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Eigenvalue and transfer function (7)

Eigenvalue and transfer function

In the Hamming's book, he repeats to mention about the merit of using trigonometric function as a basis in signal processing. Unfortunately, that is not the main point of this blog, therefore, I could not compare it with the other bases. I will just stick to this basis with assuming this is a good one. Let's see the eigenvalue of trigonometric function according to the example of Hamming's book. The first example in his book is
 A sin x + B cos x.
We apply a transformation operation. Then, let's see something is changed or not. If something doesn't change, it will be a eigenvector and we will also see its eigenvalue.

Transform T is a shift operation of the origin of coordinate like T: x → x' + h. Why someone wants to shift the coordinate? For example, signal processing usually doesn't matter when you start to measure the signal sequence. When you started to measure the signal, then, that point is the origin. Usually there is no absolute origin of time. If you want to re-set the time domain, that would be also convenient. The question is what kind of property doesn't change by transform operation. Let's apply this transform operation to the trigonometric function.

where，
A', B' are constants that is independent from x. Wow, after the transformation, the function became the almost the same form. It is a linear combination of sin x and  cos x again.
At the end, this operation has the same form of
A' and  B' looks like eigenvalues and sin and cos looks like eigenvector. (eigenfunction)

I found this point of view is fantastic. How do you think?

Once I was scared my friends talk about eigenfunction, since I don't know what it is. But, you also are not scared this anymore!

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