output signal frequency band around nwo. In short, the HTM H(s) characterizes how information moves from one frequency band to another. This process is illustrated in Fig. 4.1. Besides offering a compact frequency-domain characterization of LPTV system behav- ior, HTMs allow efficient manipulation of LPTV systems and straightforward compu- tation of the overall input-output behavior, given a system's building-block HTMs. For example, the HTMs corresponding to the parallel connection y = H1[u] + H2[u] and the series connection y = H2[H1[u]] are respectively given by (s) = H (s)+H2(s) (4.6) Hx(s) = H2(s)H (s) = (4.7) Note the resemblance with results from LTI system analysis. Furthermore, HTMs cor- responding to basic building-block behavior are straightforward to construct. An LTI system with (scalar) transfer function H(s) corresponds to a diagonal HTM whose el- ements are given by Hnm(s) = 0 (4.8) m n n The elements of a HTM corresponding to a memoryless multiplication y(t) = p(t)u(t), 4.2 PHASE-LOCKED LOOP ANALYSIS with p(1)= are specified by Hn,m(s) = Pn-m (4.9) Using (4.6)-(4.9), HTMs corresponding to more complex systems can be derived from the system's topology and the HTMs of its underlying building blocks. In summary, HTMs allow us to handle LPTV system behavior in a way that largely resembles LTI system analysis. Hereby, the traditional scalar transfer function is re- placed by a matrix of transfer functions: the harmonic transfer matrix. Understanding this basic notion is sufficient to make a beginning with practical HTM-based analysis of LPTV system behavior as occurring in PLLs and mixers. 4.2 Phase-Locked Loop analysis Phase-locked loops (PLLs) are used in both analog and digital systems for generating signals that track the phase of a given input signal. They can be used to reduce oscillator phase noise by locking it to a high-quality reference, to synthesize frequencies which are multiples of the input frequency or, in digital applications, to buffer and deskew clock signals. A rigorous introductio