- #1
juriguen
- 2
- 0
Hi there!
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
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