[600MRG] Quartile Method of Characterizing 630m Propagation Paths: Formulas. Part 2

[James Hollander]

This post shows the Variability and dB Tilt formulas and how to use them. The next post will show Variability V and dB Tilt T calculations for some 630m paths last night. I hope uploading to the reflector preserves the formulas.  If not, e-mail me to get them at mrsocion at aol.com.  
73,  Jim H   W5EST
     



  Start with a median-SNR adjacency matrix Sij withmedian SNRs (aka “SNR1/2”) to characterize overall comparisonsincluding the path loss undesirably combined with the TX, RX, tx/rx antennas over a time scaleinterval.  “Adjacency matrix” is anaccepted term for a table with entries for every pair of nodes (stations here) ina network, and accounting for direction each way between them. This median SNR matrix S is a good background or starting point, but it lacks the appealing features of the quartile method shown next.
 
  Symbolize SNR quartiles as SNR1/4, SNR1/2, SNR3/4.A “quartile” is an SNR value for which a given fraction ( ¼, ½, ¾ ) of thedecodes are less than that SNR value. The second quartile SNR1/2 isalso called the median. Sort theWSPR database by SNR for the Call ofthe transmitting station and Reporter as the receiving station.  The quartiles are evaluated for each 630m pathbetween stations indexed i and j and entered in their own adjacencymatrices—a Variability adjacency matrixVij and a dB Tilt adjacencymatrix Tij.  Or, if you like, just evaluate the quartiles for one path without making a big table for multiple paths.
 
Variability V tellshow much of a spread in SNRs exists on a path. Subtract the 1st quartile SNR from the 3rd quartile SNR for the 630m path on a given night.
   Vij = SNRij 3/4  - SNRij 1/4                                                                                                (1)
 
Tilt T tells how muchthe higher SNRs preponderate over the lower SNRs on the path.                                                                                                               
  Tij=  SNRij 3/4 + SNRij1/4 – 2 SNRij 1/2 
 
   Tij =  [SNRij 3/4  –  SNRij1/2  ] - [ SNRij 1/2   - SNRij 1/4 ]                                                (2)
 


Basically, Tilt T figuresthe difference between the dB amount the 3rd quartile SNR exceedsthe median SNR, less the dB amount the median SNR exceeds the 1st quartileSNR. Tilt tells whether one can be optimistic or pessimistic about favorable pathvariations relative to the median SNR. 

Remember--neither the Variability V nor Tilt T tell  you whether the median SNR of the path
for the path is strong, or weak, or non-existent.  Instead, look at the median SNR itself eventhough the station-specific aspects confuse it.


Practicalities ofcomputation include determination of aquartile SNR when the quartile lies between the two nearest SNR values.One should not merely average those nearest SNR dB values (dB2 higher and dB1 lower)to get quartile SNR dB.  Instead, oneshould effectively average the power levels to get quartile SNR dB using thisformula:
   SNR(dB)= 10 log10 [0.5 (1+10^(0.1(dB2 – dB1)))] + dB1.
The formulaessentially subtracts 3dB from the higher SNR “dB2” (or adds 7dB to the lowerSNR “dB1”) when the dB difference is 10dB or more.  The formula diminishes the dB adjustmentcloser to zero for smaller dB differences, e.g., adjust 2dB down for 5dBdifference, and take the arithmetic average (split the difference) for dBdifferences less than 5dB. 
 
Caveats:  These statistics are somewhat sensitive to missing decodes and the way missing decodes arehandled—such as by replacing the empty decode for the time slot by the WSPRdecode threshold -33dB or by ignoring the missing decodes altogether.  The statistical term is “censored data.”  Preferably, one works with SNRs obtained intime intervals wherein both stations have few or no missing decodes. However, I have just ignored the missing decodes and so far the results are interesting and convenient to get.  Also, multiple artifact decodes (as seen by WH2XND)yielding greatly different SNRs for a same station in the same time slot shouldbe edited out to leave only the decode with the peak SNR among them.  Because of censoring by the WSPR decoderthreshold itself, the Variability  V and dB TiltT can indeed be affected by TX, RX, and antennas if a lot of the SNRinformation lies near the WSPR decode threshold. One further caveat: Subtraction of 1st from 3rdquartile adds to amount of uncertainty inthe value of Variability V compared to amount of uncertainty in the quartileSNRs themselves. The two subtractionsinvolving the 1st, 2nd , and 3rd quartiles addeven somewhat more uncertainty in the value of Tilt T compared to the amount of uncertainty in the quartile SNRs.  
 
The special case of areciprocal path for any given pair of stations should yield no statisticallysignificant difference between values representing opposite directions, such asbetween Vij and Vji or between Tij and Tji. More work is needed to determine how to estimate statistical significance.  Statistical significance improves withthe number of decodes for each path direction and the amount of the differencebetween two values of Variability V or difference between two values of dB TiltT. 

The next post shows Variability V and dB Tilt T calculations for some 630m paths last night.