he discrete-time models in [Gard80, Hein88] do not fully recognize the mixed-signal aspects of PLLs with sampling PFDs. Mapping the PLL on a discrete- time LTI system, disregards the continuous-time nature of the PLL's output signal. This is, for example, important for spectral computations and noise analysis. These issues are readily handled using the HTM framework where sampling is recognized as a continuous-time rank-one periodically time-varying operator. Moreover, HTM-based PLL analysis is easier and more straighforward than is the conversion to a discrete-time system. This is particularly the case for the derivation of symbolic expressions. 4.2.6 Example 1: PLL with sampling PFD We illustrate and verify our analysis of time-varying PLL behavior for a system im- plementing a sampling PFD, lacking a frequency divider, i.e. N = 1, and containing VCO with constant ISF, i.e. F(t) = . HTM-based computations are compared with results extracted from time-marching simulations in Matlab/SimulinkTM. The Mat- lab/Simulink simulation model is shown in Fig. 4.12. It implements the sampling PFD using flip-flops and therefore encodes the phase error through the width of the pulses produces. This corresponds to the behavior of an actual PFD circuit realization. This allows us to test the accuracy of our approximations. loop filter sin(t+Oin) cos(t+)) Figure 4.12: Matlab/Simuling model used for time-marching simulations. Note that the sampling PFD is implemented using flip-flops. It therefore encodes the phase error through the width of the pulses that it produces. This corresponds to the behavior of an actual circuit implementation. 4.2.6 Example 1: PLL with sampling PFD Figure 4.13: Typical characteristic for A(jw) containing three poles (two at DC) and one zero. Note that the frequency-axis is normalized with respect to the unity-gain frequency WUG of A(s). 4.2.6.1 Input-output HTM With T(t) = , the HTM (4.43) -capturing the VCO behavior- is diagonal and represents an LTI system with transfer characteristic Hyco(s) = To/s