# Communication Systems/Wireless Transmission

This page will discuss Wireless EM wave propagation, and some basics
about antennas.

## 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:

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

### Received Power

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**.

## 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 Boltzmann\'s constant and all the
path losses of the transmission.

## Further reading

-   Jean-Claude Wippler. \"What if you're out of wireless
    range?\".
    2013.
