Editing Costas loop (section)

=== In the time domain ===
[[File:Costas-non-equal-freq-high-res.png|thumb|left|Time domain model of BPSK Costas loop]]
In the simplest case <math>m^2(t) = 1</math>. Therefore, <math>m^2(t) = 1</math> does not affect the input of the noise-reduction filter.
The carrier and [[voltage-controlled oscillator]] (VCO) signals are periodic oscillations <math>f_{ref,vco}(\theta_{ref,vco}(t))</math> with high-frequencies <math>\dot\theta_{ref,vco}(t)</math>.
The block <math>\bigotimes</math> is an [[analog multiplier]].

A [[linear filter]] can be described mathematically by a system of linear differential equations:
:<math>\begin{array}{ll}
\dot x = Ax + b u_{d}(t),& u_{LF} = c^*x.
\end{array}
</math>
where <math>A</math> is a constant matrix, <math>x(t)</math> is a state vector of the filter, <math>b</math> and <math>c</math> are constant vectors.

The model of a VCO is usually assumed to be linear:
:<math>
 \begin{array}{ll}
 \dot\theta_{vco}(t) = \omega^{free}_{vco} + K_{vco} u_{LF}(t),& t \in [0,T],
 \end{array}
</math>
where <math>\omega^{free}_{vco}</math> is the free-running frequency of the VCO and <math>K_{vco}</math> is the VCO gain factor. Similarly, it is possible to consider various nonlinear models of VCO.

Suppose that the frequency of the master generator is constant
<math>
 \dot\theta_{ref}(t) \equiv \omega_{ref}.
</math>
Equation of VCO and equation of filter yield
:<math>
\begin{array}{ll}
 \dot{x} = Ax + bf_{ref}(\theta_{ref}(t))f_{vco}(\theta_{vco}(t)),& \dot\theta_{vco} = \omega^{free}_{vco} + K_{vco}c^*x.
\end{array}
</math>

The system is [[Non-autonomous system (mathematics)|non-autonomous]] and rather tricky for investigation.