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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!


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


Take a look at

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

Questions & Answers

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Stoney Reply
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Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
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anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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Damian Reply
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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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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