ATION USING HTMS Figure 3.12: Parallel and series connection of two (LPTV) systems. cos(2ft) U H1(s) Figure 3.13: Block diagram of an elementary downconversion stage con- sisting of a (bandpass) pre-filter, an ideal mixing operation and a (lowpass) post-filter. Example (Downconversion stage): Consider the block diagram of the elementary downconversion stage in Fig. 3.13, consisting of a (bandpass) pre-filter, an ideal mix- ing operation and a (lowpass) post-filter. Using (3.60), (3.69) and (3.73), it is readily shown that the elements of the input-output HTM H(s) characterizing the entire down- conversion stage are given by (3.74) Downconversion of signal content from the carrier frequency to baseband is therefore captured by the harmonic transfer function V0,1(s)=H1(s+jwo)H2(s)/2 (3.75) This transfer function relates the output equivalent baseband component at DC to the input equivalent baseband component at carrier frequency. 3.4.3 Feedback systems and HTM inversions LPTV feedback connections are more difficult to handle. For example, as the HTM of the feedback system in Fig. 3.14 is concerned, straightforward manipulation yields the closed-loop characteristic (3.76) 3.4.3 Feedback systems and HTM inversions Figure 3.14: Feedback connection of two (LPTV) systems. Here, G(s) = A (s)H (s) is the open-loop gain HTM while I is the identity matrix. Hence, computing Hcl(s) involves a matrix inversion. Since, in principle, HTMs are oo-dimensional matrices, LPTV feedback requires us to invert a very large matrix. This problem is equivalent to solving very large systems of equations. Numerically, inverting very large linear systems is a hard problem to solve. Doing SO symbolically even seems impossible. Dealing with these large matrix inversions in an efficient manner is one of the main challenges in rendering HTM-based analysis suited for system and circuit design practice. Since almost any practical application contains feedback, the inversions cannot be avoided. Therefore, it is very important to develop methods that