
is the simplest and most direct of them all. doesn't even require
convolution in the frequency domain like most textbooks do.
--
r b-j [email protected]
"Imagination is more important than knowledge."
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