Clifton Laboratories 7236 Clifton Road  Clifton VA 20124 tel: (703) 830 0368 fax: (703) 830 0711

E-mail: Jack.Smith@cliftonlaboratories.com
 

Revision History
Written 07 November 2008
Revised 09 November 2008 - Added fast sample data for WJR
Revised 10 November 2008 - Added fast sample data for GTN beacon
Revised 27 November 2008 - Added excerpt from K. Davies and also statistics collected with spectrum analyzer

In working on active and short antenna designs over the last few months, I've collected quite a few signal measurements in the form of A versus B tests, comparing a test antenna with a reference antenna, generally my 80 meter inverted vee. It occurred to me that a statistical look at the measured data might be illuminating for hams who have dealt with signal fading in day-to-day operations but who have not had a chance to look at the subject quantitatively.

The data is taken with an HP 3586B selective voltmeter in 20 Hz bandwidth mode and an HP 59307A GPIB-controlled antenna relay. The 3586B and 59307A operate under GPIB control, with a program I've written using EZGPIB. I've written about GPIB instrument control at EZGPIB and Prologix GPIB Adapter so there's no need to say more about this aspect of the test setup here.

In general, the test protocol is to make five sequential measurements of the signal level at a particular frequency. It takes the 3586B about 500 ms to make a single measurement at 20 Hz bandwidth. The software then sorts the measurements and discards the strongest and weakest and then averages the remaining three measurements. This is undoubtedly a bit of a dog's breakfast statistically speaking, as it is a combination of median and mean computations. The reason I took this approach is to reduce the effect of static crashes that otherwise would bias up the data if  the mean were taken. Rather than stick with the straight median (the center value after sorting the five readings), I went with mean of the three remaining  readings with the thought that it would better represent longer term signal trends. These measurements are made for 9 stations with Antenna A and then repeated for Antenna B. This cycle, including running the 3586B's internal calibration routine, takes just under three minutes. The program waits until the start of the next minute before starting the next sequence, so the individual measurements are taken at a rate of 20 per hour.

In order to make a signal strength reading, of course, there first must be a signal, preferably a 24 hours / 7 day a week signal. The signals I'll discuss on this page are from a variety of stations:

The distance figures are rough approximations and I have not checked them against a map or made great circle computations. My receiving point, Clifton, VA, is in suburban Washington DC.

All these stations are AM and hence it's an easy matter to measure the carrier signal level with a narrow bandwidth filter.

The figure below shows the signal level for WWV at 2.5 MHz over a five day period. There's one break in the data—the straight line segment early in the morning of 11/5/2008—but otherwise it is complete. There are a few interesting features of the plot. For example, there's clear evidence of a double peak in signal level, with a maximum signal occurring before 2100, followed by a 10 db drop for an hour or two, with the signal then returning to the same level before the dip. The rather slow buildup of signal level on the 6th-7th of November is odd, as the previous four nights show rather rapid increase in signal level from noise to -80 dBm or greater.

I've more or less arbitrarily set the  threshold between skywave signal and noise at -113 dBm. Looking at the time series plot above, it might be argued  the threshold is a  bit lower, perhaps -115 or even -118 dBm. However, the exact point that divides noise from valid signal isn't the important factor in the analysis. Rather, it's the shape of the curve, or the signal level distribution.

However, there's a clear offset, in that the Gaussian distribution goes to zero (well, close to zero) as we move several distribution intervals from the mean.

We can understand why the signal strength is not Gaussian distributed if we remember first that a Gaussian distribution assumes there is some central value or mean and that the measured parameter is randomly distributed about that central value.

This is not the case in a skywave signal. Let's consider a very simple case. Suppose the signal arriving at my antenna from WWV consists of two rays or paths from the WWV transmitter. Each ray reflects from a different location in the ionosphere and combine in my antenna. For simplicity, we'll also assume that the two rays are of equal amplitude, but their phase can vary randomly. This is a reasonable assumption as a wavelength at 2.5 MHz is 120 meters and the ionospheric reflection points of the two rays may be separated by thousands of meters or more and the reflection points change with time. Hence, it's reasonable to believe that the two signals will arrive with random phase difference and that the phase difference will change over time.

What's the best case for these two signal paths? If  the phase difference is an integer multiple of 360 degrees, the two signals arrive at my antenna with identical phase and hence add. In this case, the composite signal has twice the voltage of each individual ray, and is +6 dB with respect to either ray.

The worst case is that the two rays arrive with a phase difference an integer multiple of 180 degrees and hence completely cancel each other out. In this case, the composite signal is zero, or -∞ dB with respect to either ray. In the more realistic case, it's unlikely that the two rays will have exactly equal signal strength and that the phase difference will be exactly 180° but it is easily possible for the two rays to cancel to -20 or -30 dB or more, if only for a brief period.

