Editing Costas loop (section)

== QPSK Costas loop ==
The classical Costas loop can be adapted to [[Phase-shift keying#Quadrature phase-shift keying (QPSK)|QPSK]] modulation for higher data rates.<ref>{{cite patent
| country = US
| number = 4,085,378
| status = patent
| title  = QPSK demodulator
| pubdate = 1976-11-26
| pridate = 1975-06-11
| invent1 = Carl R. Ryan
| invent2 = James H. Stilwell
| assign1 = Motorola Solutions Inc.
| url  = https://patents.google.com/patent/US4085378A/en
}}</ref>
[[File:QPSK-costas-loop2.png|thumb|left|Classical QPSK Costas loop]]
The input [[phase-shift keying#Quadrature phase-shift keying (QPSK)|QPSK]] signal is as follows
:<math>
  m_1(t)\cos\left(\omega_\text{ref} t\right) + m_2(t)\sin\left(\omega_\text{ref} t\right),
  m_1(t) = \pm 1, m_2(t) = \pm 1.
</math>

Inputs of low-pass filters LPF1 and LPF2 are
:<math>\begin{align}
  \varphi_1(t) &= \cos\left(\theta_\text{vco}\right)\left(m_1(t)\cos\left(\omega_\text{ref} t\right) + m_2(t)\sin\left(\omega_\text{ref} t\right)\right), \\
  \varphi_2(t) &= \sin\left(\theta_\text{vco}\right)\left(m_1(t)\cos\left(\omega_\text{ref} t\right) + m_2(t)\sin\left(\omega_\text{ref} t\right)\right).
\end{align}</math>

After synchronization,
the outputs of LPF1 <math>Q(t)</math> and LPF2 <math>I(t)</math> are used to get demodulated data (<math>m_1(t)</math> and <math>m_2(t)</math>). To adjust the frequency of the VCO to the reference frequency, signals <math>Q(t)</math> and <math>I(t)</math> are limited and cross-multiplied:
:<math>u_d(t) = I(t)\sgn(Q(t)) - Q(t)\sgn(I(t)).</math>

Then the signal <math>u_d(t)</math> is filtered by the loop filter and forms the tuning signal for the VCO <math>u_\text{LF}(t)</math>, similar to BPSK Costas loop. Thus, QPSK Costas can be described<ref>
{{cite journal
 | title = Tutorial on dynamic analysis of the Costas loop
 | last1 = Best      | first1=R. E.
 | last2 = Kuznetsov | first2=N. V.
 | last3 = Leonov    | first3=G. A.
 | last4 = Yuldashev | first4=M. V.
 | last5 = Yuldashev | first5=R. V.
 | journal = Annual Reviews in Control
 | volume = 42
 | pages = 27–49
 | year = 2016
 | publisher = ELSEVIER
 | doi = 10.1016/j.arcontrol.2016.08.003
| arxiv = 1511.04435
 | s2cid = 10703739 }}</ref> by a system of [[ordinary differential equations]]:
:<math>\begin{align}
               \dot{x}_1 &= A_\text{LPF} x_1 + b_\text{LPF}\varphi_1(t),\\
               \dot{x}_2 &= A_\text{LPF} x_2 + b_\text{LPF}\varphi_2(t),\\
                 \dot{x} &= A_\text{LF} x + b_\text{LF}\big(c_\text{LPF}^* x_1\sgn(c_\text{LPF}^* x_2) - c_\text{LPF}^* x_2\sgn(c_\text{LPF}^* x_1)\big),\\
 \dot{\theta}_\text{vco} &= \omega_\text{vco}^\text{free} + K_\text{VCO}\Big(c^*_\text{LF} x + h_\text{LF}\big(c_\text{LPF}^* x_1\sgn(c_\text{LPF}^* x_2) - c_\text{LPF}^* x_2\sgn(c_\text{LPF}^* x_1)\big)\Big).\\
\end{align}</math>

Here <math>A_\text{LPF}, b_\text{LPF}, c_\text{LPF}</math> are parameters of LPF1 and LPF2 and <math>A_\text{LF}, b_\text{LF}, c_\text{LF}, h_\text{LF}</math> are parameters of the loop filter.