epresents the rate of the oscillator signal that drives the LPTV system. This suggests to represent a signal u(t) by means of a vector of spectra U0(s) U(s) = U1(s) (3.1) U2(s) instead of a single scalar spectrum. In (3.1),Um(s) models the signal content of u(t) in the frequency band centered around mfo. Although not strictly necessary, the spectra Um(s) are typically chosen as bandlimited to the frequency band [-fo/2,fo/2]. A HTM-based description of LPTV behavior relies on the property that, given the vectorized signal model (3.1), input-output behavior can be captured by means of the THE STORY BEHIND THE MATH U1(s) Figure 3.2: As information can get transfered from one frequency band to another, it is natural to subdivide a signal's overall spectrum into components contained within evenly spaced frequency bands. These frequency bands are centered around multiples of fo. simple matrix-vector multiplication Yo(s) ,((s) ,((s) ,((s) U0(s) H1,0(s) H1,1(s) H1,2(s) U1 (s) (3.2) H2,0(s) H2,1(s) H2,2(s) U2 (s) The matrix is called a harmonic transfer matrix (HTM). The elements of a HTM can be any scalar transfer function. If we consider a single row of (3.2), for example (3.3) then it is noted that the spectrum Y 1 (s) is a linear combination of all input frequency band spectra Um(s). The matrix elements H1,m(s) represent the corresponding input- output weight factors. Hence, H1,m(s) characterizes the transfer of the signal content from the input signal frequency band around mfo to the output signal band at carrier frequency. This process is illustrated in Fig. 3.3. A HTM-based description of LPTV behavior therefore directly captures the signal transfers that take place between the different frequency bands. It is therefore very useful to help us understand and quantify the behavior of LPTV systems like mixers and PLLs. A different point of view on (3.2) considers each Um(s) as a separate input variable to the system and each Yn(s) as a separate output variable. The relation (3.2) can then be interpreted as the descrip