User:Renatyv

Renat V. Yuldashev
=email: renatyv@gmail.com=

=Costas Loop=

A Costas loop is a phase-locked loop based circuit which is used for carrier phase recovery from suppressed-carrier modulation signals, such as from double-sideband suppressed carrier signals. It was invented by 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 receivers

Classical implementation
In the classical implementation of a Costas loop, a local voltage-controlled oscillator (VCO) provides quadrature outputs, one to each of two phase detectors, e.g., product detectors. The same phase of the input 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.

Model of classical Costas loop in the time domain
In the simplest case $$m^2(t) = 1$$. Therefore, $$m^2(t) = 1$$ does not affect the input of noise-reduction filter. Carrier and VCO signals are periodic oscillations $$f^{1,2}(\theta(t))$$ with high-frequencies $$\dot\theta^{1,2}(t)$$. Block $$-90^{o}$$ shifts phase of VCO signal by $$-\frac{\pi}{2}$$. Block $$\bigotimes$$ is an Analog multiplier.

From the mathematical point of view, a linear filter can be described by a system of linear differential equations
 * $$\begin{array}{ll}

\dot x = Ax + b\xi(t),& \sigma = c^*x, \end{array} $$ Here, $$A$$ is a constant matrix, $$x(t)$$ is a state vector of filter, $$b$$ and $$c$$ are constant vectors.

The model of voltage-controlled oscillator is usually assumed to be linear

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

Suppose that the frequency of master generator is constant $$ \dot\theta^1(t) \equiv \omega^1. $$ Equation of VCO and equation of filter yield

\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} $$

The system is nonautonomous and rather difficult for investigation.

Model of classical Costas loop in phase-frequency domain


In the simplest case, when
 * $$\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} $$ standard engineering assumption is that the filter removes the upper sideband with frequency from the input but leaves the lower sideband without change. Thus it is assumed that VCO input is $$\varphi(\theta^1(t) - \theta^2(t))=\frac{1}{8}\sin(2\omega^1-2\omega^2)$$. This makes Costas loop equivalent to Phase-Locked Loop with phase detector characteristic $$\varphi(\theta)$$ corresponding to the particular waveforms $$f^1(\theta)$$ and $$f^2(\theta)$$ of input and VCO signals. It can be proved, that inputs $$g(t)$$ and $$G(t)$$ of VCO for phase-frequency domain and time domain models are almost equal.

Thus it is possible to study more simple autonomous system of differential equations

\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} $$ 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.

Modeling


Carrier and VCO signals are periodic oscillations $$f^{1,2}(\theta(t))$$ with high-frequencies $$\dot\theta^{1,2}(t)$$. Block $$\bigotimes$$ is an Analog multiplier.

From the mathematical point of view, a linear filter can be described by a system of linear differential equations
 * $$\begin{array}{ll}

\dot x = Ax + b\xi(t),& \sigma = c^*x, \end{array} $$ Here, $$A$$ is a constant matrix, $$x(t)$$ is a state vector of filter, $$b$$ and $$c$$ are constant vectors.

The model of voltage-controlled oscillator is usually assumed to be linear

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

Suppose that the frequency of master generator is constant $$ \dot\theta^1(t) \equiv \omega^1. $$ Equation of VCO and equation of filter yield

\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} $$

The system is nonautonomous and rather difficult for investigation.

Model of Phase Locked loop in phase-frequency domain


In the simplest case, when
 * $$\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) = -\frac{1}{2}\Big(\sin(\omega^2 t - \omega^1 t)\Big) \end{array} $$ standard engineering assumption is that the filter removes the upper sideband with frequency from the input but leaves the lower sideband without change. Thus it is assumed that VCO input is $$\varphi(\theta^1(t) - \theta^2(t))=\frac{1}{2}\sin(\omega^1-\omega^2)$$. It can be proved, that inputs $$g(t)$$ and $$G(t)$$ of VCO for phase-frequency domain and time domain models are almost equal.

Thus it is possible to study more simple autonomous system of differential equations

\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} $$ 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 Phase Locked 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 Phase Locked loop (in place of the nonautonomous one) allows one to overcome the difficulties, related with modeling Phase Locked 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.

Linearized phase domain model
Phase locked loops can also be analyzed as control systems by applying the Laplace transform. The loop response can be written as:


 * $$\frac{\theta_o}{\theta_i} = \frac{K_p K_v F(s)} {s + K_p K_v F(s)}$$

Where
 * $$\theta_o$$ is the output phase in radians
 * $$\theta_i$$ is the input phase in radians
 * $$K_p$$ is the phase detector gain in volts per radian
 * $$K_v$$ is the VCO gain in radians per volt-second
 * $$F(s)$$ is the loop filter transfer function (dimensionless)

The loop characteristics can be controlled by inserting different types of loop filters. The simplest filter is a one-pole RC circuit. The loop transfer function in this case is:


 * $$F(s) = \frac{1}{1 + s R C}$$

The loop response becomes:


 * $$\frac{\theta_o}{\theta_i} = \frac{\frac{K_p K_v}{R C}}{s^2 + \frac{s}{R C} + \frac{K_p K_v}{R C}}$$

This is the form of a classic harmonic oscillator. The denominator can be related to that of a second order system:


 * $$s^2 + 2 s \zeta \omega_n + \omega_n^2$$

Where


 * $$\zeta$$ is the damping factor
 * $$\omega_n$$ is the natural frequency of the loop

For the one-pole RC filter,


 * $$\omega_n = \sqrt{\frac{K_p K_v}{R C}}$$
 * $$\zeta = \frac{1}{2 \sqrt{K_p K_v R C}}$$

The loop natural frequency is a measure of the response time of the loop, and the damping factor is a measure of the overshoot and ringing. Ideally, the natural frequency should be high and the damping factor should be near 0.707 (critical damping). With a single pole filter, it is not possible to control the loop frequency and damping factor independently. For the case of critical damping,


 * $$R C = \frac{1}{2 K_p K_v}$$
 * $$\omega_c = K_p K_v \sqrt{2}$$

A slightly more effective filter, the lag-lead filter includes one pole and one zero. This can be realized with two resistors and one capacitor. The transfer function for this filter is


 * $$F(s) = \frac{1+s C R_2}{1+s C (R_1+R_2)}$$

This filter has two time constants


 * $$\tau_1 = C (R_1 + R_2)$$
 * $$\tau_2 = C R_2$$

Substituting above yields the following natural frequency and damping factor


 * $$\omega_n = \sqrt{\frac{K_p K_v}{\tau_1}}$$
 * $$\zeta = \frac{1}{2 \omega_n \tau_1} + \frac{\omega_n \tau_2}{2}$$

The loop filter components can be calculated independently for a given natural frequency and damping factor


 * $$\tau_1 = \frac{K_p K_v}{\omega_n^2}$$
 * $$\tau_2 = \frac{2 \zeta}{\omega_n} - \frac{1}{K_p K_v}$$

Real world loop filter design can be much more complex e.g. using higher order filters to reduce various types or source of phase noise.