) carrier. Modulating a 1 MHz carrier with an envelope that is oscillating at 1 GHz does not seem a sensible thing to do. In the frequency-domain, slow-varying um (t) correspond to bandlimited spectra Um(s). More specifically, one could require that (3.36) In words, the equivalent baseband spectra Um(jw) are strictly bandlimited to the fre- quency band [-wo/2,wo/2]. The advantages of requiring (3.36) to hold are twofold: 1. The condition (3.36) fixes all degrees of freedom in determining a linear period- ically modulated signal model for a given signal u(t). There is only one solution for the envelopes {um(t)} that satisfies both (3.31) and (3.36). This is of great help in automating computations. 2. With (3.36) being satisfied, the spectra Um(jw) have a clear cut physical inter- pretation: they represent that part of the signal content of u(t) located in the fre- quency band centered around mwo. They do not contribute to the signal content in other frequency bands. It is this interpretation that has inspired the alternate name "equivalent baseband components". It should, however, be stressed that (3.36) is a constraint that can be imposed. From a mathematical point of view, one is not required to do SO. As presented, the HTM frame- work also allows us to deal with envelopes um (t) that contain more rapid variations, i.e. the spectra Um (jw) may have content outside [-wo/2,wo/2]. But what would be the advantages in selecting these "weird" kind of envelopes? Again, there are two reasons: 1. For a single wideband input signal, it is often artificial to break it up into different components. For example, for the signal in Fig. 3.10, a decomposition of the overall spectrum U(s) into the components Um(s) depicted on the left-hand side is counterintuitive. A single logical entity is split up in an artificial manner. Selecting the envelope spectra as on the right-hand side of Fig. 3.10, i.e. U0(s) = U(s) with the other Um(s) set to 0, seems more natural. Note that, since most of the signal energy is contained in [-wn/2,