The equation for FSPL is
This equation is only accurate in the far field; it does not hold close to the transmitter.
As depicted in the figure above, it assumes that the distance between transmitting and receiving antennas 'd' is much greater than the wavelength of the EM (Electromagnetic ) waves and the wavefronts impinging the receiving antenna are planar and not spherical and this distance is large enough to neglect the curvature of earth. The relation between the transmitted and the received power (Pu and Pr ) in Watts is given as :
Derivation of the formula can be found in
A convenient way to express FSPL is in terms of dB:
where the units are as before.
For in statute miles, the constant becomes .
The FSPL expression above often leads to the erroneous belief that free space attenuates an electromagnetic wave according to its frequency. This is not the case, as there is no physical mechanism that could cause this.
The expression for FSPL actually encapsulates two effects. Firstly, the spreading out of electromagnetic energy in free space is determined by the inverse square law, i.e.
Note that this is not a frequency-dependent effect.
The second effect is that of the receiving antenna's aperture, which describes how well an antenna can pick up power from an incoming electromagnetic wave. For an isotropic antenna, this is given by
where is the received power. Note that this is entirely dependent on wavelength, which is how the frequency-dependent behaviour arises.
The total loss is given by the ratio
which can be found by combining the previous two expressions.
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