Second-order Phase-locked Loop Research Articles

Second-order LMS based wireless channel tracking: implementation under imperfect carrier synchronization

In this paper, we are concerned with the implementation of second-order LMS (SOLMS) based wireless channel tracking under imperfect carrier synchronization. First, we present a performance analysis of the SOLMS based wireless channel tracking in the presence of carrier-frequency offset. With this analysis, we study how carrier-frequency offset affects the performance of the SOLMS based tracking designed under perfect carrier synchronization. Our studies show that only when high quality oscillators are employed can we take the carrier-frequency offset as part of the fading channel gain and perform adaptive wireless channel tracking directly. Then, under the framework of extended Kalman filtering, we propose an adaptive joint wireless channel tracking and carrier-frequency offset compensation method that can be simplified to a SOLMS based channel tracking with an incorporated second-order digital phase-locked loop. Monte-Carlo simulations under an IS-136 like environment are performed to support the theoretical analysis and to demonstrate the performance of the proposed method.

Performance of Coherent Optical PSK Systems with Costas Loop in Receiver for Nonlinear Second-Order PLL Model

Using the nonlinear second-order PLL (phase-locked loop) model, the authors consider the performance of the coherent optical PSK (phase-shift keying) systems with Costas loop in receiver, for the first time. The shot noise of the corresponding PIN photodiodes is described through the signal-to-noise ratio in the loop bandwidth, and the laser phase noise is described through the normalized frequency fluctuations instead of the phase ones. The analysis is carried out for both homodyne and heterodyne types of receivers. The theory and results presented in this article can be applied to the analysis and optimization of performance in regions where the linear PLL model is not sufficient.

Third-order phase-locked loops using dual loops inserting an active filter in the second loop with improved stability

In fast mobile communication systems, a phase-locked loop (PLL) should track the signal not only with the frequency step transition but also with the frequency ramp transition caused by Doppler shift. In such an environment, since a perfect second-order PLL cannot suppress the steady-state phase error, a perfect third-order PLL is required. However, the third-order PLL using an active filter cannot be used in the low S/N environment because the PLL with small loop gain becomes unstable. One of the solutions is a third-order digital PLL using dual loops. However, since the PLL is implemented by digital equipment and is limited by the operational frequency, it cannot be used in a high-frequency band employed in satellite communication. Downconverting of the operational frequency restricts the acquisition speed. Furthermore, the recomposing of the PLL with analog equipment causes the gain imbalance of the voltage-controlled oscillator (VCO) and the PLL cannot synchronize. In this paper, we propose a new third-order PLL using dual loops inserting an active filter in the second loop, which can be implemented by analog equipment. First, we show the closed loop transfer function of the proposed PLL and investigate the steady-state phase error by linear analysis. Second, we show that the proposed PLL is always stable regardless of the value of loop gain. Third, we evaluate the transient response and show the proposed PLL has acquisition time shorter than the perfect third-order PLL. Next, we show that the steady-state phase error of the proposed PLL can be suppressed to the same level of the perfect third-order PLL. Finally, we investigate the effect of noise in PLLS, and show the proposed PLL has better jitter performance than the perfect third-order PLL. © 2001 Scripta Technica, Electron Comm Jpn Pt 3, 84(6): 75–84, 2001

Chaos from second-order PLL in presence of CW interference

The chaotic behaviour of a second-order phaselocked loop (PLL) in the presence of continuous wave (CW) interference is examined. The Lyapunov exponents and dimension of the system are calculated to confirm the chaotic phenomenon. A range of interfering signal power and frequency has been found where chaos could occur.

Stability conditions of a class of delayed PLL systems through linear and non-linear techniques

A study of the stability problems of a class of heterodyne second-order phaselocked loops incorporating an additional phase modulator has been reported in this paper. Conditions of stability of this system in terms of loop design parameters have been obtained through the Liapunov method, the Routh-Hurwitz method and the Nyquist method. Also reported are simulation results finding design parameters for stable operation of the system. Comparing the results obtained through different techniques, it can be noted that linear analytical methods provide design parameters which are consistent with simulation results for small loop phase errors, but for large loop errors design parameters predicted by the Liapunov method should be used for system design.

Performance of optical heterodyne PSK systems with Costas loop in multichannel environment for nonlinear second-order PLL model

Using the nonlinear second-order phase-locked loop (PLL) model the performance of the heterodyne coherent optical phase shift keying (PSK) systems with Costas loop in multichannel environment is considered in this paper for the first time. The shot noise of the corresponding photodiodes and adjacent channel interferences are described through the signal-to-noise ratio (SNR) in the loop bandwidth, while the laser phase noise is described through the normalized frequency fluctuations instead of the phase ones. The theory and results presented in this paper can be applied when analyzing and optimizing the performance in the region where the linear PLL model is not enough good.

