ase of an event then goes as follows: Definition (phase): The phase 0 of an event generated by a signal s(t) is the time to the occurrence of a reference event generated by a reference signal Sref(t). This time can be both positive and negative valued. Fig. 4.5(a) illustrates the definition above for s(t) and Sref(t) both being sine waves. The events generated by s(t) and Sref (t) correspond to the signals crossing a certain level. For example, Oo is the difference in time between a zero-crossing of s(t) and a zero-crossing of Sref(t). Likewise, 01 is the phase associated with s(t) and Sref (t) crossing the signal level 1/2. Fig. 4.5(b) illustrates that, when Sref(t) and s(t) are not entirely similar, e.g. when s(t) is a sine wave and Sref (t) is a square wave, it only makes sense to define phases associated with, for example, zero-crossings. Observing the figure, it is seen to be senseless to define a phase associated with the crossing of the signal level 1/2. As a final note, we mention it to be possible to associate a particular phase value 0 with an absolute instance t of time. For example, in Fig. 4.5(a), the phase value Oo can be associated with to, the time at which Sref(t) crosses zero and generates 4.2 PHASE-LOCKED LOOP ANALYSIS the reference event. This unambiguously defines the concept of a phase ((t) that varies with time. 4.2.3.2 Phase-modulated signal models The type of signals s(t) that appear in dealing with PLL small-signal behavior can best be described as oscillations whose phase varies with respect to some large-signal steady-state operating point. Hence, in modeling the small-signal behavior, we first need to specify the operating point (or reference) signals Sref(t) with respect to which small-signal phases are defined. Once Sref(t) is fixed, we assume that s(t) can be written as s(t) = Sref(t+0(t)) (4.13) The right-hand side of (4.13) constitutes a phase-modulated signal model. Here, the process ((t) is introduced to capture all kinds of small-signal phase variations. Ob- serving (4.13), it