Jaeyoung Wiki - kb

Interrupts on the 8051

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Interrupts on the 8051

I/O on the 8051

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

I/O on the 8051 The 8051 has 4 GPIO 8-bit ports. Output To write an output to a port, just move some data to that port's register. mov P1, A ; Output the contents of the accumulator on port 1. Individual bits of ports can also be set or cleared.

8051 memory access

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

8051 memory access While the 8051 has 4 kilobytes of built-in ROM (other variants may have 8 kilobytes or none at all), it is also capable of addressing 64 kilobytes of external code memory and 64 kilobytes of external data memory. In the Harvard architecture, the code memory and data memory are separate, whereas they are combined into one block of memory in the von Neumann architecture.

Intel 8051 microcontroller

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Intel 8051 microcontroller The Intel MCS-51, also known as the 8051, is an 8-bit microcontroller developed in 1980 by Intel. Derivatives of the 8051 are still widely used today. Features * Capable of addressing 64 kilobits of program memory and 64 kilobits of data memory

UART on the 8051

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

UART on the 8051 Configuration In order to configure the following registers must be initialized: SCON (Serial port control register) Configuration: 01010000 * SM0, SM1: 01 * Mode 1 (8-bit UART) * SM2: 0 * Disable multiprocessor communication (modes 2/3 only)$$ TH1 = 256 - \frac{K(f_{OSC})}{384f_{BAUD}} $$$f_{BAUD}=9600$$K=1$$f_{OSC}=11.0592MHz$$TH1=253$

Analog electronics

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Analog electronics * Operation amplifier * Transistor * amplifier * analog_filter * Transfer function * Comparator * First-order circuits * Transient response of second-order circuits * Time-averaging circuit variables * sinusoidal_steady_state

Antenna design

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Antenna design The important parameters in antenna design are: * Gain/directivity * Frequency * Bandwidth * Efficiency * Radiation resistance ($Z_{antenna}$) Efficiency The losses in an antenna can be modeled as a series resistance with the radiation resistance. The sources of losses are:$\frac{\lambda}{2}$

Bayesian statistics

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Bayesian statistics The Bayesian influence is using a likelihood function $L_n(\theta)$ that is weighted by prior knowledge. The Bayesian approach is using sample data to update prior beliefs, forming posterior beliefs. To do this, we model the parameter as a random variable, $\pi(\cdot)$$X_1, \ldots, X_n$$n$$L_n(\cdot | \theta)$$X_1, \ldots, X_n$$\theta$$\theta \sim \pi$$$\pi(\theta|X_1, \ldots, X_n) \propto L_n(X_1, \ldots, X_n|\theta) \pi(\theta)$$$$\pi(\theta|X_1, \ldots, X_n) = \frac{L_n(…

BIBO stability

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

BIBO stability A system is bounded-in bounded-out (BIBO) stable if a bounded input will result in a bounded output. A system with unit sample response $h[n]$ will be BIBO stable if and only if $h[n]$ is absolutely summable. That is: $$ \sum_{n=-\infty}^{\infty} |h[n]| < \infty $$ A BIBO stable system is guaranteed to have a $$ |H(j\omega)| = |\int_{-\infty}^{\infty} h(t) e^{-j\omega t} dt| \leq \int_{-\infty}^{\infty} |h(t)| dt < \infty $$

Common probability distributions

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Common probability distributions Gaussian/Normal * Continuous * Parameters * $\mu \in \mathbb{R}$ (mean) * $\sigma^2 > 0$ (variance) * Support: $x \in \mathbb{R}$ * PDF: $\frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$ * Mean/expectation: $\mu$ * Variance: $\sigma^2$ Calculating CDF $P(X < x)$ for $X \sim \mathcal{N}(mu, sigma)$ p = normcdf(x, mu, sigma) Calculating inverse CDF or quantile $\mathbb{P}(X \leq q) = 1 - alpha$$X \sim \mathcal{N}(mu, …

Comparator

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Comparator Comparators compare two voltages and output “high” or “low” based on which voltage is higher. A slightly modified version of the op-amp assumptions apply: $$ i_+ = i_- = 0 $$ $$ v_+ > v_- \to v_{OUT} = V_{DD} $$ $$ v_+ < v_- \to v_{OUT} = V_{SS} $$ Note the following: * The assumption $v_+ = v_-$ is no longer true because the circuit is not in negative feedback.$v_{OUT}$$V_{DD}$$V_{SS}$$v_{OUT} = V_{DD}$$v_{OUT} = V_{SS}$$v_+$$v_-$$$ v_+ = \frac{R_2}{R_1+R_2} v_{OUT} $$$$ v_- …

Notes on computer architecture

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Notes on computer architecture * Hazards * Data hazards * In a pipelined processor, an instruction may follow an instruction that changes the value of the register needed. However, the result may not get written to the register before it is accessed, which will lead to an incorrect out-of-date input for that instruction.

