tion of a multi-input multi-output LTI system. Hence, use of HTMs allows us to consider a single-input single-output LPTV system as a multivariable LTI system. Hereby, the signal content in each frequency band is to be treated as a separate variable. 3.1.3 LPTV behavior and circuit small-signal analysis Up to now, it was taken for granted that linear periodically time-varying systems, and the HTMs associated with them, are relevant to circuit design practice. But, do there 3.1.3 LPTV behavior and circuit mall-signal analysis Figure 3.3: The elements of the harmonic transfer matrix directly charac- terize the signal transfers from the different input frequency bands to the different output frequency bands. exist circuits exhibiting this type of behavior? If so, how are their HTMs related to the circuit equations? It turns out that HTMs are related to mixer and PLL circuits in much the same way as traditional (scalar) transfer functions are related to opamps and filters. More precisely, they arise by linearizing a set of circuit equations near a periodically time-varying operating point. Example (Linearized mixer behavior): Fig. 3.4 shows an elementary mixing stage. As inputs, we apply Vosc(t) cos(wot) (3.4) u(t) = up + Au(t) (3.5) respectively a (large) oscillator signal and an input signal u(t) consisting of a DC bias and a small-signal variation Au(t). If we use a first-order transistor model and if we assume the transistor to operate in the linear region, then we find for the current I(t) flowing through the transistor I(t) = (3.6) (3.7) 3.1 THE STORY BEHIND THE MATH Au(t) Figure 3.4: Elementary mixing stage to which we apply a large oscillator signal Vosc(t) and an input signal u(t). The latter consists of a DC bias and a small-signal variation Au(t). Since Au(t) is small, we can linearize (3.7) around Au(t) = 0. In this way, we obtain I(t)=1op(t)+AI(t) (3.8) Iop(t) = (3.9) AI(t) = (Bcos(wort))Au(t) (3.10) Here, Iop(t) is the operating point current. This is the current that flows through the transistor when a