This example should allow you to understand why (a) the signal distribution is limited on the high side of the mean and (b) the tail on the low side of the mean is not limited.

This simplistic example with two ray fading actually produces a slightly different statistical distribution, called Rician, after the scientist Stephen O. Rice who first analyzed the effects of two-ray fading. Incidentally, two-ray fading is a real phenomena and is often seen in microwave paths where one ray is the direct signal and the second ray is a reflected signal.

In ionospheric propagation, as in VHF/UHF mobile radio, the more common propagation mode is the sum of multiple paths, and where there is no dominant path. The statistics describing this mode are called "Rayleigh" and the term "Rayleigh fading" is often used to describe the propagation characteristics. These terms, of course, come from John William Strutt, 3rd Baron Rayleigh, who developed  these statistics before radio was invented.

One of the most noted authorities on ionospheric radio propagation, Kenneth Davies, in Ionospheric Radio Propagation, National Bureau of Standards Monograph 80, April 1 1965, describes fading statistics at pp. 242-43:

A beam of radio waves incident on the ionosphere is not reflected from a point but from an extended region. Small irregularities in electron density near  the level of reflection give rise to individual reflected wavelets and the received signal is the vector sum of the individual signals at the receiving antenna. Movements of ionospheric irregularities give rise to variations in the relative phase of the individual wavelets and thus produce interference fading. It is not uncommon for a received high-frequency signal to consist of a mixture of high- and low-angle rays, each having extraordinary and ordinary components; each such set may be combined with other sets corresponding to rays having different numbers of hops.

The resultant amplitude can vary over wide limits, the maximum value being when all the individual components are in phase. The root mean square value of the fluctuating signal is equal to the steady value of the signal had the ionosphere not broken it up into many components. Because it is impossible to determine the resultant amplitude at any given moment, the subject has to be treated on a statistical basis. Such phenomena are said to be "stochastic."

The distribution of amplitude approximates the Rayleigh law when the various components are of approximately the same amplitude and the relative phases are varying randomly. For the Rayleigh distribution, the percentage of time p(A) that the amplitude exceeds the value A is

p(A) = exp(-A²/A²_R)

where A²_R is the mean square value of A (i.e., proportional to the mean power.) Ionospheric signals are, often, better described by a Rice distribution, in which a wave of steady amplitude (specular component) is added to the randomly varying signals.

...

The amplitude distribution of a continuous wave, and of trains of long pulses, involving several paths, is usually close to Rayleigh. Individual modes resolved by short pulses often have shallower (and slower) fading corresponding to a substantial specular component ...

I've also fitted a Gaussian distribution to the data. While the fit is not perfect, it is quite decent, particularly considering the temperature change induced signal changes. The reason for the jitter or noise in the data can be easily understood when the signal level is considered. -100 dBm is a rather weak signal and background noise contaminates the signal.

There's a longer term change throughout the daytime, however, again due to temperature changes, amounting to perhaps 1 dB or so.

The plot below shows 1h:20m or so of data of WJR (760 KHz) signal level collected with the fast sample option.

The time plot shows little overall change with time, although there's clearly noise in the measurements.

The data plotted shows  both the signal distribution but also the cumulative distribution of the signal levels.

The figure below shows data for CHU's 3330 KHz signal, taken at 7 PM on November 26th, 2008.

From the cumulative probability data shown for the three signals, we may determine the 10%, 50% and 90% probability values and, after converting the dBm power levels to voltages, compute the A₉₀/A₅₀ and A₁₀/A₅₀ ratios to see how closely the match the figures provided by Davies for a Rayleigh distribution. (The microvolt levels assume 50 ohm impedance.)

The data shows rather good agreement with Rayleigh fading distribution for CHU's 3 MHz signal, as well as CHU's 7 MHz signal. However, WWV's 15 MHz signal's cumulative distribution diverges from the expected Rayleigh values. This is not unexpected, as higher frequency signals tend  to support fewer paths and therefore their fading will be closer to Rician than Rayleigh.

I should add that the lower frequency data is not a good match to a Gaussian distribution. Gaussian distributions are symmetrical and hence the ratio of A₁₀/A₅₀ should be the reciprocal of A₉₀/A₅₀. Consider CHU's 3 MHz data for example. Since A₁₀/A₅₀ = 0.376, if the signal distribution is Gaussian,  A₉₀/A₅₀ should be 1/0.376 or 2.66, but the observed value is 1.679, close to the 1.8 value expected for a Rayleigh distributed variable.

The closest fit to Gaussian is WWV 15 MHz data, where , in the Gaussian case, A₉₀/A₅₀ should be 1/0.335 or 2.98, which is not too far from the observed 2.66.