 handle the inversions efficiently. Methods for inverting HTMs are essential in bridging the gap between HTMs as an abstract theoretical framework and HTMs as a powerful tool for both numerical and symbolic computations. In what follows, we discuss three different techniques to invert HTMs. The first one truncates the oo-dimensional HTM and uses, for example, sparse-matrix techniques to handle the inversions. Although appropriate for numerical computations, this technique is very time consuming and ineffective for symbolic computations. Sections 3.4.3.2 and 3.4.3.3 introduce two methods that allow us to deal with LPTV feedback by assuming that the HTMs that are involved satisfy specific properties. The first method assumes the open-loop gain HTM G(s) to have a dominant diagonal (i.e. LTI) component. The second method is based on a rank-reduced approximation of G(s). As will be illustrated in the next chapter, most open-loop gain HTMs encountered in practice can be dealt with using one of both methods. Moreover, both approaches not only allow numerical inversion of HTMs, they also render symbolic computations possible. 3.4.3.1 Inversion by truncation A straightforward approach tackles the inversion in (3.76) by truncating the HTM (I+G(s)) to a finite N-dimensional square matrix. This amounts to neglecting all signal content beyond a particular frequency. The value of N depends on the number of frequency bands that are of importance in both the input and output signal. The ma- trix inverse can then be computed using Krylov subspace algorithms [Barr94, Saad96]. This approach is similar to harmonic balance algorithms [Kund86, Kund90]. It is adopted by most authors that use HTMs or a likewise representation in dealing with LPTV system behavior [Maas88, Molloo, Royc98, Royc99]. 3.4 LPTV SYSTEM MANIPULATION USING HTMS 3.4.3.2 Power series approximation In a lot of practical LPTV systems, a HTM H(s) that needs to be inverted (H(s) = I+G(s) in the case of the feedback example) tends to have a dominant diagonal (LTI) compo