Confidence interval

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Confidence interval A $1-\alpha$ confidence interval is an interval $[\hat{\Theta}^-, \hat{\Theta}^+]$ such that $$ \mathbb{P}^\theta(\hat{\Theta}^- \leq \theta \leq \hat{\Theta}^+) \geq 1 - \alpha $$ for all $\theta$. In other words, there is a $1-\alpha$ probability that the generated confidence interval (random value based on sampling) captures the true value (deterministic). Gaussian case $$ \hat{\Theta} \sim \mathcal{N}(\theta, \mathrm{se}^2) \sim \mathcal{N}(\theta, \hat{\mathrm{se}}^…

Control theory

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Control theory * Transfer function * Laplace transform * Z-transform * block_diagram * black_s_formula * bode_plot * loop_stability * nyquist_plot * State-space model * Observer

Convolution

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Convolution Not that convolution... Definition Discrete time convolution: $$ (f \ast g)[n] \equiv \sum_{m = -\infty}^{\infty} f[m]g[n-m] $$ Continuous time convolution: $$ (f \ast g)(t) \equiv \int_{-\infty}^{\infty} f(\tau)g(t - \tau) d\tau $$ Properties * Convolution is commutative: $$ f \ast g = g \ast f $$ * Convolution is associative: $$ f \ast (g \ast h) = (f \ast g) \ast h $$ * Convolution is distributive over addition: $$ (f + g) \ast x = f \ast x + g \ast x $$ Freque…

Crossvalidation

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Crossvalidation Crossvalidation is used to

Conversion between continuous-time and discrete-time signals

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Conversion between continuous-time and discrete-time signals Continuous-to-discrete transformation Consider a bandlimited continuous-time signal $x_c(t)$: that is, it has no frequency content for $|\omega| \geq \omega_c$. This signal can be converted to a discrete-time (DT) signal $x_d[n]$$x_c(t)$$T$$$ x_d[n] = x_c(nT) $$$$ n = \frac{t}{T} $$$$ \Omega = \omega T $$$\omega_c$$$ f_s = \frac{1}{T} \geq \frac{\omega_c}{\pi} = 2f_c $$$$ X_d(e^{j\Omega}) = \frac{X_c(j\omega)}{T} $$

Delta method

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Delta method The Delta method is used to calculate the asymptotic variance of a random variable that is a function of another random variable. The derivative of the function and the mean and asymptotic variance of the second RV are used. Let $Z_n$ be a sequence of random variables such that$$\sqrt{n}(Z_n - \theta) \xrightarrow [n\to \infty ]{(d)} \mathcal{N}(0, \sigma^2)$$$\sigma^2$$\theta \in \mathbb{R}$$Z_n$$g: \mathbb{R} \to \mathbb{R}$$\theta$$g(Z_n) \xrightarrow [n\to \infty ]{(\textbf{P}…

Digital electronics/computer engineering

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Digital electronics/computer engineering * Transistor * memory * fpga * Intel 8051 microcontroller * Notes on computer architecture SystemVerilog * Basics of SystemVerilog * Loops in SystemVerilog * Conditionals in SystemVerilog * nonsynthesizable_systemverilog

Driven response

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Driven response DT case For a SISO LTI DT state-space system: $$ \mathbf{q}[n+1] = \mathbf{A}\mathbf{q}[n] + \mathbf{b}x[n] $$ $$ y[n] = \mathbf{c}^T\mathbf{q}[n] + dx[n] $$ The state matrix $\mathbf{A}$ can be written as: $$ \mathbf{A} = \mathbf{V}\mathbf{\Lambda}\mathbf{V}^{-1} $$ where $$ \mathbf{V} = \begin{bmatrix} \mathbf{v}_1 | \mathbf{v}_2 | \dots | \mathbf{v}_L \end{bmatrix} $$ * $\mathbf{V}$ is a matrix whose columns are the eigenvectors of $\mathbf{A}$. $$ \mathbf{\Lambda} …

Applications

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Applications * designing_a_pcb * How to pick a microcontroller * ESP32 information

References

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

References * Table of Fourier transforms * laplace_transform_table

Electromagnetics and RF

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Electromagnetics and RF * impedance_matching * Transmission lines * Rectenna design * Antenna design

Energy spectral density

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Energy spectral density Definition of inner/dot product: $$ = \sum_{k = -\infty}^{\infty} x[k]v[k] $$ We can then calculate the dot product of a signal and a time-shifted signal. Let's call that dot product, which is a function of $n$, $p[n]$. $$ p[n] = \sum_{k = -\infty}^{\infty} x[k]v[k-n] $$ This formula can be rewritten as a convolution by defining a new function $\overleftarrow{v}[n] \equiv v[-n]$$v[n]$$$ p[n] = \sum_{k = -\infty}^{\infty} x[k]\overleftarrow{v}[n-k] = (x \ast \ov…

Equity (economics)

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Equity (economics) Equity is an economics term used to describe fairness.

Ergodicity

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Ergodicity For an ergodic random process, the time average of each of its instances equals the ensemble mean of the process. A process is ergodic if its autocovariance goes to zero as $n$ or $t$ goes to $\pm \infty$. Alternatively, it is ergodic if the fluctuation spectral density has no impulse at zero frequency.

