- tion 3.3.6 indicates how to extend the idea of matrix-based representations to arbitrary (non-periodic) LTV systems. 3.3.1 LPTV systems: implications of linearity and periodicity LPTV systems are characterized by signal content that is translated along the frequency axis. As illustrated in Fig. 3.6, frequency translations are linear operations1. They satisfy the principle of superposition, the basic requirement for linearity. Taking the sum of two frequency-shifted signals is equivalent with frequency shifting the sum of This linearity is seldom stressed. Since frequency-translating circuits, like mixers, implement frequency translations by means of a multiplication of an input signal with a carrier signal, e.g. a cosine, they are often thought of as nonlinear. This is a misconception that complicates the way we reason on these type of circuits. The key issue to observe is that one of the signals involved in the multiplication, the carrier, is known at the time when the circuit is implemented. It can therefore be considered as part of the system and not as an external input. Hence, with regard to the other inputs, the multiplications that give rise to upconversions are linear time-varying operations and not nonlinear ones. 3.3.1 LPTV systems: implications of linearity and periodicity the original signals. Stated in mathematical terms, we find = are (3.23) Furthermore, in LPTV systems, frequency translations are not arbitrary but occur over distances that are a multiple of a fundamental frequency fo. The corresponding car- rier waves ej2nkft in (3.23) are therefore all T-periodic with T = 1/fo. This results in a type of behavior that is termed periodically time-varying. In this section, we dis- cuss the implications of both linearity and periodic time variance on the mathematical representation of LPTV systems. 3.3.1.1 Linearity With LPTV systems being linear, we can use linear time-varying (LTV) system theory [Zad61] to obtain a template that characterizes LPTV input-output behavior. Further- more, we are only 