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This module will take the ideas of sampling CT signals further by examining how such operations can be performed in the frequency domain and by using a computer.


We just covered ideal (and non-ideal) (time) sampling of CT signals . This enabled DT signal processing solutions for CTapplications ( ):

Much of the theoretical analysis of such systems relied on frequency domain representations. How do we carry out thesefrequency domain analysis on the computer? Recall the following relationships: x n DTFT X ω x t CTFT X Ω where ω and Ω are continuous frequency variables.

Sampling dtft

Consider the DTFT of a discrete-time (DT) signal x n . Assume x n is of finite duration N ( i.e. , an N -point signal).

X ω n N 1 0 x n ω n
where X ω is the continuous function that is indexed by thereal-valued parameter ω . The other function, x n , is a discrete function that is indexed by integers.

We want to work with X ω on a computer. Why not just sample X ω ?

X k X 2 N k n N 1 0 x n 2 k N n
In we sampled at ω 2 N k where k 0 1 N 1 and X k for k 0 N 1 is called the Discrete Fourier Transform (DFT) of x n .

Finite duration dt signal

The DTFT of the image in is written as follows:

X ω n N 1 0 x n ω n
where ω is any 2 -interval, for example ω .

Sample x(Ω)

where again we sampled at ω 2 N k where k 0 1 M 1 . For example, we take M 10 . In the following section we will discuss in more detail how we should choose M , the number of samples in the 2 interval.

(This is precisely how we would plot X ω in Matlab.)

Choosing m

Case 1

Given N (length of x n ), choose M N to obtain a dense sampling of the DTFT ( ):

Case 2

Choose M as small as possible (to minimize the amount of computation).

In general, we require M N in order to represent all information in n n 0 N 1 x n Let's concentrate on M N : x n DFT X k for n 0 N 1 and k 0 N 1 numbers N  numbers

Discrete fourier transform (dft)


X k X 2 k N
where N length x n and k 0 N 1 . In this case, M N .


X k n N 1 0 x n 2 k N n

Inverse dft (idft)

x n 1 N k N 1 0 X k 2 k N n


Represent x n in terms of a sum of N complex sinusoids of amplitudes X k and frequencies k k 0 N 1 ω k 2 k N

Fourier Series with fundamental frequency 2 N

Remark 1

IDFT treats x n as though it were N -periodic.

x n 1 N k N 1 0 X k 2 k N n
where n 0 N 1

What about other values of n ?

x n N ???

Remark 2

Proof that the IDFT inverts the DFT for n 0 N 1

1 N k N 1 0 X k 2 k N n 1 N k N 1 0 m N 1 0 x m 2 k N m 2 k N n ???

Computing dft

Given the following discrete-time signal ( ) with N 4 , we will compute the DFT using two different methods (the DFTFormula and Sample DTFT):

  • DFT Formula
    X k n N 1 0 x n 2 k N n 1 2 k 4 2 k 4 2 2 k 4 3 1 2 k k 3 2 k
    Using the above equation, we can solve and get thefollowing results: x 0 4 x 1 0 x 2 0 x 3 0
  • Sample DTFT. Using the same figure, , we will take the DTFT of the signal and get the following equations:
    X ω n 0 3 ω n 1 4 ω 1 ω ???
    Our sample points will be: ω k 2 k 4 2 k where k 0 1 2 3 ( ).

Periodicity of the dft

DFT X k consists of samples of DTFT, so X ω , a 2 -periodic DTFT signal, can be converted to X k , an N -periodic DFT.

X k n N 1 0 x n 2 k N n
where 2 k N n is an N -periodic basis function (See ).

Also, recall,

x n 1 N n N 1 0 X k 2 k N n 1 N n N 1 0 X k 2 k N n m N ???


When we deal with the DFT, we need to remember that, in effect, this treats the signal as an N -periodic sequence.

A sampling perspective

Think of sampling the continuous function X ω , as depicted in . S ω will represent the sampling function applied to X ω and is illustrated in as well. This will result in our discrete-time sequence, X k .

Remember the multiplication in the frequency domain is equal to convolution in the time domain!

Inverse dtft of s(Ω)

k δ ω 2 k N
Given the above equation, we can take the DTFT and get thefollowing equation:
N m δ n m N S n

Why does equal S n ?

S n is N -periodic, so it has the following Fourier Series :

c k 1 N n N 2 N 2 δ n 2 k N n 1 N
S n k 2 k N n
where the DTFT of the exponential in the above equation is equal to δ ω 2 k N .

So, in the time-domain we have ( ):


Combine signals in to get signals in .

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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