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Will use in subsequent modules

The 2D Fourier transform has many uses. I will use the 2D Fourier transform in several future modules involving such diverse topics as:

  • Processing image pixels in the wavenumber domain
  • Advanced steganography (hiding messages in images)
  • Hiding watermarks and trademarks in images

Viewing tip

I recommend that you open another copy of this module in a separate browser window and use the following links to easily find and view the Figureswhile you are reading about them.

Figures

  • Figure 1. A standing wave on a wire.
  • Figure 2. Three example wavenumber spectra.
  • Figure 3. A three-element array with weighted sensors.
  • Figure 4. The 2D wavenumber response of a linear array.
  • Figure 5. Wavenumber response of a two-dimensional array.
  • Figure 6. Image processing in the space domain.

General discussion

Time domain and frequency domain

In my earlier modules on DSP, you learned about the relationship between the time domain and the frequency domain. For example, you learned that time hasonly one dimension. In the real world, time only goes forward.

(In the computer world, we can make it appear that time can also go backwards, but this still constitutes only one dimension.)

The important point is that time can only go forward or backwards. It cannot go sideways.

A one-dimensional Fourier transform

You learned that you can perform a one-dimensional Fourier transform to transform your data from the time domain into the frequency domain. Similarly,you can perform an inverse one-dimensional Fourier transform to transform your data from the frequency domain back into the time domain.

You learned about several characteristics of Fourier transforms. For example, you learned that a Fourier transform is both linear and reversible. You eitherhave learned or you will learn in a future module that convolution in the time domain is equivalent to multiplication in the frequency domain, and thatconvolution in the frequency domain is equivalent to multiplication in the time domain.

You learned that with enough computational power, you can easily transform a given set of data back and forth between these two domains. This makes itpossible to use the domain of your choice to perform a given signal processing operation, even if the results need to be delivered in the other domain.

Time domain data is purely real

Although it is possible to use the Fourier transform to transform a set of complex data from one domain to another domain, real-world time domain data isnot complex data. Rather, it is purely real. Assuming that the data in one domain is always purely real leads to some simplification of the computationalrequirements for performing the Fourier transform. In general, most of the previous DSP modules assumed real data in the time domain and complex data inthe frequency domain.

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir 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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