False lock in Costas loops

Results are given on the phenomenon of data-related false lock in a Costas loop which contains a perfect integrator in its loop filter. The input carrier to the loop is assumed to be biphase-modulated with a T´ periodic binary sequence constructed from Manchester symbols; there are p symbols per period. As a result, the false locked loop is described by a nonlinear system with periodic coefficients. A loop gain parameter delta appears in this system, and it is used as a perturbation parameter. A constant omega /sub f/ also appears; this constant represents the closed-loop frequency error in the false locked loop. Under some general conditions on modulation m it is shown that bifurcation occurs at delta =0 in this equation for each value of omega /sub f/ k( pi /T´), k=1, 2, . . . For each of these values the nonlinear system is shown to have four distinct periodic solutions. The loop has a false-locked state which corresponds to each solution. Finally, the theory is applied to a simple loop, and the results are displayed graphically.<>