er, these sections provide an elegant framework to manipulate LPTV systems and to compute the overall input/output behavior, given the system's building block characteristics. As an application of the material, section 3.4.4 shows how the HTM representation of an LPTV system is related to its state-space representation. 3.4.1 HTMs of elementary systems Two fundamental subclasses of LPTV systems are LTI systems and multiplications with a periodic signal p(t). Any other LPTV system can always be modeled as an interconnection of blocks belonging to these basic subclasses. The procedure to deter- mine the HTM representations for these special types of LPTV systems is straightfor- ward. In a first step, we find the kernels h(t,T) associated with either an LTI system or a multiplication. The HTM representation then follows from (3.38) and (3.45). 3.4.1.1 LTI systems For LTI systems, it holds that h(t,T) ) = h t ). This h(t) is the LTI system's impulse response. Note that it does not depend on the variable t (hence the label time-invariant). As a consequence, in the series expansion (3.38), all hk(t) = 0 except for k=0. Using (3.45), we then obtain that the HTM corresponding to an LTI operator with transfer characteristic = L{h(t)} is determined by Hn,m(s) H(s+jmwo) (3.59) Hn,m(s) m n n 3.4 LPTV SYSTEM MANIPULATION USING HTMS H(s-jwo) H(s) : (3.60) H(s+jwo) Hence, the HTM equivalent of an LTI system is a diagonal matrix with frequency- shifted copies of the transfer characteristic H(s) on the diagonal. This diagonal struc- ture makes sense since LTI systems cannot cause information to shift along the fre- quency axis. 3.4.1.2 Multiplication with a periodic signal p(t) The kernel h(t,T) corresponding to a memoryless multiplication y(t) = p(t)(u(t) of the input signal u(t) with a periodic signal p(t), where (3.61) is found by observing that y(t) : (3.62) : p(t)(u(t) (3.63) (3.64) (3.65) We therefore find hk(T) = PkS(t) with S(T) being the Dirac delta function. Using (3.45), the elements of the HTM corresponding to the ope