Eigenvalue and transfer function (8)

Last time I use sin and cos, but this relationship becomes simpler if we use Euler's formula.

Let's apply the same operator T.

Wow again. This is also eigenfunction of operator T.

This function is based on trigonometric functions. Therefore, we use these trigonometric function as the basis of the frequency domain analysis. Eigenvalues show us a brief overview of the operation and its function.

Assume we have an input x and an output y, operator T is applied to the input x, then the result is the y. If eigenvalue exists, we could write it as the following.

This means, the input is transfered to the output and how much transfered is λ. Therefore, signal processing people call this λ as a transfer function. Why it is called function? λ looks like a constant. Usually, λ is not a function of input x, but it usually has some parameter, means this is a function. For example, in the former equation, λ is not a function of x, but a function of ω.  In signal processing, x is usually time t and ω is frequency.

This means, a transfer function in signal processing is eigenvalue of linear algebra. How great it is!

This is an overview of recently I understand about eigenvalue, eigenvector, eigenfunction, and transfer function. I hope this also helps someone.

Acknowledgements

I wrote this article, but, actually first one month I didn't understand this at all. I thank Alexander B. who is so patient and answered my stupid and similar questions every time.