14.139.38.10: $\theta_{m+1}=\arg\max_{\theta}g(\theta|\theta_m)$ instead of $\theta_{m+1}=\max_{\theta}g(\theta|\theta_m)$

2013-12-23T16:43:35Z

<math> \theta_{m+1}=\arg\max_{\theta}g(\theta|\theta_m) </math> instead of <math> \theta_{m+1}=\max_{\theta}g(\theta|\theta_m) </math>

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A '''phase detector characteristic''' is a function of phase difference describing the output of the [[phase detector]].

For the analysis of [[Phase detector]] it is usually considered the models
of PD in signal (time) domain and phase-frequency domain.<ref>A. J. Viterbi,
Principles of Coherent Communication, McGraw-Hill, New York,
1966</ref>
In this case for constructing of an adequate nonlinear mathematical model of PD in phase-frequency domain it is necessary to find the characteristic of phase detector.
The inputs of PD are high-frequency signals and the output contains a low-frequency error correction signal, corresponding to a phase difference of input signals. For the suppression of high-frequency component of the output of PD (if such component exists) a low-pass filter is applied. The
characteristic of PD is the dependence of the signal at the
output of PD (in the phase-frequency domain) on the difference of phases
at the input of PD.

This '''characteristic of PD depends on the realization of PD and the types of waveforms of signals'''. Consideration of PD characteristic allows to apply averaging methods for high frequency oscillations and to pass from analysis and simulation of non autonomous models of phase synchronization systems in time domain to analysis and simulation of autonomous dynamical models in phase-frequency domain
.<ref name=2012-IEEETCASII-PLL>{{cite journal |
author = Leonov G.A., Kuznetsov N.V., Yuldashev M.V., Yuldashev R.V. |
year = 2012 |
title = Analytical method for computation of phase-detector characteristic |
journal = IEEE Transactions on Circuits and Systems Part II |
volume = 59 |
issue = 10 |
pages = 633–637 |
url = http://www.math.spbu.ru/user/nk/PDF/2012-IEEE-TCAS-Phase-detector-characteristic-computation-PLL.pdf |
doi = 10.1109/TCSII.2012.2213362}}
</ref>

== Analog multiplier phase detector characteristic ==
Consider a classical phase detector implemented with analog miltiplier and low-pass filter.

[[File:Multuplier phase detector in time domain.svg|500px|thumb|Multiplier phase detector in time domain.]]

Here <math>f^1(\theta^1(t))</math> and <math>f^2(\theta^2(t))</math> denote high-frequency signals, [[piecewise differentiable]] functions <math>f^1(\theta)</math>, <math>f^2(\theta)</math> represent [[waveforms]] of input signals, <math>\theta^{1,2}(t)</math> denote phases, and <math>g(t)</math> and denotes the output of the filter.
If <math>f^{1,2}(\theta)</math> and <math>\theta^{1,2}(t)</math> satisfy the high frequency conditions (see <ref>{{cite journal|author = G. A. Leonov, N. V. Kuznetsov, M. V. Yuldashev, R. V. Yuldashev|year = 2011|title = Computation of Phase Detector Characteristics in Synchronization Systems|journal = Doklady Mathematics|volume = 84|issue = 1|pages = 586–590|doi = 10.1134/S1064562411040223|url=http://www.math.spbu.ru/user/nk/PDF/2011-DAN-Phase-detector-characteristic-Nonlinear-analysis-PLL.pdf}}</ref><ref>{{cite journal|author = N.V. Kuznetsov, G.A. Leonov, M.V. Yuldashev, R.V. Yuldashev
|year = 2011|title = Analytical methods for computation of phase-detector characteristics and PLL design|journal = ISSCS 2011 - International Symposium on Signals, Circuits and Systems, Proceedings|pages = 7–10|doi = 10.1109/ISSCS.2011.5978639}}</ref>) then phase detector characteristic <math>\phi(\theta)</math> is calculated in such a way that time-domain model filter output
:<math>
g(t) = \int\limits_0^t f^1(\theta^1(t))f^2(\theta^2(t))dt
</math>
and filter output for phase-frequency domain model
:<math>
G(t) = \int\limits_0^t \varphi(\theta^1(t) - \theta^2(t))dt
</math>
are almost equal:
:<math>g(t) - G(t) \approx 0</math>
:[[File:Pd mult.svg|500px|thumb|Phase detector in phase-frequency domain.]]

