ns governing this averaged behavior are obtained by substituting, amongst others, the transformation of variables A=A+eh(A,T) (5.85) into the starting-point equations obtained from the perturbation analysis. 11 The theory of averaging can also be seen as a particular application of the theory of continuous transfor- mation (Lie) groups [Cohn65, Firk00] 5.6 AVERAGING A (T) eh(A(t),c) Figure 5.17: Typical harmonic oscillator startup characteristic. (a) The oscil- lation amplitude ((T) solved from the starting-point equation (5.82). (b) The amplitude startup behavior can be decomposed into a dominant slow-varying component A(T) with a small fast-varying fluctuation eh(A(t),T) on top of it. By substituting (5.84) into (5.82), we obtain (5.86) For p to meet the conditions mentioned above, an averaging transformations imposes (p,t) to satisfy (5.87) Here, the operator M[.] is called an averaging operator. M[.] acts as a lowpass filter that removes the fast-varying explicit dependence on T of h(p,t). Hence, in computing M [h] (p,t), the value of p is kept constant. With (5.87), (5.86) becomes (5.88) This relation is accurate up to first order in E. By careful selection of the lowpass av- eraging operator M[.], the transformed equation (5.88) has a right-hand side that only contains slow-varying components. As a result, the same holds for the any process p(T) that solves this equation. Equation (5.88) is called the averaged equation correspond- ing to (5.82). Because of the slow-varying nature of its right-hand side, it can be solved very efficiently using a large integration time step. Furthermore, it makes up a valuable starting point for further analysis or for the construction of compact behavioral models. Selecting the averaging operator M [ ] is a degree of freedom that can be used to opti- mize the properties of the averaged ODEs (5.88). This allows us to exploit particular properties of the right-hand side eh(p,t) of the starting-point equation (5.82). Conve- nient averaging operators are Mtraditional (5.89) Mide