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I think a better definition of the power spectrum is the following: The power spectrum of $x(t)$ is the Fourier transform of the autocorrelation function of $x(t)$, where $x(t)$ can be either a deterministic power signal, or a wide-sense stationary (WSS) random process. The definition of the autocorrelation function depends on the model for $x(t)$. If $x(...


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While I agree with almost everything @CrisLuengo wrote in his answer, there is one important thing to correct: It is no coincidence that you see a signal appear in the cross correllation between $x$ and $y$. If you look closely you'll see that this signal is exactly the horizontal mirroring (around the origin) of your system's transfer function a.k.a. ...


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