TM inversions cos(2ft) H1 (s) H2 (s) Hfb(s) Figure 3.15: Block diagram of an elementary downconversion stage with parasitic feedback from the output to the mixer's input. If (3.81) is satisfied, HTM inversions are efficiently computed by means of the Sherman- Morisson-Woodbury formula [Gol96]5. This yields (3.82) Since it was assumed that to Z(s) is invertible at low computational cost, only the com- putation of I+V(s)TZ(s)-IU(s))7 -1 might pose difficulties. However, as U(s), V(s) E CooxM, this involves inverting an M x M matrix. If M is sufficiently small, i.e. if U(s)V(s)T is of low rank, this is much less of a problem than inverting some large- dimensional matrix obtained by truncating the original oc-dimensional HTM. Example (Downconversion with feedback): Consider the system in Fig. 3.15. This is the same downconversion stage as in Fig. 3.13 but now we've added a time-invariant (parasitic) feedback connection from the output to the mixer's input. Hence, the overall input-output relation is given by (3.83) Here, the open-loop gain HTM equals (3.84) The HTMs H1(s), H2(s), Amixer(s) and Hfb(s) respectively correspond to the filters H1 (s) and H2(s), the multiplication with cos(2fot) and the feedback path Hfb(s). If the feedback loop is weak, i.e. if G(s)|<<, (3.85) then, using a first-order power series approximation, we obtain (3.86) Hence, the HTM elements H0,1(s), Ho,o(s) and ,((s) -respectively characterizing the (wanted) transfers from carrier frequency to baseband and the (unwanted) transfers Sues appendix B. 3.4 LPTV SYSTEM MANIPULATION USING HTMS from baseband to baseband and twice the carrier frequency to baseband- we find (most important terms only) (3.87) ,((s) H2(s)H2(s + jwo)Hfb(s+ jw0)H1(s (3.88) (3.89) It is observed that the presence of the feedback Hfb(s) causes (parasitic) transfers from baseband and twice the carrier frequency to baseband. Often, these transfers are unde- sired and need to suppressed. A second way to solve (3.83) builds on the observation that, with regard to the feedback lo