i.e., the square root of the -entropy of a transfer matrix is no smaller than its 2
H
norm. We also have the more complicated converse inequality
q
( ) 1 p log(1
2)
2
(5.26)
I
H
;
;
kH
k
where =
1. Thus, the relative increase in the square root of the
kH
k
=
<
1
-entropy over the 2 norm can be bounded by an expression that only depends
H
on how close the
norm is to the critical value . For example, if
2
H
kH
k
=
1
1
(
0 5), we have
2 p ( ) 1 073
2, i.e., if exceeds
by 6dB
:
kH
k
I
H
:
kH
k
kH
k
1
or more, then the square root of the -entropy and
2 cannot di er by more
kH
k
than about 0.6dB. Thus, the
norm of a transfer matrix must be near for the
H
1
square root of the -entropy to di er much from the 2 norm.
H
An important property of the entropy is that it is readily computed using state-
space methods, as we will see in section 5.6.5.
An Interpretation of the Entropy
Recall that the square of the 2 norm of a transfer function is the power of
H
H
its output when it is driven by a white noise. For the case of scalar (i.e., transfer
function) , we can give a similar interpretation of the -entropy of as the average
H
H
power of its output when it is driven by a white noise, and a certain random feedback
is connected around it, as shown in gure 5.19.
+
r
w
z
H
+
q
The -entropy of a transfer function is the average power of
Figure
5.19
H
its output when it is driven by a white noise and a random feedback
z
w
is connected around it. The values of the random feedback transfer function
are independent at dierent frequencies, and uniformly distributed on a
disk of radius 1 centered at the origin in .
=
C
The transfer function from to in gure 5.19, with a particular feedback
w
z
transfer function connected, is (1
), so the power of the output signal is
H
=
;
H
2
H
1
2 :
;
H
We now assume that is random, with ( ) and ( ) independent for = ,
j
!
j
!
6
and each ( ) uniformly distributed on the disk of radius 1 in the complex
j
!
=
5.4 IMPORTANT PROPERTIES OF GAINS
115
plane. Then we have
2
H
= ( )
(5.27)
E
1
2 I H
;
H
where denotes expectation over the random feedback transfer functions .
E
Some feedback transfer functions decrease the power in the output, while other
feedback transfer functions increase it the inequality (5.25) shows that on average,
the power in the output is increased by the feedback. The limit (5.24) shows that
if the feedback is small, then it has little e ect on the output power (indeed, (5.26)
shows that the average e ect of the feedback on the output power is small unless
the
norm of the feedback is close to the inverse of the
norm of ).
H
H
H
1
1
5.3.6
Scaling and Weights
The discussion of section 4.3.5 concerning scalings and weights for norms of vector
signals has important implications for norms of MIMO systems gains especially are
a ected by the scaling used to measure the input and output vector signals. For
example, let us consider the e ect of scaling on the RMS gain of a (square) MIMO
system. Let be a scaling matrix (i.e., diagonal). The -scaled RMS value of
D
D
H
w
is
rms =
rms the -scaled RMS value of is
rms =
rms.
kH
w
k
kD
H
w
k
D
w
kw
k
kD
w
k
D
D
Thus, the -scaled RMS gain of is
D
H
1
sup
rms
~
;
rms
kD
H
w
k
= sup kDHD wk
rms=0
rms
~
=0
~ rms
kD
w
k
rms
kw
k
kw
k
6
kw
k
6
=
1
;
kD
H
D
k
1
the
norm of the diagonally pre- and post-scaled transfer matrix.
H
1
More general transfer matrix weights can be applied, e.g., post
pre .
kW
H
W
k
5.4
Important Properties of Gains
5.4.1
Gain of a Cascade Connection
An important property of gains is that the gain of the product of two transfer
matrices can be bounded in terms of the gains of the individual transfer matrices:
if
gn denotes any gain, then
k
k
2 1 gn
2 gn 1 gn
(5.28)
kH
H
k
kH
k
kH
k
:
This inequality is easily established it can be seen from gure 5.20. The prop-
erty (5.28) does not generally hold for norms of LTI systems that are not gains. For
116