ration (3.63) are found to equal Hn,m=Pn-m (3.66) (3.67) Hence, the HTM corresponding to a memoryless multiplication with a periodic signal p(t) is a frequency-independent Toeplitz matrix. 3.4.2 HTMs of LPTV systems connected in parallel or in series Example (Ideal mixer): According to (3.67), the HTM corresponding to the elemen- tary mixing stage y(t)=cos(wot)u(t) (3.68) is given by Amixer(s) and = (3.69) which is a banded matrix. As a test, let us apply this operator on a simple input signal: u(t) = sin(wot). This is a periodic signal with its Fourier coefficients given by (3.70) In section 3.5.1, it will be shown that the Fourier coefficients of the equally periodic (steady-state) output signal y(t) satisfy Y = Hmixer( U. This evaluates as (3.71) Hence, we result is readily veri- fied by straightforward evaluation of y(t) = sin(wot) = sin(2000) 3.4.2 HTMs of LPTV systems connected in parallel or in series In constructing and manipulating linear systems, there are three types of basic connec- tions: parallel, series and feedback connections. Together, they make up a complete set that can be used to realize any interconnect pattern. Within the HTM framework, it turns out to be particularly easy to compute the HTMs corresponding to these ba- sic types of composed systems when given the HTMs of the composing systems. In what follows, we handle parallel en series connections. A separate section is devoted to feedback connections as these are harder to deal with. Computing the HTM representation for a parallel and series connection, given the HTMs of the composing LPTV systems, is straightforward. With H1 (s) and H2(s) the HTMs of two arbitrary LPTV systems, the HTM of their parallel respectively se- ries connection, as illustrated in Fig. 3.12, is given by H.(s) = HI(s)+H2(s) (3.72) Hx(s) = H2(s)HI(s) = (3.73) It is important to note that the order of multiplication in (3.73) cannot be interchanged. Contrary to single-input single-output LTI systems, LPTV systems, in general, do not commute. 3.4 LPTV SYSTEM MANIPUL