Homework: Difference between revisions

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\Rightarrow
\Rightarrow
\sum_{k=-\infty}^{\infty} \left | x(kT) \right | ^2
\sum_{k=-\infty}^{\infty} \left | x(kT) \right | ^2
=1/T\int_{-\infty}^{\infty} \left | x(t) \right | ^2\,dt
={1\over T}\int_{-\infty}^{\infty} \left | x(t) \right | ^2\,dt
</math>
</math>
==Homework #13==
==Homework #13==
Total time spent working on Wiki: 3.5 hrs
Total time spent working on Wiki: 3.5 hrs

Revision as of 09:38, 10 December 2004

Homework #9

Problem Statement:
Show that, for a bandwidth limited signal (x(t) with fmax<12T)
k=|x(kT)|2=c|x(t)|2dt
And find c.

Equations:
ϕk(t)|ϕl(t)=ϕk(t)*ϕl(t)dt
x(t)=k=x(kT)ϕk(t)
Solution:
x(t)|x(t)=x(t)*x(t)dt=|x(t)|2dt
x(t)=k=x(kT)ϕk(t)
x(t)|x(t)=k=x(kT)ϕk(t)|l=x(lT)ϕl(t)=k=l=x(kT)x(lT)ϕk(t)|ϕl(t)
By earlier work: ϕk(t)|ϕl(t)=Tδl,k
k=l=x(kT)x(lT)ϕk(t)|ϕl(t)=Tk=|x(kT)|2
k=|x(kT)|2=1T|x(t)|2dt

Homework #13

Total time spent working on Wiki: 3.5 hrs