Finding the power from the electric field amplitude

[Frov_ken]

Homework Statement

I am modelling a ray-tracing algorithm(image method), and I am in the part where I am getting the summation of all EM "rays" hitting the receiver. I come to a problem where I can't properly convert and unsure to do so.

Ideally, I should have a collection of Ei waves defined in Seidel and Rappaport's model(from below), since the summation of this would result to the total desired Electric field output. But I don't have the converted components working for E-field such as the f_r , f_t, and L_i(d). Furthermore, I don't know if I could use S=E^2/Zo to get the power density to get the power received by the antenna. Would power densit($W/m^2$)y for Friis also be the Power(W)?

In contrast, Friis Transmission Equation states the power received already by the antenna already from the equation (below). I do have the G_T, G_R, and R necessary for the equation, but the polarization and phase details I have yet to convert(from E-field factor) and include. This is where the extended version comes into guide me how I can try to add more variables in the Friis equation to account for these two.

I'm working at free space with certain wall permittivity.

Homework Equations

Here are some equations that serves as my basis.

• From the paper "Site-Specific Propagation Prediction for Wireless In-Building Personal Communication System Design" Scott Y. Seidel and Theodore S. Rappaport, the model for the complex field amplitude of a wave path is as shown:
  $E_i = E_o f_{ti} f_{ri} L_i(d)\prod_{j} Γ(θ_{ji}) e^{-j*2*pi*d/lambda}$
  
  where f_ti and f_ri are the field amplitude radiation pattern of the transmitter and the receiver antenna respectively,
  L_i(d) is the path loss dependence, with d being the distance.
  Γ is just the reflection coefficient.
  and the exponential tells the phase
  E_o is the reference field strength
  
  This will just be my guide. The reflected rays are the ones that I want, so the reflection coefficients would not need the (1 -)
• 
  
  this is from http://www.antenna-theory.com/basics/transmission.bmp . This explains how the friis equation above is formulated. I will be using its tutorial onto how to do my own conversions for my need.

The Attempt at a Solution

First, I tried using the Friis Transmission Equation model and try to convert the components of the polarization and phase to power. I used the power density function(S = (reflection_coeff or Γ*E)^2/Zo) and tried to simulate reflection coefficients and found that Γ² would be the power-equivalent factor, just like in the EXTENDED model above.

For the phase, I tried simulating a sum of two E-waves.
$\frac{(E_1*e^{(-j2*pi*d_1/lambda)} + E_2*e^{(-j2*pi*d_2/lambda)} )^2}{Zo}$ and used $P1 = \frac{E_1^2}{Zo}$ and $P2 = \frac{E_2^2}{Zo}$.
I got. $P_1 e^{(-4j*pi*d_1/lambda)} + 2*P_1*P_2*Z_o* e^{(-2j*pi/lambda*(d_1+d_2))} + P_2 e^{(-4j*pi*d_2/lambda)}$.
I noticed that this is a bit too complex or inefficient for the code to run through. And I'm not sure it it's even right.

I decided that Seidel and Rappaport model's works better for the summation. So, I have to get the equivalent factor for f_r, f_t, and L_i from the power gain I already have (it's in power notation, like in Friis equation)

To get this, I tried to relate the power density

with the E shown in Seidel and Rappaport's model and Friis' equation. I get
f_r to be $\sqrt{G_T}$
and f_t to be $\sqrt{G_T}$
, and Li(d) to be $\sqrt{\frac{1}{4*pi*d^2}}$.

Is this right? I think it isn't because I just related power density to power itself.
After this, I have to convert the consolidated E-wave summed output to power to display the power delay profile of the receiver. How is this "power" defined? would

work or would $P=0.5*ε_oE^2$ work?

Any reference to my problem would really make me understand my situation here more.

 

[Frov_ken]

I think I'm overthinking in this problem. I just want to know if this equation is applicable or $P=0.5*ε_oE^2$ should be used here? So i could say that when I have the gain of an antenna
$Ga P = 0.5 *ε_o(E*f_a)^2$
$f_a = \sqrt{G_a}$

What about the path loss term L_i?
$\frac{1}{4πR^2}P = 0.5 *ε_o(E*L_i)^2$
$L_i = \sqrt{\frac{1}{4πR^2}}$
will this be right?

and how can I get the $E_o$ of the Seidel- Rappaport model when I'm only given the Power transmitted to the antenna and it's Gain(dBi) pattern. Is it the relation between

and

?