A notion closely related to the size of a signal is the size of a transfer function or
LTI system. In this chapter we explore some of the ways this notion of size of a
system can be made precise.
5.1
Paradigms for System Norms
In this chapter we discuss methods of measuring the size of an LTI system with input
, output , and transfer matrix , shown in gure 5.1. Many commonly used
w
z
H
norms for LTI systems can be interpreted as examples of several general methods
for measuring the size of a system in terms of the norms of its input and output
signals. In the next few subsections we describe these general methods.
w
z
H
A linear system with input w and output z.
Figure
5.1
We leave as an exercise for the reader the veri cation that the norms we will
encounter in this chapter do satisfy the required properties (i.e., de nition 4.1).
5.1.1
Norm of a Particular Response
The simplest general method for measuring the size of a system is to measure the
size of its response to a particular input signal part, e.g., a unit impulse, a unit
w
step, or a stochastic signal with a particular power spectral density. If we use the
norm output to measure the size of the response, as shown in gure 5.2, we de ne
k
k
part =