ck loops, can be tackled efficiently by exploiting the properties of LPTV systems as they occur in design practice. These methods bridge the gap between HTMs as an abstract theoretical frame- work and HTMs as a powerful tool for both numerical and symbolic computa- tions. In the next chapter, the theory will be illustrated for different telecommunication- related examples like mixers and PLLs. The intent is to demonstrate the practical use of HTMs for analyzing telecommunication systems and their building blocks. Firstly, however, we discuss the theoretical foundations. Laplace-domain modeling of LPTV systems using Harmonic Transfer Matrices This chapter builds on HTMs to model and analyze different aspects of LPTV system behavior. Hence, in a first step, we spend some time to elaborate this concept in depth. The approach here presented in obtaining the HTM representation is to some extent similar to the ones in [Moll00, Were91a]. However, although starting from the same integral representation, it stays closer to the idea of an LPTV system as a system that transfers signal content from one frequency band to another. LAPLACE-DOMAIN MODELING OF LPTV SYSTEMS USING HARMONIC TRANSFER MATRICES U1(s) (s)+U2(s) U2(s) Figure 3.6: Frequency translations are linear operations as the principle of su- perposition holds. Taking the sum of two frequency-shifted signals is equiv- alent with frequency shifting the sum of the original signals. Firstly, section 3.3.1 stresses the linear nature of the frequency translations induced by LPTV systems. This allows us to describe LPTV behavior using an integral rep- resentation. Next, section 3.3.2 formalizes the idea of "the signal content contained in a particular frequency band". In section 3.3.3 both results are combined, yielding a matrix-based representation for capturing LPTV system behavior. Section 3.3.4 dis- cusses some of the structural properties that characterize this matrix representation. Finally, section 3.3.5 comments on the co-dimensional nature of the matrix while sec