Comparator

2014-01-09T17:45:46Z

12.226.68.244: /* Op-amp voltage comparator */

A '''Costas loop''' is a [[phase-locked loop]] based circuit which is used for [[carrier wave|carrier]] phase [[Carrier recovery|recovery]] from suppressed-carrier [[modulation]] signals, such as from double-[[sideband]] suppressed carrier signals. It was invented by [[John P. Costas (engineer)|John P. Costas]] at [[General Electric]] in the 1950s. Its invention was described as having had "a profound effect on modern digital communications".
The primary application of Costas loops is in wireless receivers. Its advantage over the PLL-based detectors is that at small deviations the Costas loop error voltage is sin(2(''θi''−''θf'')) vs sin(''θi''−''θf''). This translates to double the sensitivity and also makes the Costas loop uniquely suited for tracking doppler-shifted carriers esp. in [[OFDM]] and [[GPS receiver]]s
<ref>{{cite journal
|url = http://ieeexplore.ieee.org/iel5/5/22249/01037569.pdf?arnumber=1037569
|title = Introduction to `Synchronous Communications', A Classic Paper by John P. Costas
|author = D. Taylor
|journal = Proceedings of the IEEE
|volume = 90
|issue = 8
|pages = pp. 1459–1460
|year = 2002
|month = August
}}</ref>

== Classical implementation ==

{| class="wikitable"
|[[File:Costas loop before sync.svg|340px|thumb|Costas loop, before synchronization]]
|[[File:Costas loop after sync.svg|290px|thumb|Costas Loop after synchronization]]
|}

{| class="wikitable"
|[[File:Costas loop signals before sycnhronization.svg|290px|thumb|Carrier and VCO signals before synchronization]]
|[[File:Costas loop trainsient process.svg|160px|thumb|VCO input during synchronization]]
|[[File:Costas loop after synchronization.svg|290px|thumb|Carrier and VCO signals after synchronization]]
|}

In the classical implementation of a Costas loop,<ref>{{cite journal
|url = http://rfdesign.com/images/archive/0102Feigin20.pdf
|format=PDF|title = Practical Costas loop design
|author = Jeff Feigin
|date = January 1, 2002 |journal = RF Design
|pages = pp. 20–36
}}</ref> a local [[voltage-controlled oscillator]] (VCO) provides [[Quadrature phase|quadrature]] outputs, one to each of two [[phase detector]]s, ''e.g.'', [[Product detector|product detectors.]] The same phase of the input [[signal (information theory)|signal]] is also applied to both phase detectors and the [[output]] of each [[phase detector]] is passed through a [[low-pass filter]]. The outputs of these low-pass filters are inputs to another phase detector, the output of which passes through noise-reduction filter before being used to control the voltage-controlled oscillator. The overall loop response is controlled by the two individual low-pass filters that precede the third phase detector while the third low-pass filter serves a trivial role in terms of gain and phase margin.

== Mathematical models of Costas loop ==

=== Model of Costas loop in the time domain ===
[[File:Costas loop general siangls.svg|thumb|left|Time domain model of 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 noise-reduction filter.
Carrier and VCO signals are periodic oscillations <math>f^{1,2}(\theta(t))</math> with high-frequencies <math>\dot\theta^{1,2}(t)</math>.
Block <math>-90^{o}</math> shifts phase of VCO signal by <math>-\frac{\pi}{2}</math>.
Block <math>\bigotimes</math> is an [[Analog multiplier]].

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

The model of [[voltage-controlled oscillator]] is usually assumed to be linear
:<math>
\begin{array}{ll}
\dot\theta^2(t) = \omega^2_{free} + LG(t),& t \in [0,T],
\end{array}
</math>
where <math>\omega^2_{free}</math> is a free-running frequency of voltage-controlled oscillator and <math>L</math> is an oscillator gain. Similar it is possible to consider various nonlinear models of VCO.

Suppose that the frequency of master generator is constant
<math>
\dot\theta^1(t) \equiv \omega^1.
</math>
Equation of VCO and equation of filter yield
:<math>
\begin{array}{ll}
\dot{x} = Ax + bf^1(\theta^1(t))f^2(\theta^2(t)),& \dot\theta^2 = \omega^2_{free} + Lc^*x.
\end{array}
</math>

The system is nonautonomous and rather difficult for investigation.

