s. Hence, topics like noise folding, i.e. high-frequency noise components ending up at baseband, are left unaccounted by CT-LTI analysis. As a consequence of the first condition, stability as predicted by the LTI model is only valid as long as the unity-gain frequency WUG of the open-loop gain, i.e. the relevant frequency for stability computations, is well below wo/2. It will be demonstrated in section 4.2.6 that for fast feedback loops, i.e. WUG/WO > 0.05, time-varying effects do need to be taken into account for stability predictions to be accurate. 4.2.5.3 PLL input-output HTM for sampling-PFDs While (4.54) is an approximation that holds for all types of PFDs, exact expressions can be derived for PLLs built on sampling PFDs. This is done by means of sim- ilar techniques as in the previous section and by making use of the fact that, for sampling PFDs, HPFD (s) = (wo/2 is by nature rank one. As a reminder, I = define (4.61) (4.62) then we can write G(s)=V(s)-W(s)T (4.63) 4.2.6 Example 1: PLL with sampling PFD Using the Sherman-Morisson-Woodbury formula, we then find (4.64) (4.65) (4.66) In words, X(s) equals the sum of all elements of Substitut- ing (4.65) in (4.45) and using the fact thatHycoss(s)T the input-output relation now becomes (4.67) This relation is exact as long as (4.36) accurately models the sampling-PFD's behavior. 4.2.5.4 The discrete-time LTI PLL model and its limitations Observing (4.67), it is seen that the stability of a PLL with sampling PFD is gov- erned by a single scalar quantity N(s). Using its definition (4.66) and the structural properties of HTMs as discussed in section 3.3.4, it is readily shown that X(s) is periodic in the direction of the imaginary axis, or 2(s+jwo)= : N(s). This implies that X(s) corresponds to a discrete-time LTI system with z-domain transfer function ND(2) = a (woln( /2). Hence, the stability of a PLL with sampling PFD is deter- mined by that of a discrete-time system. This is in correspondence with discrete-time (LTI) PLL models in [Gard80, Hein88]. However, t