 Symbolic computations become intractable. Sections 3.4 and 3.5 THE STORY BEHIND THE MATH cos (2ft) H (s) H2(s) Hfb(s) Figure 3.1: The characterization the behavior of a downconversion stage with (parasitic) feedback requires understanding and quantifying all signal trans- fers that occur, both wanted and unwanted. The wanted signal components are indicated in white, the unwanted ones are in gray. Only the signal trans- fers to the output frequency band of interest (at DC) are drawn. therefore elaborate approximation strategies that render HTM-based computations ef- ficient. As will be illustrated in this and the next chapter, these strategies allow us to obtain both numerical results and symbolic expressions. 3.1 The story behind the math Telecommunication frontend building blocks like mixers and PLLs involve up- and downconversion of signal content. A GSM transmitter frontend, for example, is re- quired to shift the input signal content, located at baseband, to the 900 MHz band. This is accomplished using one or more mixing stages. Such a mixing stage is a typ- ical example of an LPTV system. In what follows, we present the basic ideas behind the HTM framework that is introduced as a means to model and analyze all aspects of LPTV system behavior. Firstly, however, we consider a simple example that helps us to identify the kind of information a designer needs when dealing with this type of system behavior. 3.1.1 What's of interest: A designer's point of view Let us consider the (LPTV) downconversion stage illustrated on the right-hand side of Fig. 3.1. It consists of a bandpass pre-filtering stage, a mixer responsible for the downconversion and a lowpass filter that suppresses high-frequent residues. Further- more, we've added a (parasitic) feedback from the output to the mixer's input. Such parasitic couplings are, for instance, due to transistor gate-source or gate-drain capac- itances and are typically highpass in nature. This downconversion stage is fed with 3.1.2 Using harmonic transfer matrices to characte