Intuitive understanding of convolution?

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In summary, the conversation discusses the topic of convolutions, which is a mathematical concept used in solving problems related to Fourier transform. The speaker had a negative experience with their adjunct professor in ODEs and is now trying to understand the use and theory behind convolutions. They ask for resources to gain an intuitive understanding of convolutions and are recommended to look into books on Fourier Transform or watch lectures on Stanford's Youtube channel.
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filter54321
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I had a terrible adjunct professor in ODEs and got little or no theory. I'm not in PDEs and my much better professor just (re)introduced convolutions while generalizing the heat equation to Rn - unfortunately it was not a reintroduction for me.

Later chapters in the book deal with transforms, which are, I think "special" convolutions where you mix the subject function with a specific kernel function. Apparently I need to understand this concept.

Any resources for getting an intuitive understanding of what a convolution *is* and why one would want to do such a thing? I played around on Youtube and Wikipedia and I see the mechanics but not the theory.

Thanks
 
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Convolution was first derived somewhere along the progress of when mathematicians were trying to solve the problems associated with Fourier transform.

It would be best if you could find a book on Fourier Transform. You could also find the explanation on Stanford's Youtube channel but I think you would need to start from lecture 1 to understand it.
 

1. What is convolution?

Convolution is a mathematical operation that combines two functions to produce a third function. It is commonly used in signal processing and image processing to analyze and manipulate data.

2. How does convolution work?

Convolution works by multiplying each data point in one function by the corresponding data point in the other function, summing the products, and shifting the resulting function along the x-axis. This process is repeated for each data point, resulting in a new function that represents the combined effects of the two original functions.

3. Why is convolution important in science?

Convolution is important in science because it allows us to analyze and manipulate data in a way that takes into account the relationships and interactions between different data points. It is particularly useful in fields such as signal processing, image processing, and machine learning.

4. What is the difference between discrete and continuous convolution?

Discrete convolution is used for discrete data sets, where the data points are separated by equal intervals. Continuous convolution is used for continuous data sets, where the data points are not necessarily separated by equal intervals. The mathematical equations for the two types of convolution differ slightly, but the basic principles are the same.

5. Can you give an example of how convolution is used in real life?

Convolution is used in many real-life applications, such as digital signal processing, image recognition, and natural language processing. For example, when a voice command is given to a virtual assistant, the audio signal is convolved with a filter to remove background noise and enhance the speech signal for better recognition. In image recognition, convolution is used to identify patterns and features in an image, such as edges and textures, to classify objects.

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