<< Chapter < Page
  Digital signal processing - dsp     Page 15 / 15
Chapter >> Page >

The deepest part of the trough in the center is relatively narrow with respect to the folding wave numbers at the edges. However, it is somewhatbroader than the peak in the spectrum at the lower left.

The result of convolution

The image in the upper right shows the result of convolving the space domain surface in the upper left with the convolution operator in the upper center.This output space domain surface has a green square area in the center that is at the same level as the green background. In this case, green represents anelevation of 0, which is about midway between the lowest elevation (black) and the highest elevation (white).

Positive and negative fences

Surrounding the green square is a yellow and white fence representing very high elevations. Surrounding that fence is a black and blue fence, representingvery low elevations consisting of large negative values.

Thus, as you move from the outside to the inside of the square in the output surface, the elevation goes from a background level of zero, to a large negativevalue, followed immediately by a large positive value, followed by zero.

Edge detection

This is one form of edge detection. The edges of the square in the input surface have been emphasized and the flat portion of the inputsurface has been deemphasized in the convolution output.

Wavenumber spectrum of the convolution output

The wavenumber spectrum of the output from the convolution operation is shown in the lower right. The spectrum indicates that this surface is made up mostlyof wavenumber components having mid range to high values.

If you are familiar with digital signal processing, you will know that inorder for a space (or time) function to contain very rapid changes in value (such as the elevation changes at the fences described above) the function must contain significant high wavenumber (or frequency) components. That appears to be the case here indicated by the red areas on the four sides of the wavenumberspectrum.

Although this spectrum was produced by convolution in the space domain followed by a 2D Fourier transform on the convolution output, you should be ableto see that the shape of the spectrum on the bottom right approximates the product of the spectrum of the original surface on the bottom left and thespectral response of the convolution operator in the bottom center.

Thus, the same results could have been produced using multiplication in the wavenumber domain followed by an inverse Fourier transform to produce the spacedomain result. Convolution in the space domain is equivalent to multiplication in the wavenumber domain and vice versa.

Hidden watermarks and trademarks

Another interesting application that I can demonstrate online is using 2D Fourier transforms to hide secret trademarks and watermarks in images. Thepurpose of a hidden trademark or watermark is for the owner of the image to be able to demonstrate that the image may have been used inappropriately by someoneelse. Once again, this application can be satisfied by treating the space domain data as purely real. I plan to demonstrate how this is done in a future module.

Summary

I began by explaining how the space domain and the wavenumber domain in two-dimensional analysis are analogous to the time domain and the frequencydomain in one-dimensional analysis.

Then I introduced you to some practical examples showing how 2D Fourier transforms and wavenumber spectra can be useful in solving engineering problemsinvolving antenna arrays.

What's next?

In Part 2 of this two-part series, I will provide and explain a Java class that can beused to perform forward and inverse 2D Fourier transforms, and can also be used to shift the wavenumber origin from the upper left to the center for a morepleasing plot of the wavenumber spectrum.

In addition, I will provide and explain a program that is used to:

  • Test the forward and inverse 2D Fourier transforms to confirm that the code is correct and that the transformations behave as they should
  • Produce wavenumber spectra for simple surfaces to help the student gain a feel for the relationship that exists between the space domain and thewavenumber domain

Miscellaneous

This section contains a variety of miscellaneous information.

Housekeeping material
  • Module name: Java1490-2D Fourier Transforms using Java
  • File: Java1490.htm
  • Published: 07/12/05

Learn how the space domain and the wavenumber domain in two-dimensional analysis are analogous to the time domain and the frequency domain in one-dimensional analysis. Learn about some practical examples showing how 2D Fourier transforms and wavenumber spectra can be useful in solving engineering problems involving antenna arrays.

Disclaimers:

Financial : Although the Connexions site makes it possible for you to download a PDF file for thismodule at no charge, and also makes it possible for you to purchase a pre-printed version of the PDF file, you should beaware that some of the HTML elements in this module may not translate well into PDF.

I also want you to know that, I receive no financial compensation from the Connexions website even if you purchase the PDF version of the module.

