nd compo- nent) Um(t) used to modulate the carrier wave Ym(t) = ejmwot. Due to the periodic nature of the carrier waves, waveforms obtained using (3.31) are called linear period- ically modulated waveforms. Also important is the fact that, for any signal u(t), it is always possible to construct a set of envelopes such that (3.31) is satisfied. Such a set constitutes a linear periodically modulated signal model. Example (Modeling a sinewave): Consider the signal u(t) = sin(wo(t)) (3.33) For this simple case, the most obvious linear periodically modulated signal model is specified as (3.34) i.e. a set of constant envelopes. This model is illustrated on the left-hand side in Fig. 3.9. A second linear periodically modulated signal model for u(t) is depicted on the right- hand side in Fig. 3.9. Here, we select (3.35) Admittedly, this second solution is counterintuitive. Nevertheless, it is mathematically correct since (3.31) is satisfied. 3.3.2.2 On the non-uniqueness of linear periodically modulated signal models The example above illustrates that, for a given signal u(t), the set of envelopes {um(t)} for which (3.31) holds is not unique. There are always an infinite number of possibil- ities for which (3.31) is satisfied. However, if we look at Fig. 3.9, we see only one set of envelopes that intuitively seems "right". This solution is the one on the left with all envelopes slow-varying. When talking about modulated sinewaves, it seems natural 3.3 LAPLACE-DOMAIN MODELING OF LPTV SYSTEMS USING HARMONIC TRANSFER MATRICES 10(t) u1(t) Figure 3.9: A sinewave u(t) = sin(wot) can be represented using several dif- ferent linear periodically modulated signal models. The most obvious model is the one on the left encircled in bold. Here, all envelopes are slow-varying. Technically, however, the model on the right is also correct although it is somewhat counterintuitive. 3.3.2 Linear periodically modulated signal models to demand that an envelope um (t) varies at a rate that is slower than that of the cor- responding (sinusoidal