10 CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
Forp ersistents ignals,wher e the abovenorm is unbounded, we can define a power norm,
kxi(t)k= lim
T→∞
1
2TZT
−T
|xi(t)|2dt!1/2
.(2.1)
The above norms haveb een defined in terms of a single component,xi(t), of a vector
valued signal, x(t). The choice of spatial norm determines how we combine these
components to calculate kx(t)k. We can mix and match the spatial and time parts of the
norm of a signal. In practice it usually turns out that the choice of the time norm is
more important in terms of system analysis. Unless stated otherwise, kx(t)kimplies the
Euclidean norm spatially and the 2-norm in the time direction.
Certain signal spaces can be defined in terms of their norms. For example, the set of
signals x(t), with kx(t)k2finite is denoted by L2. The formal definition is,
L2=x(t)kx(t)k<∞.
A similar approach can be takenin the discrete-time domain. Considera sequence,
{x(k)}∞
k=0, with 2-norm given by,
kx(k)k2= ∞
X
k=0
|x(k)|2!1/2
.
A lower case notation is used to indicate the discrete-time domain. All signals with finite
2-norm are therefore,
l2=x(k),k=0,...,∞
kx(k)k
2<∞.
Weca nessentially split the space L2into two pieces, H2and H⊥
2.H2is the set of
elements of L2which are analytic in the right-half plane. This can be thought of as
those which have their poles strictly in the left half plane; i.e. all stable signals.
Similarly, H⊥
2are all signal with their poles in the left half plane; all strictly unstable