One of the factors that need to be accounted for when making the collision decision is the attenuation that signals undergo in the Ethernet cable. In general, this attenuation is a function of the frequency but this study will be limited to the case of a distortionless cable, in which the atenuation is constant and the phase shift is linear for all frequencies. Under this assumption, the only transmission line effect to consider is the line attenuation.
This project presents a simulation of the collision detection algorithm. The line parameters can be adjusted to simulate the waveforms along a transmission line when two stations are transmitting at the same time, and determine whether a collision occured or not.
Let 
be the sinusoidal source that
excites the
transmission
line as shown in Figure 1.
Figure 1.
For a sinusoidal excitation, v(z,t) and i(z,t) may be expressed as
Substituting equations (3) and (4) into equations (1) and (2), and
canceling out the 
terms from both sides of the equations,
one obtains
Here we have assumed a source of time variation of the form
because it simplifies the equations by getting rid of the derivative
with respect to time.
Differentiating equation (5), once more with respect to z, and substituting equation (6) into the result, we obtain
Similarly, one obtains
Equations (7) and (8) are second order differential equations which have solutions
where 
and the amplitude of the forward
and backward voltage and current waves are related as follows:
Finally, it is important to point out that in the case of a distortionless
transmission line, 
and 
where
the attenuation factor and 
the velocity factor.
It is important to point out that in this project we deal with transmission
lines of characteristic impedance 
that are terminated with resistors
with the same impedance. In our case, we assume 
ohms. This
implies that we don't have any reflections in the line. Therefore, Equations (9)
and (10), become for the case of a forward traveling wave:
and for a backward traveling wave:
It is known from Fourier analysis, that we can represent a periodic wave as a sum of complex exponentials, as indicated in Equation (17)
where the coefficients 
are complex in general and are given by:
Now consider, sending one sinusoidal of frequency 
down the line. Then Equations (13), (14), (15),
and (16) still are applicable and the only thing that changes is
which becomes 
. It is easy to see
that the same thing will hold true for 
.
Thus, we obtain that for a periodic wave, the previous equations become:
Notice that 
, where is entirely independent of the frequency
(under our assumptions). Therefore, we can factor out 
or 
and have
it multiply the expression inside of the summation sign. Then, since we assumed a
distortionless line (one in which 
is linear with 
) we have the same pulse train that
we had in the beginning attenuated over the length of the line.