 the corresponding Gershgorin circle provides a worst-case upperbound for the approximation error. Gershgorin circles can therefore be used to estimate how badly system stability is (potentially) deteriorated by the presence of time-varying components in the feedback path. 3.5.2 Stability analysis Rank-reduced approximations Suppose that there exists a pair of finite-rank matrices U(s), V(s) E ( Coox M such that G(s) ~ U (s) V(s) 7 (3.116) constitutes a valid approximation, i.e. 1. In that case, the eigenvalues of the oo-dimensional HTM G(s) that can have a magnitude larger than 1 are roughly equal to the eigenvalues of the finite-dimensional matrix Greduced(s) = V(s) U(s) (3.117) Here,Greduced(s)ECMXM.Basically, this approach amounts to a subspace projection method whereby U(s) spans the M-dimensional subspace containing the eigenvectors associated with the largest eigenvalues of G(s). Example (Downconversion with feedback): Consider again the downconversion stage with (parasitic) feedback in Fig. 3.15. The closed-loop input-output behavior is characterized by (3.83) with the open-loop gain HTM G(s) given by (3.84). Based on the Gershgorin theorem, we find the magnitude of the eigenvalues Nk(s) of G(s) to be bounded by (3.118) The hatched region in Fig. 3.20(a) marks the region bounded by (3.118) for the filters H2(s) and Hfb(s) in Fig. 3.15 respectively a second-order lowpass filter and a first-order highpass filter. As is observed, this region does not include -1+ j0. As a consequence, it is guaranteed that the time-varying feedback path does not cause instabilities to occur. More precise results are obtained using rank-reduced approximations. According to the rank-one approximation (3.90)-(3.91), Greduced (s) = 0. Therefore, as long as the rank-one approximation is valid, the feedback loop only has marginal impact on the system's dynamic behavior. However, when the rank-one approximation starts to break down, e.g. due to a growing bandwidth of H2(s), we need to increase the number of rows of G(s) that we take