Costas loop

A Costas loop is a phase-locked loop (PLL) based circuit which is used for carrier frequency recovery from suppressed-carrier modulation signals (e.g. double-sideband suppressed carrier signals) and phase modulation signals (e.g. BPSK, QPSK). It was invented by John P. Costas at General Electric in the 1950s .^[1] Its invention was described^[2] 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 $\sin(2(\theta_i-\theta_f))$ as compared to $\sin(\theta_i-\theta_f)$. This translates to double the sensitivity and also makes the Costas loop uniquely suited for tracking Doppler-shifted carriers especially in OFDM and GPS receivers.^[2]

Classical implementation

Costas loop

In the classical implementation of a Costas loop,^[3] 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.

Mathematical models

In the time domain

Time domain model of Costas loop

In the simplest case $m^2(t) = 1$. Therefore, $m^2(t) = 1$ does not affect the input of noise-reduction filter. Carrier and voltage-controlled oscillator (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 a VCO 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. Similarly, 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 non-autonomous and rather difficult for investigation.

In phase-frequency domain

Equivalent phase-frequency domain model of Costas loop

VCO input for phase-frequency domain model of Costas loop

In the simplest case, when

$\begin{array}{l} f^1\big(\theta^1(t)\big)=\cos\big(\omega^1 t\big), f^2\big(\theta^2(t)\big)=\sin\big(\omega^2 t\big) \\ f^1\big(\theta^1(t)\big)^2 f^2\big(\theta^2(t)\big) f^2\big(\theta^2(t) - \frac{\pi}{2}\big) = -\frac{1}{8}\Big( 2\sin(2\omega^2 t) +\sin(2\omega^2 t - 2\omega^1 t) +\sin(2\omega^2 t + 2\omega^1 t) \Big) \end{array}$

the 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 $\varphi(\theta^1(t) - \theta^2(t))=\frac{1}{8}\sin(2\omega^1 t - 2\omega^2 t).$ This makes a Costas loop equivalent to a 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.^{[4] [5] [6]}

Thus it is possible ^[7] to study more simple autonomous system of differential equations

$\begin{array}{ll} \dot{x} = Ax + b\varphi(\Delta\theta), & \Delta\dot\theta = \omega^2_{free} - \omega^1 + Lc^*x, \\ \Delta\theta = \theta^2 - \theta^1. & \end{array}$.

The Krylov–Bogoliubov averaging method allows one to prove that solutions of non-autonomous 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 non-autonomous 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.

Frequency acquisition

Costas loop before synchronization

Costas loop after synchronization

Carrier and VCO signals before synchronization

VCO input during synchronization

Carrier and VCO signals after synchronization

References

1. Costas, John P. (1956). "Synchronous communications". Proceedings of the IRE 44 (12): 1713–1718. doi:10.1109/jrproc.1956.275063.
2. Taylor, D. (August 2002). "Introduction to `Synchronous Communications', A Classic Paper by John P. Costas" (PDF). Proceedings of the IEEE 90 (8): 1459–1460. doi:10.1109/jproc.2002.800719.
3. Feigin, Jeff (January 1, 2002). "Practical Costas loop design" (PDF). RF Design: 20–36.
4. Leonov, G. A.; Kuznetsov, N. V.; Yuldashev, M. V.; Yuldashev, R. V. (August 2012). "Differential equations of Costas loop" (PDF). Doklady Mathematics 86 (2): 723–728. doi:10.1134/s1064562412050080.
5. Leonov, G. A.; Kuznetsov, N. V.; Yuldashev, M. V.; Yuldashev, R. V. (2012). "Analytical method for computation of phase-detector characteristic" (PDF). IEEE Transactions on Circuits and Systems Part II 59 (10): 633–637. doi:10.1109/tcsii.2012.2213362.
6. Leonov, G. A.; Kuznetsov, N. V.; Yuldashev, M. V.; Yuldashev, R. V. (2015). "Nonlinear dynamical model of Costas loop and an approach to the analysis of its stability in the large". Signal processing (Elsevier) 108: 124–135. doi:10.1016/j.sigpro.2014.08.033.
7. Kuznetsov, N. V.; Leonov, G. A.; Neittaanmaki, P.; Seledzhi, S. M.; Yuldashev, M. V.; Yuldashev, R. V. (2012). "Nonlinear mathematical models of Costas Loop for general waveform of input signal". IEEE 4th International Conference on Nonlinear Science and Complexity, NSC 2012 - Proceedings (IEEE Press) (6304729): 75–80. doi:10.1109/NSC.2012.6304729.

•  This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C".