reason, we call them 2 For this to hold, the frequency of the input signal must be within a sufficiently small range (the capture range) of that of the divided free-running VCO signal. 4.2 PHASE-LOCKED LOOP ANALYSIS loop filter frequency divider Figure 4.3: Continous-time linear time-invariant PLL model. This model neglects all time-varying behavior. Its validity is limited to slow feedback loops. sampling PFDs. A second PFD implementation builds on the multiplication of the frequency-divided VCO signal and the reference signal by means of a mixer . The phase error is then computed as the DC component of the mixing product. This type of PFD is called a mixing (or multiplying) PFD and is currently still often used in microwave applications. 4.2.2 Prior art In an attempt to capture a PLL's behavior, different models have been proposed in the past. The one best known from textbook analyses is the continuous-time linear time- invariant (CT-LTI) model [Cran98, Gard79, Lee98]. For PLLs with a sampling PFD, a discrete-time linear time-invariant (DT-LTI) model has been developed [Gard80, Hein88]. Other existing techniques for analyzing PLL behavior are simulation-based [Perr02]. 4.2.2.1 Continuous-time, linear time-invariant model Classical textbook treatments typically capture a PLL's small-signal behavior using CT- LTI feedback theory [Gard79, Lee98]. Analysis builds on the loop model illustrated in Fig. 4.3. In the frequency domain, input-output behavior is governed by the relation Health) (4.10) Here, (4.11) is the PLL's open-loop gain. The quantities Oin and 0 respectively represent the phase of the input and the VCO signal. HLF is the loop filter's transfer function. Furthermore, the VCO behavior is modeled by means of an integrator. This stems from (4.12) 4.2.2 Prior art loop filter frequency divider Figure 4.4: PLL implemented using a sampling PFD. Here, the PFD samples the instantaneous phase error once (or twice) per period of the input signal. Sampling is modeled as a multiplication of the phase error with a t