[ns] Explanation of propagation crossover distance

[Leonard Miller <lmiller@antd.nist.gov>]

In the ns file tworayground.cc, the formula for received power is
switched between two different formulas, depending on the link
distance.  For shorter distances, the free space propagation model is
used [function "Friss"] and for longer distances, the asymptotic (small
reflection angle) form of the two-ray (flat-earth) model is selected
[function "TwoRay"].

The general form of the received power equation is

Pr = Pt * Gt * Gr * [1 / path_loss] / L

where Pr = received power, Pt = transmitted power, Gt = transmitter
antenna gain, Gr = receiver antenna gain, and L represents various
system losses (antenna cable, building penetration, orientation, etc.).
Note that L is (or should be) common to whatever path loss model is
used.

For the free space propagation model, the path loss equation is

path_loss = (4 * pi * distance / lambda)**2

and for the two ray propagation model, the path loss equation is

path_loss = (distance)**4 / (ht * hr)**2

where ht = transmitter antenna height and hr = receiver antenna height. 
Equating the two path loss equations gives the crossover distance as

crossover distance = 4 * pi * ht * hr / lambda

The formula in tworayground.cc is equivalent to this expression except
that the factor L has crept in, evidently because L was not held common
to the received power under either path loss model.  Apparently, since L
= 1 is the value used in the examples (which is not realistic--it can be
much higher, depending on the scenario), this inappropriate appearance
of L in the formula for the crossover distance does not seem to have
been noticed before.

Actually, the switching between the two propagation models that is
performed in tworayground.cc is an idealization.  As the graph in the
attached PDF file (proploss.pdf) illustrates, when the link distance is
less than the crossover value discussed above, if the two-ray theory
applies (i.e., the terrain is flat), then the received signal is subject
to positive or negative reinforcement of a direct path by a reflected
path
of nearly equal strength.  The effective path loss with respect to free
space can be 3 dB less (positive reinforcement) or infinite
(cancellation).  The nulls in the received signal occur at the
boundaries of the Fresnel zones.  By opting for the free space model at
small distances AND the two-ray model for larger distances, the
calculation in tworayground.cc ignores the reflected path altogether. 
Probably this is a good policy for now, until the ns physical model is
ready for a more sophisticated approach.

--Len Miller 
________________________________________________________
*._..*.*_.***__*..*._..*._..*.*._.***._*_***_.*..*...*_*
--------------------------------------------------------
Leonard E. Miller, Wireless Commun. Technologies Group
    National Institute of Standards & Technology (NIST)
    100 Bureau Drive Stop 8920, Bldg. 820, Room 445
    Gaithersburg, Maryland 20899-8920
Phone: 301-975-8018        Fax: 301-590-0932
email: lmiller@antd.nist.gov, Len_Miller@compuserve.com
http://w3.antd.nist.gov/wctg/

proploss.pdf