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Continuous Time Periodic Signals

By Michael Haag, Justin Romberg

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 TT-periodic), continuous-time signals in L2 L2 to infinite-length, discrete-frequency signals in l2 l2.

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:

ft=ft+mT m :mZ f t f t m T m m
1
where T>0 T 0 represents the fundamental period of the signal, which is the smallest positive value of T for the signal to repeat. Because of this, you may also see a signal referred to as a T-periodic signal. Any function that satisfies this equation is said to be periodic with period T.

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

  1. as functions on all of R
    Figure 1: Continuous time periodic function over all of R where f t 0 =f t 0 +T f t 0 f t 0 T
  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 ft f t is undefined outside 0 T 0 T .

An aperiodic CT function ft f t , on the other hand, does not repeat for any TR T ; i.e. there exists no T T such that this equation holds.

Demonstration

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

Figure 3: Interact (when online) with a Mathematica CDF demonstrating a Periodic Sinusoidal Signal with various frequencies, amplitudes, and phase delays.

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

Khan Lecture on Periodic Signals
Download Khan_Lecture_On_Periodic
Figure 4: video from Khan Academy

Conclusion

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