r d/dt[.]. Here IN E RNxN is the identity matrix. The numerical extraction of HTMs or related quantities can now be accomplished by means of the solution of (3.94) whereby all matrices are properly truncated [Phil98, Royc98]. Typically, one needs Krylov subspace solvers [Barr94, Saad96] to handle the large-scale matrix inversions that arise. Example (Downconversion stage): Let us consider the downconversion circuit in Fig. 3.16. This circuit implements a mixing stage followed by a lowpass filtering oper- ation. By linearizing the circuit equations in the neighborhood of the periodic operating point, we obtain the linear periodically time-varying system of DAEs (3.96) y(t) = x(t) (3.97) Here, we assumed the opamp to be ideal and we used the results of the example in section 3.1.3 (see equation (3.10)) to model the transistor's time-varying small-signal behavior. If we map each of the terms in (3.96)-(3.97) on their HTM equivalent, then 7 Since E(t) is a matrix, multiplication with E(t) represents a multi-input multi-output (MIMO) LPTV system. The HTM corresponding with such a MIMO system has the same structure as a HTM corresponding to a single-input single-output (SISO) system. The only difference is that the HTM elements are now matrix valued instead of scalar valued. 3.5 LPTV SYSTEM ANALYSIS USING HTMS Figure 3.16: Downconversion circuit that implements a mixing stage fol- lowed by a lowpass filtering operation. we obtain that HE = C.I, with I the unity matrix, HA = - (1/R).I,H=I,H = 0 (3.98) Substituting this into (3.94) yields the input-output relation 2(1+(s-jwo)RC) Y(s) = 2(1+sRC) 2(1+sRC) (3.99) 2(1+(s+jwo)RC) The matrix in the equation above is the HTM associated with the state-space description (3.96)-(3.97). If we take a look at the HTM element that models the transfer of signal content from the carrier frequency to baseband, i.e H0,1(s) = -BR/2(1+sRC), then, as expected, we clearly observe a lowpass filtering action. 3.5 LPTV system analysis using HTMs Having discussed HTMs, their basic properties and 