Euclid's "Extreme and Mean Ratio" in RF Sidebands
by Sean Logan, KG7ARW
Summer 2017
Say you have two waves, with frequencies F and KF, where K is some ratio.
F = fundamental
K = real number > 1
If these two waves heterodyne, we will get two sidebands, a lower and an upper sideband.
LSB = KF - F
USB = KF + F
Let's plot all four frequencies in the frequency domain.
A = LSB
B = F
C = KF
D = USB
Now, it is possible to choose a value for K, such that:
B/A = C/B = D/C
This value for K would be a solution to the equation:
X^2 - X - 1 = 0
K = ( 1 + sqrt(5) ) / 2
K = 1.618033...
Euclid called this the "Extreme and Mean Ratio".
It has the property:
K^n + K^(n+1) = K^(n+2)
K^n - K^(n-1) = K^(n-2)
Therefore, it is useful to use this number as a base of logarithms.
Consider the series:
..., K^-3, K^-2, K^-1, K^0, K^1, K^2, K^3, ...
K = 1.618033...
If we add together any two consecutive terms in the series, we will get the next term.
If we take the difference between any two consecutive terms, we will get the previous term.
Now, if we begin with two frequencies, F and KF, which are two consecutive terms in this series, say,
F = K^5
KF = K^6
Then when F and KF heterodyne, we get,
KF - F = K^4 LSB
KF + F = K^7 USB
Which are the next and previous terms in the series.
If the LSB heterodynes with F, we get,
K^4 + K^5 = K^6 = KF
K^5 - K^4 = K^3
It reinforces the frequency at KF, and also gives us a new term in the series.
If the USB heterdynes with KF, we get a similar phenomenon.
Hypothesis:
Antennas or other resonant structures, whose physical geometry is based on the Extreme and Mean Ratio, may preform well in Spread Spectrum applications.
More specifically, they may be useful for generating and receiving sets of frequencies which form a logarithmic pattern in the frequency domain.
73
KG7ARW
2017