From Robert Bogdan Staszewski Group’s Paper Phase Noise of Charges-Sharing Locking Timestamps of Oscillator and Reference [\begin{align} t_{ref}[n] &= n T_{ref} + \Delta t_{ref}[n] t_{osc}...

Paper Multirate Timestamp Modeling for Ultralow-Jitter Frequency Synthesis: A Tutorial High-bandwidth PLL (\(\ge 30 \% f_{ref}\)). ADPLL with TDC and digitial loop filter. Injection locki...

Heaviside step function [u(t) = \begin{cases} 1 & \text{ for } \quad t > 0, 0 & \text{ for } \quad t \le 0 \end{cases}] \(H(s)\) \(h(t)\) \(\text{init co...

Caratheodory’s Extension Theorem We already know given a \(\sigma\)-additive \(\mu: \mathscr{S} \to [0,\infty]\), there is an unique extension to an algebra with \(\sigma\)-additive \(\nu: \math...

Set Functions Additive Given \(\mathscr{C} \subseteq \mathscr{P}(\Omega)\) and \(\emptyset \in \mathscr{C}\). The function \(\mu: \mathscr{C} \to \mathbb{R}_+ \cup \{\infty\}\) is additive, if ...

Classes of Subsets Semi-Algebras Definition: \(\mathscr{S} \subseteq \mathscr{P}(\Omega)\) is a semi-algebra, if and only if: (i) \(\Omega \in \mathscr{S}\) (ii) \(A, B \in \mathscr{S} \impli...

Introdution: A Non-Measurable Set Can we define a measure on \(\mathbb{R}\), such that it works on all the subsets of \(\mathbb{R}\)? (i) \(\lambda: \mathfrak{P}(\mathbb{R}) \to \mathbb{R}_+ \c...

From book_probability_theory_and_examples Measure Theory Probability Spaces Here and throughout the book, terms being defined are set in boldface. Here and in what follows, countable means fini...

From book_topics_in_the_theory_of_random_noise Chapter 1: Random Functions and Their Statistical Characteristics 1. Random Variables A random variable $\xi$ should have definition mean of any fu...

From Ali Hajimiri’s Paper [h_{\phi}(t,\tau) = \dfrac{\Gamma(\omega_0 \tau)}{q_{max}} u(t-\tau)] [\mathcal{L}(\Delta \omega) = \dfrac{\dfrac{\overline{i_n^2}}{\Delta f} \Gamma_{rms}^2}{2 q_{max}^2...