Second-order Phase-locked Loop Research Articles

Acquisition time of second-order phase-locked loop by two-variable method

This paper describes a new method, namely, the two-variable perturbation procedure, for the determination of acquisition times of phase-locked loops in the absence of noise. Two types of loop filters are considered : (i) perfect integrator and (ii) imperfect integrator. In both cases initial frequency deviation was assumed to be large. The results are compared with those already known in the literature. In the Case of a passive filter loop, in addition to obtaining an expression for acquisition time, the existence of limit cycles was also observed.

Interferences in Phase-Locked Loops

An analysis of the behavior of a second-order phase-locked loop is presented when an unwanted signal is added to the useful signal. Both signals are sinusoidal and unmodulated, and the analysis is made in the absence of additive noise. When the loop remains locked on the useful signal, a parasitic signal exists at the phase detector output. This signal produces a parasitic phase modulation of the VCO and a static phase error in the loop. The parasitic signal amplitude, the parasitic phase modulation index, and the static phase error are calculated. A necessary condition for the loop to remain in lock is derived. When the loop is initially unlocked, locking can occur either on the useful signal or on the unwanted signal, depending on the amplitude ratio and the frequency difference of the two signals. A formula allowing one to compute the pull-in time is obtained. When the loop locks on the useful signal, acquisition can be slower or faster than in the absence of an unwanted signal. The same phenomenon is observed when the loop locks on the unwanted signal.

Stochastic Lyapunov stability of the phase-locked loop†

In this paper stability bounds for the stochastic phase-locked loop are studied, using the second method of Lyapunov. The fundamental configurations for the phase-locked loop and the related equations describing loop operation are given. The principles necessary to establish the idea of a stochastic Lyapunov function are presented, with special emphasis on the Ito differential form. An examination of the stochastic stability of a second-order phase-locked loop is carried out qualitatively. The stability analysis technique employed in this paper yields probabilistic data for cycle-slipping in the phase-locked loop, and thus represents a significant departure from the traditional approach based on Fokker—Planck techniques. The concepts presented, however, are applicable to a much wider variety of control system stochastic stability problems.

Second-Order Phase-Lock-Loop Acquisition Time in the Presence of Narrow-Band Gaussian Noise

Phase-lock acquisition time of a perfect (proportional plus integral control) second-order phase-lock loop (PLL) in the presence of narrow-band Gaussian noise is studied by means of computer simulation. 100 nonoverlapping sequences of pseudorandom noise are generated. In the presence of each noise sequence, phase-lock acquisition is simulated at each of five pairs of initial phase error φ 0 and initial frequency error φ 0 . For each (\phi_{0},\phi_{0}) with each noise sequence, phase-lock acquisition is simulated at nine different ratios of signal-to-noise energy at the PLL input. If the PLL phase detectors were linear rather than sinusoidal, the range of PLL output signal-to-noise ratios simulated would extend from 4 to 50 dB. The 4500 acquisitions of phase lock simulated are represented by 45 distribution functions that explicitly indicate the effect of PLL output signal-to-noise ratio on PLL phase-lock acquisition time.

Injection synchronization and the second-order phase-locked loop

A type of injection synchronized oscillator, having an additional arrangement for its gain control, has been studied and its locking performance has been compared to that of a phase-locked loop (PLL) incorporating passive filter networks having no limiting and finite high-frequency gain. Also presented are the experimental results in support of the conclusion.

Some Simulation Results for the Time to Indicate Phase Lock

A digital simulation of a second-order phase-lock loop (PLL) acquiring lock in the presence of noise is reported. Since the system simulated includes lock-indicating circuitry, a distinction is made between the time to phase lock and the time to indicate lock. The parameter of interest is the time to indicate lock. The major results of the simulation are estimates of 1) the mean time to indicate lock, and 2) the probability distribution function for the mean time to indicate lock. Results are presented for a wide range of signal-to-noise ratios and for input frequency offsets ranging from one-half to three times the phase-lock loop noise bandwidth. In each case 100 independent trials of the simulation were conducted to obtain the estimate. It is found that the mean time to indicate lock is relatively independent of signal-to-noise ratio ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S/N</tex> ) down to about <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S/N = 10</tex> dB. For greater values of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S/N</tex> the mean time to indicate lock may be closely approximated by the time to indicate lock in the absence of noise. The results are compared with Viterbi's time to frequency lock and found in good agreement for the high <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S/N</tex> cases. Most of the difference is attributed to the time constant of the lock indicating circuitry. The system simulated is considered representative of many practical phase-lock receiving systems where there is a requirement for indicating acquisition of lock. Hence, the data presented in this paper can be used to good advantage for estimating the probability of acquiring and indicating lock in a specified time.

