6.1: Continuous Time Periodic Signals

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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 \(l^2\).

Figure \(\PageIndex{1}\): Mapping \(L^2([0, T))\) in the time domain to \(l^2(\mathbb{Z})\) in the frequency domain.

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:

\[ f(t)=f(t+m T) \forall m:(m \in \mathbb{Z}) \label{6.1} \]

where \(T>0\) 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 \(T\)) two different ways:

as functions on all of \(\mathbb{R}\)

Figure \(\PageIndex{1}\): Continuous time periodic function over all of \(\mathbb{R}\) where \(f(t_0) = f(t_0+T)\)
or, we can cut out all of the redundancy, and think of them as functions on an interval \([0,T]\) (or, more generally, \([a,a+T]\)). If we know the signal is \(T\)-periodic then all the information of the signal is captured by the above interval.

Figure \(\PageIndex{2}\): Remove the redundancy of the period function so that \(f(t)\) is undefined outside \([0,T]\).

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 Equation \ref{6.1} holds.

Demonstration

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

sinDrillDemo — Figure \(\PageIndex{3}\): Interact (when online) with a Mathematica CDF demonstrating a Periodic Sinusoidal Signal with various frequencies, amplitudes, and phase delays. To download, right click and save file as .cdf.

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

Khan Lecture on Periodic Signals

video

from

Khan Academy

Conclusion

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