uctured as (4.47) This shows that, at least for s|<00/2,G will be dominated by the rows closest to n = 0. This in turn suggests to try and approximate G(s) by its projection on the rows with indices n R with R the rank-reduction parameter. Mathematically, this is written as (4.48) ~ P.V(s)T (4.49) Here, with (4.50) 2R+1 columns In words, (4.48) and (4.50) express that, for an input signal with low-frequency equiv- alent baseband components, the output of the PLLs time-varying open-loop gain is dominated by the signal content contained in the lowest frequency bands. Note from the general theory outlined in section 3.4.3.3 that the approximation (4.48) only holds valid for those values of SEC for which I (I [-PP7)G(s)|| 1 12. Using a similar argument, we equally find P Vin(s)T (4.51) With (4.49), the factor (I+G(s))-1 in the input-output relation (4.45) can be elabo- rated using the Sherman-Morisson-Woodbury formula (see also section 3.4.3.3 of the previous chapter). This yields (4.52) 12 Due to the presence of poles at DC in a typical design of the loop filter HLF (s), this approximation may fail for values of S very close to 0. In such a case, rank-reduced approximations are still possible, but it is somewhat more involved to construct them. However, since this only makes a difference for signal transfers from or to frequency bands beyond Wyco = NWO, it is unimportant for most practical purposes. 4.2.5 PLL closed-loop input-output HTM Here, (4.53) represents the (2N+1) x (2N + 1) submatrix of G(s) centered around its origin (n,m) = (0,0). Using (4.51) and (4.52), the input-output relation (4.45) turns into H(s)~P(I+Gp(s))-1Vin(s)T (4.54) Note from the structure of P in (4.50) that this rank-reduced approximation neglects all transfers of signal content to frequency bands beyond RWO. Example (Rank-one approximation): The most elementary rank-reduced approxi- mation to G(s) is a matrix of rank one, i.e. we set R=0. In this case, the projector PPT only takes the 0-th row of G(s) into account. We illustrate the resul