Generalized Phase Comparators for Improved Phase-Locked Loop Acquisition

Phase-locked loops (PLL) with pulse-type (sawtooth) phase comparators find many applications in digital transmission systems. In this paper, the results available on the pull-in performance of the second-order PLL with sawtooth phase comparator are summarized. Then a generalized phase comparator, called a phase-and-frequency comparator, is shown to improve PLL acquisition performance significantly. Several circuit implementations are discussed in detail.

Acquisition time of second-order phase-locked loop†

The Krylov-Bogoliubov asymptotic method has been used for solving the differential equation of the second-order phase-locked loop to calculate the acquisition time in the absence of noise. The acquisition time ta . is defined as the smallest value of time t for which the frequency error ta with ∊1 and ∊2 properly chosen. The results are valid for small initial phase and frequency errors.

First Slip Times Versus Static Phase Error Offset for the First- and Passive Second-Order Phase-Locked Loop

Static phase error offsets occur in first-order and passive second-order phase-locked loops, whenever the rest frequency of the local oscillator and the carrier are different prior to acquisition. The effect of the offset is to degrade performance and in particular to decrease the mean time to first slip. This paper studies first slip times via computer simulation. The mean time to first slip is displayed graphically as a function of signal-to-noise ratio (SNR) for various static phase error offsets and values of the damping parameter.

The Effect of Loop Stress on the Performance of Phase-Coherent Communication Systems

In phase-coherent communication systems which use phase-locked loops to provide synchronization of the data detector, the communications engineer is frequently faced with the problem of determining the effects which noisy timing has upon detection efficiency. This paper is concerned with determining these effects when second-order phase-locked loops, operating in the presence of frequency detuning, are used as a means of providing phase synchronization. The results are also applicable to the problem of establishing noisy reference losses in a broad class of coherent PCM telemetry systems, e.g., PCM/PSK/PM. Also presented are results which can be used to determine steady-state statistical dynamics of second-order loops.

On a Solution to the Second-Order Phase-Locked Loop

A solution of the stationary, mod-2π-reduced phaseerror probability density function for the second-order phase-lock loop is derived, based on a mean square fit to a certain unknown conditional expectation term. The resulting phase variance obtained from this density function yields an error that is negligible for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sigma_{\phi}^{2} \leq 1</tex> and an error no greater than 0.06 rad <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> for signal-to-noise ratios (SNRs)≥ 2 dB by comparison with actual measurements, and it appears to be the most accurate to date. The usefulness of the result lies in the fact that an accurate estimate of the phase-error variance for arbitrary parameter values is easily obtained without the use of a computer.

Pull-in time in second-order phase-locked loops with a sawtooth comparator

A phase-locked loop incorporating a sawtooth phase comparator and an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RC</tex> or phase-lag filter is considered in a noisefree environment. The pull-in time problem during initial synchronization or abrupt change of the input frequency is studied in detail and equivalently the out-of-lock time, when the loop is "knocked out of lock and then brought back to its normal operation." An exact and convenient graphical method is presented for the evaluation of the number of "skipped" cycles and the pull-in time (or out-of-lock time) for any input mistuning within the pull-in range and any initial state (i.e., any initial phase and frequency offset) of the system. For large initial mistuning, a simple approximation that avoids the graphical procedure is given. These graphical methods provide qualitative insight and quantitative information as to the pull-in performance and beat mode of operation.

