The mechanisms behind wireless radio propagation can be classified into a large-scale and small-scale effects. The large-scale effects, such as path loss, determine the average received signal power. The small-scale effects, in particular multipath fading determine the channel properties, such as the received amplitude and phase for a specific frequency and time instant. A commonly used small-scale channel model is that of Rayleigh fading. Its mathematical foundation dates back from 1880, when Rayleigh commented on the statistical properties of the acoustical signal resulting from (infinitely) many violins playing in unison. This model leads to an exponentially distributed received signal power. The 'local-mean' p is determined by the large-scale effects. Further tayloring of this random model towards the application of mobile radio, showed that the received power becomes uncorrelated if the carrier frequency is shifted over 2p /T_RMS, where T_RMS is the rms delay spread of multipath waves, or if the antenna position is moved over l /2, with l the wavelength. If one denotes the velocity of the portable terminals as v, this implies that the channel is relatively constant if the transmission time T_L satisfies vT_L << l /2.

Particularly when a block of data is transmitted within a time-frequency window which is significantly shorter than l /(2v) and much narrower in bandwidth than 2p /T_RMS, it often is a sufficiently good approximation to assume that a message is received successfully if and only if the signal-to-interference-plus-noise ratio exceeds a certain threshold z. Generally speaking, modulation methods which achieve a large number of bit/s per Hz are vulnerable to interference, these require a large z.

Based on this model, the probability that the wanted signal power p₀ sufficiently exceeds the joint interference plus noise power p_t is

where we insert the appropriate pdfs of received signal power. In the special case of a Rayleigh-fading wanted signal, its pdf of signal power is an exponential one. Hence, for probabilities conditional on the local-mean p₀ , the integral over y can be solved analytically. An elegant mathematical framework has been developed by interpreting the result as a Laplace transform of the pdf of joint interference power, namely

where L{f,s} denotes the one-sided Laplace transform of the function f at the point s. For more sophisticated models of fading, it is known that the probability can we written as a series expansion of derivatives of Laplace images. The joint interference signal p_t is the incoherent sum of multiple individual signals. For independent fading, the pdf of the joint interference power is the convolution of the pdf of individual interference powers. So if one simplifies the large-scale propagation to a plane-earth loss model with a 4-th power path loss law,

with r the propagation distance between the base station and the mobile terminal, one finds

Here i indexes the interfering signals. For a practical situation it appears that the path loss law inaccurately estimates the local mean power. In generic system studies it is undesirable to use detailed terrain databases, so the inaccuracies of the path loss model are usually taken into account by assuming a log-normally distributed 'shadow' attenuation. In such case the received local-mean power in dBm varies as a Gaussian distributed random variable around an 'area-mean' determined by the path loss law. A precise mathematical treatment of this effect would not significantly enhance the intuition and insight needed here, so it is omitted. Nonetheless in some of the following sections we take the liberty to quote results from papers that consider shadowing.