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

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

Periodic signals in discrete time repeats themselves in each cycle. However, only integers are allowed as time variable in discrete time. We denote signals in such case as x[n], n = ..., -2, -1, 0, 1, 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:

$$f\left[n\right]=f\left[n+mN\right]\forall m:\left(m\in \mathbb{Z}\right)$$

1

discrete-time periodic signal

We can think of periodic functions (with period $N$) 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 $\left[0,N\right]$ (or, more generally, $\left[a,a+N\right]$). If we know the signal is N-periodic then all the information of the signal is captured by the above interval.

An aperiodic DT function
$f\left[n\right]$
does not repeat for **any**
$N\in \mathbb{R}$;
i.e. there exists no $N$ such that this equation holds.

SinDrillDiscrete Demonstration

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

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