# What does zero crossing period of a Auto correlation signify?

What does the first and second zero crossing time of the autocorrelation function signify anything for a signal?

How do we get the period of the signal from its autocorrelation? Im trying it for a periodic signal with some low frequency component in it.

• What autocorrelation function? – user17592 Jun 6 '13 at 10:45
• I think it's a reasonable assumption the question is about autocorrelation in signal processing. – Phil Frost Jun 6 '13 at 11:38

For a signal $x(t)$ of finite duration (say nonzero only for $t \in [0,T]$), the (unnormalized) autocorrelation function is $$R_x(\tau) = \int_0^T x(t)x(t-\tau)\,\mathrm dt, ~\tau \geq 0$$ and of course $R_x(\tau) = R_x(-\tau)$ for $\tau < 0$. Since $x(t-\tau)$ is nonzero only when $\tau \in [\tau, \tau+T]$, the lower limit on the integral can be increased to $\tau$. Note that $R_x(\tau) = 0$ for $|\tau| \geq T$. If $t_1 < T$ is the smallest positive real number such that $R_x(t_1) = 0$, then this means that the signals $x(t)$ and $x(t-t_1)$ are orthogonal over the interval $[\tau,T]$, (or over $[0,T]$ if you like).
If $x(t)$ consists of $n \geq 1$ periods of a single-frequency sinusoid, that is, $x(t) = \cos(2\pi nt/T + \theta)$, then $R_x(\tau) = \frac{1}{2}(T-|\tau|)\cos(2\pi n\tau/T)$ for $0 \leq \tau \leq T$ and so the zero-crossings are at times $t_i = \frac{i}{T}, 1 \leq i \leq n$. If$x(t)$ also contains signals other than the single-frequency sinusoid mentioned, there can be other zero-crossings too.