find power of the signal

Question

I have this signal: $$x(t)=\sum \mathrm{rect}\left(\frac {t-kT_0} {20}\right)+\mathrm{rect}\left(\frac {t-kT_0} {40}\right)+\mathrm{rect}\left(\frac {t-kT_0} {80}\right)$$ with $T_0=140$, pass it through the filter $H(f)={\mathrm{rect}}\left(\frac f F\right)$. What is the value of $F$ such that power is $P=1$?

Fourier Trasform is:

$\sum C_n\delta(f-nf_0)$

$C_n=\frac 1 {T_0}[sinc(20nf_0)+sinc(n40f_0)+sinc(n80f_0)]$

if call y(t) the signal output, I have:

$Y(f)=X(f)H(f)$.

the power spectral density is:

$P(f)= \sum (|C_n|^2\delta(f-nf_0))|H(f)|^2$

with $P_x(f)=\sum (|C_n|^2\delta(f-nf_0) $

don't know find F

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if don't error for n=0

$C_n=1$

so for $0<F<\frac 1 {T_0}$ the power should be 1. but I'don't sure

Answer 1

Since this is clearly homework, and the limits on the $\sum$ are crucial to figuring out whether there's an answer:

Hint: Parseval's Theorem states that power in time domain is equivalent to power in frequency domain. What is the power in $x(t)$? Is $x(t)$ periodic? If so, what's the period?

As usual, it often helps to make a drawing (that's, by the way, the minimum I'd have expected from your question!) to get a feeling for the signal you're dealing with.