GPS Signal Tracking Using Maximum-Likelihood Parameter Estimation

This paper considers the problem of GPS carrier tracking in the presence of spurious modulation components in amplitude and phase caused by rotating antenna elements in a reentry body. The commonly used phaselock loop approach is not adequate since the disturbances are not specifically modeled. This is a nonlinear estimation problem that is attacked here as a linear estimation problem with an unknown modulation parameter. The modulation parameter is estimated using a maximum-likelihood method in an architecture that uses two Kalman filters of similar structure, one for parameter estimation and one for state estimation. This architecture uncouples the state and parameter estimation processes and reduces the tendency to build up incorrect correlations in the estimator. The performance of the estimator is studied using a Monte Carlo simulation. Maximum-likelihood results are found to be superior to those obtained using a second-order phaselock loop and a nonadaptive fourth-order Kalman filter.

A per-survivor phase acquisition and tracking algorithm for detection of TCM signals with phase jitter and frequency error

In this letter we investigate the performance of a per-survivor phase estimation algorithm for detection of trellis-coded modulation (TCM) signals in the presence of phase jitter and frequency error. Phase and frequency estimates are derived based on a second-order phase-locked loop (PLL)-like algorithm. A simplified transient and steady-state analysis of the algorithm recursive equations is derived. A heuristic blind phase acquisition scheme is proposed and shown to achieve good results. Computer simulations are used to compare the algorithm's performance to other methods described in the literature.

Reduced-complexity detection and phase synchronization of CPM signals

Based on an extension of Laurent (1986) decomposition of continuous phase modulation (CPM) signals into a sum of linearly modulated components, we derive a class of reduced-complexity maximum-likelihood (ML) coherent detection and closed-loop phase synchronization schemes. The complexity of the resulting detection schemes, expressed in terms of the number of matched filters and states of a Viterbi algorithm, is significantly reduced with respect to that of optimal coherent receivers with negligible performance loss. This result extends a known one valid for the binary case to multilevel CPM. The proposed synchronization schemes do not require an increased number of matched filters, and are perfectly suitable to be used in conjunction with these receivers. Based on the phase-locked loop (PLL) equivalent linear model, a method for optimizing the parameters of digital second-order PLL's is presented. Numerical examples, based on theoretical analysis and computer simulation, are provided for two specific formats in the CPM class: tamed frequency modulation (TFM) and a quaternary raised-cosine (RC) modulation.

The half-plane pull-in range of a second-order phase-locked loop

The pull-in range Ω p of a second-order, Type I phase-locked loop (PLL) is defined as the maximum value of loop detuning ω os for which pull-in occurs from anywhere on the PLL's phase plane. That is, pull-in is guaranteed from anywhere on the phase plane if |ω os | < Ω p . Simple approximations are available for computing Ω p . The concept is expanded here, and a definition is given for the PLL's half-plane pull-in range Ω 2 . Simply stated, pull-in is guaranteed from anywhere on the phase plane's lower-half if 0 < ω os < Ω 2 . Unlike the parameter Ω p , a simple approximation for Ω 2 is not available. However, a Galerkin based algorithm is presented for computing the PLL's half-plane pull-in range Ω 2 , and it is applied to a simple example.

Piecewise-linear analysis of high-damping chaotic phase-locked loops using Melnikov's method

Studies the chaotic dynamics observed from a widely used second-order phase-locked loops (PLLs) operating as a frequency modulated demodulator. The authors apply Melnikov's method to high-damping PLLs to obtain the parameter range where there exist homoclinic points. This implies the PLL is chaotic in such a parameter range in view of the Smale-Birkhoff theorem. In particular, since the current PLL used in practice has a triangular phase detector as its nonlinearity (i.e., a periodic triangular shaped function), one can use piecewise-linear methods to derive the Melnikov integral analytically even for the practical high-damping (hence, high dissipation) case. The advantage of this method compared to the previous numerical integration method is that all integrations are made analytically, and therefore reliable results can be obtained for all parameter values. The authors have obtained many boundary curves for homoclinic tangency for a wide range of damping coefficients. As a result, some inaccuracies in previous numerical integration results have been detected. >

Jitter analysis of a phase-locked digital timing recovery system

An analytical approach is presented for the jitter performance of a timing recovery circuit consisting of a hard-limiter, an exclusive-or gate, followed by a second-order phase-locked loop. Exact expressions for the autocorrelation function and the power spectral density of the zero-crossing jitter in the data signal are derived, and then the phase jitter variance of the timing wave, generated at the output of the timing circuit, is obtained analytically as a function of various system parameters. Finally, numerical results and a comparative performance analysis show that considerable improvement can be achieved in jitter performance in addition to having the advantage of low cost and simple hardware implementation.

