Hi guys,

i am going to calculate the following integral:
\int_0^{f_c+f_m} |y(f)|^2\ df where:

y(f)=\frac{\pi}{2} \alpha_m \sum_{l=1}^{l} \sqrt{g_l}\left [ e^{-j(\omega \tau_l - \theta_m)} \delta(\omega - \omega_0) + e^{-j(\omega \tau_l + \theta_m)} \delta(\omega + \omega_0) \right ]

with \omega_0= 2\pi (f_c + f_m), \ \ \alpha_m=constant, \ \ f_c, f_m: frequencies, \ \ \theta_m: initial \ phase .

then, the integral we are looking for will get the following form:

\int_0^{f_c+f_m} |y(f)|^2 df= \int_o^{f_c + f_m} (\pi \alpha_m)^2 \big|\sum_{l=1}^l \sqrt{g_l}e^{-j \omega \tau_l} \big|^2 cos^2[2 \pi (f_c + f_m) + \theta_m]df =\\
(\pi \alpha_m)^2\int_0^{f_c+f_m} \sum_{l=1}^l g_l e^{-2j \omega \tau_l} \big[cos^2[2 \pi (f_c + f_m) + \theta_m]\big]df =\\
(\pi \alpha_m)^2 \big(\sum_{l=1}^l g_l e^{-j2(2\pi) \tau_l}\big) \big[cos^2[2 \pi (f_c + f_m) +\theta_m] \big] \int_0^{f_c+f_m}e^f df

using a delta's dirac property: \delta(\omega - \omega_0)f(\omega)= f(\omega - \omega_0) ( correct me if it is wrong, because i have doubts about it), i got:

y(f)=\frac{\pi}{2} \alpha_m \sum_{l=1}^{l} \sqrt{g_l}\left [ e^{-j[(\omega - \omega_0 )\tau_l - \theta_m)]} + e^{-j[(\omega - \omega_0) \tau_l + \theta_m)]} \right ] =\\
=\frac{\pi}{2} \alpha_m \sum_{l=1}^{l} \sqrt{g_l} e^{-j \omega \tau_l} \left [ e^{j(\omega_0\tau_l + \theta_m)} + e^{-j( \omega_0 \tau_l + \theta_m)]} \right ] =\\
=(\pi \alpha_m) \big(\sum_{l=1}^{l} \sqrt{g_l} e^{-j \omega \tau_l} \big) cos [2 \pi (f_c + f_m)\tau_l + \theta_m]

so, finally:

|y(f)|^2=(\pi \alpha_m)^2 \big|\sum_{l=1}^l \sqrt{g_l}e^{-j \omega \tau_l} \big|^2 cos^2[2 \pi (f_c + f_m) + \theta_m].

being \int_0^{f_c+f_m}e^f df = e^{f_c+f_m} - 1\approx e^{f_c+f_m}, then:

\int_0^{f_c+f_m} |y(f)|^2 df= (\pi \alpha_m)^2 \big(\sum_{l=1}^l g_l e^{-j4 \pi (f_c + f_m) \tau_l}\big) \big[cos^2[2 \pi (f_c + f_m) +\theta_m] \big]

my supervisor told me i am supposed to find a solution proportional to: \big|\sum_{l=1}^l \sqrt{g_l}e^{j 2 \pi (f_c + f_m)\tau_l} \big|^2.

could you me to find the right solution and where the error is?

you so much for your , i would really appreciate that!