Digital Signal Processing Reference
In-Depth Information
a
1
a
2
b
1
b
2
a
3
a
4
b
3
b
4
a
n
−
1
a
n
b
n
−
1
b
n
Figure 8-19
Power waves for an
n
-port system.
Hermitian transpose, where the complex conjugate of each term is taken, and
then the matrix is transposed. For example, consider matrix
A
:
a
−
jb
c
+
jd
A
=
e
+
jf
g
−
jh
where the Hermitian transpose is
A
H
:
a
+
jb
e
−
jf
A
H
=
c
−
jd
g
+
jh
This allows equation (8-22) to be written in terms of the power wave matrices
that will produce a real value for the power absorbed by the network. A system
is passive if
a
H
a
b
H
b
−
≥
0
(8-23)
where
a
is a matrix that contains all the power waves incident to each port and
b
contains the power waves coming out of each port. The product of a matrix
with complex values and its Hermitian transpose produces a real value. Therefore,
equation (8-23) simply ensures that the total power absorbed in a network is
greater than or equal to zero.
In the frequency domain, (8-23) is evaluated at each frequency point. In the
time domain, the passivity requirement is essentially the same, except that the
function must be integrated:
t
a
(τ )
T
a
(τ )
−
b
(τ )
T
b
(τ ) dτ
≥
0
(8-24)
−∞
where the transpose is used instead of the Hermitian transpose because time-
domain signals are always real. Equations (8-21), (8-23), and (8-24) represent
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