Digital Signal Processing Reference
In-Depth Information
which is more efficiently expressed in matrix form as
b
2
=
S
22
·
a
2
b
1
S
11
S
12
a
1
(9-16)
S
21
More generally, the elements of a scattering matrix are described in equa-
tion (9-17) for an arbitrary number of ports,
power measured at port
i
power injected into port
j
b
i
a
j
=
S
ij
=
(9-17)
and an arbitrary-sized scattering matrix takes the form
b
=
Sa
(9-18)
If the scattering matrix of a system is known, the response of the system can be
predicted for any input.
Return Loss
Consider the circuit depicted in Figure 9-8. In this case there will
be no reflections from the far end because the line is perfectly terminated with
the characteristic impedance. However, the source impedance is not equal to the
characteristic impedance, indicating that a portion of the power wave incident to
port 1 will be reflected. This scenario allows the simplest definition of
S
11
, which
is simply the reflection coefficient between the source resistor and the impedance
of the transmission line. Note that
a
2
=
0 because there is no source at port 2.
v
1
/
√
R
v
1
/
√
R
=
a
2
=
0
=
v
1
v
1
=
b
1
a
1
v
reflected
v
incident
Z
0
−
R
S
11
=
=
0
=
(9-19)
Z
0
+
R
The term
S
11
is often referred to as the
return loss
, because it is a measure of
power reflected, or returned to the source.
The calculation of
S
11
becomes more complex when the far end of the network
is not perfectly terminated because the reflection arriving at the source will have
V
Z
0
R
I
=
Z
0
R
∼
Z
0
R
Z
0
+
R
−
Γ =
0
S
11
=
Figure 9-8
Return loss for the special case when the network is perfectly terminated in
its characteristic impedance.
Search WWH ::
Custom Search