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Important discrete time signals

The signals and relations presented in this module are quite similar to those in the Analog signals module. So do compare and find similarities and differences!

Sequences

Generally a time discrete signal is a sequence of real or complex numbers. Each component in the sequence is identifiedby an index: ...x(n-1),x(n), x(n+1),...

[x(n)] = [0.5 2.4 3.2 4.5]is a sequence. Using the index to identify a component we have x 0 0.5 , x 1 2.4 and so on.

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Manipulating sequences

  • Addition

    Add individually each component with similar index
  • Multiplication by a constant

    Multiply every component by the constant
  • Multiplication of sequences

    Multiply each component individually
  • Delay

    A delay by k implies that we shift the sequence by k. For this to make sense the sequence has to be of infinite length.

Given the sequences [x(n)] = [0.5 2.4 3.2 4.5]and [y(n)]= [0.0 2.2 7.2 5.5].

a)Addition. [z(n)]=[x(n)]+[y(n)]=[0.5 4.6 10.4 10.0]

b)Multiplication by a constant c=2. [w(n)]= 2 *[x(n)]= [1.0 4.8 6.4 9.0]

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Elementary signals&Relations

The unit sample

The unit sample is a signal which is zero everywhere except when its argument is zero, thenit is equal to 1. Mathematically

n 1 n 0 0
The unit sample function is very useful in that it can be seen as the elementary constituent in any discrete signal.Let x n be a sequence. Then we can express x n as follows (using the unit sample definition and the delay operation)
x n k x k n k

The unit step

The unit step function is equal to zero when its index is negative and equal to one for non-negative indexes,see for plots.

u n 1 n 0 0
Unit step function, no delay.
Unit step function, delayed by 5.
Two unit step functions.

Trigonometric functions

The discrete trigonometric functions are defined as follows. n is the sequence index and is the angular frequency. 2 f , where f is the digital frequency.

x n n
x n n
A discrete sine with digital frequency 1/20.

The complex exponential function

The complex exponential function is central to signal processing and some call it the most important signal. Remember that it is a sequence and that 1 is the imaginary unit.

x n n

Euler's relations

The complex exponential function can be written as a sum of its real and imaginary part.

x n n n n
By complex conjugating and add / subtract the result with we obtain Euler's relations.
n n n 2
n n n 2
The importance of Euler's relations can hardly be stressed enough.

Matlab files

unit_step_discrete.m

Take a look at

  • Introduction
  • Analog signals
  • Discrete vs Analog signals
  • Frequency definitions and periodicity
  • Energy&Power
  • Exercises
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Questions & Answers

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
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
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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Source:  OpenStax, Information and signal theory. OpenStax CNX. Aug 03, 2006 Download for free at http://legacy.cnx.org/content/col10211/1.19
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