m-th carrier wave 4m(t) = ejiwwot to the n-th carrier wave Yn(t) = einwot In a lot of applica- tions, this is the knowledge that is of greatest interest. 3.3.4 Structural properties of HTMs As is observed from (3.45), the elements of a HTM H(s) can not be chosen indepen- dently. These structural properties of HTMs arise due to non-uniqueness of the signal model (3.31). This non-uniqueness manifests itself as m=-00 (um-1(t)e-john) (3.46) = um(t)ejimoo(t) m=-00 As illustrated in Fig. 3.11, a tone located at a frequency fo + of in the spectrum of um. 1 (t) can equally well be captured as a tone at Af in the spectrum of um(t). This implies that the vectors U0(s) U-1(s+jwo) U0(s+jwo) U(s) : PU(s+j@o)=U's (3.47) U2(s) U1(s+jwo) U3 (s) U2(s+jwo) 3.3.4 Structural properties of HTMs Figure 3.11: A linear periodically modulated signal model is non-unique. A tone located at a frequency fo +Af in the spectrum of Um-1(t) can equally well be captured as a tone at Af in the spectrum of um(t). represent the same signal u(t). Here, P denotes the permutation operator shifting the elements of a matrix downwards, i.e. xm = Xm-1- It can be represented as a matrix with elements Pn,m=S(m-n+1) (3.48) where S(k) EZ-{0,1} is the Kronecker delta function. It equals 1 for k = 0 and 0 otherwise. If both vectors in (3.47) represent the same (input) signal, the same must hold for the (output) signal models obtained by multiplying U(s) and U'(s) with H(s), the HTM corresponding to some arbitrary LPTV operator. If one applies the same input signal to a given system, the same output signal will result, no matter the representation that is used for either signal. As a consequence, we obtain PH(s+jon)Ustjon) = (3.49) Y'(s) (3.50) H(s)U'(s) (3.51) = H(s)PU(s+jwo) (3.52) Since U(s) is arbitrary, this implies that (3.53) In terms of the HTM elements Hn,m(s), this yields An,m(s+jw)n+1,m+1(s) (3.54) Hence, all elements on a same HTM diagonal are frequency-shifted copies of each other. Although (3.54) can also be obtained from (3.45), the derivation abov