Prove the convolution of f and g

Thread starter abstracted6
Start date Aug 2, 2012
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Convolution

In summary, the convolution of two functions f and g is a mathematical operation where the two functions are combined to create a new function. It is represented as f * g or f ∗ g and has significant applications in signal and image processing, as well as other fields of science. The operation can be visualized using a convolution diagram and has real-life examples in image and audio processing, physics, and economics.

Aug 2, 2012

abstracted6

This was the bonus question on my test, I couldn't really figure out how to begin.

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Aug 2, 2012

LCKurtz

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Start by writing the Fourier transform of$$
\int_{-\infty}^{\infty}f(x-y)g(y)dy$$
It will be an integral of that, so it will be a double integral. See if you can find a change of variables that allows you to separate the integrals.

Related to Prove the convolution of f and g

1. What is the definition of convolution of f and g?

The convolution of two functions f and g is a mathematical operation that combines the two functions to create a new function. It is defined as the integral of the product of the two functions, where one of the functions is reversed and shifted. This operation is commonly used in signal processing, image processing, and other fields of science.

2. How is the convolution operation represented mathematically?

The convolution of two functions f and g is represented as f * g or f ∗ g.

3. What is the significance of convolution in science?

The convolution operation is widely used in various fields of science, such as physics, engineering, and mathematics. It is used to model real-world systems, filter signals, and analyze data. It also has applications in image and audio processing, pattern recognition, and solving differential equations.

4. Can you prove the convolution of f and g using a visual representation?

Yes, the convolution operation can be visualized using a graphical representation called a convolution diagram. This diagram shows how the two functions f and g are multiplied and integrated to create the resulting function, which is the convolution of f and g.

5. What are some real-life examples of convolution?

Convolution is commonly used in image processing, such as blurring and edge detection. It is also used in audio processing for effects like reverb and echo. In physics, it is used to model systems like heat diffusion and wave propagation. In economics, it is used to model stock prices and predict market trends.