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Chapter: Wireless Channels

Free Space Propagation

For propagation distances d much larger than the antenna size, the far field of the generated Electromagnetic wave dominates all other components. In free space, the energy radiated by an omni-directional antenna is spread over the surface of a sphere. The surface area of a sphere of radius d is 4 pi d^2.

The power density w at distance d from a transmitter with power p_T and antenna gain G_T is

w =p_T G_T/ (4 pi d^2)..

The available power p_R at a receive antenna with gain G_R is



where A is the effective area or `aperture' of the antenna, with G_R = 4pi A / ^2. The wavelength is c / f_c with c the velocity of light and f_c the carrier frequency. The product G_T p_T is called the effectively radiated power (ERP) of the transmitter.

While cellular telephone operator mostly calculate in received powers, in the planning of the coverage area of broadcast transmitters, the CCIR recommends the use of the electric field strength E at the location of the receiver. The conversion is

.

Exercise:

Show that for a reference transmitter with ERP of 1 kwatt in free space,

As the propagation distance increases, the radiated energy is spread over the surface of a sphere of radius d, so the power received decreases proportional to d^-2. Expressed in dB, the received power is


Exercise:

Show that the path loss L between two isotropic antennas (G_R = 1, G_T = 1) can be expressed as L dB= - 32.44 - 20 log fM_c/ 1MHz - 20 log d / 1km.

Engineers speak about a "20 log d" path loss law..



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