From Hao Zhang’s Lecture 08 Nonlinear vs. Gaussian One Random Variable For \(Y \sim N(0, \sigma^2)\), let’s calculate \(E(Y^n)\) [E(Y^n) = \begin{cases} 0, \quad n = 2k-1 (2k-1)!! \sigma^{2k}, ...

From Hao Zhang’s 2023 Lecture 08 Gaussian Processes Sample Mean and Variance   \(X_1, X_2, \dots, X_n \overset{\text{i.i.d.}}{\sim} N(\mu, \sigma^2)\), we look at \(\overline{X} = \dfrac{1}{n} \...

Review inner product: an inner product \(\langle x, y \rangle \to \mathbb{R}\) if and only if it satisfies   \(\langle x, y \rangle = \langle y, x \rangle\)   \(\langle x, x \ran...

陈纪修老师的课程 P1 第一章 集合与映射 集合 集合（集），具有某种特定性质，具体的或抽象的对象汇集的总体。集合一般用大写字母，例如\(S,T,A,B,X,Y\)表示。集合中的一个元素一般用小写字母标识，例如\(s,t,a,b,x,y\)。一些记号 [x \in S] [y \notin S] 正整数集合 整数集合 ...

From Hao Zhang’s 2023 Lecture 07 Gaussian Processes Definition   \(n=1, \quad X \sim N(\mu, \sigma^2)\) [f_X(x) = \dfrac{1}{\sqrt{2\pi} \sigma} \exp\Big( -\dfrac{(x-\mu)^2}{2\sigma^2} \Big)...

From Hao Zhang’s Lecture 03 Spectral Analysis Let’s review spectral analysis of deterministic signal. If \(x(t)\) is periodic signal with period \(T\), we have Fourier series. [x(t) = \sum_{k=-...

From Hao Zhang’s Lecture 02 Correlation Functions Correlation functions are special functions, they have many interesting and good properties. [R_{X}(t,s) = E(X(t)X(s))] as we know correlation ...

From Hao Zhang’s Lecture 01 Introduction In stochastic process we study when there are multiple (possibily infinite) random variables. For example, two random variables \(X, Y\). For random vari...

From TCAS-2017: Design Methodology for Phase-Locked Loops Using Binary (Bang-Bang) Phase Detectors [\begin{align} H_{LP}(s) &= H_0 \dfrac{1}{1 + \dfrac{s}{\omega_0 Q} + \dfrac{s^2}{\omega_0^2}...

[H(s) = \mathrm{tf(a,b)} = \dfrac{a_0 s^m + a_1 s^{m-1} + \dots + a_m}{b_0 s^n + b_1 s^{n-1} + \dots + b_n}] [H(z) = \mathrm{tf(a,b,T_s)} = \dfrac{a_0 z^m + a_1 z^{m-1} + \dots + a_m}{b_0 z^n + b_...