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Minimum Achievable Phase Noise of RC Oscillators

From Reza Navid’s Paper Analytical Formulation of Phase Noise Time-Domain Phase Noise Analysis for Switching-Based Oscillators The transfer function from \(I_{n}(s)\) to \(V_c(s)\) is [\dfrac...

Probability and Stochastic Processes

Stochastic Processes A random variable maps the output of a random experiment, $\zeta$, to a real number. A stochastic process maps the output of a random experiment, $\zeta$, ...

Signal And System Chapter

Signal and System Signal Signal is numbers with index. For example the signal \(x(t-t_0)\) actually means def signal(t): return x(t-t_0) or signal = lambda t: x(t-t_0) Linear System Th...

Signal And System Explain

Signal and System convolution is commutative: proof from [\sum_{k=-\infty}^{\infty} x[k]h[n-k]] let \(k=n-r\) [\sum_{r=-\infty}^{\infty} x[n-r]h[r]] LTI system eigen function: proof [\begin{...

Signal And System 01

Signal and System How to understand signal \(x(t-t_0)\)? If \(t_0\) is positive, it is a delayed version of \(x(t)\); If \(t_0\) is negative, it is a advanced version of \(x(t)\). You can use the...

VCO 01

[\mathcal{L}{VCO}(f_m) = \dfrac{10^{\mathrm{FOM}{\mathrm{VCO}}/10}}{P_{\mathrm{VCO}}/1 \mathrm{mW}} \cdot \dfrac{f_{\mathrm{VCO}}^2}{f_m^2}] for \(LC\) oscillator, take the simple model shown belo...

Paper A Low Noise Sub-Sampling PLL in Which Divider Noise is Eliminated and PD/CP Noise is Not Multiplied by $$N^2$$

From Xiang Gao’s Paper Low Noise Phase Detection Classical Three-Stage PFD/CP The in-band phase noise caused by CP can be calculated as [\begin{align} H_{CP}(s) &= \dfrac{\phi_{out,n}}{i_...

TikZ Tutorial

From link. The doc can also be invoked by texdoc tikz. In LaTex you can use inline code, e.g., \tikz \draw (0pt,0pt) -- (20pt,6pt); to produce inline figures. Tutorial: A Picture for Karl’s Stud...

Introduction to Probability 001

Probability Models And Axioms Lecture 01.1 Youtube Lecture 01.2 Youtube Sample Space Sample space is a list (set) of possible outcomes, \(\Omega\) Event: a subset of the sample space. Prob...

Fourier Analysis 002 Proof of Fejer's Theorem

From book Fourier Analysis Proof of Fejer’s Theorem Theorem 1.5 (i) If \(f: \mathbb{T} \to \mathbb{C}\) is Riemann integrable then, if \(f\) is continuous at \(t\), [\sigma_{n}(f...