) U2(s) ... Tr (3.42) and using a similar definition for Y(s), (3.41) can be rewritten as a matrix-vector mul- tiplication Y(s)=H(s)U(s) . (3.43) Here, H(s) is a doubly co-dimensional matrix H_1,-1(s) H_1,0(s) H_1,1(s) H0,-1 Ho,o(s) (3.44) H1,-1( H1,0(s) with its elements equal to (3.45) The Hk (s) represent the Laplace transforms of the signals hk(T) obtained from the Fourier series expansion (3.38) of the kernel h(t,t). 3.3 LAPLACE-DOMAIN MODELING OF LPTV SYSTEMS USING HARMONIC TRANSFER MATRICES At this point, we introduce some names and notational conventions that will be used throughout this and the next chapter. The matrix H(s) is called a harmonic transfer matrix (HTM) [Moll00, Were91a, Vana02c]. Its elements Hn,m(s) are termed harmonic transfer functions (HTFs). The latter choice differs, for example, from [Royc98] where the name harmonic transfer function is reserved for the Hk(s). However, the quantities Hn,m (s) are the ones that directly model signal transfers from the m-th input to the n-th output frequency band. Hence, they are the ones having physical meaning and therefore deserve the name harmonic transfer functions. Furthermore, a tilde on top of a quantity denotes that that quantity is a structured matrix or an element of a structured matrix. In a structured matrix, the elements cannot be chosen independently of each other. As follows from (3.45), HTMs are structured with their elements related as Hn+k,m+k (s) = Hnm(s+jkwo). We refer to the next section for a more detailed discussion on this structure. The elements of matrices that have no tilde on top of them can freely be chosen. The physical meaning of HTMs and their elements follows directly from (3.41). As was outlined in section 3.1.2 and as illustrated in Fig. 3.3, the harmonic transfer func- tion ,(m(s) directly captures the transfer of signal content from the input frequency band centered around mfo to the output frequency band centered around n.fo. Stated in a more abstract manner: ,((m(s) models the transfer of signal content from the 