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Distribution of S/N ratio under fading on HF

This article reports an experiment to record S/N over an ionospheric path on 7MHz and to explore the statistical distribution.

Knowledge of the distribution may suggest that some parametric statistical methods are applicable.

The experiment initially sought to capture data over two paths simultaneously on 29/12/2012:

In the event, interference prevented acquisition of useful data from W4HBK, always a risk with this type of experiment.

Keyer

A simple QRSS keyer was used with a quite standard Kenwood TS2000 in CW mode adjusted for 5W output.

Fig 1:

Fig 1 shows the internals of the QRSS keyer. The keyer is described in detail at Another simple Morse beacon keyer - basic, but accurate.

The message is structured to send VK1OD at 12WPM then a period of key-down for the remainder of a 10min period, then to recycle. The timing is quite accurate, allowing time based analysis of the recorded measurements.

Measurements were recorded at ZL2IK using Spectrum Lab's Watch List facility and the measurements exported to a text file for further analysis. Over the hour, S/N ranged from 0 to 20dB, or 1 to 100 as a numeric ratio. The measurement technique is likely to provide poor accuracy for S/N lower than perhaps 3dB.

Data analysis

A period of one hour was selected for analysis as a plot showed no tendency for longer term variation in propagation. A little later there was a general lift that was probably caused by Es propagation reported on local ionosondes.

Additionally, 30s of observations around the transmission of the 12WPM ID were excised, so that all the remaining observations were for a steady transmitted signal.

Fig 2:

Fig 2 shows a time plot of S/N in dB, there remain 3254 observations.

Is S/N normally distributed

A lot of natural phenomena exhibit a normal distribution, and we can characterise the distribution by its mean and standard deviation.

Fig 3:

Fig 3 shows the probability distribution of S/N indicated by the measurement set. Clearly it does not exhibit the bell shaped curve of a normal distribution, but the long tailed distribution does hint that it might be a log-normal distribution.

Is S/N a log-normal distribution

A log-normal distribution is one where the log of a variable is normally distributed. As it happens, the log of S/N is related to S/N expressed in dB (S/N(dB)=10*log(S/N)dB).

Fig 4:
 

Fig 4 compares the probability distribution of measured S/N in dB with a normal distribution with the same mean (m) (7.927) and standard deviation (σ) (3.111). To the eye, they are fairly close, and it is reasonably accurate to assume that it is a normal distribution, and inferences made using the mean and standard deviation. The slightly asymmetric lower tail may be cause by the way in which Spectrum Lab calculated S/N, and might render application of a normal distribution unsuited to small S/N data sets.

Applying the normal distribution with m 7.927 and σ 3.111, we would expect that 97.725% of observations would be <= m+2σ=14.15dB. The 0.5dB frequency distribution reconciles well, it gives 97.36% of observations <= 14.0dB, so 6.2dB fade margin would capture almost 98% of observations.

Another application of the distribution would be in comparing two sets of measurements (eg antenna A and antenna B) and testing the null hypothesis that the means are the same using Student's t-test.

Conclusions

The experiment was conducted with a quite ordinary HF transceiver, a keyer to provide the required ID cycle, and Spectrum Lab for measurement. Interesting experiments can be designed around simple equipment.

S/N over an ionospheric propagation path with fading may be well represented as a log-normal distribution, or more simply the S/N expressed in dB as a normal distribution.

Where this applies, it allows application of a parametric statistical model that might for instance provide upper and lower bounds for S/N to a given confidence level (in other words, to predict a fade margin).

The standard deviation is unlikely to be a constant, but rather will depend on the propagation mechanisms and phenomena at the time and could easily be much greater than the path / time explored in this article.

Links / References

Changes

Version Date Description
1.01 30/12/2012 Initial.
1.02    
1.03    

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