ESP32 information

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

ESP32 information

Methods of estimation

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Methods of estimation Given some data, we may want to estimate the parameter(s) of the true probability distribution they come from. There are three methods: plugin, feature matching, and maximum likelihood. Plugin estimator For the plugin estimator, simply plug in the data, weighting each data point by its associated probability.$$ \mu = \mathbb{E}[X] $$$$ \hat{M} = \frac{1}{n} \sum_{i=1}^{n} X_i = \hat{\mathbb{E}}[X] $$$$ v = \mathbb{E}\left[\left(X - \mathbb{E}[X] \right)^2 \right] $$$$ \h…

Sampling distribution of estimator

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Sampling distribution of estimator The estimator for a parameter based on a sample could be different from the true value of the parameter, depending on what the sample is. We can estimate the sampling distribution for the estimator in order to predict how good the estimator is.$\hat{\Theta} - \theta$$b^\theta = \mathbb{E}^\theta[\hat{\Theta} - \theta] $$v^\theta = \mathbb{E} \left[ (\hat{\Theta} - \mathbb{E}^\theta[\hat{\Theta}])^2 \right]$$\mathrm{se}^\theta = \sqrt{v^\theta}$$\mathrm{MSE}^\t…

/etc/sudoers files

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

/etc/sudoers files /etc/sudoers controls who can use the sudo command and how it is used. Whenever editing /etc/sudoers, the visudo command, which verifies the file before writing, should be used in order to prevent a situation where access to sudo

First-order circuits

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

First-order circuits First-order circuits are circuits that, when all independent sources are turned off, can be simplified to either a circuit with a single resistor and a single capacitor or inductor. These are all examples of first-order circuits.$ i_C = C \frac{dv_C}{dt} $$ v_C(0^+) = v_C(0^-) $$v_C(0^-) = 0$$ i_C(\infty) = 0 $$ \tau = RC $$ v_L = L \frac{di_L}{dt} $$ i_L(0^+) = i_L(0^-) $$i_L(0^-) = 0$$ v_L(\infty) = 0 $$ \tau = L / R $$\tau$$$ X(t) = X(0^+) e^{-t/\tau} + X(\infty) \left(…

Fisher information

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Fisher information The Fisher information roughly describes how much information a random variable gives about an unknown parameter of its distribution. It is defined as: $$I(\theta) = {\rm Var}[\ell'(\theta)] = -\mathbb{E}[\ell''(\theta)]$$ From the Fisher information, we can derive the asymptotic variance of the parameter.$$\sqrt{n}(\hat{\theta}_n^{MLE} - \theta^*) \xrightarrow[n \to \infty]{(d)} \mathcal{N}(0, \frac{1}{I(\theta^*)})$$$\theta^*$$\hat{\theta}_n^{MLE}$

Fourier transform

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Fourier transform Discrete-time Fourier transform $$ x[n] \leftrightarrow X(e^{j\Omega}) $$ $$ X(e^{j\Omega}) = \sum_{k = -\infty}^{\infty} x[k] e^{-j\Omega k} $$ $$ x[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(e^{j\Omega}) e^{j\Omega n} d\Omega $$ Continuous-time Fourier transform $$ x(t) \leftrightarrow X(j\omega) $$ $$ X(j\omega) = \int_{-\infty}^{\infty} x(t)e^{-j\omega t} dt $$ $$ x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(j\omega) e^{j\omega t} d\omega $$ Properties of Fourier t…

Table of Fourier transforms

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Table of Fourier transforms CTFT $$ x(t) \leftrightarrow X(j\omega) $$ $x(t)$ (CT signal) $X(j\omega)$ (CTFT) $\delta(t)$ $1$ $\delta(t - t_0)$ $e^{-j\omega t_0}$ $1$ $2\pi \delta(\omega)$ $e^{j\omega_0 t}$ $2\pi \delta(\omega - \omega_0)$ $e^{-at}u(t), \mathrm{Re}\{a\} > 0$ $\frac{1}{\alpha + j\omega}$ $u(t)$ $\frac{1}{j\omega} + \pi \delta(\omega)$ $\frac{\sin \omega_c t}{\pi t}$ $\left\{ \begin{array}{ll} 1, & -\omega_c < \omega < \omega_c \\ 0, & \mathrm{otherwise} \end{arr…

Goodness of fit test

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Goodness of fit test Let $X$ be a random variable. Given i.i.d. copies of $X$, the goodness of fit test determines whether $X$ has a certain distribution (e.g. normal, uniform, Student's T). Multinomial distribution The multinomial distribution is a generalization of the binomial distribution. A multinomial distribution with $K$$K$$K = 2$$n'$$p_i$$i$$\sum_{i=1}^Kp_i$$\textbf{p} = [p_1 \ p_2 \ \ldots \ p_K]^T$$\textbf{N} \in \mathbb{Z}$$N^{(i)}$$i$$\textbf{n}$$\sum_{i=1}^{K}n^{(i)}=n', n^{(i)}…

