# 1.4 Common discrete time signals

 Page 1 / 1
Presents several useful discrete time signals.

## Introduction

Before looking at this module, hopefully you have an idea of what a signal is and what basic classifications and properties a signal canhave. In review, a signal is a function defined with respect to an independent variable. This variable is often timebut could represent any number of things. Mathematically, discrete time analogsignals have discrete independent variables and continuous dependent variables. This module will describe some useful discrete time analog signals.

## Sinusoids

One of the most important elemental signal that you will deal with is the real-valued sinusoid. In its discrete-timeform, we write the general expression as

$A\cos (\omega n+\phi )$
where $A$ is the amplitude, $\omega$ is the frequency, and $\phi$ is the phase. Because n only takes integer values, the resulting function is only periodic if $\frac{2\pi }{\omega }$ is a rational number. A discrete-time cosine signal is plotted as a stem plot. Note that the equation representation for a discrete time sinusoid waveform is not unique.

## Complex exponentials

As important as the general sinusoid, the complex exponential function will become a critical part of your study of signals and systems. Its general discrete form iswritten as

$z^{n}$
where $z$ , is a complex number. The set of complex exponentials for which $\left|z\right|=1$ are a special class, expressed as $e^{j\omega n}$ , (where $\omega$ is the angular position on the unit circle, in radians).

The discrete time complex exponentials have the following property.

$e^{j\omega n}=e^{j(\omega +2\pi )n}$
Given this property, if we have a complex exponential with frequency $\omega +2\pi$ , then this signal "aliases" to a complex exponential with frequency $\omega$ , implying that the equation representations of discrete complex exponentials are not unique.

## Unit impulses

The second-most important discrete-time signal is the unit sample , which is defined as

$\delta (n)=\begin{cases}1 & \text{if n=0}\\ 0 & \text{otherwise}\end{cases}$ The unit sample.

More detail is provided in the section on the discrete time impulse function. For now, it suffices to say that this signal is crucially important in the study of discrete signals, as it allows the sifting property to be used in signal representation and signal decomposition.

## Unit step

Another very basic signal is the unit-step function defined as

$u(n)=\begin{cases}0 & \text{if n< 0}\\ 1 & \text{if n\ge 0}\end{cases}$ Discrete-Time Unit-Step Function

The step function is a useful tool for testing and for defining other signals. For example, whendifferent shifted versions of the step function are multiplied by other signals, one can select a certain portion of thesignal and zero out the rest.

## Common discrete time signals summary

Some of the most important and most frequently encountered signals have been discussed in this module. There are, of course, many other signals of significant consequence not discussed here. As you will see later, many of the other more complicated signals will be studied in terms of those listed here. Especially take note of the complex exponentials and unit impulse functions, which will be the key focus of several topics included in this course.

#### Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

### Read also:

#### Get Jobilize Job Search Mobile App in your pocket Now!

Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications? By By OpenStax By OpenStax By Brooke Delaney By Cath Yu By OpenStax By OpenStax By Sandy Yamane By Brooke Delaney By Joli Julianna