In a signals and systems class in engineering school, I was shown a proof that a sinusoidal signal goes through a linear time-invariant (LTI) system undistorted; its shape and frequency are not changed - only its amplitude and phase (potentially) are.
In other words, if the input is of the form $v_i = V\sin\left(\omega t + \phi\right)$, then in general the output is of the form $v_o = V^{\prime}\sin\left(\omega t + \phi^\prime\right)$.
This was an "aha! moment" for me in understanding the usefulness and power of the Fourier series/transform.
Now I wonder, however, is a sinusoidal signal the only type of (periodic) signal that goes through an LTI system undistorted?
- If so, I would love to be directed to a proof, and to hear any insight about why this is so.
- If not, is it possible to define a new transform based on the alternative kernel?