al [h] (p,t) (5.90) Figure 5.18: Frequency-domain characteristic corresponding to (a) the av- eraging operator (5.89) and (b) the averaging operator (5.90). Here, fm = 1/TM. Here, in both cases, TM is the time constant of the (lowpass) averaging operator (fil- ter). The operator (5.89) corresponds to a time average over the time span TM. This time average coincides with our intuitive notions on averaging. The operator (5.89) is the one that typically appears in traditional literature on averaging [Bogo61, Verhu96]. Fig. 5.18(a) shows its frequency-domain characteristic. The second operator (5.90) represents the ideal lowpass filtering operation over the range f = -1/2TM,1/2TM] Its frequency characteristic is shown in Fig. 5.18(b). This operator is ideal in a sense that it realizes a perfect separation of slow- and fast-varying signal components. Nu- merically, however, (5.90) is sometimes more difficult or expensive to evaluate than (5.89). We conclude this discussion with some notes on the term eh(p,t) in the transformation (5.84). This term captures the fast fluctuations in p(T) and is solved from (5.87). With M[.] chosen SO that the right-hand side of (5.87) has no DC component with respect to the variable T, eh(p,t) is guaranteed to remain bounded At. However, since (5.87) is a partial differential equation, we need a boundary condition to fix eh(p,t) uniquely. Bearing in mind the interpretation of eh(p,t) as the small fast-varying fluctuations of p(T), it is most sensible to impose the term eh(p,t) to be void of a DC component with respect to the variable T, or (5.91) Equations (5.87) and (5.91) uniquely fix the term eh(p,t) and, therefore, the averaging transformation (5.84). Fig. 5.19 summarizes the steps needed to compute the compo- 5.6 AVERAGING h (p,T) M[h](p,T) h(p,T) Figure 5.19: Steps needed to compute the components of the averaging trans- formation and the associated averaged equations. nents of the averaging transformation and the associated averaged equations. Example (Voltage-controlled harmonic