Help with interpolating function

[hxluo]

please help, i don't understand what the question is asking, please click on the thumbnail for the question.

 

[rbj]

hxluo, you need to learn how to use LaTeX here so you can post math equations that we can cut, copy, or paste in our discussion with you.

i think there is a mistake in the question. it needs to read:

_________

Let x(t) be a bandlimited signal such that

X(j \omega) = 0 \quad \forall \quad |\omega| \ge \frac{\pi}{T}
_________

i think the little t should be a capital T in the denominator of that fraction.

 

[hxluo]

yes you are right, so how do i solve this problem?

 

[rbj]

okay, you know about Laplace or Fourier Transformation, right?

X(s) = \int_{-\infty}^{+\infty} x(t) e^{-st} dt

this is the same X(.) as in

X(j \omega) = 0 \quad \forall \quad |\omega| \ge \frac{\pi}{T} .

and, from the nature of the problem, you know about the Nyquist/Shannon Sampling and Reconstruction Theorem, right? so if the differentiator wasn't there, how would you come up with an interpolation relationship for reconstructing

x(t) = \sum_{n=-\infty}^{\infty} x(nT) h(t - nT) ?

what is "h(t)"?

 

[hxluo]

i just don't understand the concept of interpolation that much.