Convolution of a gaussian function and a hole

In summary: Yes, I understand the relationship between convolution algebra and C_0( \mathbb R^n). Convolution is a function that takes two inputs and produces an output. The input that is not convolved is called the "kernel". The convolution algebra is a set of algebraic structures that help us understand what a convolution is and how it works. The Fourier transform is a continuous mapping from the convolution algebra to the real line and is a continuous but not injective function.
  • #1
Newser
5
0
Hello,

I want to do the convolution of a gaussian function and a hole. If I want to use Fourier transform which functions should I use? Can I use rms? I want to calculate the spot size of a gaussian signal after a circular aperture.

Thanks!
 
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  • #2


Newser said:
Hello,

I want to do the convolution of a gaussian function and a hole. If I want to use Fourier transform which functions should I use? Can I use rms? I want to calculate the spot size of a gaussian signal after a circular aperture.

Thanks!

rms has nothing to do with either convolutions or Fourier transforms. Use the Fourier transform for whatever function you mean by a "hole". Fourier transform of a Gaussian is also a Gaussian.
 
  • #3


Thanks for the reply. Yes, I know rms has nothing to do with Fourier transform, I was asking if I could use it instead...
 
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  • #4


Newser said:
Thanks for the reply. Yes, I know rms has nothing to do with Fourier transform, I was asking if I could use it instead...

Instead of what ?

It has nothing to do with the Fourier transform. It has nothing to do with convolution.

You question involved convolution.

Do you understand what convolution is ? And do you understand the relationship between the convolution algebra [tex] L^1(\mathbb R^n)[/tex] and the algebra [tex]C_0( \mathbb R^n)[/tex] ?

Both are Banach algebras and the Fourier transform is a continuous homomorphism from [tex] L^1(\mathbb R^n)[/tex] to [tex]C_0( \mathbb R^n)[/tex] which is injective but not surjective. Thus the Fourier transform can sometimes be used to calculate convolutions that are difficult to calculate directly. It takes convolutions (hard to understand) to pointwise multiplication (easy to understand).
 
  • #5


Yes, I do understand what a convolution is, I think you are the one not understanding my question. I am not asking how to calculate a convolution, obviously if I am talking about Fourier transform I know how to do this...I was wondering about the most suited way to model this physical phenomenon (beam after aperture). But I guess I shouldn't have posted in this section of the forum, I thought I would found multi-skilled people...
 
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Related to Convolution of a gaussian function and a hole

1. What is a convolution?

A convolution is a mathematical operation that combines two functions to produce a third function. It is often used in signal processing and image processing to analyze the relationship between two signals.

2. What is a gaussian function?

A gaussian function, also known as a normal distribution, is a mathematical function that represents a bell-shaped curve. It is commonly used in statistics to describe the distribution of a set of data.

3. What is a "hole" in this context?

In this context, a "hole" refers to a discontinuity or gap in the gaussian function. This can occur when the function is defined over a limited range or when there is a sudden change in the function's value.

4. Why would you want to convolve a gaussian function and a hole?

Convolution of a gaussian function and a hole can be useful in image processing to create a blurring effect. It can also be used in signal processing to filter out specific frequencies or noise from a signal.

5. How is the convolution of a gaussian function and a hole calculated?

The convolution is calculated by multiplying the two functions together, shifting one of the functions by a specified amount, and then integrating the product over the entire range of the shifted function. This process is repeated for different values of the shift, resulting in a new function that represents the convolution of the two original functions.

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