# Administrative Information - **Class Meeting**: Friday, September 11, 2026, 1:00–1:50pm — **Coolbaugh 212**. *(No class Wednesday 9/16 — Career Day.)* - Previous meeting: [[MATH310F26(Day 7) - The Fourier transform, intuitively]] - 📄 [[Teaching/MATH310/MATH310F26/Administration/MATH310_F26_Syllabus_1.0.pdf|MATH 310 F26 Syllabus, v1.0]] > [!info]- Admin. Notes > Expect an LR and WB for each lecture. WBs target 15-minute engagements (longer early on); LRs are short, ~5 minutes. > > - 🎯 **[LR] [Day 8 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121215)** — 4 questions drawn from today's material, unlimited attempts, highest kept — due **Monday 9/14, 11:59pm** > - 🎯 **[LR] [Day 8 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/121216)** — one free response, no wrong answers — due **Monday 9/14, 11:59pm** > - 🎯 **[LR] [Day 8 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121215)** — 4 questions drawn from today's material, unlimited attempts, highest kept — due **Monday 9/14, 11:59pm** > - 🎯 **[LR] [Day 8 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/121216)** — one free response, no wrong answers — due **Monday 9/14, 11:59pm** > - 🎯 **[LR] [Day 7 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121181)** — 4 questions drawn from Wednesday's material, unlimited attempts, highest kept — due **Monday 9/14, 11:59pm** > - 🎯 **[LR] [Day 7 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/121183)** — one free response, no wrong answers — due **Monday 9/14, 11:59pm** > - 🎯 **[CA] [Yahtzee Dice Data](https://elearning.mines.edu/courses/81632/assignments/586037)** — due **today, 10:00am**; late sheets still wanted > - **Individual project feedback** lands on Canvas today — recorded, transcribed, lightly sighed > - 🎯 **[WB] [Workbook 6](https://elearning.mines.edu/courses/81632/assignments/586618)** — conjugate pairs and the exponential representations of the sinusoids; upload one PDF > - 📄 [[MATH310F26_WB6.pdf|Workbook 6 (blank, 1 page)]] > - Start with the companion: [[MATH310F26(Workbook 6 Companion) - What a conjugate pair extracts]] > - 🎯 **[WB] [Workbook 7](https://elearning.mines.edu/courses/81632/assignments/586619)** — the Fourier transforms of $\cos(\omega t)$ and $\sin(\omega t)$, checked with the inverse; upload one PDF > - 📄 [[MATH310F26_WB7.pdf|Workbook 7 (blank, 1 page)]] > - Start with the companion: [[MATH310F26(Workbook 7 Companion) - Two tones and a constant, transformed]] > [!example]- 🗃️ COMAP Corner — three from the archive > Three from the archive — one data, one computational, one weird — for a day of pure tones: > > - **Data/stats — The Influence of Music (2021 ICM D).** Audio features for ~98,000 songs and influencer/follower links among ~5,900 artists across 90 years: build a network model of musical influence and detect evolutionary versus revolutionary genre shifts. The *Earworm* thread, industrialized. 📄 [[2021_ICM_Problem_D_The_Influence_of_Music.pdf|problem PDF]] · [comap.com](https://www.comap.org/component/zoo/?task=callelement&format=raw&item_id=3342&element=71992483-3d3b-47dc-a146-3c901bdea04f&method=download) > - **Computational — Multi-hop HF Radio Propagation (2018 MCM A).** A 100-watt HF signal skips between the ionosphere and the ocean, losing strength with every hop off calm or turbulent seas. How many hops before it drops below 10 dB SNR — and what changes over rugged terrain or for a moving ship? 📄 [[2018_MCM_Problem_A_Multi-hop_HF_Radio_Propagation.pdf|problem PDF]] · [comap.com](https://www.comap.org/component/zoo/?task=callelement&format=raw&item_id=3375&element=71992483-3d3b-47dc-a146-3c901bdea04f&method=download) > - **Just weird — The Leaves of a Tree (2012 MCM A).** Nobody knows offhand how much the leaves on a tree weigh, or why leaves come in the shapes they do. Connect leaf shape and mass to branching structure — do arrangements minimize shadow overlap? — and write it up for a journal editor. 