requency band to another. In a first step, this requires the formalization of the concept of "signal content stored in a particu- lar frequency band". This is accomplished by means of a linear periodically modulated signal model u(t)== (4.1) Here, the Um(t) are termed envelopes or equivalent baseband components. They model the signal content of u(t) in the frequency band around mmo. When dealing with LPTV system behavior, it turns out to be more efficient to represent a signal u(t) using the vector of spectra U(s)= = [ U-2(s) U-1(s) U0(s) U1(s) (4.2) rather than the single scalar spectrum U(s) = L{u(t)}. Here, Um(s) =L{um(t)} is the Laplace transform of the equivalent baseband component Um(t)1. Let us be given an input signal u(t) and an LPTV system (4.3) Using the signal representation (4.2) for both u(t) and y(t), it can then be shown that the operation of this LPTV system is described by frequency-domain matrix-vector multiplication, or Y(s)=H(s)U(s) (4.4) The matrix H(s) is called the harmonic transfer matrix (HTM) [Moll00, Vana02c, Were91a] corresponding to the LPTV system H|.]. When truncated and evaluated at S = jw, it corresponds to the harmonic conversion matrix introduced in [Maas88]. The meaning of the HTM H(s) and its elements ,((m) becomes clear when we consider a single row (with index n) in (4.4), i.e. (4.5) The output signal content in the frequency band around nwo is seen to be a linear combination of contributions coming from all possible input frequency bands. The 1t often makes good sense to require the equivalent baseband spectra Um(s) to be bandlimited to [-wo/2,000/2]. However, as was discussed in section 3.3.2 in the previous chapter, this is not strictly neces- sary. Figure 4.1: The HTM elements H1,0(s), H1,1(s) and 1,2(s) characterize the transfer of signal content from different input frequency bands to the output frequency band around the carrier frequency. HTM element ,(m(s) hereby characterizes the transfer of the signal content from the input signal frequency band around mwo to the 