In an LTI system, consider the following:
The input signal: $$ x(t)= \begin{cases} 16 \quad & ; -7<t<0 \\ 0 & ; \text{otherwise} \\ \end{cases} $$ And the unit impulse response: $$ h(t)=e^{7-t} u\left( t-7 \right) $$ Find the energy of the output signal and the energy spectral density of that same output signal.
Remark: A related question is Energy spectral density in an LTI system .
First we find the output signal:
$$ y(t)=(x*h)(t)=
\begin{cases}
0 \quad & ; t<0 \\
16 \left( 1-e^{-t} \right) & ; 0<t<7
\\
16\, e^{-t} \left( e^7 -1 \right) & ; 7<t
\\
\end{cases}
$$
Then the energy will be: $$ E_y =\int_{-\infty}^{\infty} \mid y(t) \mid ^2 dt = \int_{0}^{\infty} \mid y(t) \mid ^2 dt $$
And I think I can solve that integral by hand. Now, the energy spectral density will be: $$ S_{yy}= \mid \hat{y} (\xi) \mid ^2 $$
So, first we find: $$ \hat{y}(\xi)= \int_0^{7} 16(1-e^{-t} )e^{-2\pi i\xi t} dt + \int_{7}^{\infty} 16(e^7 -1)e^{-t}e^{-2\pi i\xi t} dt $$
To solve that by hand, I think I will have to go to WolframAlpha first and solve integrals of the form $e^{(a+bi)t} $.
Are the steps above correct? Is there any way of solving this with less steps?