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I am having a bit of a trouble when I try to work out a demonstration involving Dirac delta functions. I know, they are not real functions, and all that, but it only makes my life more difficult :)

Lets begin by the beginning to see if anyone can help. The first equation I will write I think comes straight from the definition of the Dirac distribution:

[tex]

\int_{-\infty}^{\infty} f(t) \delta(t-nT) \mathrm{d}t = f(nT)

[/tex]

Ok, so far so good. But now I want to evaluate a more complicated expression:

[tex]

\sum_{n=0}^{N-1} f(nT) g(nT)

[/tex]

I guess, there should be no problem to rewrite each sampled function by means of the integral involving the Dirac distribution, so that the equation becomes:

[tex]

\sum_{n=0}^{N-1} f(nT) g(nT) = \sum_{n=0}^{N-1} \int_{-\infty}^{\infty} f(t) \delta(t-nT) \mathrm{d}t \int_{-\infty}^{\infty} g(\tau) \delta(\tau-nT) \mathrm{d}\tau

[/tex]

But now it comes when I don't know how to continue. For me the following demonstration would be just right, but the result is quite surprising:

[tex]

\sum_{n=0}^{N-1} f(nT) g(nT) = \int_{-\infty}^{\infty} f(t) \int_{-\infty}^{\infty} g(\tau) \sum_{n=0}^{N-1} \delta(t-nT) \delta(\tau-nT) \mathrm{d}t \mathrm{d}\tau = \int_{-\infty}^{\infty} f(t) \int_{-\infty}^{\infty} g(\tau) \sum_{n=0}^{N-1} \delta(t-\tau) \mathrm{d}t \mathrm{d}\tau = N \int_{-\infty}^{\infty} f(t) g(t) \mathrm{d}t

[/tex]

There's almost definitely something wrong there, since I believe the above result should only hold in the limit when N tends to infinity. I say this because the equality basically resembles the interpretation of the integral as a Riemann sum.

Any help would be really appreciated!

Thanks in advance

Jose

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# Double integral of product of Diracs

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