Discrete time periodic signals

Introduction

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

Relevant spaces

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, ...

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:

Discrete-time periodic signal

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

1. as functions on all of $\mathbb{R}$

   

2. 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 $f(n)$ does not repeat for any $N\in \mathbb{R}$ ; i.e. there exists no $N$ such that this equation holds.

Sindrilldiscrete demonstration

Conclusion

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