Why care? A heavy diaphragm misses fast attacks and high-frequency detail. A lighter one follows them more faithfully.
01 — THE SOUND SOURCE
Sound Represented
In Waveform
Signal Shrew sings the simplest signal: one steady tone. Use W/S for amplitude, A/D for wavelength, then hold Space to push the sound into the air.
WAVELENGTH · λ0.78 m
FREQUENCY · f440 Hz
AMPLITUDE · A0.80 Pa
AMPLITUDE
WAVELENGTH
f = 343 / λ Combine Space + W/S + A/D to shape the note live.
READYA4 · 440 Hz
LIVE PRESSURE WAVE · p(t)440 HZ / 0.80 PA
02 FROM NOTE TO TUNE
Twinkle Twinkle
Little Star
Signal Shrew now sings “Twinkle, Twinkle, Little Star.” Each syllable selects a new frequency and envelope; the oscilloscope traces the resulting pressure wave live.
p(t) = Ak(t) sin(2πfkt + φk)
fkthe frequency of note k
Ak(t)each note's attack and release
tkwhen each note begins and ends
Listen + look Faster wiggles are higher notes. The playhead shows which part of the tune is currently pushing the air.
SIGNAL SHREW · TWINKLE TWINKLELIVE MODEL 02
READYC4
CURRENT NOTEC4
FREQUENCY262 Hz
AMPLITUDE0.80 Pa
TEMPO96 BPM
AMPLITUDE
TEMPO · 96 BPM
03 PRESSURE → VOLTAGE
Converting audio
to voltage.
Twinkle’s pressure moves a diaphragm and its coil through a fixed magnetic field. The moving wire separates charge, creating voltage; faster motion makes more voltage, and reversing motion flips its polarity.
mẍ + bẋ + kx = Δp · S
v(t) = NBℓ · ẋ(t)
INTUITION The coil sweeps across magnetic field lines. Faster motion crosses more field per second, giving charges a stronger push. With B, N, and ℓ fixed: 2× speed → 2× voltage.
Why care? Too little damping colors the recording with resonance; too much makes the microphone quiet and dull.
TWINKLE ENTERS A DYNAMIC MICLIVE MODEL 03
READYC4 · 262 Hz
DIAPHRAGM±0.00 mm
COIL VELOCITY±0 mm/s
COIL OUTPUT±0.0 mV
MASS · 1.00×
DAMPING · 0.80×
COIL SPEED · |ẋ|
× NBℓ =
VOLTAGE · |v|
THE HANDOFF ANALOG → DIGITAL
We have voltage.
How do we send it?
The microphone gives us a continuously changing voltage. A computer cannot store or transmit every instant of that curve—it needs a finite sequence of numbers.
ANALOG IN
continuous voltage · v(t)
DIGITAL OUT
B-bit integer codes · [11, 15, 10, 4, 2, 7, 14, 12]
1 · SAMPLE TIMEMeasure the voltage at regular moments.
2 · QUANTIZE VOLTAGERound each measurement to an available binary value.
04 ANALOG → DIGITAL
How many dots
recover the curve?
Twinkle is one continuous voltage. The ADC turns it into an ordered array: every measurement becomes one B-bit integer. How often must we measure for a receiver to reconstruct the curve?
It has d + 1 unknown coefficients, so d + 1 exact points determine it.
quadratic → 3 points
It needs more than two measurements per cycle of its fastest frequency.
fs > 2fmax
STROBE-LIGHT INTUITION
Take more than two snapshots of the fastest cycle.
Sample more slowly and a different, slower wave can hit every same dot. The receiver cannot tell them apart: that ambiguity is aliasing.
OUR TOY TWINKLE
A4 = 440 Hz → sample above 880 Hz
Total samples are N = fs × T. A real voice has higher harmonics, so preserving audio to about 20 kHz calls for 44.1 or 48 kHz.
TWINKLE: ONE VOLTAGE → MANY MEASUREMENTSLIVE MODEL 04
1 · ANALOG INPUT
+0.74 V
height at one instant
2 · NEAREST LEVEL
13 / 15
represents +0.73 V
3 · ONE SAMPLE WORD
1101₂
integer 13 · 4 bits
4 · OUTPUT STREAM
[1000, 1110, 1101, 0100, …]
ordered 4-bit binary words
PHRASE10.0 s
SAMPLE RATE2.0 kHz
TOTAL SAMPLES20,000
VOLTAGE PRECISION4 bit · 16 levels
RATE · 2.0 kHz
BIT DEPTH / VOLTAGE PRECISION · 4 BIT
RECOVERABLE · 2.0 kHz is above the 0.88 kHz minimum
ADC COMPLETE YOU’RE DIGITAL NOW
These numbers
are the audio.