Hypothesis testing

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Hypothesis testing Hypothesis testing involves deducing the quantity of a hypothesis $H$, which takes on one of the values $H_0, H_1, \dots$ from a measurement $R=r$. Maximum a posteriori rule We can do this by making the decision that minimizes the probability of error *conditional* on the measurement $R = r$$P(H_1|R = r) > P(H_0|R = r)$$H = H_1$$H = H_0$$R = r$$'H_1'$$P(H_1|R = r) < P(H_0|R = r)$$H = H_1$$H = H_0$$R = r$$'H_0'$$$ P(\mathrm{error}|R = r) = \min\{1 - P(H_0|R = r), 1 - P(H_1|R…

Kolmogorov-Lilliefors test

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Kolmogorov-Lilliefors test The Kolmogorov-Lilliefors Test tests if a random variable follows a certain family of distributions (e.g. Gaussian). Donsker's theorem is no longer valid, which means that Kolmogorov-Smirnov values will no longer work. We also cannot plug in estimators from the data into the potential distribution that we are testing against, because that would result in conclusions that are too conservative.$X_1, \ldots, X_n$$F$$\hat{F^0}$$\hat{F^0}$$\mu$$\sigma^2$$X_i$$\hat{\mu}$$\h…

Kolmogorov-Smirnov test

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Kolmogorov-Smirnov test The Kolmogorov-Smirnov test is used to test if an empirical_cumulative_distribution follows a particular distribution. Glivenko-Cantelli Theorem (Fundamental theorem of statistics) Let $F(t)$ be the true CDF of $X_1, \ldots, X_n \stackrel{iid}{\sim} X$. Let $F_n(t)$ be the empirical cdf of $X_1, \ldots, X_n$. Then, $$\sup_{t\in \mathbb{R}}|F_n(t)-F(t)|\xrightarrow[n\rightarrow\infty]{a.s.}0$$ This tells us that as the number of samples increases, the empirical and t…

Laplace transform

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Laplace transform Laplace transforms turn time-domain functions, where $t$ is the variable (time), into frequency-domain functions, where $s$ is the variable (complex frequency). Formal definition $$F(s) = \int_{0}^{\infty} f(t)e^{-st} dt$$ Inverse Laplace transform Similar to the Z-transform, we usually calculate the inverse Laplace transform by reorganizing the Laplace representation into a form we recognize with partial fractions and then pattern matching. Again, the time-domain represen…

Linear regression

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Linear regression Single-variable linear regression Given two random variables X and Y, we can use regression to predict Y from X and estimate the error bars around the prediction. Assume that $(X_i, Y_i), i = 1, \ldots, n$ are i.i.d. from some unknown joint distribution $\mathbb{P}.$ $\mathbb{P}$ can be described by one of the following:$h(x,y)$$X$$h(x) = \int h(x,y) dy$$h(y|x) = \frac{h(x,y)}{h(x)}$$h(y|x)$$Y$$X$$Y$$X=x$$\mathbb{E}[Y|X=x] = \int_{-\infty}^{\infty}yh(y|x)dy$$Y$$X=x$$\int_{-…

Running MacOS in Proxmox

Anonymous (anonymous@undisclosed.example.com) — 2025-03-07T08:04:29+00:00

Running MacOS in Proxmox Follow the steps here: To pass through an RX 6600: * Add PCI passthrough (must be raw device, not mapped) * Disable “Above 4G decoding” and “Resizeable BAR” in host BIOS * Add “agdpmod=pikera” as a boot argument in OpenCore

Market failures

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Market failures Market failures are nonidealities in markets that cause them to behave differently from models. Some examples are: * Possibility of fraud * Imperfection information * Externalities

Microeconomics

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Microeconomics Microeconomics is the study of how individuals and firms make decisions in a world of scarcity. It is a series of constrained optimizations. Individuals and firms try to maximize how well off they are given some constraints. In order to maximize this, they have to deal with tradeoffs, which are driven by opportunity costs. An opportunity cost is the cost of every action (or inaction), which is some other action that could have been performed instead. Every action that can be per…

Minicom

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Minicom Minicom is a TUI program used, among other things, for serial communications.

Minimum mean square error (MMSE) estimator

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Minimum mean square error (MMSE) estimator Consider two random variables $X$ and $Y$ with a joint pdf $f_{X,Y}(x,y)$. Let's say we want an estimator for $Y$ given $X = x_0$ that minimizes the mean squared error: $$ \mathrm{min} E[(Y-\hat{y})^2|X = x_0] $$ This is satisfied by the choosing $\hat{y}$ to be the conditional expectation of $Y$$$ \hat{y} = E[Y|X=x_0] $$$X$$\hat{Y}$$$ \hat{Y} = E[Y|X] $$$\hat{y}_l(x) = ax + b$$$ \underset{a, b}{\min} E[(Y - \hat{y}_l (X))^2] $$$b$$$ b = \mu_Y - a\m…

Motor

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Motor DC motor model $$ u = Ri + L \frac{di}{dt} + E $$ $$ i = \frac{1}{K_t} \tau_m $$ $$ E = K_t \omega_m $$ Where: * $u$ is the voltage applied to the armature of the motor. $[V]$ * $R$ is the armature resistance. $[\Omega]$ * $E$ is the back EMF. $[V]$ * $K_t$ is the torque constant. $[Nm/A]$ or $[Vs]$ * $\omega_m$ is the angular velocity of the motor. $[\mathrm{rad/s}]$ * $\tau_m$ is the motor torque. $[Nm]$$$ \frac{K_t}{R} u = \tau_m + T_e \frac{d\tau_m}{dt} + \frac{K_t^2}…

