2D delta function fourier transform

In summary, the conversation discusses using the delta function and its properties to find the Fourier transform of f(x,y) = DeltaFunction(y - x*tan(theta)). The approach involves using Fubini's theorem and the sifting property of the delta function to express the double integral as a single integral.
  • #1
quasartek
2
0

Homework Statement


Given f(x,y) = DeltaFunction(y - x*tan(theta))

a) Plot function.
b) Take Fourier transform.
c) Plot resulting transform.


Homework Equations


Delta function condition non-zero condition DeltaFunction(0) = Infinity
Sifting property of delta functions


The Attempt at a Solution


I am using Mathematica and can plot a 1D version, DeltaFunction(x), however, I am having trouble extending it to 2D for part a. Part b, I tried using the sifting property integral(deltafunction(y-x*tan(theta))*f(x,y)=f(y-tan(theta)) but could not progress after that point.

Any help or suggestions would be greatly appreciated.
 
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  • #2
Remember that one of the defining properties of the delta function is that
[tex]\int_{-\infty}^\infty f(x)\delta(x)\,dx=f(0)[/tex]​
for sufficiently chosen functions f. Using http://mathworld.wolfram.com/FubiniTheorem.html" [Broken], you can show that this is also true for double integrals.

Can you think of a way you might use this to find your Fourier transform?
 
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  • #3
I now tried to incorporate that property and Fubini's theorem
where

DoubleIntegral[ f(x,y)*Delta(y - x*tan(theta)), x,y,-Inf,Inf] = f(0,0) and plugging f(0,0) into Delta(0-0) gives Delta(0,0) which is Infinity. The Fourier Transform of that is 1?
 
  • #4
quasartek said:
I now tried to incorporate that property and Fubini's theorem
where

DoubleIntegral[ f(x,y)*Delta(y - x*tan(theta)), x,y,-Inf,Inf] = f(0,0)

No,
[tex]\iint_{\mathbb{R}^2} f(x,y)\delta(y-x\tan\theta)\,dA[/tex]​
equals the sum of all the f(x,y)'s with [tex]y-x\tan\theta=0[/tex]. Try expressing that as a (single) integral.
 

1. What is a 2D delta function fourier transform?

A 2D delta function fourier transform is a mathematical tool used in signal processing and image analysis to represent a two-dimensional signal or image in the frequency domain. It essentially decomposes a signal into its constituent frequencies, allowing for analysis and manipulation in the frequency domain.

2. How is a 2D delta function fourier transform calculated?

A 2D delta function fourier transform is calculated by taking the two-dimensional signal or image and multiplying it by a complex exponential function. This multiplication is then integrated over the entire domain of the signal or image. The result is a complex number representing the amplitude and phase of each frequency component present in the signal or image.

3. What is the significance of the delta function in a 2D delta function fourier transform?

The delta function, denoted as δ(x,y), is used to represent a point or location in the two-dimensional plane. In the context of a fourier transform, it is used to locate the specific frequencies present in the signal or image. This allows for a precise representation of the signal or image in the frequency domain.

4. Can a 2D delta function fourier transform be applied to non-rectangular signals or images?

Yes, a 2D delta function fourier transform can be applied to any two-dimensional signal or image, regardless of its shape or size. However, the transform result may be more complex and difficult to interpret for non-rectangular signals or images.

5. What are the practical applications of a 2D delta function fourier transform?

A 2D delta function fourier transform has a wide range of practical applications, such as image filtering, noise reduction, and feature extraction in image analysis. It is also commonly used in signal processing for tasks such as audio and video compression, pattern recognition, and data encryption.

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