rb-j, you wrote
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
>
>
Precisely, and one way to get there is by starting from the Poisson
Summation Formula and taking f(n) = T dirac(t-nT) (thus the distributional
hand waving requirement). This is what I meant by PSF + hand waving. I
think we're on the same page, basically.
cheers,
E
On Fri, Mar 28, 2014 at 1:32 PM, robert bristow-johnson <
[email protected]> wrote:
> 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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