# Workbook 10 — Companion: reading a sound's fingerprint > [!abstract] What this page is for > Workbook 10 hands you pictures — a live power spectrum, three waveforms with their spectra, two mystery spectrograms, and ten bursts of audio — and asks for written observations. This page practices **the same reading on different pictures**, so the sheet's images stay yours to decode. Three skills, one per step: reading a power spectral density, reading a spectrogram, and asking what a sound's *structure* is for. If you want a live one to poke at first, Google's [Chrome Music Lab: Spectrogram](https://musiclab.chromeexperiments.com/spectrogram/) runs in the browser with your microphone. > > 📄 [[MATH310F26_WB10.pdf|Workbook 10 (blank, 2 pages)]] · [[MATH310F26(Day 11) - Cooking the coefficients|Day 11 note]] · [[MATH310F26(Workbook 9 Companion) - The even reduction|WB9 companion]] > > 🎓 [Submit on Canvas](https://elearning.mines.edu/courses/81632/assignments/587683) --- ## 1. A power spectrum is a list of coefficients you can see Monday's boxed formula $c_{n}=\int_{-L}^{L}f(x)\,e^{-i\omega_{n}x}\,dx$ assigns every frequency a coefficient, and a **power spectral density** is nothing more than those coefficient sizes drawn as spikes — frequency along the bottom, power up the side. So reading one is a two-question exercise: *which* harmonics carry energy, and *how fast* do the amplitudes fade as frequency climbs? The sheet gives you a sine, a square, and a sawtooth. Here is a different pair — the **square against the triangle**, both at a 440 Hz fundamental: ![[wb10_guide_square_triangle.png]] Same skeleton, different flesh. Both waves carry **only odd harmonics** — the evens are silent in each — so the spike *positions* cannot tell them apart. What distinguishes them is the *decay*: the square's amplitudes fade like $1/n$, the triangle's like $1/n^{2}$, and on the dB axis that is the difference between the third harmonic sitting $9.5$ dB below the fundamental (square) and $19.1$ dB below it (triangle). Your ear reads that decay as **timbre**: slow decay means lots of high-frequency energy — the buzzy, icky square — while fast decay leaves mostly the fundamental, the mellow triangle. You have met these exponents before: the sawtooth's $c_{n}\sim 1/n$ came off Monday's board, and the triangle's $c_{n}\sim 1/n^{2}$ is exactly [[MATH310F26(Workbook 9 Companion) - The even reduction|WB9's tent]] — jumps decay slowly, corners decay fast. Carry both questions to the sheet's three waveforms, and to the live capture: a real instrument's spikes ride on a noise floor, and its decay pattern is the instrument's signature. ## 2. A spectrogram is that list, replayed in time A spectrogram stacks one power spectrum per time-slice, colors the power, and lets you watch the coefficients move. Here is a signal that is *not* on your sheet — two plucks, the second an octave above the first — with the reading marked on it: ![[wb10_guide_pluck.png]] The checklist, in the order worth writing it: The **ridges** come first: horizontal bright lines are sustained frequencies, and *equally spaced* ridges are a fundamental with its harmonic stack — one pitched sound source, with the spacing equal to the fundamental. Then **motion**: these ridges hold steady (a held note); ridges that slide are pitch bends, and if the whole stack slides *together*, it is still one source changing pitch — at the second pluck every ridge doubles its frequency at once, which is what an octave *is*. Then **brightness**: power fades going up the stack (the $1/n$-type decay from Step 1, now visible as dimming) and fades left to right (the note dying away). Last, the **attack**: at each pluck onset there is a brief vertical smear across all frequencies — a sudden event is broadband, the uncertainty tradeoff from the rectangular-pulse lecture wearing its audio costume. Sudden in time, wide in frequency. ## 3. Structure is a design decision The sheet's Step 3 shows you engineered sound and asks *why* someone would build it that way. To calibrate what "engineered" looks like, here is a different scheme — one the sheet does **not** use: ![[wb10_guide_morse.png]] One carrier at 600 Hz, keyed on and off: all the information lives in the *durations* — short, long, and the gaps between them. (This one spells a five-letter word Mines students should approve of.) It is the oldest trick in telecommunication: make the sound trivially easy for a receiver to detect (one known frequency) and put the message somewhere else (time). When you reach the sheet's ten bursts, ask the same design questions — where does *its* information live, in time or in frequency? how many frequencies per burst? why would a machine on the other end prefer that? — and say what you see before you guess what it is. ## 4. What to take to the sheet Step 1 is the two PSD questions: which harmonics, how fast the fade — and what the real instrument's spectrum has that the ideal waveforms lack. Step 2 is the checklist: ridges and spacing, motion, brightness in both directions, attacks. Step 3 is the design question: find where the information lives, then hypothesize who is meant to listen. Observations first, identification last — the sheet grades the reading, not the guess. > [!success]- Numbers checked in one sandbox run > The harmonic levels were measured off the synthesized signals' FFTs, not asserted: the square's 3rd and 5th harmonics landed at $-9.5$ and $-14.0$ dB relative to the fundamental (theory $20\log_{10}(1/n)$: $-9.5$, $-14.0$), the triangle's at $-19.1$ and $-28.0$ dB (theory $20\log_{10}(1/n^{2})$: $-19.1$, $-28.0$). The pluck spectrogram's brightest ridges read back $215.3$ Hz and $441.4$ Hz for the synthesized $220\to 440$ Hz octave (the offsets are the FFT's $\sim 10.8$ Hz bin width). The Morse carrier's peak bin returned $602.9$ Hz for the $600$ Hz tone, same bin-width story.