Costas Loop Behavior

The Costas loop can be considered as a special version of the phase-locked loop (PLL). The PLL is designed to generate an electrical signal (voltage), the phase of which is automatically tuned to the phase of the input (reference) signal. Dynamic behavior of the PLL and the Costas loop has been described extensively in the literature [1–11], and a number of key parameters has been defined that describe its lock-in and lock-out characteristics. When the PLL is initially out of lock, two different types of acquisition processes can occur, either the so-called lock-in process or the so-called pull-in process. The first of those is a fast process, i.e. the acquisition takes place within at most one beat note of the difference between reference

The time required to get acquisition is called pull-in time T P . In case of the PLL all these acquisition parameters can be approximated by characteristic parameters of the PLL, i.e. from natural frequency ω n and damping factor ζ.

To the authors knowledge such acquisition parameters have not been analytically derived for the different types of Costas loops. It seems that most authors only described the static properties of the Costas loop such as the derivation of the phase error in the locked state and the like. Based on methods developed earlier for the PLL, the authors could now derive similar expressions for all relevant acquisition parameters of the Costas loop. This enables the designer to determine the lock-in and pull-in ranges, and to estimate the duration of the corresponding processes.

Because the systems considered are highly non linear, exact computation of such parameters is very difficult or even impossible. It has shown necessary therefore to introduce a number of simplifications. This implies that the obtained results are only approximations, in some cases rather crude

u 1 (t) = m 1 (t) cos(ω 1 t) + m 2 sin(ω 1 t), where m 1 and m 2 can both have two equal and opposite values +c and −c. It is obvious that in both cases the input signal is a real quantity. In the following these two types of Costas loop will be referred to as

“conventional Costas loops”.

Much later, Costas loops have been developed that operate not on real signals, but on pre-envelope signals [16]. These types of Costas loops will be referred to as “modified Costas loop” in the following sections. The block diagram shown in Figure 5.2 explains how the pre-envelope signal is obtained. The

2real input signal u 1 (t) is applied to the input of a Hilbert transformer [2], [5]. The output of the Hilbert transformer û 1 (t) is considered to be the imaginary part of the pre-envelope signal, i.e the pre-envelope signal is obtained from u +

1 (t) = u 1 (t) + j û 1 (t).

The Costas loops operating with pre-envelope signals will be referred to as “modified Costas loops”, cf. sections 4 and 5. Because there are different types of Costas loops the acquisition parameters must be derived separately for each of these types. This will be performed in the following sections. In order to