Alias Lock Behavior of Sampled-Data Costas Loops

The potential false lock problem associated with the sampled data type of Costas loop implementation is addressed in this paper. The term "alias lock" is used here to differentiate this type of false lock behavior from the data sideband false lock behavior of analog Costas loops [1]-[8]. It is shown that the sampled-data version of the conventional Costas loop, sampled at a rate
1/T_{s}
, can alias lock at frequencies that are multiples of
1/2T_{s}
away from the carrier frequency. It is also shown that the alias lock problem of the sampled data version of the Costas loop with hard-limited in-phase channel is further Compounded by the potential occurrence of false lock frequencies at rational multiples of
1/2T_{s}
, away from the carrier. The false lock
S
-curves of
I-Q
loops and decision-directed
I-Q
loops are investigated in detail, with and without additive noise. Close agreement between theory and earlier experimental results is also demonstrated.