8) HLTI(s+2jw)-HLT(2jw0)) (4.79) These expressions were obtained using the power series method discussed in sec- tion 3.4.3.2 of the previous chapter. In fig. 4.19, we plot HTM-element Ho,o(s) that characterizes transfers from baseband to baseband. We do SO for several values of the unity-gain frequency WUG of the LTI open-loop gain A(s) (see Fig. 4.13). Here, the solid lines are obtained from (4.78) while the marks follow from the spectral analysis of time-marching simulation results in Matlab/Simulink. The figure illustrates that, even for high-speed loops, the HTM- element Ho,o(s) 22 HLTI(s), i.e. it deviates little from results predicted by time-invariant analysis. Note, however, that there is also a significant transfer of signal content from twice the carrier frequency to baseband. The latter transfer is modeled by H0,2(s) and is not predicted by LTI analysis. Similarly as was accomplished for sampling-PFD-based PLLs, we could go on and analyze the stability and noise behavior for PLLs built using mixing PFDs. This, how- 4.2 PHASE-LOCKED LOOP ANALYSIS 10° 0 Figure 4.19: Baseband to baseband signal transfer for a PLL implemented with a mixing PFD. The HTM element Ho,o(s) is plotted for WUG/WO = 0.1, 0.2, 0.3, 0.4 and 0.5. The solid lines are obtained by evaluating (4.78). The marks are extracted from Matlab/Simulink simulations. ever, does not contribute new elements to our HTM-based analysis of a PLL's behavior. We therefore do not discuss this any further. 4.2.8 Conclusions The small-signal behavior of all current PLL implementations is inherently time-varying. As a PLL's steady-state operating point varies periodically with time, the linearization of its behavior in the neighborhood of this periodic operating point will result in an LPTV system. HTMs provide a natural framework for a frequency-domain description of this time-varying PLL small-signal behavior. It turns out that time-varying effects, especially for PLLs with a fast feedback loop and a sampling PFD, can have a significant impact on PLL syste