# Communication Systems/Wireless Transmission

## Isotropic Antennas

An isotropic antenna radiates it's transmitted power equally in all directions. This is an ideal model; all real antennas have at least some directionality associated with them. However, it is mathematically convenient, and good enough for most purposes.

### Power Flux Density

If the transmitted power is spread evenly across a sphere of radius R from the antenna, we can find the power per unit area of that sphere, called the Power Flux Density using the Greek letter Φ (capital phi) and the following formula:

$\Phi = \frac{P_T}{4 \pi R^2}$

Where $P_T$ is the total transmitted power of the signal.

### Effective Area

The effective area of an antenna is the equivalent amount of area of transmission power, from a non-ideal isotropic antenna that appears to be the area from an ideal antenna. For instance, if our antenna is non-ideal, and 1 meter squared of area can effectively be modeled as .5 meters squared from an ideal antenna, then we can use the ideal number in our antenna. We can relate the actual area and the effective area of our antenna using the antenna efficiency number, as follows:

$\eta = \frac{A_e}{A}$

The area of an ideal isotropic antenna can be calculated using the wavelength of the transmitted signal as follows:

$A = \frac{\lambda^2}{4 \pi}$

The amount of power that is actually received by a receiver placed at distance R from the isotropic antenna is denoted $P_R$, and can be found with the following equation:

$P_R = \Phi_R A_e$

Where $\Phi_R$ is the power flux density at the distance R. If we plug in the formula for the effective area of an ideal isotropic antenna into this equation, we get the following result:

$P_R = \frac{P_T}{(4 \pi R / \lambda)^2} = \frac{P_T}{L_P}$

Where $L_P$ is the path-loss, and is defined as:

$L_P = \left( \frac{4 \pi R}{\lambda} \right)^2$

The amount of power lost across freespace between two isotropic antenna (a transmitter and a receiver) depends on the wavelength of the transmitted signal.

## Directional Antennas

A directional antenna, such as a parabolic antenna, attempts to radiate most of its power in the direction of a known receiver.

Here are some definitions that we need to know before we proceed:

Azimuth Angle
The Azimuth angle, often denoted with a θ (Greek lower-case Theta), is the angle that the direct transmission makes with respect to a given reference angle (often the angle of the target receiver) when looking down on the antenna from above.
Elevation Angle
The elevation angle is the angle that the transmission direction makes with the ground. Elevation angle is denoted with a φ (Greek lower-case phi)

### Directivity

Given the above definitions, we can define the transmission gain of a directional antenna as a function of θ and φ, assuming the same transmission power:

$G_T(\theta, \phi) = \frac{\Phi_{\theta,\ \phi}}{\Phi_{isotropic}}$

### Effective Area

The effective area of a parabolic antenna is given as such:

$A_e = \eta \frac{\pi D^2}{4}$

### Transmit Gain

$G_{max} = \frac{4 \pi A_e}{\lambda^2}$

If we are at the transmit antenna, and looking at the receiver, the angle that the transmission differs from the direction that we are looking is known as Ψ (Greek upper-case Psi), and we can find the transmission gain as a function of this angle as follows:

$G(\Psi) = \left( \frac{2J_1((\pi D / \lambda) sin(\Psi))}{sin(\Psi)} \right)^2 \left( \frac{\lambda}{\pi D} \right)^2$

Where $J_1(\ )$ denotes the first-order bessel function.

### Friis Equation

The Friis Equation is used to relate several values together when using directional antennas:

$P_R = \frac{P_T G_T G_R}{L_P}$

The Friis Equation is the fundamental basis for link-budget analysis.

If we express all quantities from the Friis Equation in decibels, and divide both sides by the noise-density of the transmission medium, N0, we get the following equation:

$C/N_0 = EIRP - L_P + (G_R/T_e) - k$

Where C/N0 is the received carrier-to-noise ratio, and we can decompose N0 as follows:

$N_0 = kTe$

k is Boltzmann's constant, (-228.6dBW) and Te is the effective temperature of the noise signal (in degrees Kelvin). EIRP is the "Equivalent Isotropic Radiated Power", and is defined as:

$EIRP = G_T P_T$

To perform a link-budget analysis, we add all the transmission gain terms from the transmitter, we add the receive gain divided by the effective temperature, and we subtract boltzman's constant and all the path losses of the transmission.