Editing Costas loop (section)

=== In the phase-frequency domain ===
[[File:Costas-phase-pll-ics-high-res.png|thumb|left|Equivalent phase-frequency domain model of Costas loop]]
[[File:PLL trainsient process phase domain.svg|thumb|left|VCO input for phase-frequency domain model of Costas loop]]
In the simplest case, when
: <math>\begin{align}
  f_{ref}\big(\theta_{ref}(t)\big) = \cos\big(\omega_{ref} t\big),\ 
    f_{vco}\big(\theta_{vco}(t)\big) &= \sin\big(\theta_{vco}(t)\big) \\
  f_{ref}\big(\theta_{ref}(t)\big)^2 f_{vco}\left(\theta_{vco}(t)\right)
    f_{vco}\left(\theta_{vco}(t) - \frac{\pi}{2}\right) &=
    -\frac{1}{8}\Big(
       2\sin(2\theta_{vco}(t)) +
        \sin(2\theta_{vco}(t) - 2\omega_{ref} t) +
        \sin(2\theta_{vco}(t) + 2\omega_{ref} t)
     \Big)
\end{align}</math>

The standard engineering assumption is that the filter removes the upper sideband frequency from the input but leaves the lower sideband without change. Thus it is assumed that the VCO input is <math>\varphi(\theta_{ref}(t) - \theta_{vco}(t)) = \frac{1}{8}\sin(2\omega_{ref} t - 2\theta_{vco}(t)).</math> This makes a Costas loop equivalent to a [[phase-locked loop]] with [[phase detector characteristic]] <math>\varphi(\theta)</math> corresponding to the particular waveforms <math>f_{ref}(\theta)</math> and <math>f_{vco}(\theta)</math> of the input and VCO signals. It can be proved that filter outputs in the time and phase-frequency domains are almost equal.<ref>
{{cite journal
 | title   = Differential equations of Costas loop
 | url = http://www.math.spbu.ru/user/nk/PDF/2012-DAN-Nonlinear-analysis-Costas-loop-PLL-simulation.pdf
 |first1= G. A. |last1=Leonov
 |first2=N. V.  |last2=Kuznetsov
 |first3=M. V.  |last3=Yuldashev
 |first4=R. V.  |last4=Yuldashev
 |journal = Doklady Mathematics
 |volume  = 86
 |issue  = 2
 |pages   =723–728
 |date=August 2012
 |doi=10.1134/s1064562412050080
| s2cid = 255276607
 }}</ref><ref>{{cite journal
 |title   = Analytical method for computation of phase-detector characteristic
 |url     = http://www.math.spbu.ru/user/nk/PDF/2012-IEEE-TCAS-Phase%20detector-characteristic-computation-PLL.pdf
 |last1   = Leonov
 |first1  = G. A.
 |last2   = Kuznetsov
 |first2  = N. V.
 |last3   = Yuldashev
 |first3  = M. V.
 |last4   = Yuldashev
 |first4  = R. V.
 |journal = IEEE Transactions on Circuits and Systems Part II
 |volume  = 59
 |issue   = 10
 |pages   = 633–637
 |year    = 2012
 |doi     = 10.1109/tcsii.2012.2213362
|s2cid = 2405056
 }}{{Dead link|date=July 2019 |bot=InternetArchiveBot |fix-attempted=yes }}</ref><ref>
{{cite journal
 | title   = Nonlinear dynamical model of Costas loop and an approach to the analysis of its stability in the large
 | last1=Leonov    | first1=G. A.
 | last2=Kuznetsov | first2=N. V.
 | last3=Yuldashev | first3=M. V.
 | last4=Yuldashev | first4=R. V.
 | journal   = Signal Processing
 | volume    = 108
 | pages     = 124–135
 | year      = 2015
 | publisher = Elsevier
 | doi       = 10.1016/j.sigpro.2014.08.033
| url=https://jyx.jyu.fi/bitstream/123456789/50667/1/leonovetal1s2.0s0165168414003971main.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://jyx.jyu.fi/bitstream/123456789/50667/1/leonovetal1s2.0s0165168414003971main.pdf |archive-date=2022-10-09 |url-status=live
 | doi-access=free
 | bibcode=2015SigPr.108..124L }}</ref>

Thus it is possible<ref>{{cite book
 |last1=Kuznetsov |first1=N. V.
 |last2=Leonov |first2=G. A.
 |last3=Neittaanmaki |first3=P.
 |last4=Seledzhi |first4=S. M.
 |last5=Yuldashev |first5=M. V.
 |last6=Yuldashev |first6=R. V.
 |title=2012 IEEE 4th International Conference on Nonlinear Science and Complexity (NSC) |chapter=Nonlinear mathematical models of Costas Loop for general waveform of input signal |number = 6304729
 |pages =75–80
 |year = 2012
 |publisher = IEEE Press
 |doi = 10.1109/NSC.2012.6304729
 |isbn  = 978-1-4673-2703-9
|s2cid=5812970 }}</ref> to study the simpler [[Autonomous system (mathematics)|autonomous system]] of differential equations
:<math>\begin{align}
            \dot{x} &= Ax + b\varphi(\Delta\theta), \\
 \Delta\dot{\theta} &= \omega_{vco}^{free} - \omega_{ref} + K_{vco}c^*x, \\
       \Delta\theta &= \theta_{vco} - \theta_{ref}.
\end{align}</math>.

The [[Krylov–Bogoliubov averaging method]] allows one to prove that solutions of non-autonomous and autonomous equations are close under some assumptions.
Thus, the Costas loop block diagram in the time domain can be asymptotically changed to the block diagram on the level of phase-frequency relations.

The transition to the analysis of an autonomous dynamical model of the Costas loop (in place of the non-autonomous one) allows one to overcome the difficulties related to modeling the Costas loop in the time domain, where one has to simultaneously observe a very fast time scale of the input signals and slow time scale of signal's phase. This idea makes it possible<ref>
{{cite journal
 |title  = Nonlinear model of the optical Costas loop: pull-in range estimation and hidden oscillations
 |last1  = Kuznetsov |first1=N. V.
 |last2  = Leonov |first2=G. A.
 |last3=Seledzhi |first3=S. M.
 |last4=Yuldashev |first4=M. V.
 |last5=Yuldashev |first5=R. V.
 |journal =  IFAC-PapersOnLine
 |volume = 50
 |pages = 3325–3330
 |year = 2017
 |publisher = ELSEVIER
 |issn = 2405-8963
 |doi = 10.1016/j.ifacol.2017.08.514
|doi-access=free
 }}</ref> to calculate core performance characteristics - [[Phase-locked loop ranges|hold-in, pull-in, and lock-in ranges]].