- #1
Mik256
- 5
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Homework Statement
Hi guys,
I have the following transmitted power signal:
$$x(t)=\alpha_m \ cos[2\pi(f_c+f_m)t+\phi_m],$$
where: ##\alpha_m=constant, \ \ f_c,f_m: frequencies, \ \ \theta_m: initial \ phase.##
The multipath channel is:
$$h(t)=\sum_{l=1}^L \sqrt{g_l} \ \delta(t-\tau_l).$$
The received signal is the convolution between ##x(t)## and ##h(t)##, i.e.:
$$y(t)=x(t)*h(t)$$
where ##*## represent the convolution symbol.
I need to calculate the received power, which is:
$$R=(f_c+f_m) \int_0^{\frac{1}{f_c+f_m}} |y(t)|^2 dt.$$
Homework Equations
In order to simplify the calculation of ##y(t)##, I decided to calculate the Fourier-transform of ##x(t)## and ##h(t)##, so that the time-domain convolution will become a multiplication in frequency-domain; then I got:
$$Y(f)= \frac{\pi}{2} \alpha_m \sum_{l=1}^{L} \sqrt{g_l}\left [ e^{-j(\omega \tau_l - \theta_m)} \delta(\omega - \omega_0) + e^{-j(\omega \tau_l + \theta_m)} \delta(\omega + \omega_0) \right ],$$
with: ##\omega_0= 2\pi (f_c + f_m).##
The Attempt at a Solution
According the Parseval's theorem, the integral ##\int_0^{\frac{1}{f_c+f_m}} |y(t)|^2 dt## will be equivalent to ##\int_0^{f_c+f_m} |Y(f)|^2 df##, so that:
$$R=(f_c+f_m) \int_0^{f_c+f_m} |Y(f)|^2 df=$$
$$=(f_c+f_m) \Big(\frac{\pi}{2} \alpha_m\Big)^2 \int_0^{f_c+f_m} \Big| \sum_{l=1}^{L} \sqrt{g_l}\left [ e^{-j(\omega \tau_l - \theta_m)} \delta(\omega - \omega_0) + e^{-j(\omega \tau_l + \theta_m)} \delta(\omega + \omega_0) \right ] \Big|^2 df =$$
$$=(f_c+f_m) \Big(\frac{\pi}{2} \alpha_m\Big)^2 \sum_{l=1}^{L} {g_l} \int_0^{f_c+f_m} \left [ \Big( e^{-j(\omega \tau_l - \theta_m)} \delta(\omega - \omega_0) \Big)^2 + \Big( e^{-j(\omega \tau_l + \theta_m)} \delta(\omega + \omega_0) \Big)^2 +\underbrace{2 \Big| e^{-j(\omega \tau_l - \theta_m)} e^{-j(\omega \tau_l + \theta_m)}\delta(\omega - \omega_0)\delta(\omega + \omega_0)}_{=0} \Big| \right ] df $$
The last term is equal to zero because I have the multiplication of 2 delta; then:
$$R=(f_c+f_m) \int_0^{f_c+f_m} |Y(f)|^2df =$$
$$= (f_c+f_m) \Big(\frac{\pi}{2} \alpha_m\Big)^2 \sum_{l=1}^{L} {g_l} \Big[ \int_0^{f_c+f_m} \Big( e^{-j(\omega \tau_l - \theta_m)} \delta(\omega - \omega_0)\Big)^2 df + \int_0^{f_c+f_m} \Big(e^{-j(\omega \tau_l + \theta_m)} \delta(\omega + \omega_0) \Big) ^2 df \Big] $$
We can rewrite ##R## in this form:
$$R=(f_c+f_m) \Big(\frac{\pi}{2} \alpha_m\Big)^2 \sum_{l=1}^{L} {g_l} \Big[ \Big|e^{-j(\omega_0 \tau_l - \theta_m)} \Big|^2 + \Big| e^{j(\omega_0 \tau_l - \theta_m)} \Big|^2 \Big]$$
My supervisor suggested told me the solution is about correct and it should be proportional to ##|H(f_c+f_m)|^2##.
Actually I am stucked here and I cannot find the errors; so, your help will be really appreciated.
Thanks so much.