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Periodic Signals

Module by: Michael Haag, Justin Romberg. E-mail the authors

Summary: This module defines a periodic function and describes the two common ways of thinking about a periodic signal.

Note: You are viewing an old version of this document. The latest version is available here.

Recall that a periodic function is a function that repeats itself exactly after some given period, or cycle. We represent the definition of a periodic function mathematically as:

f⁢t=f⁢t+m⁢T ∀m :m∈Z f t f t m T m m (1)
where T>0 T 0 represents the period. Because of this, you may also see a signal referred to as a T-periodic signal. Any function that satisfies this equation is periodic.

We can think of periodic functions (with period TT) two different ways:

#1) as functions on all of R

Figure 1: Function over all of R where f⁢ t 0 =f⁢ t 0 +T f t 0 f t 0 T
Figure 1 (per_fxn1.png)

#2) or, we can cut out all of the redundancy, and think of them as functions on an interval 0 T 0 T (or, more generally, a a+T a a T ). If we know the signal is T-periodic then all the information of the signal is captured by the above interval.

Figure 2: Remove the redundancy of the period function so that f⁢t f t is undefined outside 0 T 0 T .
Figure 2 (per_fxn2.png)

An aperiodic CT function f⁢t f t does not repeat for any T∈R T ; i.e. there exists no T T s.t. this equation holds.

Question: DT definitions?

Continuous-Time

Discrete-Time

Note: Circular vs. Line

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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