# Convolution Integral Explained - Understand Fundamentals

• barksdalemc
In summary, convolution is a mathematical operation used to find the density of the sum of two random variables. It involves finding the repartition function and differentiating it, resulting in the convolution of the two individual densities. This can provide insights into the relationship between the two random variables.
barksdalemc
Can someone explain convolution to me. I have read three different books and gone to office hours and am not getting the fundamentals.

In what context? Do you mean you don't understand some of the "applications"?

I'm trying to understand in the context of probability distributions. What the convolution of the sum of two random variables represents.

Oh.

I never really took time to ponder about this. The way it was presented to me was that the convolution appeared kind of coincidentally:

We set out to find the density f of Z=X+Y by finding it's repartition function F and then differentiating it. So we proceed from definition

$$F_{Z}(z)=P(X+Y<z)=\int_{-\infty}^{+\infty}\int_{-\infty}^{z-y}f_X(x)f)Y(y)dxdy=\int_{-\infty}^{+\infty}F_X(z-y)f_Y(y)dy$$

This is the convolution $F_X$ and $f_Y$. The density of Z is found simply by differentiating $F_Z$ wrt z and it gives the convolution of $f_X$ and $f_Y$.

There is probably a way to understand something from this and gain some insights about the relation btw the sum of two random variables.

Let me know if you find something interesting.

## 1. What is a convolution integral?

A convolution integral is a mathematical operation that combines two functions to produce a third function. It is commonly used to represent the output of a system when given an input, and is often used in signal processing and image processing applications.

## 2. How is a convolution integral calculated?

A convolution integral is calculated by multiplying one function by a reversed and shifted version of the other function, and then integrating the result over the entire range of the variables. This process is repeated for every possible shift of the functions, and the resulting values are added together to produce the final output function.

## 3. What are the applications of convolution integrals?

Convolution integrals have a wide range of applications, particularly in signal and image processing. They are used to model linear systems, such as filters and detectors, and are also useful in solving differential equations and analyzing the behavior of complex systems.

## 4. How is convolution different from correlation?

While convolution and correlation are similar operations, they have different applications. Convolution is used to find the output of a system given an input, while correlation is used to measure the similarity between two signals. Additionally, convolution involves multiplying one function by a reversed and shifted version of the other, while correlation only involves shifting the functions.

## 5. Can convolution integrals be applied to non-linear systems?

No, convolution integrals can only be applied to linear systems. This is because they rely on the principle of superposition, which states that the output of a system is equal to the sum of the outputs produced by each individual input. Non-linear systems do not follow this principle and therefore cannot be represented using convolution integrals.

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