Multidimensional data

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Multidimensional data

Multivariate linear regression

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Multivariate linear regression Setup $$\textbf{Y}_i = \textbf{X}_i^T \boldsymbol \beta^* + \varepsilon_i, i = 1, \ldots, n$$ where * $\textbf{X}_i$ is the vector of explanatory variables or covariates * $\textbf{Y}_i$ is the response/dependent variable * $\boldsymbol \beta^* = (a^*, \textbf{b}^T)^T$ ($\beta_1^*$ is the intercept) * $\varepsilon_{i=1, \ldots, n}$: noise terms Then, the least squares estimator (LSE) of $\hat{\boldsymbol \beta}$ is the minimizer of the sum of errors s…

Nonparametric hypothesis testing

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Nonparametric hypothesis testing Goodness of fit test

Observer

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Observer Motivation In general, we can't know the values of states $\mathbf{q}$ of a state-space system. We only have access to the input $x$ and output $y$. Observers are used to estimate the values of states $q$ based on the input and output. A realistic state-space model of a system includes some extra terms:$$ \mathbf{q}[n+1] = \mathbf{A}\mathbf{q}[n] + \mathbf{b}x[n] + \mathbf{w}[n] $$$$ y[n] = \mathbf{c}^T\mathbf{q}[n] + dx[n] + \zeta[n] $$$\mathbf{w}$$\zeta$$$ \hat{\mathbf{q}}[n+1] = \…

Operation amplifier

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Operation amplifier An operational amplifier, or op-amp for short, is a circuit with a variety of uses in analog and digital electronics. Op-amp models Ideal The ideal model of an op-amp assumes that the output voltage is proportional to the difference in voltage between the two input nodes, and that it responds instantly to changes in input voltages. Additionally, the input resistance is infinite (no current flows into or out of the input terminals) and the output resistance is zero (ideal …

Power spectral density

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Power spectral density The power spectral density $S_{xx}$ is the Fourier transform of the autocorrelation $R_{xx}$ of a wide-sense stationary process $x$. For CT process $x(t)$: $$ S_{xx}(j\omega) \leftrightarrow R_{xx}(\tau) $$ For DT process $x[n]$: $$ S_{xx}(e^{j\Omega}) \leftrightarrow R_{xx}[m] $$ Instantaneous power Instantaneous power is defined: $$ x^2(t) $$ The expectation of instant power is the autocorrelation with zero time shift:$$ E[x^2(t)] = R_{xx}(0) $$$$ E[x^2(t)] = R_…

Principal component analysis

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Principal component analysis Principal component analysis is essentially boiling down multidimensional data with a lot of dimensions (aka columns) into a few dimensions while keeping most of the information. Given $n$ $m$-dimensional vectors, steps to find the top $k$$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i $$m \times m$$S = \frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})(x_i - \bar{x})^T $$m$$k$$S$$v_1, \ldots v_k$$\lambda_1, \ldots, \lambda_k$$k$$x_i$$\hat{x}_i = (x_i^Tv_1, \ldots x_i^Tv_k)$$$ \min \…

Probabilistic models

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Probabilistic models Let $A$ and $B$ be some event in the sample space $\Psi$. * $P(\Psi) = 1$ * $P(A) \geq 0$ - probability is nonnegative * $P(A \cup B) = P(A) + P(B)$ if $A$ and $B$ are mutually exclusive Conditional probability $$P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{P(AB)}{P(B)} = P(B|A)P(A)$$ Bayes' theorem $$ P(B|A) = \frac{P(A|B)P(B)}{P(A)} $$ Independence $A$ and $B$ are independent if: $$ P(A|B) = P(A) $$ $$ P(AB) = P(A)P(B) $$ Which means that knowing $B$$A$$$ f_X…

Using Python as a math shell

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Using Python as a math shell Python can be used as a MATLAB-like shell for math and stuff. The following components are needed (in order of decreasing importance): * IPython * NumPy * Matplotlib * SciPy * SymPy IPython makes the Python shell behave more like tools such as MATLAB.

QQ plots

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

QQ plots A quantile-quantile plot, or QQ plot, is a visual goodness-of-fit test. It checks if two distributions $F$ (distribution we're testing against) and $F_n$ (empirical distribution) are close. This is done by plotting $F^{-1}$ on the x-axis against $F_n^{-1}$ on the y-axis. If the points are close to the line $y=x$$X_i$$X_i$$i$$$(F^{-1}(\frac{1}{n}),X_{(1)}), (F^{-1}(\frac{2}{n}),X_{(2)}), \ldots , (F^{-1}(\frac{i}{n}),X_{(i)}), \ldots , (F^{-1}(\frac{n-1}{n}),X_{(n-1)})$$$(F^{-1}(\frac{n…