=== Model of Costas loop in phase-frequency domain ===

[[File:Costas loop pd model.svg|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{array}{l}
f^1\big(\theta^1(t)\big)=\cos\big(\omega^1 t\big),
f^2\big(\theta^2(t)\big)=\sin\big(\omega^2 t\big)
\\
f^1\big(\theta^1(t)\big)^2
f^2\big(\theta^2(t)\big)
f^2\big(\theta^2(t) - \frac{\pi}{2}\big)
=
-\frac{1}{8}\Big(
2\sin(2\omega^2 t)
+\sin(2\omega^2 t - 2\omega^1 t)
+\sin(2\omega^2 t + 2\omega^1 t)
\Big)
\end{array}
</math>
standard engineering assumption is that the filter removes the upper sideband
with frequency from the input but leaves the lower sideband without change.
Thus it is assumed that VCO input is <math>\varphi(\theta^1(t) - \theta^2(t))=\frac{1}{8}\sin(2\omega^1-2\omega^2)</math>.
This makes Costas loop equivalent to [[Phase-Locked Loop]] with [[phase detector characteristic]] <math>\varphi(\theta)</math> corresponding to the particular waveforms <math>f^1(\theta)</math> and <math>f^2(\theta)</math> of input and VCO signals. It can be proved, that inputs <math>g(t)</math> and <math>G(t)</math> of VCO for phase-frequency domain and time domain models 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
|author = G.A. Leonov, N.V. Kuznetsov, M.V. Yuldashev, R.V. Yuldashev
|journal = Doklady Mathematics
|volume = 86
|issue = 2
|pages = pp. 723–728
|year = 2012
|month = August
}}</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
|author = Leonov G.A., Kuznetsov N.V., Yuldashev M.V., Yuldashev R.V.
|journal = IEEE Transactions on Circuits and Systems Part II
|volume = 59
|issue = 10
|pages = pp. 633–637
|year = 2012
}}</ref>
<ref>{{cite journal
|title = Nonlinear analysis of the Costas loop and phase-locked loop with squarer
|url = http://www.math.spbu.ru/user/nk/PDF/2009-SIP-Nonlinear-analysis-Costas-loop-PLL-squarer.pdf
|author = N.V. Kuznetsov, G.A. Leonov, and S.M. Seledzhi
|journal = Proceedings of Eleventh IASTED International Conference Signal and Image Processing
|volume = 654
|pages = pp. 1–7
|year = 2009
|publisher = ACTA Press
}}</ref>

Thus it is possible
<ref>{{cite journal
|title = Nonlinear mathematical models of Costas Loop for general waveform of input signal
|author = Kuznetsov N.V., Leonov G.A., Neittaanmaki P., Seledzhi S.M., Yuldashev M.V., Yuldashev R.V.
|journal = IEEE 4th International Conference on Nonlinear Science and Complexity, NSC 2012 - Proceedings
|number = 6304729
|pages = pp. 75–80
|year = 2012
|publisher = IEEE Press
|doi = 10.1109/NSC.2012.6304729
}}</ref>
to study more simple autonomous system of differential equations
:<math>
\begin{array}{ll}
\dot{x} = Ax + b\varphi(\Delta\theta), &
\Delta\dot\theta = \omega^2_{free} - \omega^1 + Lc^*x,
\\
\Delta\theta = \theta^2 - \theta^1. &
\end{array}
</math>
Well-known [[Krylov–Bogoliubov averaging method]] allows one to prove that solutions of nonautonomous and autonomous equations are close under some assumptions.
Thus the block-scheme of Costas Loop in the time space can be asymptotically changed to the block-scheme on the level of phase-frequency relations.

The passage to analysis of autonomous dynamical model of Costas loop (in place of the nonautonomous one)
allows one to overcome the difficulties, related with modeling Costas loop in time domain where one has to simultaneously observe very fast time scale of the input signals and slow time scale of signal's phase.

==References==
<references/>
*{{FS1037C}}

[[Category:Oscillators]]
[[Category:Communication circuits]]

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