Second-order Loop Research Articles

Global bifurcation and chaos from automatic gain control loops

The dynamics of the automatic gain control (AGC) loops which employ variable-gain amplifiers with a square control law are studied. The selected loop filters are of second-order low-pass type. The existence of global homoclinic bifurcation and Smale horseshoe chaos are proved without the traditional small-signal model assumption. To investigate systems that are monitored during discrete-time intervals, sampled-data AGC loops are also considered. Under the same square gain-control law, the first-and second-order loop filters are studied. The period-doubling routes to chaos are verified for the first-order systems, and Hopf bifurcations for the second-order filters. Simulations are performed and the resultant homoclinic tangles, chaotic time waveforms, bifurcation diagrams, and Lyapunov exponents are illustrated to demonstrate the theoretical results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Self-calibration techniques for a second-order multibit sigma-delta modulator

An oversampled modulator design that uses a second-order loop and a three-bit quantizer to exhibit low quantization noise is reported. Second-order loops are attractive because they are unconditionally stable and require only a third-order sinc filter in the decimation filter. Tone energy is significantly reduced in a multibit loop. In addition, a multibit quantizer avoids increased noise generation with input levels near full scale. The modulator uses digital self-calibration of the DAC (digital-to-analog coverter) capacitors to reduce their mismatch. High linearity and low noise are then simultaneously possible when random dynamic capacitor matching is used. The modulator is clocked at 6.144 MHz, and a 0.5-VRMS low-distortion sine wave at 2 kHz is applied at the input. The decimation filtering, with an oversampling ratio of 128-to-1, is performed by an external processor. >

A wideband heterodyne optical phase-locked loop for generation of 3-18 GHz microwave carriers

Experimental results of a wideband heterodyne second-order optical phase-locked loop with 1.5- mu m semiconductor lasers are presented. The loop has a bandwidth of 180 MHz, a gain of 181 dBHz, and a propagation delay of only 400 ps. A beat signal of 8 MHz linewidth is phase locked to become a replica of a microwave reference source close to carrier with a noise level of -125 dBc/Hz. The total phase variance of the locked carrier is 0.04 rad/sup 2/ and carriers can be generated in a continuous range from 3 to 18 GHz. The loop reliability is excellent with an average time to cycle slip of 10/sup 11/ s and an acquisition range of 640 MHz. >

Absorptive optical bistability due to impurity nonlinearity in CdS:Cu

With the excitation of a ps pulse laser we found both the optical bistability and the multi-order bistability in CdS: Cu+ crystals. The nonlinear response time is as fast as 150 ps, and the switching time of the second-order bistable loop approaches 80 ps. The technique of the degenerate four wave mixing (DFWM) has been used to investigate the optical nonlinearity related to the Cu+ impurity. The third-order nonlinear susceptibility of χ(3) was measured to be 9 × 10−9 esu.

Joint blind equalization, carrier recovery and timing recovery for high-order QAM signal constellations

Two existing blind equalization tap update recursions for 64-point and greater QAM (quadrature amplitude modulation) signal constellations are studied, along with existing and novel carrier and timing recovery techniques. It is determined that the superior tap update recursion is the one known as the constant modulus algorithm. Carrier recovery requires a modified second-order decision-directed digital phase-locked loop. An all-digital implementation of band-edge timing recovery is used. With 14.4 kb/s outbound transmission using CCITT V.33 trellis-coded 128-QAM signals having 12.5% excess bandwidth, a prototype blind retrain procedure is developed to demonstrate the feasibility of the techniques for high-speed multipoint modems. A WE DSP32-based real-time digital signal processor was employed to test the retrain over a set of severely impaired channels. For each channel in the set, the retrain succeeded at least 90% of the time. >

Chaos from third-order phase-locked loops with a slowly varying parameter

The dynamic behavior of a third -order PLL (phase-locked loop) is studied by using a second-order loop filter for tracking frequency variable signals. The authors prove the existence of horseshoe chaos in the three-dimensional nonautonomous systems by the perturbation methods based on the ideas of Melnikov. This approach makes it possible to treat three-dimensional, periodically forced, slowly varying oscillators. The Lyapunov exponents and Lyapunov dimension are also calculated to confirm the theory. Theoretical results indicate that the parameter ranges where the chaos could occur are realistic in the typical designs. Computer simulations are performed to obtain the actual chaotic attractors.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

High-stability 1.5 mu m external-cavity semiconductor lasers for phase-lock applications

Applications using phase-locked semiconductor lasers, such as homodyne detection, require lasers with narrow linewidth and high-frequency stability. The design and operating characteristics of two 1.5 mu m external-cavity semiconductor lasers built for such applications are described. The measured beat linewidth is 4 kHz, and the spectral density of relative frequency noise deviates significantly from the intrinsic white spectrum only at frequencies below 4 kHz. It is estimated that this frequency jitter will induce approximately 1.1 degrees RMS phase error in a second-order homodyne optical phase-lock loop that is optimized for the present beat linewidth. >

Acquisition problem of class of second-order digital phase-locked loops

Analysing the acquisition characteristics of a ZC1-DPLL by a new technique, the letter presents computer simulation results showing the variation of frequency acquisition ranges of the loop with different loop design parameters. The results obtained modify the existing understanding on the problem towards a realistic direction.