In the past, unknown individuals have copied my modules from cnx.org, converted them to Kindle books, and placed them for sale on Amazon.com showing me as the author. Ineither receive compensation for those sales nor do I know who does receive compensation. If you purchase such a book, please beaware that it is a copy of a module that is freely available on cnx.org and that it was made and published withoutmy prior knowledge.

Affiliation : I am a professor of Computer Information Technology at Austin Community College in Austin, TX.

-end-

Questions & Answers

what is the meaning of extended family
Shellyann Reply
extended family is two are more nuclear family living together under one roof
Nordalee
what is genetic engineering
Ebot Reply
good day please what is a cell?
Colota Reply
cell is the basic unit of all living things/organism
Hassan
differenciate b/w cilia and flagellum
Abshira
Cilia: 1. They are smaller in number. 2. Cilia usually occur throughout or major part of the surface of a cell. 3. They beat oar like and in a co-ordinated rhythm 4. Cilia help in locomotion, feeding, circulation, etc.. 5. Example: Paramecium
iti
Flagella: 1. They are larger in size. ADVERTISEMENTS: 2. Flagella are commonly found at one end of the cell. 3. The flagella beat whip-like and independently. 4. Flagella help in locomotion. 5. Example: Euglena
iti
smallest structural and functional unit of an organism!
zidane
cell is the basic unit of all living things
Lukman
what is homeostasis
alex Reply
the ability of an organism to maintain stable internal environment, even when the external environment changes
Bee
what's the function of the hypothalamus
Agyekum
what is homeostasis
Edward Reply
constant maintenance of internal body temperature
Agyekum
What is the function of the Pituitary Gland?
WeNdlovu
how can homeostasis be maintained?
alex
pituitary gland also known as the master gland secretes hormones which triggers other gland to secretes their hormones
Agyekum
what's the full meaning of DNA and RNA
Deborah Reply
DNA is Deoxyribonucleic acid and RNA means Ribonucleic acid
Christian
sha is mitochondria?
Sparrow Reply
what is biology
Femi Reply
biology is the study which deals with the study of the biological components or animal and plants.
Juhi
biology is the study of life.
Roger
biology is the branch of science which deals with study of two parts 1.botany&2zoology these is known as biology....
Chetan
biology is the study of human beings ...
Chetan
Biology is a branch of science that deals with the study of nature
Agyekum
biology is the science of living organism
Abshira
biology is the branch of science that deals with study of living organisms
Lukman
what is neculer cell?
is it nuclear cell?
iti
asthma occur in which part of the body organ?
Festus Reply
Lungs
NICHOLAS
lungs
marian
heart
Cynthia
can you explain to me why you say heart is the answer
marian
what is hemoglobin
Lucky Reply
a protein that carries oxygen, it is found within red blood cells
EmmyOrangePeel
Apart from endocrinology, what other alternative should be done to a D/m client
Deborah Reply
good day please what is excretion
Blaise Reply
excretion is the process by which metabolic wastes and other none-useful materials are eliminated from an organism.
Petrene
its the process of eliminating or removal of metabolic waste and non useful materials from an organ.
eyong
Its the removal of metabolic waste product from the body
Boo
it is the removal from the body of living organisms, toxic waste products of metabolism which if allowed to accurate can cost harm to the body..
Renzo
how are protons formed?
Brian Reply
What is isotopes?
Brian
isotopes is a phenomenon in which atoms of the same element has the same atomic number but different mass number due to a difference in their neutron number or relative abundance in the earth crust
Efosa
give me more mcqs there r just two mcqs
Salma Reply
Define the term Disease in one paragraph.
Sutanya Reply
Disease is any disorders that counteract with the normal performance of the immune system.
Agyekum
what is cell?
Colota
Cell is the basic unit of life It is the building block of an organism
Agyekum
do agree with me that if someone needs to truely be a biologist he should be mathematician physicist & chemist in oreder to acheive the level #biologist
Assila
what is a protoplasm?
Colota
colourless living part of the cell
Jenny
what is protoplasm made up of?
Jack
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Digital signal processing - dsp. OpenStax CNX. Jan 06, 2016 Download for free at https://legacy.cnx.org/content/col11642/1.38
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Digital signal processing - dsp' conversation and receive update notifications?

Ask