Experimental signal/noise-ratio comparison of the second-order phase-locked loop and the second-order frequency-locked loop

The signal/noise ratio performance of a novel f.m. threshold extension demodulator, the frequency-locked loop, is experimentally compared with that of the phase-locked-loop demodulator and the conventional f.m. discriminator. Experimental results indicate that the f.l.l. represents a significant improvement over the p.l.l. and the conventional discriminator for f.m. systems subject to large carrier deviations.

Effects of CW Interference on Narrow-Band Second-Order Phase-Lock Loops

This paper describes an experimental study of the effect of continuous wave (CW) interference and white noise on a second-order phase-lock loop. The reciprocal of the loop mean-square phase error is used as an index of performance, and the effect of interference levels that do not cause cycle skipping or loss of lock is described in terms of this index. Loop thresholds are determined by measurement of cycle-skipping rates. Stationary or slowly-sweeping CW interference caused a degradation in loop threshold of roughly 3 dB for every 6 dB of interference power above the noise power level. The effective loop signal-to-noise ratio was decreased approximately 1 dB at interference-to-noise power ratios of -3 dB. Interference levels equal to the signal level consistently caused loss of lock, regardless of the loop signal-to-noise ratio.

A Model Distribution for the Phase Error in Second-Order Phase-Locked Loops

This paper describes a model distribution for the phase error in the second-order phase-locked loop (PLL) of greatest practical interest. Experimental results relative to the distribution of the phase error and its properties are given. The experimental measurements are used to complement the analytical model in a signal-to-noise region where the linear PLL theory does not apply. Compared to the actual measurements of the second-order loop performance, it is shown that the model distribution provides excellent agreement over the useful range of most phase-locked receivers.

Some analytical and experimental phase-locked loop results for low signal-to-noise ratios

This paper is concerned with the nonlinear behavior of the second-order phase-locked loop (PLL) in the presence of noise. The loop filter is of the proportional-plus-integral control type. This filter corresponds to the one generally employed for carrier tracking purposes in the implementation of phase-coherent communication systems. The paper is composed essentially of two parts: the first part presents analytical results which pertain to the probability distribution of the phase-error. Since these analytical results are approximations, valid only for certain regions of signal-to-noise ratio, they are complemented by experimental results obtained from simulation of the PLL system in the laboratory. The experimental techniques used to measure the statistical properties of the loop behavior and the corresponding results comprise the second part of the paper. Approximate analytical expressions for the distribution of the system phase-error are first obtained by using the Fokker-Planck apparatus and, secondly, by assuming that the PLL behaves as a very narrow band-pass filter. The range of signal-to-noise ratios for which these approximations are valid is obtained by graphically comparing the analytical expressions to experimentally derived phase-error distributions. In addition, measurements relative to the variance of the phase-error are compared to those predicted by the linear PLL theory and the variance as computed from the approximate solutions. Finally, experimental results relative to the probability distribution of the time intervals between cycle-slipping events are given for signal-to-noise ratios in a range where the linear PLL theory does not apply. In particular, the maximum length of time the loop may be expected to remain in-lock is illustrated graphically as a function of signal-to-noise ratio in the loop bandwidth.

Optimum Detection of Random Phase Modulation with Phase-Lock Loops

A study was conducted to determine the optimum design for a second-order phase-lock loop when used to detect a randomly phase-modulated sine wave of unknown average frequency in a white-noise background. The optimum physically realizable loop was also found for the case when the average frequency is known a priori. A quasilinear model was used to account for the nonlinear characteristic of the phaselock loop and to predict the threshold effect. The results of the study give formulas for calculating the root-mean-square loop-phase error and detector-output signal-to-noise ratio as a function of the input signal-to-noise ratio and the power and bandwidth of the assumed modulating function. The results show a strong dependence of the loop threshold on the characteristics of the modulating function. Careful consideration must be given these parameters in system designs employing phase-lock loops for the processing of this class of signal.

Oscillator Stability and the Second-Order Phase-Locked Loop

The causes of short-term oscillator instability are discussed. The effects of the instability of the received signal on the phase-locked loop performance are analyzed in terms of resultant mean-squared loop phase error. The results of the analysis permit the calculation of the effects of oscillator instability in the transmitter on the loop bandwidth and power requirement. They also furnish a systematic approach to specifying the oscillator stability requirements.