havior. The method is applicable to arbi- trary types of PLL architectures, implemented using either a sampling- or mixing-type PFD. The models are stated in terms of the HTM formalism. Hence, it is a frequency- domain approach, extending ideas and intuitions on LTI systems to include a PLL's time-varying behavior. This method can be used to obtain both numerical results and symbolic expressions. Furthermore, it is shown that the traditional CT-LTI model is ob- tained from the more complete time-varying one through rank reduction of the open- loop gain HTM. This establishes solid mathematical underpinnings for the CT-LTI model with its shortcomings clearly identified. Finally, it is outlined how a HTM- based analysis unifies the derivation of both continuous-time and discrete-time PLL models. The theory is illustrated by means of several examples. 3 This of course when the building block models are sufficiently accurate. 4.2.3 Signal phases and phase-modulated signal models Sref(t) Figure 4.5: (a) The phase Oo associated with a zero-crossing of s(t) is the time to some zero-crossing of some the reference signal Sref(t). Likewise, the phase 01 is defined via the events whereby s(t) and Sref(t) cross the level 1/2. (b) When s(t) and Sref are not entirely similar, it only makes sense to define phases that are, for example, associated with zero-crossings. 4.2.3 Signal phases and phase-modulated signal models The fundamental task of a PLL is to align the phase of the (frequency-divided) VCO to that of the input signal. In order to elaborate mathematical models for these type of systems, we therefore first need a sound grasp on the concept of the phase of an oscillating signal. This then brings us to phase-modulated signal models and their use in describing PLL small-signal behavior. 4.2.3.1 Signal phase By nature, a phase, or phase difference, is not associated with a continuous signal as a whole, but with discrete events generated by those signals. Such an event could, for example, be a zero-crossing. Defining the ph