period apart, output signals may have different shapes (compare first and second input-output combination). Fig 3.7 illustrates this definition for Dirac impulses applied at several instances in time. As is observed, if an input signal u(t) is applied at time T, the system produces the same output y(t) as if u(t) were applied at time 0. The only difference of course is that the output is delayed in time by the period T. Note that this invariance does not necessarily hold when the same u(t) is applied at time instances that are not an integer multiple of T apart. To understand the implications of the definition above on the properties of a periodi- cally time-varying linear system with kernel h(t,T), we consider a simple Dirac pulse being applied at a time instance to, i.e. u(t) = 8 (t - to) (3.25) If the system is linear, it immediately follows from (3.24) that y(t)=HS(t-to)]=h(t,t-to) (3.26) 3.3.2 Linear periodically modulated signal models The definition of periodic time variance in turn implies that h(t-T,t-to-T)=DrHS(t-to)] = = H[8(t-to-T)] = h(t,t-to-T) (3.27) Hereby, Dr[.] denotes the delay operator DT[u(t)] =u(t-T). Since to is arbitrary, (3.27) implies h(t-T,t)=h(t,T) (3.28) i.e. the kernel h(t,T) is T-periodic in t. Hence, by pursuing both the implications of the system's linearity and periodicity, we have obtained a first, integral, representation characterizing LPTV system behavior. 3.3.2 Linear periodically modulated signal models The LPTV system representation derived above does not tell us much about their signal processing characteristics. More specifically, it does not clearly reveal how LPTV systems transfer signal content from one frequency band to another. However, in order to describe this kind of LPTV system behavior, we must first formalize the notion of "the signal content that is stored in a particular frequency band". 3.3.2.1 Storing signal content on carriers Storing signal content in a particular frequency band relates to the more general ques- tion on how information is stored on wavef