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==Homework #9==
==Homework #9==
Problem Statement:
<b>Problem Statement:</b>
<br>
<br>
Show that, for a bandwidth limited signal (<math> x(t) </math> with <math> f_{max} < {1\over {2T}} </math>)
Show that, for a bandwidth limited signal (<math> x(t) </math> with <math> f_{max} < {1\over {2T}} </math>)
Line 10: Line 10:
<br>
<br>
And find c.
And find c.
<br>
<br>
<b> Equations: </b>
<br>
<math>
\left \langle \phi_k(t) \vert \phi_l(t) \right \rangle=\int_{-\infty}^{\infty} \phi_k(t)^{*} \phi_l(t)\,dt
</math>
<br>
<math>
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>
<br>
By earlier work:
<math>
\left \langle \phi_k(t) \vert \phi_l(t) \right \rangle
=T\delta_{l,k}
</math>
<br>
<math>
\Rightarrow
\sum_{k=-\infty}^{\infty}\sum_{l=-\infty}^{\infty} x(kT)x(lT)
\left \langle \phi_k(t) \vert \phi_l(t) \right \rangle
=T\sum_{k=-\infty}^{\infty} \left | x(kT) \right |^2
</math>
<br>
<math>
\Rightarrow
\sum_{k=-\infty}^{\infty} \left | x(kT) \right | ^2
={1\over T}\int_{-\infty}^{\infty} \left | x(t) \right | ^2\,dt
</math>
<br>
<math>
\Rightarrow
c={1\over T}
</math>
==Homework #13==
Total time spent working on Wiki: 3 hrs

Latest revision as of 11:02, 10 December 2004

Homework #9

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

And find c.

Equations:


Solution:



By earlier work:


Homework #13

Total time spent working on Wiki: 3 hrs