Inside Collection (Course): Digital Signal Processing

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

An operator for which *bounded operator*.

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

Such a system satisfies

One can show that

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!

Consider

Since

A very important class of linear operators are those for which

Suppose that *eigenvector* is
a vector *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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