For general references on vector spaces, norms of signals, or norms in general, see the
Notes and References for the next chapter. The text
] by Wong and Hajek covers
WH85
stochastic signals.
The bold in the symbols 1, 2, and
stands for the mathematician H. Lebesgue.
L
L
L
L
1
Our Definition of Norm
Our denition 4.1 diers from the standard denition of a norm in two ways: rst, we
allow norms to take on the value + and second, we do not require v > 0 for nonzero
1
k
k
v (which is called the deniteness property). The standard term for what we call a norm
is seminorm. We will not need the niteness or deniteness properties of norms in fact,
the only property of norms that we use in this book is convexity (see chapter 6). Our less
formal usage of the term norm allows us to give a less technical discussion of norms of
signals.
The mathematically sophisticated reader can form a standard norm from each of our
seminorms by rst restricting it to the subspace of all signals for which u is nite, and
k
k
then forming the quotient space, modulo the subspace of all signals for which u is zero.
k
k
This process is discussed in any mathematics text covering norms, e.g., Kolmogorov and
Fomin
] and Aubin
]. As an example,
ss is a standard norm on the
KF75
A
ub79
k
k
1
vector space of equivalence classes of eventually bounded signals, where the equivalence
classes consist of signals that dier by a transient, i.e., signals that converge to each other
as t
.
!
1
Some Mathematical Notes
There are signals for which the RMS value (4.1) or average-absolute value (4.2) are not de-
ned because the limits in these expressions fail to exist: for example, u(t) = coslog(1+t).
These norms will always be dened if we substitute limsup for lim in the denitions (4.1)
and (4.2). With this generalized denition of the RMS and average-absolute norm, many
but not all of the properties discussed in this chapter still hold. For example, with limsup
substituted for lim in the denition of the RMS norm of a vector signal (given in (4.16)),
equation (4.17) need not hold.
We also note that the integral dening the power spectral density (equation (4.5)) need
not exist. In this case the process has a spectral measure.
Spectral Factorization of Weights
Youla
] developed a spectral factorization analogous to (4.13) for transfer matri-
You61
ces, which yields an interpretation of the W-weighted RMS norm as the square root of the
total average power dissipated in a passive n-port admittance G(s). In
], Ander-
And67
son showed how this spectral factorization for transfer matrices can be computed using
state-space methods, by solving an algebraic Riccati equation. See also section 7.3 of
Francis
].
Fra87
