## Sunday, June 9, 2013

### Friis Transmission Formula and S/N Ratio Limit on Communication Range

The signal power that can be transmitted between two points in space is dependent on the flux tube connecting two antennas. The transmitting antenna creates a beam of radiation for which the radiation intensity, I, or power per unit solid angle is a constant. The receiving antenna intercepts part of this beam and collects a fraction of the power in it. The received power and the transmitted power are related by the Friis transmission formula. A derivation of the relation between antenna aperture and gain can be found in Papas, Theory of Electromagnetic Wave Propagation (1965). The Friis transmission formula is derived as follows.

If there is little or no noise in the transmitted signal then we only have to consider the receiver noise which is just Nyquist noise. Dividing the received signal power from the Friis transmission formula by this noise we get the S/N ratio.

One sees that the inverse square degrades the signal to the point where the received signal is less that the noise level and this imposes a limit on the range of communications. There are a few tricks that one can use to boost the signal like summing over a period of time to let some of the random noise cancel itself out. If one sums over a number of cycles then this effectively increases the wavelength of the signal but lowers the channel capacity. A competing signal in the receiver's angular aperture will be mixed with the transmitted signal and also degrades the signal. This would be the case if the signal had to compete with the thermal noise from a star and would limit one's ability to detect a civilization on an alien planet since it would likely be near a star. To increase the range and speed of a channel one might consider a relay network connecting two points in space.