op, the lowpass filter H2(s) throws out most of the high-frequent information. If H2(s) rolls off sufficiently fast, we can use a rank-one approximation for G(s) (valid in the frequency interval (-wo/2,00/2), i.e. (3.90) Here, Go, s)[0H2(sHfb(s-jwo) 0 H2(s)Hb(s+jwo) 0 ... ] (3.91) is the 0-th row of G(s). Using (3.82), it is then straightforward to obtain the same result as in (3.87)-(3.89). This illustrates that, often, both power series approximation and a rank-reduced approximations can be used to obtain (approximate) expressions for the input-output HTM elements of feedback systems6. Accuracy of these approximations will be verified for some practical cases in the next chapter. 3.4.4 Relating HTMs to state-space representations A final topic related to LPTV system manipulation using HTMs concerns the link be- tween an LPTV system's HTM representation and its state-space equations. The latter are, for example, obtained by linearizing a set of circuit equations in the neighborhood 6This, however, does not hold true for all cases. For analyzing PLLs with sampling phase-frequency detectors (see the next chapter), only the rank-reduced approximation turns out to be successful. 3.4.4 Relating HTMs to state-space representations of a time-varying operating point. This typically results in a set of linear differential- algebraic equations (DAEs) that are structured as (3.92) c(t)x+d(t)u (3.93) Here, E(t), A(t) E RNxN, b(t) E R N x 1, c(t) E RNx1 and d(t) E R are T-periodic ma- trices (with T = 2n/wo). Using the expressions for constructing elementary HTMs and the rules for addition and concatenation, it is straightforward to obtain (3.94) as the HTM relating y(t) to u(t). Here, HE represents a block Toeplitz matrix structured like the one in (3.67)7. The elements of HE equal the matrix-valued Fourier coefficients of E(t) = k Ekejkwot HA, Hb, Hc and H are likewise obtained from A(t), b(t), c(t) and d(t). Furthermore, - jwo)IN Hdiff(s) : (3.95) (s+jwo)IN is the (block) HTM representation for the differentiation operato