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The space domain

In this module, we will extend the concept of the Fourier transform from the time domain into the space domain. In making this extension, we will encountersome significant additional complexity. For example, while time is one-dimensional, space is three-dimensional. While you can only move forward andbackwards in time, you can move up, down, forward, backward, and from side to side in space.

(In order to keep the complexity of this module in check, we will assume that space is only two-dimensional, allowing movement up, down, andfrom side to side only. This will serve us well later for such tasks as image processing. Three-dimensional Fourier transforms are beyond the scopeof this module.)

It is also possible and very common to combine time domain signal processing with space domain signal processing. However, that also is beyond the scope ofthis module.

Time and space are analogous

We will consider the space domain to be analogous to the time domain, with the stipulation that the space domain has two dimensions. The unit of measure inthe time domain is usually seconds, or some derivative thereof. The unit of measure in space is usually meters, or some derivative thereof.

As with the time domain, we will assume that all space domain surfaces are purely real (as opposed to being complex) . This will allow us to simplify our computations when performing the 2D Fourier transform to transformour data from the space domain into the wavenumber domain.

(I will point out that from a practical viewpoint this assumption is much more limiting in the space domain than in the time domain. Complexspace domain functions are quite common in such areas as antenna array processing.)

Frequency and wavenumber are analogous

We will consider the wavenumber domain to be analogous to the frequency domain. The unit of measure in the frequency domain is cycle per second, or somederivative thereof. The unit of measure in the wavenumber domain is cycles per meter or some derivative thereof.

Period and wavelength are analogous

The reciprocal of the typical unit of measure in the frequency domain is seconds per cycle, commonly referred to as the period. The reciprocal of thetypical unit of measure in the wavenumber domain is meters per cycle, commonly referred to as the wavelength.

Some real world examples

With all of this as background, I will begin by discussing some real world engineering problems for which the solution lies in an understanding of thewavenumber domain. I will use these examples to show some of the practical uses of 2D Fourier transforms.

Following that (in Part 2 of this series) , I will present and explain a class that you can copy and use to perform 2D Fourier transforms. Then I will presentand explain a program that exercises and tests the 2D Fourier transform class for some common 3D surfaces.

A commercial radio station

Assume that you have just acquired an FCC license to build and operate a new commercial radio station in a small town in west Texas. As is frequently thecase in west Texas, your town is situated at the intersection of two highways. One highway runs northeast and southwest. The other highway runs northwest andsouthwest. The two highways are generally perpendicular to one another. Like many highways in west Texas, each of these highways is straight as an arrow withvery few curves.

Questions & Answers

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
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
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
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
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Source:  OpenStax, Digital signal processing - dsp. OpenStax CNX. Jan 06, 2016 Download for free at https://legacy.cnx.org/content/col11642/1.38
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