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

%% Heavy tails (t3) nu = ones(1,100) * 3; r = trnd(nu); x = -5:0.01:5; figure; hold on; plot(x, tpdf(x, 3)); plot(x, normpdf(x)); legend("t_3", "N(0,1)"); figure; qqplot(r); %% Light tails (unif [0,1]) r = 2 .* rand(1,100) - 1; figure; hold on; plot(x, unifpdf(x, -1, 1)); plot(x, normpdf(x)); legend("Unif([-1,1])", "N(0,1)"); figure; qqplot(r); %% Right skewed (Exp(1)) mu = ones(1,100); r = exprnd(mu); figure; hold on; plot(x, exppdf(x, 1)); plot(x, normpdf(x)); legend("Exp(1)"…

Random processes

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Random processes Properties of random processes * Expected value of signal at time $t$: $$ \mu_X(t) = E[X(t)] $$ * Autocorrelation of signal at times $t_1$ and $t_2$: $$ R_{XX}(t_1, t_2) = E[X(t_1)X(t_2)] $$ * Autocovariance of signal at times $t_1$ and $t_2$: $$ C_{XX}(t_1, t_2) = E[\tilde{X}(t_1)\tilde{X}(t_2)] = R_{XX}(t_1, t_2) - \mu_X(t_1)\mu_X(t_2) $$ where $\tilde{X}(t) = X(t) - \mu_X(t)$ Properties of two random processes * Cross-correlation of $X(t_1)$$$ R_{XY}(t_1, t_…

Rectenna design

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Rectenna design Overall system The components in a rectenna system are: * Antenna * Matching network * Rectifier The antenna receives the power from RF signals. The rectifier turns the incoming AC voltage into a useful DC voltage. The magnitude of the AC voltage input into the rectifier should be as large as possible in order to maximize the time in which the voltage is greater than the turn-on voltage of the diode. In order to accomplish this, the input impedance of the rectifier shou…

Region of convergence

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Region of convergence RoC in the Z domain The region of convergence (RoC) is defined as the set of $z$ for which the z-transform (and by extension, its infinite series representation) of the signal converges/exists. The ROC cannot include a pole, so any possible regions of convergence exclude the circles $|z| = |p|$$p$$z$$h[n]=0$$n < 0$$z=0$$|z| = 1$$\operatorname{Re}(s) = \operatorname{Re}(p)$$p$$\operatorname{Re}(s) = 0$

Kinematics (robotics)

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Kinematics (robotics) Forward kinematics Jacobian The Jacobian matrix consists of the partial derivatives of the end-effector position with respect to joint angles/lengths. 2D case with 2 revolute joints: $$ \mathbf{J} = \begin{bmatrix} \frac{\partial x_e}{\partial \theta_1} & \frac{\partial x_e}{\partial \theta_2} \\ \frac{\partial y_e}{\partial \theta_1} & \frac{\partial y_e}{\partial \theta_2} \end{bmatrix} $$ This can be used to convert joint speeds into end-effector velocity:$$ \begin…

Statics (Robotics)

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Statics (Robotics) Converting end-effector force to joint torques/forces Just multiply the forces by the transpose of the kinematics Jacobian matrix. $$ \mathbf{\tau} = \mathbf{J}^T \mathbf{F} $$

Supply-demand model

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Supply-demand model The demand curve shows how much of a good people are willing to buy at a given price. It is generally downward sloping because people are willing to buy more of a good if it is cheaper. The supply curve shows how much of a good firms are willing to sell for a given price. It is generally upward sloping.$$ \varepsilon = \frac{\Delta Q / Q_0}{\Delta P / P_0} \leq 0 $$$\Delta Q$$Q_0$$\Delta P$$P_0$$\varepsilon = 0$$\varepsilon = \infty$$$ \gamma = \frac{\Delta Q / Q_0}{\Delta …

Transient response of second-order circuits

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Transient response of second-order circuits The solutions of all time-dependent circuits look like this: $$ x(t) = x_p + x_h(t) $$ where $x_p$ is the steady state or particular solution, and $x_h(t)$ is the transient or homogeneous solution. We can get the steady-state solution by replacing capacitors with opens and replacing inductors with shorts, just as in the first-order case. Finding the transient solution takes a bit more effort and is the subject of these notes.$$ s^2 + 2 \alpha s + \o…

Signal detection

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Signal detection Let $r[n]$ be a noisy signal that is either: $$ H_0: R[n] = W[n] $$ $$ H_1: R[n] = s[n] + W[n] $$ where $s[n]$ is the signal that we are trying to detect, and $W[n]$ is an i.i.d. zero-mean Gaussian process with variance $\sigma^2$. The maximum a posteriori rule can be written as: $$ \frac{f(r[0], r[1], \dots, r[L-1] | H_1)}{f(r[0], r[1], \dots, r[L-1] | H_0)} \overbrace{\gt}^{'H_1'} \underbrace{\lt}_{'H_0'} \frac{p_0}{p_1}$$ Given that $W[n]$ is Gaussian, this can be rewr…

