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Linear Systems

Module by: Mark A. Davenport. E-mail the author

In this course we will focus much of our attention on linear systems. When our input and output signals are vectors, then the system is a linear operator.

Suppose that L:X→YL:X→Y is a linear operator from a vector space XX to a vector space YY. If XX and YY are normed vector spaces, then we can also define a norm on LL. Specifically, we can let

L L ( X , Y ) = max x ∈ X L x Y x X = max x ∈ X : x X = 1 L x Y L L ( X , Y ) = max x ∈ X L x Y x X = max x ∈ X : x X = 1 L x Y
(1)

An operator for which LL(X,Y)<∞LL(X,Y)<∞ is called a bounded operator.

Example 1

BIBO (bounded-input, bounded-output) stable systems are systems for which

x ∞ < A ⟹ L x ∞ < B . x ∞ < A ⟹ L x ∞ < B .
(2)

Such a system satisfies L∞<BAL∞<BA.

One can show that ·L(X,Y)·L(X,Y) satisfies the requirements of a valid norm. In fact L(X,Y)={ bounded linear operators from X to Y}L(X,Y)={ bounded linear operators from X to Y} is itself a normed vector space! If YY is a Banach space, then so is L(X,Y)L(X,Y)!

Bounded linear operators are common in DSP—they are “safe” in that “normal” inputs are guaranteed to not make your system explode.

Are there any common systems that are unbounded? Not in finite dimensions, but in infinite dimensions there are plenty of examples!

Example 2

Consider L2-π,πL2-π,π. For any kk, fk(t)=12πe-jktfk(t)=12πe-jkt is an element of L2-π,πL2-π,π with fk(t)2=1fk(t)2=1. Consider the system D=ddtD=ddt, and note that

d d t f k ( t ) = - j k 2 π e - j k t ⟹ D f k ( t ) 2 = k . d d t f k ( t ) = - j k 2 π e - j k t ⟹ D f k ( t ) 2 = k .
(3)

Since fk(t)∈L2-π,πfk(t)∈L2-π,π for all kk, we can set kk to be as large as we want, so DD cannot be bounded.

A very important class of linear operators are those for which X=YX=Y. In this case we have the following important definition.

Definition 1

Suppose that L=X→XL=X→X is a linear operator. An eigenvector is a vector xx for which Lx=αxLx=αx for some α∈Kα∈K (i.e. α∈Rα∈R or α∈Cα∈C). In this case, αα is called the corresponding eigenvalue.

Eigenvalues and eigenvectors tell you a lot about a system (more on this later!). While they can sometimes be tricky to calculate (unless you know the eig command in Matlab), we will see that as engineers we can usually get away with the time-honored method of “guess and check”.

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