All NewsGroup readers appear to agree that rho is the reflection
coefficient.

Beyond that it seems that there are dissenters...

Here's some thoughts which will cause readers to pause... er... delay, when
it comes
to complex Zo and complex rho.

rho is the ratio of the reflected voltage wave "b" to the incident voltage
wave "a".

rho = b/a = (ZL - Zo)/(ZL + Zo)

rho is then in general a complex function of angular frequency w = 2*pi*f
where f
is "cyclic" frequency in Hz simply because both ZL and Zo can themselves be
complex.

rho may of course be considered to be a"transfer function" which yeilds the
reflected
voltage "b" or voltage echo, sometimes known as the "talker echo", which
results from
the application of the incident voltage "a".

rho(jw) = [(ZL(jw) - Zo(jw))/(ZL(jw) + Zo(jw))] = |rho(w)|*exp[j*phie(w)]

Where phie(w) is widely known in the literature as the "echo phase" and
|rho(w)| is
the magitude of the reflection coefficient.

Following the literature and taking natural logarithm's of the inverse of
rho one finds that:

ln[1/rho(jw)] = -ln|rho(w)| - j*phie(w) = Ae(w) - j*phie(w)

Where Ae(w):

Ae(w) = -ln|rho(jw)| "Echo Attenuation or Return Loss"

Measures the echo attenuation in Nepers and is the so-called "Echo
Attenuation" or "Return Loss", in Nepers (Np).

Ae(w) measures how much the talker echo or reflected voltage echo "b" is
attenuated below the incident voltage wave "a".

Aside: Europeans seem to prefer the term Echo Attenuation and the symbol
Ae(w) and Np units [which is also my personal preference] however North
American authors seem to prefer the term "Return Loss" and symbol RL and
prefer to use dB units rather that Np units. For those who need to convert
from Np to dB that conversion is simply dB = 20 log(e) = 8.686 Np.

Of more interest to those fans of complex Zo is phie(w), the so-called
"Echo Phase" or "Return Phase" measured in radians. The Echo Phase
measures the phase lag of the voltage echo as compared to the phase
angle of the incident voltage.

Now for some stuff that should really interest Cecil.

:-)

Differentiating the echo phase phie(w) with respect to w, one obtains the
well-known "Echo Group Delay", this echo group delay or, as it is sometimes
called, the "Return Delay", is always a real function and is given by.

taugre(w) = d[phie(w)]/dw

and has units of seconds.

taugre(w) measures the group delay of the returned voltage echo as a
function of frequency. In a differential bandwidth dw the value of
taugre(w) measures the average delay of a "packet" of energy launched by the
incident wave "a" in that differential frequency band dw and returned to the
source in the voltage echo.

Wow, sort of like transmission line radar. Question for Time Domain
Reflectometer
fans:

Is this taugre(w) the delay that would be measured by a TDR?

An even more interesting quantity is the echo group velocity or vge(w) which
is simply the reciprocal of taugre(w)

vge(w) = 1/taugre(w)

which has units of inverse seconds.

The vge(w) may never be faster than the speed of light in the Zo media.
In general the velocity of frequency groups is not constant with frequency!

vge(w) gives the actual group velocity of packets or groups of frequencies
in the echo as it passes from the source back around to the source again.
It is a function of frequency and illustrates that in transmission lines
with
complex Zo not all frequencies in the echo travel with the same velocity and
some frequencies actually arrive back at the source before the others.

Should designers of echo cancellers for complex Zo lines care about this?

Of course no such thing occurs with real Zo lines!

Real Zo lines are sooooo dull and unintersting!

Another interesting quantity which I am sure Cecil will enjoy is the
so-called echo phase delay or tauphe(w)

tauphe(w) = phie(w)/w

The echo phase delay is an interesting function and I will leave it to the
reader to work out its' physical significance. This quantity leads to the
very curious echo phase velocity or vpe(w)

vpe(w) = 1/tauphe(w)

It is interesting to note that vpe(w) can actually exceed the speed of light
in the Zo media. In a transmission system with a real Zo of course
vpe = vge. But in the widely deployed complex Zo telephone twisted
pair plant it is not.

Things are really dull and uninteresting when Zo is real, huh?

Finally the arcane "echo signal front delay" tausfe(w), when it exists
equals the limit.

tausfe(w) = limit as w - infinity of the ration phie(w)/w

And of course the reciprocal of the echo signal front delay gives the echo
signal front velocity vsfe(w) when it exists, thus:

vsfe(w) = 1/tausfe(w) when it exists.

It is left as an exercise for the reader to find out under what physical
circumstances this exotic velocity exists and just what it's physical
meaning might be...

A quantity of great great interest to Cecil will be the value of the echo
group delay evaluated at the origin (DC)!

taugre(w) evaluated at w = 0 [DC] gives the first "time moment" of the echo!

taugre(0) = T*area of voltage echo = Integral from 0 to infinity of
t*b(t)*dt

taugre(0) when divided by the area under the echo voltage gives the average
time delay T by which the energy in the echo is delayed.

T = taugre(0)/area under b(t).

It turns out that this is an extremely useful value to know if one is
interested in designing or operating an effective and feasible analog or
digital talker echo canceller to suppress echos on the line and allow the
receiver to receive remotely generated signals in the face of extremely
strong incident signals.

Circuitry or algorithms that can make fast real time estimates of T are
highly desirable and are held as trade secrets by certain companies.

The design and operation of talker echo cancellers for transmission systems
with complex Zo is quite a complicated and challenging task.

That's enough titilating information for fans of complex Zo.

And so... until the next time...

I'd like some thoughts and comments these arcane and exotic functions and
about
comparisons between such functions for transmission systems with real Zo and
those with complex Zo.

Thoughts, comments?

--
Peter K1PO
Indialantic By-the-Sea, FL.