=== Sine waveforms case ===
Consider a simple case of harmonic waveforms <math>f^1(\theta)=\sin(\theta),</math> <math>f^2(\theta)=\cos(\theta)</math> and integration filter.
:<math>\sin(\theta^1(t))\cos(\theta^2(t)) = \frac{1}{2}\sin(\theta^1(t) + \theta^2(t)) + \frac{1}{2}\sin(\theta^1(t) - \theta^2(t))</math>
Standard engineering assumption is that the filter removes
the upper sideband <math>\sin(\theta^1(t) + \theta^2(t))</math> from
the input but leaves the lower sideband <math>\sin(\theta^1(t) - \theta^2(t))</math>
without change.

Consequently, the PD characteristic in the case of sinusoidal waveforms is
:<math>
\varphi(\theta) = \frac{1}{2}\sin(\theta).
</math>

=== Square waveforms case ===

Consider high-frequency square-wave signals <math>f^1(t) = \sgn(\sin(\theta^1(t)))</math> and <math>f^2(t) = \sgn(\cos(\theta^2(t)))</math>.
For this signals it was found<ref>{{cite journal|author = G. A. Leonov|year = 2008|title = Computation of phase detector characteristics in phase locked loops for clock synchronization|journal = Doklady Mathematics|volume = 78|issue = 1|pages = 643–645|doi = 10.1134/S1064562408040443}}</ref> that similar thing takes place.
The characteristic for the case of square waveforms is
:<math>
\varphi(\theta) = \begin{cases}
1+\frac{2\theta}{\pi}, & \text{if }\theta \in [-\pi,0],\\
1-\frac{2\theta}{\pi}, & \text{if }\theta \in [0,\pi].\\
\end{cases}
</math>

=== General waveforms case ===

Let us consider general case of piecewise-differentiable waveforms <math>f^{1}(\theta)</math>, <math>f^2(\theta)</math>.

This class of functions can be expanded in Fourier series.
Denote by
:<math>
a^p_i=\frac{1}{\pi}\int\limits_{-\pi}^{\pi} f^p(x)\sin(ix)dx,
</math><math>b^p_i=\frac{1}{\pi}\int\limits_{-\pi}^{\pi} f^p(x)\cos(ix)dx,
</math><math>c^p_i=\frac{1}{\pi}\int\limits_{-\pi}^{\pi} f^p(x)dx, p = 1,2
</math>
the Fourier coefficients of <math>f^1(\theta)</math> and <math>f^2(\theta)</math>.
Then the phase detector characteristic is
<ref name=2012-IEEETCASII-PLL />

:<math>
\varphi(\theta) = c^1c^2 + \frac{1}{2}\sum\limits_{l=1}^{\infty}\bigg((a^1_la^2_l + b^1_lb^2_l)\cos(l\theta) + (a^1_lb^2_l - b^1_la^2_l)\sin(l\theta)\bigg).
</math>

Obviously, the PD characteristic <math>\varphi(\theta)</math> is periodic, continuous, and bounded on <math>\mathbb{R}</math>.

Modeling method based on this result is described in <ref>Patent RU 2011113212/08(019571)</ref>

=== Examples ===
{| cellpadding="2" style="border: 1px solid darkgray;"
|+ Multiplier phase detector characteristics
! width="200" | Waveforms <math>f^{1,2}(\theta)</math>
! width="200" | PD characteristic <math>\varphi(\theta)</math>
|- border="0"
| [[File:Cosine waveform.svg|255px]]
| [[File:Cosine waveforms pd characteristic.svg|255px]]
|-
| [[File:Square waveform.svg|255px]]
| [[File:Square waveforms pd characteristic.svg|255px]]
|-
| [[File:Saw waveform.svg|255px]]
| [[File:Saw waveform pd characteristic.svg|255px]]
|}

== References ==
{{reflist}}

[[Category:Detectors]]
[[Category:Electronic circuits]]

MM algorithm - Revision history

14.139.38.10: \theta_{m+1}=\arg\max_{\theta}g(\theta|\theta_m) instead of \theta_{m+1}=\max_{\theta}g(\theta|\theta_m)

14.139.38.10: $\theta_{m+1}=\arg\max_{\theta}g(\theta|\theta_m)$ instead of $\theta_{m+1}=\max_{\theta}g(\theta|\theta_m)$