Digital Signal Processing Reference
In-Depth Information
Cascading with the T -Matrix
Another method commonly used to cascade
S
-parameters is the
T-matrix
, sometimes called
transmission parameters
.
The
T
-parameters are derived simply by rearranging the equations for the
S
-parameters. Equation (9-18) describes the relationship between the incident
power waves
a
and the exiting power waves
b
:
b
1
b
2
S
11
a
1
a
2
S
12
=
S
21
S
22
To facilitate the cascading of networks by simple matrix multiplication, the
equation needs to be rearranged so that the ingoing and outgoing waves of port 1
can be described in terms of the waves at port 2. This is done with the
T
-matrix.
A two-port
T
-matrix is
a
1
b
1
T
11
T
22
b
2
a
2
T
12
=
(9-47)
T
21
If the output of circuit A is attached to the input of circuit B, the total response
can be calculated simply by multiplying the
T
-matrices, because the power wave
exiting circuit A is
b
2
, which feeds into the input of circuit B, which is
a
3
.
Therefore,
b
2
=
a
3
and
a
2
=
b
3
, as shown in Figure 9-33.
a
1
b
1
T
11
T
22
b
2
a
2
T
12
=
T
21
A
a
3
b
3
T
11
b
4
a
4
T
12
=
T
21
T
22
B
a
1
b
1
T
11
T
22
T
11
T
22
B
b
4
a
4
T
12
T
12
=
T
21
T
21
A
Therefore, S-parameters can be cascaded by converting to T
-
parameters and
multiplying
. The cascaded scattering matrix is then calculated by converting the
product of the
T
-matrices back to
S
-parameters.
Conversion between the
T
- and
S
-parameters (and vice-versa) requires simple
algebraic manipulation of the equations, which can be done for any number of
b
2
a
3
b
4
a
1
[
T
]
circuit
1
[
T
]
circuit
2
b
1
a
2
b
3
a
4
Figure 9-33
T
-parameters are cascaded through multiplication.
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