7.1: Discrete Time Periodic Signals

Last updated
Save as PDF

Page ID: 22879

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

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 \(x[n]\), \(n=\ldots,-2,-1,0,1,2, \dots\)

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:

\[f[n]=f[n+m N] \forall m:(m \in \mathbb{Z}) \label{7.1} \]

where \(N > 0\) represents the fundamental period of the signal, which is the smallest positive value of N for the signal to repeat. Because of this, you may also see a signal referred to as an N-periodic signal. Any function that satisfies this equation is said to be periodic with period N. Here's an example of a discrete-time periodic signal with period N:

Figure \(\PageIndex{2}\): Notice the function is the same after a time shift of N

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

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

Figure \(\PageIndex{3}\): discrete time periodic function over all of \(\mathbb{R}\) where \(f[n_0]=f[n_0+N]\)
or, we can cut out all of the redundancy, and think of them as functions on an interval \([0,N]\) (or, more generally, \([a, a+N]\)). If we know the signal is N-periodic then all the information of the signal is captured by the above interval.

Figure \(\PageIndex{4}\): Remove the redundancy of the period function so that \(f[n]\) is undefined outside \([0,N]\).

An aperiodic DT function \(f[n]\) does not repeat for any \(N \in \mathbb{R}\); i.e. there exists no \(N\) such that Equation \ref{7.1} holds.

SinDrillDiscrete Demonstration

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

Conclusion

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