

Tutorial on PLLs: Part 1
Tutorial on PLLs: Part 1
Few topics in electrical engineering have demanded as much attention over the years as the phaselocked loop (PLL). The PLL is arguably one of the most important building blocks necessary for modern digital communications, whether in the RF radio portion of the hardware where it is used to synthesize pristine carrier signals, or in the baseband digital signal processing (DSP) where it is often used for carrier and timerecovery processing. The PLL topic is also intriguing because a thorough understanding of the concept embraces ingredients from many disciplines including RF design, digital design, continuous and discretetime control systems, estimation theory and communication theory. The PLL landscape is naturally divided into (i) low signaltonoise ratio (SNR) applications like Costas carrierrecovery and timerecovery applications and (ii) high SNR applications like frequency synthesis. Each of these areas is further divided between (a) analog/RF continuoustime implementations versus (b) digital discretetime implementations. The different manifestations of the PLL concept require careful attention to different usage, analysis, design and implementation considerations. With so many good tutorials about PLLs available on the Internet and elsewhere today, a theoretically unifying development will be presented in this article with the intention of providing a deepened understanding for this extremely pervasive concept. In Part 1 of this series we'll look at PLL basics as well as different perspectives on PLL theory. In Part 2, we'll continue our look into PLL theory and then provide some realworld PLL design examples. PLL Basics "While recovering from an illness in 1665, Dutch astronomer and physicist Christiaan Huygens noticed something very odd. Two of the large pendulum clocks in his room were beating in unison, and would return to this synchronized pattern regardless of how they were started, stopped or otherwise disturbed. An inventor who had patented the pendulum clock only eight years earlier, Huygens was understandably intrigued. He set out to investigate this phenomenon, and the records of his experiments were preserved in a letter to his father. Written in Latin, the letter provides what is believed to be the first recorded example of the synchronized oscillator, a physical phenomena that has become increasingly important to physicists and engineers in modern times." (See http://www.globaltechnoscan.com/20thSep26thSep/out_of_time.htm It should come as no surprise that modern researchers would later find that the behavior of such injectionlocked oscillators can be closely modeled based upon PLL principles.^{6,7,8,9}. Anyone who has tried to colocate RF oscillators running at different but nearly the same frequency has experienced how incredibly sensitive this coupling phenomenon is! In 1840, Alexander Bain proposed a fax machine that used synchronized pendulums to scan an image at the transmitting end and send electrical impulses to a matching pendulum at the receiving end to reconstruct the image. The device, however, was never developed. "The phaselock concept as we know it today was originally described in a published work by de Bellescize in 1932^{1} but did not fall into widespread use until the era of television where it was used to synchronize horizontal and vertical video scans. One of the earliest patents showing the use of a PLL with a feedback divider for frequency synthesis appeared in 1970.^{2} The PLL concept is now used almost universally in many products ranging from citizens band radio to deepspace coherent receivers."^{1} A PLL consists of three basic components that appear in one form or another:^{4,5}
Loop "type" refers to the number of ideal poles (or integrators) within the linear system. A voltagecontrolled oscillator (VCO) is an ideal integrator of phase for example. Loop "order" refers to the polynomial order of the describing characteristic equation for the linear system. Looporder must always be greater than or equal to the looptype. Although the term "settling time" is frequently used in the literature, a specified settling time is meaningless unless the definition for settling is also provided. A properly rigorous statement would be for example, "The settling time for the PLL is 1.5 ms to within +/5 degrees of steadystate phase." ContinuousTime Versus DiscreteTime Systems
where T_{s} is the time interval between samples. The lefthand side of Equation 1 is by definition the ztransform of h(t) weighted by the quantity T_{s}. It is insightful to look at this statement for the classic type2 thirdorder PLL shown in Figure 1 for which the openloop gain is given by:
where K_{V} is the VCO tuning sensitivity (rad/sec/V), K_{d} is the phase detector gain (A/rad.), N is the feedback divider ratio, and τ_{p} and τ_{2} are the time constants associated with the leadlag loop filter. In this form, the loop natural frequency and loop damping factor are given respectively by Equations 3 and 4. The discreteequivalent ztransform for G_{OL}(s) can be computed as:
As developed at length in Chapters 4 and 5 of Reference 3, sampling control system factors adversely affect PLL stability, settling time, and phase noise performance as the closedloop bandwidth is permitted to exceed approximately 1/10th of the phase comparison frequency. Sampling effects on the openloop and closedloop transfer functions can be assessed by either going to the trouble to first compute the ztransform of the openloop gain function as in Equation 7, or the Poisson Sum formula can be used to compute the closedloop transfer function much more conveniently as:
Only a very few of the aliased G_{OL}(s) gain terms need to be retained in the denominator in order to very accurately capture the sampling effects of interest. The openloop gain functions with and without the inclusion of sampling effects are shown in Figure 2 assuming a sampling rate of 100 kHz, a natural frequency of 5 kHz and damping factor of 0.90. The closedloop response for this same system is shown in Figure 3 using Equation 8.
If the PLL natural frequency is increased to 12.5 kHz (representing 1/8th of the sampling rate), stability problems become readily apparent as an excessive amount of gainpeaking that appears as shown in Figure 4 and the almost nonexistent gainmargin as shown in Figure 5.
