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

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

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:

$$f\left(t\right)=f\left(t+mT\right)\forall m:\left(m\in \mathbb{Z}\right)$$

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

- as functions on
**all**of $\mathbb{R}$ - 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
$f\left(t\right)$, on the other hand,
does not repeat for **any**
$T\in \mathbb{R}$;
i.e. there exists no $T$ such that this equation holds.

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

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