📄 [[2012_MCM_Problem_A_The_Leaves_of_a_Tree_Problem_B_Camping_along_the_Big_Long_River.html|problem page (archive)]] · [comap.com](https://www.contest.comap.com/undergraduate/contests/mcm/contests/2012/problems/) --- # Deliverables - 🎯 **[LR] [Day 8 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121215)** — 4 questions drawn from today's material, unlimited attempts, highest kept - 🎯 **[LR] [Day 8 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/121216)** — one free response, no wrong answers - 🎯 **[WB] [Workbook 6](https://elearning.mines.edu/courses/81632/assignments/586618)** and **[Workbook 7](https://elearning.mines.edu/courses/81632/assignments/586619)** — upload one PDF each Questions and statement are due Monday 9/14 at 11:59 pm. --- # Lecture Boards + Transcript + GenAI **Eighth meeting, September 11 — the spectrum of a pure tone.** One clock on the watch, mini-blinds misbehaving, and a plan: sinusoids today, rectangular pulses in the third slot if time allows. (Narrator: time did not allow. "Nice happy-dappy integrals" awaited Monday the 14th.) **No class next Wednesday** — Career Day. ## Housekeeping Individual **project feedback** went out through the Canvas email channels — responses to your Day 1 what's-out-there questions. How the sausage is made: about an hour of reading and speaking produces ~7,000 transcribed words, cut up and sent per person, with a final pass listening to everything at 4X speed to catch anything crazy ("normally that would just cook my brain"). If a stray "sigh" — or heaven forbid an "eye roll" — made it through, report it. Where things stand: many are headed in a direction, and brainstorming stays open for a couple more weeks, with planning hammered down near the end of the month. The email thread works for follow-ups if you're an email person; otherwise **office hours: MWF 2:00pm**, plus 3:00–4:00pm by appointment (some Wednesday 3pm slots are meetings — schedule ahead and they're yours). Dice data: a look happens over the weekend — "I'm betting on weekend Scott wanting to continue to do work-week Scott stuff" — with at least a minor report Monday. ## The pair, and two things you can do with it Equation one and equation two, our working kit: $\text{(1)}\quad\hat f(\omega)=\mathcal{F}\{f\}=\int_{\mathbb{R}}f(t)\,e^{-i\omega t}\,dt,\qquad\qquad\text{(2)}\quad f(t)=\mathcal{F}^{-1}\{\hat f\}=\frac{1}{2\pi}\int_{\mathbb{R}}\hat f(\omega)\,e^{i\omega t}\,d\omega.$ New physics word for the inverse: **superposition**. The inverse transform takes the primitive waves $e^{i\omega t}$, weights them by $\hat f$, and superimposes — crest meeting trough fills in (**destructive interference**), crest meeting crest embellishes (**constructive interference**) — and by exactly that interference the original data reassembles. Edit $\hat f$ before coming back and you have engineered the signal: lift the low-$\omega$ amplitudes and you've boosted the bass. > [!tip]- Where your music went: MP3 > MP3 is a compression algorithm that begins life in $\hat f$-land. If the room is blasting one pitch and a quieter voice sings a *nearby* frequency, you hear the room, not the voice — so the algorithm **deletes the quiet neighbor outright** and keeps the loud one. Frequencies whose $\hat f$ is dwarfed by a close-by competitor are swept aside, and what remains compresses far smaller. (The scheme has grown more sophisticated since, but that is the founding idea — psychoacoustics doing your data compression.) Streaming has hidden the files from view — "my children just press buttons on Spotify, and then I have to listen to the consequences" — but the transform is still in there, working. ## The interrobang integral, resolved Wednesday left the transform of the primitive wave itself stuck at $\mathcal{F}\{e^{i\omega_{0}t}\}\;=\;\int_{\mathbb{R}}e^{i\omega_{0}t}e^{-i\omega t}\,dt\;=\;\text{‽}$ — an integral that ends by interrogating sinusoids at temporal infinity. (The board awarded it the **interrobang**, a real glyph living unused in your font libraries; never standardized, also answering to *exclamaquest*; both offered free of charge as band names.) Two preliminary observations set up the rescue. **The tone has modulus one.** With $f(t)=e^{i\omega_{0}t}$, conjugation flips every $i$, so $|f|^{2}=f\bar f=e^{i\omega_{0}t}e^{-i\omega_{0}t}=e^{0}=1$ — Workbook 6's arithmetic in one line: the tone lives *on* the unit circle, spinning at frequency $\omega_{0}$, never growing, never decaying. And that is exactly why the classical integral can't converge: nothing decays. **But the pairing makes sense.** The same $f$ happily *induces a distribution*, because the induced pairing borrows its decay from the test function: $\langle T_{f},\varphi\rangle=\int_{\mathbb{R}}e^{i\omega_{0}t}\,\varphi(t)\,dt$ converges beautifully — $\varphi$ is "nice," $|\varphi|\to 0$ faster than any polynomial as $|t|\to\infty$, so when the sinusoids come asking their questions about infinity, $\varphi$ answers: *don't worry about it; I'm going to zero fast enough that you'll never have to know.* **The move.** So don't transform the function — transform the distribution. And here Day 6's lesson replays note for note. There, the step function couldn't accept a derivative, so integration by parts slid the derivative onto the glorious $\varphis. Here, the tone can't accept a Fourier transform, so *the transform slides onto the test function instead*: working in $\omega$-land, the transformed distribution pairs as $\langle\hat T_{f},\varphi\rangle_{\omega}=\int_{\mathbb{R}}\Big(\int_{\mathbb{R}}e^{i\omega_{0}t}e^{-i\omega t}\,dt\Big)\varphi(\omega)\,d\omega=\int_{\mathbb{R}}e^{i\omega_{0}t}\Big(\int_{\mathbb{R}}e^{-i\omega t}\,\varphi(\omega)\,d\omega\Big)dt,$ swapping the order so the yucky green inner integral wraps the test function instead of standing alone. (One board slip caught live: the inner kernel wanted to be $e^{+i\omega t}$ and had to be corrected in red to $e^{-i\omega t}$ — going from $\omegas against a test function, equation one's minus sign does the work.) Now the three-minute stare: *what is the parenthesized quantity?* Compare it to equation one. The integration variable is $\omega$ — but the integration variable is a **stand-in variable** (a 2014 class objected to "dummy"), an emoji you will integrate out and never see again — so the parentheses hold exactly $\hat\varphi$ evaluated at $t$: $\langle\hat T_{f},\varphi\rangle_{\omega}=\int_{\mathbb{R}}e^{i\omega_{0}t}\,\hat\varphi(t)\,dt.$ One maneuver left. That is *almost* equation two read at the point $\omega_{0}$ — it's missing its $2\pi$. Deploy the classic multiply-by-one: $\int_{\mathbb{R}}e^{i\omega_{0}t}\,\hat\varphi(t)\,dt=2\pi\cdot\frac{1}{2\pi}\int_{\mathbb{R}}\hat\varphi(t)\,e^{i\omega_{0}t}\,dt=2\pi\,\mathcal{F}^{-1}\{\hat\varphi\}(\omega_{0})=2\pi\,\varphi(\omega_{0}).$ Stare at what happened: paired against any test function, the transformed distribution **grabs exactly one value of $\varphi$, at $\omega_{0}$, scaled by $2\pi$.** Of all the objects we know, exactly one grabs single values like that. So $\langle\hat T_{f},\varphi\rangle_{\omega}=2\pi\,\varphi(\omega_{0})\qquad\Longrightarrow\qquad\hat T_{f}=2\pi\,\delta(\omega-\omega_{0})\ \ \text{in the sense of distributions,}$ or, in the abuse of notation we run under the hood's protection: $\mathcal{F}\{e^{i\omega_{0}t}\}=2\pi\,\delta(\omega-\omega_{0}).$ **Ideally localized amplitude at the frequency $\omega_{0}$** — the green ink's phrase. Two perspectives on one object: left of the transform, a thing that repeats; right of the transform, the statement *this repeats at exactly this frequency*. That is the lens: point it at repetitious things, read off the frequencies of the repetitions. ## Accosted on the street: $\mathcal{F}\{\cos\}$ Distributions are what's under the hood — but if someone whips out a switchblade in Denver and demands the Fourier transform of cosine, you should be ready. Direct assault on $\int\cos(\omega_{0}t)e^{-i\omega t}dt$ re-earns the sad face, so use Wednesday's machine. Write the conjugate pair (Euler at $\pm\omega_{0}t$, acted out at the board: cosine folds across the vertical axis — **even** — while sine tosses its negative out front — **odd**): $e^{i\omega_{0}t}=\cos(\omega_{0}t)+i\sin(\omega_{0}t),\qquad e^{-i\omega_{0}t}=\cos(\omega_{0}t)-i\sin(\omega_{0}t).$ Add the rows: the sines cancel, two cosines remain, divide by two — the exponential representation. Then the transform, being an integral, splits over the sum and passes the halves through: $\mathcal{F}\{\cos(\omega_{0}t)\}=\tfrac{1}{2}\,\mathcal{F}\{e^{i\omega_{0}t}\}+\tfrac{1}{2}\,\mathcal{F}\{e^{-i\omega_{0}t}\}=\pi\,\delta(\omega-\omega_{0})+\pi\,\delta(\omega+\omega_{0}),$ the halves cancelling against the $2\pis, and the $-\omega_{0}$ tone landing its spike at $+\omega_{0}s mirror. The dual picture: ![[day8_cos_spectrum.png]] No amplitude, no amplitude, spike of weight $\pi$ at $-\omega_{0}$, nothing, spike of weight $\pi$ at $+\omega_{0}$, nothing — and it pays now that we *weight* the spikes rather than think of them as infinities. Fold the spectrum on the vertical axis and the points touch: the **even** cosine has an **even** spectrum, its two spikes reading as the clockwise and counterclockwise polarities of one physical frequency. And the sine? That is [[MATH310F26_WB7.pdf|Workbook 7]], where you run these same steps — the class got as far as "somebody say odd so I can get out of this — *odd*, yes!" Position complete. > [!success]- Numbers checked in one sandbox run > The engine of the derivation — $\int_{\mathbb{R}}e^{i\omega_{0}t}\hat\varphi(t)\,dt=2\pi\,\varphi(\omega_{0})$ — was run numerically with $\varphi(\omega)=e^{-(\omega-1)^{2}/3}$ and $\omega_{0}=3$: both sides returned $1.656230$ (imaginary part $10^{-16}$). The modulus computation returned $|e^{i\omega_{0}t}|^{2}=1.0$ exactly, and the $\mathcal{F}\{\cos\}$ spike weights were confirmed spike-area style in Thursday's workbook run: $+\pi,+\pi$, real and even, at $\omega=\pm\omega_{0}$. ## The animation: watching the delta get built The closer — cued while fielding non-technical questions and any random swearing that needed to take place — was the FFT animation now also embedded in the [[MATH310F26(Workbook 7 Companion) - Two tones and a constant, transformed|Workbook 7 companion]]. Two functions are drawn live, a blue sinusoid and a higher-frequency red one; a second pane adds them; a third pane runs the numerical Fourier transform of the sum as it grows:  > [!quote] Attribution > **"Fast Fourier Transform (FFT) Animation using Matlab"** — meyavuz: [youtube.com/watch?v=-GYB7khbIA0](https://www.youtube.com/watch?v=-GYB7khbIA0). Watch what the class watched for. While the blue sinusoid is *incomplete* — a partial window — many frequencies show excited amplitudes; the moment complete cycles fill the window, **all the energy localizes into one spike**: the delta being built before your eyes, the finite-window smear collapsing toward ideal localization. When red joins blue, a second spike stands up — higher frequency, and (red being louder) taller: which tones, and how much. And then the animation throws in a step function — "what you doing, buddy?" — and a whole ladder of frequencies lights up above the fundamental: what the music business calls **harmonics**, and what Monday's rectangular-pulse integrals would explain. Have a good weekend. > [!example]- Board scans — September 11, 2026 > ![[MATH310F26-Day8-Board-1.jpg]] > ![[MATH310F26-Day8-Board-2.jpg]] > ![[MATH310F26-Day8-Board-3.jpg]] > ![[MATH310F26-Day8-Board-4.jpg]] > ![[MATH310F26-Day8-Board-5.jpg]] > > 📄 [[MATH310F26-Day8-Boards-2026-09-11.pdf|Full board capture (5 pages, PDF)]]