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Given two CT signals $x_1(t)$ (even signal) and $x_2(t)$ (odd signal). If $x_1(t)+x_2(t)$ is an even signal then what is the energy and power of $x_2(t)$.

My Attempt

$\int \limits _{-\infty}^{+\infty}$ $\left| x_1(t)+x_2(t)\right|^2 dt$

Since the functions are real we can

= $\int \limits _{-\infty}^{+\infty}$ $\left( x_1(t)+x_2(t)\right)^2 dt$

= $\int \limits _{-\infty}^{+\infty}$ $x_1^2(t)dt$ + $\int \limits _{-\infty}^{+\infty}$ $x_2^2(t)dt$ + $\int \limits _{-\infty}^{+\infty}$ $2 x_1(t) x_2(t)dt$

This one was useful for the next step

= $\int \limits _{-\infty}^{+\infty}$ $x_1^2(t)dt$ + $\int \limits _{-\infty}^{+\infty}$ $x_2^2(t)dt$

Now the energy of $x_2(t)$ is the difference between the energy of $x_1(t) +x_2(t)$ and that of $x_1(t)$.

We know that $x_1(t)$ is even , $x_1(t)+x_2(t) $ is even , $x_2(t) $ is odd (however its energy function $x_2(t)^2$ is even since its product of two odds)

~~ And I'm stuck here ~~~

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  • $\begingroup$ I think I figured this out (: I'll try to write the answer asap $\endgroup$ – AcademicalResearcher Aug 16 '20 at 15:22

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