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written 4.2 years ago by |

The free space propagation model is used to predict received signal strength when the transmitter and receiver have a clear, unobstructed line-of-sight path between them. Satellite communication systems and microwave line-of-sight radio links typically undergo free space propagation.

As with most large- scale radio wave propagation models, the free space model predicts that received power decays as a function of the T-R separation distance raised to some power (i.e. a power law function). The free space power received by a receiver antenna which is separated from a radiating transmitter antenna by a distance d, is given by the Friis free space equation,

$$P_{r}(d)=\frac{P_{t} G_{t} G_{r} \lambda^{2}}{(4 \pi)^{2} d^{2} L}$$

where $P_t$ is the transmitted power, $P_r (d)$ is the received power which is a function of the T-R separation, $G_t$ is the transmitter antenna gain, $G_r$ is the receiver antenna gain, d is the T-R separation distance in meters and $\lambda$ is the wavelength in meters. The gain of an antenna is related to its effective aperture, $A_e$ by,

$$G=\frac{4 \pi A_{e}}{\lambda^{2}}$$

The effective aperture $A_{e}$ is related to the physical size of the antenna, and $\lambda$ is related to the carrier frequency by

$$\lambda=\frac{c}{f}=\frac{2 \pi c}{\omega_{c}}$$

where f is the carrier frequency in Hertz, $\omega_{c}$ is the carrier frequency in radians per second, and c is the speed of light given in meters/s.

An isotropic radiator is an ideal antenna which radiates power with unit gain uniformly in all directions, and is often used to reference antenna gains in wireless systems. The effective isotropic radiated power $(E I R P)$ is defined as

$$E I R P=P_{t} G_{t}$$

and represents the maximum radiated power available from a transmitter in the direction of maximum antenna gain, as compared to an isotropic radiator.

In practice, effective radiated power (ERP) is used instead of EIRP to denote the maximum radiated power as compared to a half-wave dipole antenna (instead of an isotropic antenna).

The path loss, which represents signal attenuation as a positive quantity measured in dB, is defined as the difference (in dB) between the effective transmitted power and the received power, and may or may not include the effect of the antenna gains. The path loss for the free space model when antenna gains are included is given by

$$P L(\mathrm{dB})=10 \log \frac{P_{t}}{P_{r}}=-10 \log \left[\frac{G_{t} G_{r} \lambda^{2}}{(4 \pi)^{2} d^{2}}\right]$$

When antenna gains are excluded, the antennas are assumed to have unity gain, and path loss is given by

$$P L(\mathrm{dB})=10 \log \frac{P_{t}}{P_{r}}=-10 \log \left[\frac{\lambda^{2}}{(4 \pi)^{2} d^{2}}\right]$$

The Friis free space model is only a valid predictor for $P_r$ for values of d which are in the far-field of the transmitting antenna. The far-field, or *Fraunhofer* region, of a transmitting antenna is
defined as the region beyond the far-field distance $d_f$, which is related to the largest linear
dimension of the transmitter antenna aperture and the carrier wavelength. The *Fraunhofer*
distance is given by

$$d_{f}=\frac{2 D^{2}}{\lambda}$$

where D is the largest physical linear dimension of the antenna. Additionally, to be in the far-field region, $d_f$ must satisfy

$$d_{f} \gg D$$