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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

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris 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
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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