Last Updated on May 13, 2026 by John Berry
Amateur radio often exploits signal strengths in the tails of various probability distributions. The term ‘DX’ gives the idea of communication for low, or even very low, percentages of time and locations. If we are to exploit DX, we need to understand the various propagation effects and their associated probability distributions.
Probability distributions
There are two primary distributions that combine to describe affairs in propagation – the log-normal distribution and the Rayleigh distributions.
Log normal distribution
The normal or Gaussian distribution is often known as the ‘bell curve’ since it looks a bit like the profile of a church bell. The log-normal distribution is the Gaussian distribution expressed in terms of natural logarithms (since we express power and signal level in decibels). The ordinate (the x-axis) is often in terms of frequency of occurrence.
The log-normal distribution is often used to describe a path where the path loss (and hence received signal) is log-normally distributed about a median. Examples include free-space or obstructed tropospheric paths, meteor trail paths, or sporadic E paths. The spread or variance around the median value is described by the standard deviation in dB.
So, you might say that there’s a median path loss, a loss exceeded for 90% of time, and a loss not exceeded for 10% of time. It’s the low percentage that’s of interest since it might describe DX. It’s then when atmospheric phenomena conspire to reduce the path loss. Suddenly (for 10% of time) the path works.
Also of significance is the rate of change of the signal. Paths through the F Region perhaps change (fade) slowly whereas paths via meteor trails change extremely quickly.
Rayleigh distribution
The result of many signal arrivals at an antenna, each of differing amplitude and phase, is described by the Rayleigh distribution. These arrivals result from multiple paths between transmitter and receiver such as is found when mobiles transit towns and cities and during certain lunar phases in EME. The result is Rayleigh fading.
Rayleigh fading is characterised by deep signal nulls and broad cusps with a period of half a wavelength between nulls.
Joe Taylor, K1JT, has illustrated his understanding of this on EME paths from reflections from the lunar surface. In his algorithms in WSJT-X he has implemented significant error correction in the coding that effectively flywheels over the nulls. Ordinarily, engineers add margin to reduce the effects of the nulls – for example, to achieve a 99% availability demands a 20dB margin above median. Joe has achieved the same effect in coding.
Combined log-normal and Rayleigh
The log-normal and Rayleigh distributions occur together though one may dominate. The log-normal distribution describes (slow) path fading, while the Rayleigh distribution describes (fast) multi-path effects.
The idea is shown below.
The main path loss vector (of some dB loss and phase) varies as the characteristics of the path vary. The multi-path vector varies as the degree of multi-path arrivals in amplitude and phase. Sometimes the multi-path vector will perfectly add, and sometimes perfectly subtract from the path loss, resulting in the nulls and cusps.
The path loss vector is often relatively steady, while the multi-path is experienced as flutter.
Combined distribution
The essence of the combined distribution is summed up in the curves below.
Values on the x and y axes can be read for values of standard deviation of the log-normal component of the distribution (defined by the nine curves).
For high percentages
For a steady state log-normal case (such as a free-space path), the 0dB curve is used. This describes the Rayleigh distribution alone. For a standard deviation typical of path to path variations around a repeater, for example, the 10dB curve might be appropriate. To make sense of the values, ask yourself how certain you want to be of communication within the repeater service area. If you want to be 90% certain, you’ll need to receive a median that is about 12dB above the receiver threshold. For 99% certainty, you’ll need about 30dB. These values are appropriate when planing the repeater service area and publishing coverage plots. It feels counter intuitive but the 0dB point on the y-axis is at the fifty-percentile or median. The median is the planning value and must be above threshold.
For low percentages
For DX and small percentages of chance, you’ll need to work at the other ends of the curves. We can read off a value of a margin for a path availability of 1%. For a well-diffracted path with a standard deviation of 12dB, we’d need around +20dB. This means that the path would need to improve by 20dB above it’s median state for us to receive the DX station. Consider first an obstructed tropospheric path. Inverse beam bending such as seen during tropospheric lifts would be needed to dispense with the normally horrendous diffraction loss. Similarly, some very advantageous sporadic E patches would need to form for the path from UK to South America to fade up by 20dB. Both effects are rare – 1% rare in my example here.
Summary
We might want to be certain of a communication. Or we might want to know our chances of DX. Either way, we must have some knowledge of statistics. All radio propagation is described by probability – by chance, expressed statistically.
There are two main probability distributions in propagation. First, there’s the log-normal (or Gaussian after German mathematician Carl Friederich Gauss). It’s named after Gauss, but it is based on the work of Abraham de Moivre! Second, there’s the Rayleigh (after John William Strutt, 3rd Baron Rayleigh, a British physicist). Log-normal describes slow fading of a path. Rayleigh describes fast fading resulting from multi-path.
For anyone interested in this further, see ITU-R Rec. P.1057-7 in the Bibliography.