Simple and Exact Closed-Form Expressions for the Expectation of the Linn-Peleg M-PSK Lock Detector

Exact closed-form expressions for the expectation of the Linn-Peleg M-PSK lock detector [1] are derived through usage of Fourier series analysis. It is shown that the expectation can be expressed as a summation of a finite number of terms which are composed of exponentials and polynomials. These results are then used to model the behavior of the lock detector under conditions of frequency-flat slow fading, and closed-form results are derived for Nakagami-m fading. The results derived in this paper thus allow for lock detector threshold values, lock probabilities, and false alarm rates to be determined easily and accurately without the need to perform numerical integrations or any other complicated mathematical operation.