On a Solution to the Second-Order Phase-Locked Loop

A solution of the stationary, mod-2π-reduced phaseerror probability density function for the second-order phase-lock loop is derived, based on a mean square fit to a certain unknown conditional expectation term. The resulting phase variance obtained from this density function yields an error that is negligible for
\sigma_{\phi}^{2} \leq 1
and an error no greater than 0.06 rad
²
for signal-to-noise ratios (SNRs)≥ 2 dB by comparison with actual measurements, and it appears to be the most accurate to date. The usefulness of the result lies in the fact that an accurate estimate of the phase-error variance for arbitrary parameter values is easily obtained without the use of a computer.