# Probability distributions

Last Updated on November 6, 2023 by John Berry

Amateur radio often exploits signal strengths in the tail of various probability distributions. The term ‘DX’ gives the idea of communication for low, or even very low, percentages of time. If we are to exploit DX, we need to understand the various propagation effects.

## Multiple arrivals

A single wave launched from a transmitting antenna, to arrive at a receiving antenna having only encountered atmospheric diffusion, is rare. To a greater or lesser extent, the signal at the receiving antenna at any point in time is the vector sum of multiple arrivals. The figure below shows the phenomenon.

The concept is simple. That all propagation paths occur simultaneously. There is no notion of a single ray from transmit antenna to receive antenna.

Of course, some are more likely than others. At 3.8MHz there may be a significant ground wave path between two stations 30km apart that dominates to support communications. The sky wave may be redundant because the bulk of the skyward energy is refracted to Earth at say 200-400km away – beyond the distant station.

The same is true for all paths. Some dominate, others are minimal. Some are perhaps equally probable. That’s why it’s overly simplistic to declare a VHF path as ‘tropospheric’ in the middle of an Es opening – or vice versa. Talk must be of likelihood from knowledge of the geometry rather than certainty.

It’s possible to calculate the path loss for all paths, whatever their route from TX to RX. There is significant path-by-path variation.

Just when one would think that all was stable, paths fade. This idea of path fading is exemplified by the phenomenon of a meteor burst. The path via the meteor trail only exists for a few hundred milliseconds. Then it’s gone. All paths behave somewhat like this – it’s just that often the time for which the path is supported is longer – minutes, hours, or even days.

## Probability distributions

Fading and multiple path propagation discussed above are understood by way of probability distributions. Probability distributions are an engineer’s way of expressing the whole idea of chance. I’ve discussed chance and probability in one of the Core radio concepts pages. Now it’s time to elaborate.

### 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 y-axis) is often in terms of frequency of occurrence, and we don’t express this as logarithm.

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 burst 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.

For anyone interested in this further, see ITU-R Rec. P.1057-7 in the Bibliography.