o obtained as a special case of the results presented in section 5.7 of chapter 5. 8 Having a VCO that settles as slow as the PLL as a whole bears stability problems with it. The VCO setling behavior then introduces extra poles near the PLL's unity-gain frequency, degrading the phase mar- 9 This relation follows from the fact that, for all small-signal excursions u(t), d0/dt = r(t+Oop(t)+0(t))(uop(t) + u(t)) - doop/dt By considering the PLL's steady-state behavior, i.e. ((t) u(t) = 0, we obtain that Substituting this into the previous equation then yields the desired result. 4.2 PHASE-LOCKED LOOP ANALYSIS Note that the latter approximation only holds valid when dT(t)/dt. Oop(t) is sufficiently small, something which is typically true. Hence, observing (4.41), we find that O(t) is related to u(t) via an LPTV operator consisting of a multiplication with T(t) followed by an integration. This model is depicted by the left-hand block in Fig. 4.11. The HTM corresponding to this operator equals s-jwo s-jwo N-1bands Hyco(s) = (4.42) s+jwo ++jwo N-1 bands and is readily computed using (4.7), (4.9) and (4.41). Here, the Tk are the Fourier coefficients of T(t), or (4.43) Since the VCO, and hence the ISF T(t), oscillates at Wyco = NWO and since all HTMs are constructed with respect to the carrier waves Ym(t) = ejmwot Hyco(s) contains a great number of super- and sub-diagonal bands filled with zeros. This readily follows from the fact that, when expanding T(t) with respect to 4m(t) = ejiwwot only the coef- ficients associated with the carriers (t) = e.jkk)wot can assume non-zero values. As was already mentioned, often I (t) 22 To in which case Hyco(s) reduces to a diagonal matrix. This means that the high-frequency components of the small-signal control voltage u(t) do (almost) not influence the oscillator's instantaneous frequency. Note that, as illustrated in Fig. 4.11, the VCO's phase-noise behavior can be captured by introducing a noise source just before the integrator 10. We can then use the cyclo- stationary noise theo