Last Updated on May 8, 2023 by John Berry

If we draw a path profile with an effective Earth radius of 4/3, we can draw a straight line between stations and this will be the state, or better, for 50% of time. We can then term the resulting loss over this path with this geometry as the median. Signals received are then normally distributed in time about this median.

Many of the pages here discuss incidence of k-factor fading – slow fading about the median caused by changes in the refractivity of the troposphere. Typically, in UK and western Europe, the refractivity gradient, dN/dh (∆N), is -40 N/km. N is in units of refractivity. This corresponds to an Earth radius factor of 4/3 and an effective radius of 8500km. For reference, the actual Earth radius is 6731km.

## k exceeded

Professionals talk of ‘k exceeded’ for 50% of the time, and indeed some design systems for the ninety-percentile – k exceeded for 90% of time. The signal received will be this or better.

Mostly however, k-factor fading is an issue in interference. If k rises, and the Earth flattens, interference might be incident from distant stations that would not normally be heard. Interference degrades wanted services. This is the core of frequency planning – ensuring that interference will only be experienced for small percentages of time.

But radio amateurs want to know the value of k for low percentages of time – at the five percentile, say – the value exceeded for 5% of the time or less.

Recommendation ITU-R P.453-14 gives the month median value of ∆N at around -40.

Rec. P.453 also gives values for 1% (-300), 5% (-100), 90% (-20) and 99% (+30) of time, allowing the frequency distribution below to be assembled.

Then k_{e}, k effective, can be calculated from the formula, k_{e} = 157/(157-∆N). As noted in Recommendation ITU-R P.452-16, ∆N is a positive quantity in this formula (otherwise the result is nonsense).

## Application

k_{e} can be used in the calculation of diffraction losses and distance estimated in the nomograph in the page on this site on Diffraction.

If the value for 5% of the time is to be estimated, for example, the corrected frequency is f/k_{e}^{2} or about 16MHz for a 144MHz operational frequency. This gives a diffraction loss over Free Space Loss of about 65dB. For reference, the diffraction loss over Free Space Loss for 50% of the time is around 130dB. A loss of 65dB will allow a 300km path to work. At 130dB it won’t.

- RECOMMENDATION ITU-R P.452-16 Prediction procedure for the evaluation of interference between stations on the surface of the Earth at frequencies above about 0.1 GHz available at https://www.itu.int/dms_pubrec/itu-r/rec/p/R-REC-P.452-16-201507-S!!PDF-E.pdf.
- RECOMMENDATION ITU-R P.453-14 The radio refractive index: its formula and refractivity data available at https://www.itu.int/dms_pubrec/itu-r/rec/p/R-REC-P.453-14-201908-I!!PDF-E.pdf.