A digital phase-locked loop for stuffing synchronization systems

Stuffing synchronization systems are now being applied actively to new fields such as digital cross-connect and SONET. In stuffing synchronization systems, however, the analog PLLs are still used for the receive destuffmg circuitry. This paper describes a method of applying a complete second-order digital phase-locked loop (PLL) to a stuffing synchronization system. In the PLL used for the stuffing synchronization systems, a buffer memory consisting of an N-stage frequency divider is provided in order to suppress jitter. This paper analyzes first the effects on the DPLL resulting from the incorporation of an N-stage frequency divider based on a multilevel quantized phase comparator. The characteristics parameters of the complete second-order DPLL when the N-stage frequency divider is added are determined. Significant low-frequency jitter is present in the narrow-band complete second-order DPLL. The phenomenon of low-frequency jitter generation is analyzed and procedures for determining the coefficient values of the complete second-order DPLL that will result in tolerable levels of low-frequency jitter are described. Finally, the DPLL parameters are evaluated theoretically and experimentally to determine the suitability of the DPLL design for stuffing synchronization systems.

Adaptive phase-locked loop operating in the pseudo-linear region

An adaptive second-order phase-locked loop is proposed which keeps the loop within the pseudo-linear region for a wide range of input frequency steps. FETs are used to vary the filter transfer function as well as the overall loop gain, thus yielding almost linear behaviour. Theoretical analysis of the proposed system has been carried out, and the system has been constructed to show that it is possible to increase this region for the loop from a frequency step of 353.5 rad/s to 3535 rad/s without affecting its noise-rejection property.

Performance analysis of phase-locked loops in the frequency search mode of operation

A method is proposed to evaluate the acquisition statistics of phase-locked loops in the frequency search mode of operation. The frequency sweep rate limit and the relevant statistical characteristics of a high-gain second-order phase-locked loop are given. Loop performance is optimised with respect to minimum acquisition time, and a suboptimum loop is reported.

An optical PSK homodyne transmission experiment using 1320 nm diode-pumped Nd:YAG lasers

A second-order optical phase-locked loop was constructed using 1320-nm diode-pumped miniature Nd:YAG ring lasers. Using the loop, a 140-Mb/s PSK homodyne transmission experiment was demonstrated over 28.6 km of single-mode fiber. With a loop natural frequency of 13 kHz and a damping factor of 0.6, the receiver sensitivity was -62.8 dBm, or 25 photons/bit. The authors believe this is the highest sensitivity obtained to date with any optical communication system. >

On the Frequency Acquisition Performance Evaluation of Phase-Locked Loops

The statistical definition acquisition range (FAR) is introduced. The superiority of the newly performance measure over the traditional pull-in limits is demonstrated, and the frequency acquisition performance of second-order phase-locked loops is evaluated via the FAR. The results obtained using a simple approximate method are confirmed by the exact time domain analysis of the acquisition process.

A 1320 nm experimental optical phase-locked loop

An experimental balanced optical second-order phase-locked loop constructed using 1320 nm diode laser pumped miniature Nd:YAG lasers is discussed. The loop is stable and has a phase error of less than 1.8 degrees when the received signal power is -65 dBm or more. The phase error appears to be dominated by the lasers' frequency noise as long as the signal power is more than -60 dBm.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Chaos from phase-locked loops

The chaotic dynamics observed for the second-order phase-locked loop (PLL) operating as a frequency-modulated (FM) demodulator are studied theoretically as well as experimentally. The existence of horseshoe chaos in this circuit is proved using Melnikov's method. It is demonstrated that chaos can indeed occur in a practical FM demodulator circuit made of an integrated circuit module and using realistic parameter values. Explicit expressions that specify the possible region of chaos are derived.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Investigation of Synchronization Parameters in a Digital Subscriber Loop Transmission System

The degree of echo reduction achieved in a digital subscriber loop (DSL) transceiver employing adaptive echo cancellation has previously been shown to be very sensitive to timing jitter. This paper reports the experimental study, by computer simulation, of a 144 kbit/s DSL timing recovery system incorporating a prefilter, nonlinearity, and second-order phase-locked loop. The simulations were carried out for a wide variety of line lengths, bridged-tap conditions, loop and filter parameters, and choice of nonlinearity. The results showed that a properly designed timing recovery system with a 20 khzwide prefilter allows satisfactory echo reduction performance, even for long loops with bridged taps. For shorter loops the achievable echo reduction is less, but is still sufficient to yield satisfactory performance.