# If an autocorrelation function of a process is a rectangular function, then is the process deterministic?

I would like to clarify one doubt regarding autocorrelation function $R_x (\tau)$ of a process $X(t)$. Suppose $R_x (\tau) = \Pi (\tau)$, where $\Pi (\tau)$ is the rectangular function. Then is the process $X(t)$ deterministic.

• Why do you think it would be deterministic? – Jason R Dec 3 '13 at 4:15
• A friend of mine told that this autocorrelation cannot hold true for a random process. So I was thinking that it would be deterministic. – dexterdev Dec 3 '13 at 4:45
• As Jazzmaniac's answer implies, the rectangular pulse cannot be the autocorrelation function of anything, random process or deterministic signal or whatever. Your friend's statement is correct but incomplete and you are choosing an inapporpriate negation. "OK, the rectangular pulse is not the autocorelation of a random process and so the process must be deterministic" Not so. The rectangular pulse is not the autocorrelation of any process. – Dilip Sarwate Dec 3 '13 at 14:02

## 1 Answer

An autocorrelation function like that is not possible for ordinary signals. The autocorrelation function is the Fourier transform of the power spectral density, which is a strictly positive quantity. However, to get a rectangular function as Fourier transform you need to have negative values. That is in conflict with the strictly positive power spectrum.