o interest. Also, a larger number of terms makes the symbolic results harder to interpret. However, combining results in Fig. 4.30 and 4.31, it is observed that a satisfactory degree of accuracy can be obtained for relatively simple expressions. 4.4 Conclusions and directions for further research Both PLL analysis and the SymbolicHTM algorithm demonstrate that the use of the HTM formalism goes beyond shear theory. They show that it is possible to obtain detailed results -both numerical and symbolic- that characterize the behavior of real- life systems. Hereby, it is possible to automate part of the analysis. This contributes to the power and flexibility of HTMs as a frequency-domain framework for capturing LPTV system behavior. As such, it is made a valuable extension to the well-known techniques for LTI system analysis. HTMs were proven to be an elegant framework for an exact mathematical treatment of (linearized) PLL behavior. HTMs support a detailed description of the time-varying characteristics inherent to all practical PLL implementations. Existing (LTI) techniques for PLL modeling were shown to be specific approximations of the exact time-varying description. For the first time, these traditional models were given solid mathematical groundings with their shortcomings clearly identified. The systems to which HTM-based analysis applies, are by no means exhausted. For ex- ample, an oscillator's impulse sensitivity function (see section 4.2.4.3) can be obtained using HTM-based techniques. The HTMs involved turn out to have a rank-degenerate structure at DC. This can be exploited to streamline computations. A first analysis of oscillator behavior using HTM-based techniques shows promising results for both the efficient computation of steady-state and phase noise behavior. Furthermore, having established methods that capture time-varying linearized PLL be- havior, we can proceed by taking time-varying weak nonlinearities into account. In a PLL, for instance, the PFD and the oscillator make up sources of time-var