Which signals or set of signals is the most predictable or in other words most self-correlated (autocorrelation)?

I think one of them is $x(t)=c$, for all $t$, where $c$ is a scalar constant. The other one can be $x(t)=c$, for $t=t_0$ and $x(t)=0$, otherwise (a discrete signal with only one value). $t_0$ is a constant.

Here the book says: "the most predictable signal" is the sum of discrete cosine functions with random phases. Why? My two examples above seem to me more predictable!


Sums of discrete cosines are good candidates because they're bound to repeat after a finite number of samples. In fact, anything that repeats after some finite number of samples is the most predictable thing in the world because you can predict the entire thing from just a handful of samples. In fact, the converse of this statement is true, i.e. anything that repeats after a finite number of samples can be written down as a sum of discrete cosines with different phases. In fact, so can your own candidate: $$c = \sum_{k=0}^{\infty}c_k\cos(\omega_k + \phi_k)$$ where $c_k=c\delta(k)$, $\omega_0=0$, $\phi_0=0$. In other words a constant signal is just a cosine of zero frequency. So there's no tautology here.

You may say of course, that your example is still better, but if we're trying to establish classes of predictability, then both your example and the generic form fall into the same class. I don't see a contradiction here.

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