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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 $TT$-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:

$f⁢t=f⁢t+m⁢T ∀m :m∈Z f t f t m T m m$
1
where $T>0 T 0$ represents the 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 f⁢t f t is undefined outside 0 T 0 T .

An aperiodic CT function $f⁢t f t$, on the other hand, does not repeat for any $T∈R T$; 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:

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

Khan Lecture on Periodic Signals

# Conclusion

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