orms. As outlined in chapter 2, the informa- tion contained in a set of signals {um(t) E R - > C} is stored by using them to modulate the properties of a set of carrier waveforms. The most straightforward way to do SO is by means of a linear modulation scheme [Proa01]. Here, the overall signal u(t) is composed as the sum of a set of weighted carriers Ym(t) E R C. The information signals Um(t) are used as the weights, or (3.29) The principles of this linear modulation scheme are illustrated in Fig. 3.8. It can be used to transmit several information signals Um(t) simultaneously. Hereby, each carrier wave 4m(t) can be considered as a separate transmission channel³. The signals m(t) are called the envelopes or also the equivalent baseband components associated with the carriers Ym(t). The harmonic functions (3.30) with wo = 2fo, constitute a set of carrier waves that frequently occur in practice. For (3.30), the transmission channels correspond with evenly spaced frequency bands 3 Of course, in order to avoid excessive interchannel interference, the Um(t) must be restricted to vary on a time scale that is much larger than the time scale over which the different carriers Ym(t) are correlated. 3.3 LAPLACE-DOMAIN MODELING OF LPTV SYSTEMS USING HARMONIC TRANSFER MATRICES "00(t) u1(t) u2(t) Figure 3.8: the information contained in a set of signals {um(t)} (fat solid lines) is stored by using them to modulate the properties of a set of carrier waveforms {Ym(t)} (fast-oscillating lines). 3.3.2 Linear periodically modulated signal models centered around multiples of fo. Hence, given a set of envelopes Um(t), the overall waveform u(t) becomes (3.31) With Um(s) = L L{um(t)} = the Laplace transform of the envelope Um(t), the Laplace transform of the overall signal then equals U(s)= (3.32) As was illustrated in Fig. 3.2, the component Um(t) is upconverted to the frequency band centered around mfo. The notion of the signal content contained in a particular frequency band is therefore formalized as the envelope (equivalent baseba