A linear time-invariant system has finite impulse response when its impulse response is nonzero at only finitely many indices. For a causal filter supported at indices $0,\dots ,M-1$,This is a finite discrete convolution. The coefficients $h[j]$ specify the complete operation; they need not be positive or equal. Such a filter has a simple bounded-input estimate, $|y[n]|\le \Vert x{\Vert }_{\infty }{\sum }_j|h[j]|$, so bounded inputs remain bounded. The moving average chooses $h[j]=1/M$.

A causal filter uses only present and past input values to produce its current output. The trailing moving average is causal; a centered average that uses future samples is not. Future samples can be available when processing a finished recording, but they are unavailable to a live measurement at the current time.

By Gilbert Strang.

Online text: ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/resources/lecture-6-column-space-and-nullspace/

Cited passages: Gilbert Strang, Lecture 6: Column space and nullspace, opening lecture description: the distinct roles of Ax=0 and Ax=b.

The Life blinker alternates between horizontal and vertical lines of three live cells. Start with $B_0=\{(-1,0),(0,0),(1,0)\}$ and every other cell dead. The two endpoints each have one live neighbor and die; the center has two and survives. The dead cells $(0,-1)$ and $(0,1)$ each have three live neighbors and are born. No other dead cell has three. ThereforeThe same neighbor-count argument with horizontal and vertical exchanged gives the second step. Since $B_1\ne B_0$, the period is exactly two, rather than merely a divisor of two.

The Life blinker is a minimal hands-on check of synchronous updating. Mark both endpoints for death and both births before changing anything. Removing an endpoint early would make a prospective birth see only two neighbors; adding a birth early would alter other counts. A correct implementation reproduces both sets above exactly, rather than merely keeping the live population equal to three.

An ångström is a unit of length equal to $10^{-10}\mathrm{m}$.

Online text: gordanz.github.io/stochastic-book/more-about-random-walks.html

The notes homepage gives a last-updated date of 1 May 2024.

Cited passages: §4.3, Problem 4.10 and its solution; §4.3, Problem 4.8 and its solution.

Online text: www.ch.nat.tum.de/fileadmin/w00bzu/tec/Teaching/PC-I_LabCourseScript_WS2021_2.9_B_W.pdf

Cited passages: Chapter 8, printed pp. 82–84, §§8.2.5–8.3.2, and p. 85, §8.5.1: reaction, apparatus, temperatures, endpoint and baseline.