Signal processing

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Signal processing * Transfer function * Fourier transform * Laplace transform * analog_filter * Z-transform * Energy spectral density * Convolution * BIBO stability * Region of convergence * Conversion between continuous-time and discrete-time signals * Power spectral density * Wiener filter * Signal detection

Knowledge base

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Knowledge base Books Electrical engineering Economics Finance Mechanical engineering Probability and statistics Computer Fun facts

State-space model

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

State-space model A state-space model models a system as a set of inputs, outputs, and state variables. * The change in the state variables is a function of state variables and inputs. This change is described by the state evolution equations. * $$ \dot{\mathbf{q}}(t) = \mathbf{A}(t)\mathbf{q}(t) + \mathbf{B}(t)\mathbf{x}(t) $$$$ \mathbf{y}(t) = \mathbf{C}(t)\mathbf{q}(t) + \mathbf{D}(t)\mathbf{x}(t) $$$\mathbf{q}(t) = \begin{bmatrix} q_1 \\ q_2 \\ \vdots \\ q_L \end{bmatrix}$$L$$\mathbf{y…

Stationarity

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Stationarity Strict-sense stationarity A random process is strict-sense stationary if the joint density function of the random variables obtained by sampling that process is invariant under arbitrary time shifts: $$ f_{X(t_1), \dots X(t_\ell)}(x_1, \dots, x_\ell) = f_{X(t_1 + \alpha), \dots X(t_\ell + \alpha)}(x_1, \dots, x_\ell) $$ Wide-sense stationarity A random process is strict-sense stationarity if:$\mu_X(t)$$$ \mu_X(t) = \mu_X $$$R_{XX}(t_1, t_2)$$C_{XX}(t_1, t_2)$$(t_1 - t_2)$$$ R_{…

Basics of SystemVerilog

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Basics of SystemVerilog SystemVerilog is a hardware description language (HDL) used to describe Register-Transfer Level (RTL), which models digital logic as data flows between hardware registers. Synthesis tools convert RTL into a hardware implementation composed of combinational and sequential logic elements. It is an extension of Verilog, an older HDL.

Conditionals in SystemVerilog

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Conditionals in SystemVerilog There is a number of ways to

Loops in SystemVerilog

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Loops in SystemVerilog Loops exist in SystemVerilog, but they do not behave like in computer programs. Loops in computer programs are executed sequentially, but it's not possible to define a sequential loop in synthesizable SystemVerilog. Instead, when the loop is synthesized, each

The Moon Is a Harsh Mistress

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

The Moon Is a Harsh Mistress Plot summary Chapter summary Chapter 1 The narrator describes the “High-Optional, Logical, Multi-Evaluating Supervisor, Mark IV, Mod. L,” or HOLMES FOUR, which is the computer that runs various aspects of the lunar settlement. He has nicknamed the computer

Time-averaging circuit variables

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Time-averaging circuit variables Finding the time average of circuit variables is commonly done in power electronics. It is usually very difficult to calculate time averages explicitly by finding time-varying expressions for the variable in questions and integrating. Instead, we can use the constitutive relations of capacitors and/or inductors to greatly simplify the math.$$ i_C(t) = C \frac{dv_C}{dt} $$$$ v_L(t) = L \frac{di_L}{dt} $$$i_C(t)$$t = 0$$t = T$$$ = \frac{1}{T} \int_0^T i_C(t)…

Transfer function

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Transfer function The transfer function of a system represents the ratio of its output to its input. Transfer functions are written in the Laplace transform or Z transform. $$H(s) = \frac{Y(s)}{X(s)}$$ Examples of transfer functions For a capacitor with capacitance $C$, let $v$, the voltage across the capacitor, be the input, and $i$$$H(s) = \frac{I(s)}{V(s)} = sC$$$\frac{I(s)}{V(s)}$$\frac{1}{sC}$$m$$F$$x$$$H(s) = \frac{X(s)}{F(s)} = \frac{1}{ms^2}$$$$F = ma = m\ddot{x}$$$$F(s) = ms^2X(s)…

Transistor

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Transistor According to Professor Max Shulaker, the invention of fire is what allowed apes to evolve into humans, and transistors are the next invention that is allowing humans to evolve into its next form..? Anyway, the point is, transistors are very important, and all of modern electronics are built with them.$$I_C = I_S(1+\frac{V_{CE}}{V_A})exp(\frac{V_{BE}}{V_{th}})$$$$I_B = \frac{I_C}{\beta_F}$$$$\beta_F = \beta_{F0}(1+\frac{V_{CB}}{V_A})$$$$r_o = \frac{V_A}{I_C}$$$I_S$$V_{th}$$\frac{KT}{q…

Transmission lines

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Transmission lines Wavelength of EM waves in a vacuum $$\lambda=\frac{c}{f}$$ where $c$ is the speed of EM waves and $f$ is the frequency. Wavelength of EM waves in a medium The speed of EM waves depends on the substrate material because it takes time for the charges in the substrate to align to the waves.$$D_k=\epsilon_r$$$$\lambda=\frac{c}{f\sqrt{\epsilon_{eff}}}$$$\frac{1}{20}$$$Z_0=\sqrt{\frac{R+j\omega L}{G+j\omega C}} \approx \sqrt{\frac{L}{C}}$$$$\alpha_t=\alpha_c+\alpha_d+\alpha_r$$…

Trimming videos with FFMPEG

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Trimming videos with FFMPEG To trim video with FFMPEG, use the following command: ffmpeg -i input.mp4 -ss xx:xx:xx -to yy:yy:yy -c copy output.mp4 where * input.mp4 is the input file * xx:xx:xx is the start time * yy:yy:yy is the end time

Uncorrelatedness and independence

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Uncorrelatedness and independence Two random variables $X$ and $Y$ are independent if their joint PDF $f_{XY}(x,y)$ can be separated into a product of their individual PDFs: $$ f_{XY}(x,y) = f_X(x) f_Y(y) $$ For any functions $g(\dot)$ and $h(\dot)$: $$ E[g(X)h(Y)] = E[g(X)]E[h(Y)] $$ Two random variables $X$ and $Y$ are independent if: $$ E[XY] = E[X]E[Y] $$ Alternatively:$$ Cov(X, Y) = 0 $$

Utility

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Utility Utility is an ordinal concept that allows a consumer to rank choices. Consumers behave to maximize their utility. They have preferences, which is what they want, and they get as much as possible within a certain budget. Preference assumptions $ A > B, B > C \implies A > C $$$ U = \sqrt{XY} $$$$ \frac{\partial U}{\partial X} = \frac{1}{2}\sqrt{\frac{Y}{X}} $$$X$$$ MRS = \frac{\Delta Y}{\Delta X} = -\frac{MU_X}{MU_Y}$$$MU_X$$$ \mathrm{Budget} = \mathrm{Income} $$$$ I = X p_X + Y p_Y $$$…

Random variables as vectors

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Random variables as vectors Random variables can be seen as vectors. Inner product The inner product is the generalization of dot products for vectors. The inner product of two random variables is the expected value of their product. The inner product of two random variables $X$$Y$$$ = E[XY] $$$X$$Y$$$ cos(\theta) = \frac{}{\sqrt{}} = \frac{E[XY]}{\sqrt{E[X^2]E[Y^2]}} $$$X$$Y$$E[XY] = 0$$X$$Y$$E[XY] = E[X]E[Y]$$E[\tilde{X}\tilde{Y}] = 0$$\tilde{X} = X - \mu_X$$\tilde…

Wiener filter

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Wiener filter Unconstrained Wiener filter Wiener filtering allows us to estimate the value of one wide-sense stationary random process from measurements of another WSS random process that is jointly WSS. Essentially, a Wiener filter is an LMMSE estimator that uses the values of one process to estimate the values of the other. This can be written in the form:$$ \hat{y} [n] = \mu_y + \sum_{j = 0}^{L-1} h[j] \underbrace{(x[n-j] - \mu_x)}_{\tilde{x}[n-j]} $$$x$$h$$$ \hat{y}[n] = (h \ast x)[n] $$$…

LTI filtering of WSS processes

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

LTI filtering of WSS processes Let $x(\dot)$ be a wide-sense stationary process with: * Mean $\mu_x$ * Autocorrelation $R_{xx}(\tau)$ * Autocovariance $C_{xx}(\tau)$ * $E[x^2(t)] \lt \infty$ Let $y(t) = h \ast x(t)$. Then, the following relations are true: $$ E[y(t)] = H(j0) \mu_x $$ $$ R_{yx}(\tau) = h \ast R_{xx}(\tau) $$ $$ C_{yx}(\tau) = h \ast C_{xx} (\tau) $$ $$ R_{xy}(\tau) = \overleftarrow{h} \ast R_{xx}(\tau) $$ $$ C_{xy}(\tau) = \overleftarrow{h} \ast C_{xx}(\tau) $$ $…

Z-transform

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Z-transform The Z-transform converts a discrete-time signal into a complex frequency-domain representation. It is the discrete-time equivalent of the Laplace transform. Definition Bilateral/two-sided Z-transform: $$ x[n] \leftrightarrow X(z) $$ $$ X(z) = \mathcal{Z}\{x[n]\} = \sum_{n = -\infty}^{\infty} x[n]z^{-n} $$ $$ x[n] = \mathcal{Z}^{-1}\{x[n]\} = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(z) z^n d\Omega |_{z = \bar{r}e^{j\Omega}} $$ Inverse Z-transform Usually, we will compute the invers…

Zero-input response

Anonymous (anonymous@undisclosed.example.com) — 2024-04-30T04:03:45+00:00

Zero-input response A zero-input response (ZIR), or the undriven response, of a state-space system is its output when the input $\mathbf{x} = 0$. In other words, the ZIR is the response of the system to its initial conditions. CT case To find the ZIR of a CT state-space system: * $$ \mathbf{x} \equiv 0 $$$$ \dot{\mathbf{q}}(t) = \mathbf{A}\mathbf{q}(t) $$$$ \mathbf{q}(t) = \mathbf{v}e^{\lambda t} $$$$ e^{\lambda t} \lambda \mathbf{v} = \mathbf{A}\mathbf{v} e^{\lambda t} $$$\lambda$$\mathbf{…