Continuum-regularized loop-space equation

We project large- N continuum-regularized gauge theory onto loop-space, obtaining a regularized form of the second-order loop equation. In this form, the regularization is seen as an invariant all-order generalization of conventional one-loop proper-time methods. The result may be expressed as a relation between laplacians on field-space and loop-space.

Quantum Chromodynamics in 1+3 Dimensions at High Temperature: Dimensional Reduction Revisited

The gluon sector of QCD in 1+3 dimensions in analyzed at high temperature (much larger than the critical ones) thereby generalizing previous results by other authors. The imaginary time formalism is used. The analysis is carried out to all orders in an improved perturbation theory which includes all second-order internal quark loops in the “free” gluon propagators. General results are given for the leading high temperature contributions to all renormalized connected gluon Green's functions (for fixed external threemomenta, much smaller than the temperature). The latter are generated by a new (dimensionally reduced) high-temperature partition function ZHT, which corresponds to: i) the Yang-Mills (“magnetic”) gluon field coupled to a massive scalar (“electric”) gluon field, all in 3 spatial dimensions and at zero imaginary time, ii) the quark field, which continues to depend on imaginary time, coupled to the above gluon fields ZHT also depends on the renormalized quark masses and gauge coupling constant at zero temperature, the second order quark-loop contributions to the zero-temperature renormalization constants for the gluon field and the three and four gluon vertices and on new gluon mass terms. The latter correspond to a finite number of diagrams in the improved perturbation theory at high temperature. ZHT could be useful as the starting point for further non-perturbative studies. For the pure Yang-Mills plus ghosts theory (no quarks), it is conjectured that contributions to Green's functions depending on external momenta due to internal electric gluons could be regarded, as subdominant. Arguments are given in order to justify that conjecture. Then, the above ZHT can be simplified and another high-temperature generating functional depending only on magnetic gluon fields is given. For the full theory including quarks, the possibility of neglecting contributions due to internal quark loops is discussed: certain infrared divergences beyond the oneloop level appear to imply that such a simplification, although not discarded, is rather hard to establish.

Nonlinear dynamics of a digital phase locked loop

A second-order digital phase-locked loop may exhibit unusual behavior for some parameters due to a fractal boundary between the basin of attraction of the locked fixed point and the attracting basins of coexisting periodic orbits. The usual optimization criterion of the loop parameters using linearized analysis is insufficient, due to coexisting periodic orbits. A new optimization procedure is presented which is based on numerical bifurcation studies. The authors obtain an analytical estimate of average lock time based on the rate of contraction of the phase space in a neighborhood of the fixed point and the size of the phase space that is regular for the underlying Hamiltonian dynamics. The analytical estimate is verified by numerical calculations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Exact calculation of phase-locked loop lock-in frequency

Using a novel cycle-slip criterion, two methods of computing the lock-in frequency of phase-locked loops are reported. The lock-in frequency of a second-order loop with sinusoidal phase detector is presented as a function of system parameters, and useful estimates by the pull-out and pull-in frequencies are given.

Second-order PLL loop filters with independent adjustment of ω n and ζ

Second-order type-one and type-two PLL loop flltets are given in which the values of ωn and ζ can be set independently.

Phaselock system for millimeter-wave Gunn oscillators used in radio astronomy and laboratory spectroscopy

A frequency-agile heterodyne phaselock loop (PLL) system for millimeter-wave Gunn-effect oscillators between 40 and 110 GHz is described. The Gunn oscillators are phase-locked via the bias in an active second-order servo loop. A facility for fast frequency switching with a maximum rate of 10 kHz and a frequency separation up to 80 MHz is provided. Measurements on the spectral characteristics of a phase-locked Gunn oscillator are presented. The described PLL system is used in radio astronomy and laboratory molecular spectroscopy.

Performance analysis and laser linewidth requirements for optical PSK heterodyne communications systems

An optical PSK heterodyne communications receiver is investigated. The receiver is based on the decision-directed phase-locked loop. The performance of the phase-locked loop subsystem is analyzed taking into account both shot noise and laser phase noise. It is shown that for reliable phase locking (rms phase error less than 10°), heterodyne second-order loops require at least 6771 electrons/s per volt every hertz of the laser linewidth. This number corresponds to the limit when the loop dumping factor η is infinitely large; if \eta = 0.7 , then the loop needs 10 157 electrons/(s . Hz). If the detector has a unity quantum efficiency and \lambda = 1.5 \mu m, the above quoted numberers give 0.9 pW/ kHz for \eta \rightarrow \infty and 1.35 pW/kHz for \nu = 0.7 . The loop bandwidth required is also evaluated and found to be 155 \Delta\nu , where \Delta\nu is the laser linewidth. Finally, the linewidth permitted for PSK heterodyne recievers is evaluated and found to be 2.26 \cdot 10^{-3} R_{b} where R b is the system bit rate. For R_{b}=100 Mbit/s, this leads to \Delta\nu kHz. Such and better linewidths have been demonstrated with laboratory external cavity lasers. For comparison, ASK and FSK heterodyne receivers are much more tolerant to phase noise,-they can tolerate \Delta\nu up to 0.09 R b . At the same time, homodyne receivers impose much more stringent requirements on the laser linewidth ( \Delta\nu ).

