Homework: Difference between revisions

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<math>
<math>
x(t)=\sum_{k=-\infty}^{\infty} x(kT)\phi_k(t)
x(t)=\sum_{k=-\infty}^{\infty} x(kT)\phi_k(t)
</math>
<br>
<b>Solution:</b>
<br>
<math>
\begin{matrix}
\left \langle x(t) \vert x(t) \right \rangle & = & \int_{-\infty}^{\infty} x(t)^{*} x(t)\,dt
\\ \ & = & \int_{-\infty}^{\infty} \left | x(t) \right |^2\,dt
\end{matrix}
</math>
<br>
<math>
x(t)=\sum_{k=-\infty}^{\infty} x(kT)\phi_k(t)
</math>
<br>
<math>

\begin{matrix}
\Rightarrow \left \langle x(t) \vert x(t) \right \rangle & = &
\left \langle \sum_{k=-\infty}^{\infty} x(kT)\phi_k(t) \vert \sum_{l=-\infty}^{\infty} x(lT)\phi_l(t) \right \rangle
\\ \ & = & \sum_{k=-\infty}^{\infty}\sum_{l=-\infty}^{\infty} x(kT)x(lT)
\left \langle \phi_k(t) \vert \phi_l(t) \right \rangle
\end{matrix}

</math>
</math>

Revision as of 09:25, 10 December 2004

Homework #9

Problem Statement:
Show that, for a bandwidth limited signal ( with )

And find c.

Equations:


Solution: