Continuous time periodic signals

Introduction

This module describes the type of signals acted on by the Continuous Time Fourier Series.

Relevant spaces

The Continuous-Time Fourier Series maps finite-length (or $T$ -periodic), continuous-time signals in $L^2$ to infinite-length, discrete-frequency signals in $l^2$ .

Periodic signals

When a function repeats itself exactly after some given period, or cycle, we say it's periodic . A periodic function can be mathematically defined as:

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

1. as functions on all of $\mathbb{R}$

   

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

   

An aperiodic CT function $f(t)$ , on the other hand, does not repeat for any $T\in \mathbb{R}$ ; i.e. there exists no $T$ such that this equation holds.

Demonstration

Here's an example demonstrating a periodic sinusoidal signal with various frequencies, amplitudes and phase delays:

To learn the full concept behind periodicity, see the video below.

Conclusion

A periodic signal is completely defined by its values in one period, such as the interval [0,T].