# 1.1 Discrete time signals

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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.

## Manipulating sequences

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]

## 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

$(n)=\begin{cases}1 & \text{if n=0}\\ 0 & \text{otherwise}\end{cases}$
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)
$x(n)=\sum_{k=()}$ 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.

$u(n)=\begin{cases}1 & \text{if n\ge 0}\\ 0 & \text{otherwise}\end{cases}$

## Trigonometric functions

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

$x(n)=\sin (n)$
$x(n)=\cos (n)$

## 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 $i=\sqrt{-1}$ is the imaginary unit.

$x(n)=e^{in}$

## Euler's relations

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

$x(n)=e^{in}=\cos (n)+i\sin (n)$
By complex conjugating and add / subtract the result with we obtain Euler's relations.
$\cos (n)=\frac{e^{in}+e^{-(in)}}{2}$
$\sin (n)=\frac{e^{in}-e^{-(in)}}{2i}$
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
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#### Questions & Answers

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
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
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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