PLL Theory Perspectives In a similar fashion, different analysis must be used to study PLL operation under low signaltonoise ratio (SNR) cases (e.g., customarily found in receiver applications) as compared to high SNR cases (e.g., like those encountered in frequency synthesizer usage). Several different perspectives that all help expand the phaselocked loop concept are discussed in the material that follows. Control Theory Perspective (High SNR)
Figure 6: Classical type2 secondorder PLL with sampleandhold phase detector.
Several firstorder approximations are helpful to keep in mind when dealing with this classical PLL system based upon simple Bode diagramming techniques. The openloop gain diagram of interest is Figure 7 whereas Figure 8 pertains to the closedloop characteristics. In both figures, the unitygain radian frequency ω_{u} is given by Equation 11.
As noted elsewhere, the behavior of realworld sampled systems matches the continuoustime behavior very closely if the system bandwidths are small relative to the sampling rate. Therefore, it is very convenient to use the results from continuoustime theory to approximate useful quantities for both types of systems. A number of these helpful results for the continuoustime case are provided in Table 1.
In moving beyond the strictly continuoustime domain so that we can include digital dividers and phase detectors, we now include the zeroorder sampleandhold in the openloop gain formula as given by Equation 12. In this formulation, K_{d} now has dimensions of V/rad. and T_{s} is the time between sampling instants. The closedloop natural frequency and damping factor are still given by Equations 9 and 10 respectively.
In the case where the continuoustime openloop gain is given by Equation 12, full sampling effects can be included by computing the equivalent ztransform for this openloop gain function which is:
The system gainmargin G_{M} based upon Equation 13 can be shown to be:
PhaseLocked Loops for Low SNR Applications
where s(t)= A cos(ω_{o}t + θ) and the frequency and phase are considered constant. In the phaselock condition, we can further assume that the frequency ω_{o} is known whereas the system is attempting to track the phase θ, which is assumed to be quasistatic relative to the bandwidth of the PLL tracking system. It can be shown that the probability density function for the θ estimate can be written as:
where γ is the receive SNR. The cumulative pdf using Equation can be numerically computed to create the traditional "Scurve" for the ideal phase error metric. Example probability density functions and their associated Scurves are shown in Figures 10 and 11.
FokkerPlanck techniques can be used to solve the ensuing closedloop tracking performance question for type1 PLLs.^{10,11,12}. The classic result that follows is the wellknown Tikhonov probability density function for the closedloop phase error given as:
where ρ is the SNR within the closedloop bandwidth and I_{o}() is the modified Bessel function of order zero. A more insightful exploration into the tracking performance of the type1 PLL can be made by using the Scurve results that were just presented along with a firstorder Markov model for the system. In the firstorder Markov model for a type1 PLL^{13,14}, the phase error range (π,+π] is quantized across N states. Particularly nice closedform results occur^{14} if the state transitions are limited to strictly nearestneighbor transitions as shown in Figure 12. Since the use of N states divides the total phase range of 2π into N equallyspaced phase intervals, the closedloop bandwidth is inversely proportional to N. The statetransition probabilities denoted by the p_{i} and q_{i} are directly obtained from the Scurve at the SNR of interest.
The Markov steadystate probability equations can be formulated as:
in which the S_{k} denote the steadystate occupancy probabilities for each state with k=1...N. This set of equations can be solved as:
The mean tracking point and tracking error variance can be directly computed from the steadystate probabilities as:
The steadystate probabilities results are shown for two SNR cases with N=64 in Figure 13. The tracking error standard deviation for the SNR= 2dB case is 14.7 degrees rms whereas it is 9.9 degrees rms for the SNR= +2dB case (Figure 14).
Another important quantity related to low SNR PLL operation is the quantity known as "meantime to cycleslip". This can be directly computed from the transition probabilities in a similar fashion as described in References 13 and 14. Minimum Variance Estimator
in which n(t) represents complex Gaussian channel noise and s(t) represents a complex sinusoid as
If the received signal is discretized in time (t_{k} = kT_{s}), noise samples at t_{k} are assumed to be uncorrelated, and the estimates for the sinusoid's parameters are given by , the variance for the joint estimate is given by:
This can be expanded as:
Assuming that the PLL has already achieved frequencylock, we will assume that and there is no frequency error present. Minimizing the estimator variance with respect to each individual parameter separately results in the following partial derivatives:
where is always a real quantity. The estimators that minimize the tracking error variance are then given as:
in which K is the total number of signal samples involved and z_{k}= exp(jω
MaximumLikelihood Estimator
where and represent the Kdimensional measurement and signal estimate, and R is the KxK correlation matrix. In this real case being considered, s_{k} = A cos(ω_{o}kT_{s}+θ). We can equivalently seek to maximize the loglikelihood function of θ which is given by:
Assuming that the noise samples have equal variances and are uncorrelated, R= σ_{n2}I, where I is the KxK identity matrix. In order to maximize Equation 34 with respect to θ, a necessary condition is that the derivative of Equation 34 with respect to θ be zero, or equivalently:
Simplifying this result further and discarding the doublefrequency terms that results, the maximumlikelihood estimate for θ is that value that satisfies the constraint:
The topline indicates that doublefrequency terms are to be filtered out and discarded. This result is equivalent to the minimumvariance estimator derived earlier in Equation 31. Wrap Up In Part 2 of this article, we will close out the theoretical discussions by looking at (i) the maximum a posteriori (MAP) estimator PLL form, (ii) the CramerRao bound which provides helpful insights into achievable theoretical performance, and finally (iii) the PLL derived based upon Kalman filtering concepts. The balance of the article will look at several realworld applications using the PLL concept. Editor's Note: To view Part 2, click here. References
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