New Phase-Lock Loop Circuit Providing Very Fast Acquisition Time

We present a new circuit configuration for second-order phase-lock loops that provides, for large initial frequency offsets, acquisition times several orders of magnitude shorter than those achieved using conventional phase-lock loops. This new circuit also provides frequency locking almost instantaneously when the time delay around the loop is small. Furthermore, it can, without losing lock, sustain frequency changes several hundred times faster than those which can be sustained by a conventional circuit.

Digital Phase-Locked Loop Behavior with Clock and Sampler Quantization

This paper deals with the analysis of digital phase-locked loop (DPLL) models that take into account the discrete nature of the analog-to-digital converter (ADC) and the numerical controlled oscillator (NCO). The models are general enough to allow for different kinds of noise distributions and nonuniform qnantization. Weak assumptions on the nature of the loop parameters reduce the models to finite Markov chain problems. It is shown that the resulting probability transition matrices are lumpable, which translates into convenient Computational savings. In this manner, an exact statistical analysis is possible not only for first-order but also for second-order loops. Numerical results give evidence that for appropriate ADC and NCO characteristics, few quantization levels are needed to match the performance of models that ignore qnantization. However, when the values of these characteristics are not proper, a substantial change in performance can occur. This change in the loop behavior cannot be analyzed or predicted by those models that neglect the quantization effects. It is also found that the use of a nonlinear ADC improves the loop behavior when the noise density distribution is heavy-tailed, e.g., atmospheric noise.

New design approach to precision thermostats at liquid-helium temperature

A precision liquid-helium temperature proportional thermostat designed for use with a superconducting cavity stabilized oscillator is described, where temperature of the X-band superconducting Nb cavity mounted inside a vacuum can is regulated at the temperature below 2 K. By noticing the fact that the time for establishment of thermal equilibrium is very short at liquid-helium temperature and thermal transfer properties can, therefore, be measured in the frequency domain, the frequency domain technique rather than the conventional time domain technique is used to design and measure feedback loops of a thermostat. The application of a second-order loop to the design of thermostats is discussed in order to obtain better transient and steady-state performance. In the thermostat constructed, temperature is measured with a germanium resistance thermometer in the ac Wheatstone bridge, a commercial lock-in amplifier is used as a null detector, and the second-order loop is used. The loop characteristics were confirmed by measuring the Bode plot of the feedback loop which also contains the frequency response of the thermal interface, and the gain crossover frequency and the phase margin were 26 Hz and 52°, respectively, at the operating temperature near 1.7 K. The thermostat has achieved the long-term temperature drift of less than 3.2 μK, where the drift caused by the instability of the bridge itself is excepted, in the presence of the liquid-helium temperature fluctuation of 35 mK for periods more than 10 h; it has also shown very fast and good transient response. In addition, the Bode plots of the closed loops have shown that the loop is stable to the variation of the gain crossover frequency of more than four octaves. It has been shown that the method of designing a precision thermostat by measuring its thermal properties in the frequency domain can be useful and can give very reliable results at liquid-helium temperature.

Servo-controlled air pump for calibration of respiratory measurement systems

We describe a servo-controlled piston pump driven by a stepping motor. The analogue controller is a single non-linear second-order feedback loop with adjustable speed and acceleration limits. This system, designed to simulate slowly-moving active and non-linear systems, can be used as a low (0–6.5 litre s −1) flow volume generator in calibration procedures.

Soft-momentum quark-model predictions for the pseudoscalar-meson decays into lepton pairs

The pseudoscalar decays $P\ensuremath{\rightarrow}{l}^{+}{l}^{\ensuremath{-}}$ are driven by second-order electromagnetic loop diagrams which are rendered finite by use of appropriate off-shell $P\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma}$ form factors determined in the quark model. The theoretical predictions using the soft-momentum limit are in reasonable agreement with the experimental rates and branching ratios, the latter being approximately twice the unitarity bounds for the decays ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}}$, $\ensuremath{\eta}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$, and ${K}_{L}^{0}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$. We also investigate the radiative decays ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}}\ensuremath{\gamma}$ and $\ensuremath{\eta}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}\ensuremath{\gamma}$ in the same model, and deduce that one cannot extract unambiguously the constituent quark masses from the experimental data.