The ADC emits one fixed-width binary word on every clock tick. Keep those words in order and we have PCM: a digital recording of how the voltage changed over time.
- PCM
- Pulse-Code Modulation: a regular sampling pulse produces a binary code whose value follows—or modulates with—the waveform.
- WORD
- One fixed-width group of bits representing one voltage measurement from one channel at one sample time.
SAMPLE CLOCK
2.0 kHz
ORDERED PCM SAMPLE WORDS · x[n]
n=01000
n=11111
n=21011
n=30011
Each word is one amplitude measurement. Its position supplies the time.
Store it, transmit it, or send it directly toward a DAC.
Group samples into blocks to measure their frequency content.
LOCATION A → LOCATION B SEND / RECEIVE
The waveform leaves
as bits.
A physical link—wire, fiber, or radio—carries symbols for 0 and 1. The receiver restores their order, groups them back into sample words, and hands the same PCM numbers to a DAC.
LOCATION A
PCM SENDER
100011101010
PHYSICAL LINK · 0 / 1 SYMBOLS IN TIME
2.0 kHz · 4-bit · mono
LOCATION B
RECEIVER
1000
bits regrouped into one word
100011101010Same values, same timing.
One held voltage per word.
Reconstructed audio · p(t)
RX LOCATION B · RECEIVE + REGROUP
Parse the Binary
Into Voltages
The playback device detects each incoming 0 or 1, uses the shared sample clock to keep time, then groups every B bits back into one PCM sample word.
BIT SYMBOLS→WORDS→PCM BUFFER
Format agreement Sender and receiver must agree on sample rate, bit depth, channel count, and where each word begins.
REBUILD THE SAMPLE STREAMREMOTE PLAYER
One physical 0/1 decision at a time.
1000
WORD = 1000
At 4-bit depth, every four bits complete one word.
100011101010Word n must reach the DAC before word n + 1.
2.0 kHz · 4-bit · mono · one word every 0.5 ms
05 DIGITAL → ANALOG
Each code becomes
a held voltage.
The DAC maps each received integer code to one voltage level and holds it until the next sample arrives. A reconstruction filter then smooths the rapid steps into a continuous, band-limited voltage.
x(t) = Σ x[n] sinc((t−nTs)/Ts)
ONE WORD → ONE VOLTAGELIVE MODEL 05
RECEIVED WORD1000
→INTEGER CODE8 / 15
→HELD OUTPUT+0.07 V
held voltage reconstructed
06 THE SPEAKER
Voltage moves.
Air sings.
The reconstructed voltage drives current through a voice coil. Magnetic force moves the attached cone; the cone compresses and rarefies the air, rebuilding the pressure wave for Signal Shrew to hear.
F = Bℓi
VOLTAGE→FORCE→AIR
Full circle The microphone’s mass-spring-damper model returns, now driven in reverse.
SIGNAL SHREW · LISTENING
THE MICROPHONE PATH · RUN IN REVERSELIVE MODEL 06
CONE DAMPING · 0.70×
FREQUENCY · 220 Hz
07 POLYPHONY · SUPERPOSITION
Many notes.
One waveform.
Real sampled recordings of a trumpet on C4, piano on E4, and guitar on G4 form a C-major harmony. The faint sine guides show each note’s fundamental; the jagged colored traces add overtones and attack—the shape we hear as timbre. The microphone receives their instantaneous sum, shown at right, as one more complicated voltage.
ptotal(t) = Σi pi(t)
AIRall pressure waves overlap
→MICone summed voltage v(t)
→ADCone number x[n] per instant
The ownership is gone The summed waveform preserves every contribution, but it does not label which instrument made which frequency. An FFT can reveal frequency content; separating instruments is a harder source-separation problem.
THREE RECORDED INSTRUMENTS · ONE MICROPHONELIVE MODEL 07
ACTIVE SOURCES3 NOTES
ADD →MIC VOLTAGE NOW+0.00 V
SAMPLE →ONE PCM WORD1000
08 THE FOURIER LENS
Playback skips it.
Analysis needs it.
We freeze 4,096 PCM samples from the trumpet–piano–guitar waveform you just saw. The 3D Fourier cube places that waveform on its time face, the spectrum on its frequency face, and a few sinusoidal components on slices inside. Hold the arrow keys to orbit it through 360°, then hover a component or bin to hear it by itself.
X[k] = Σ x[n]e−j2πkn/N
x[n]N samples in one short time window
X[k]N frequency-bin coefficients, each with magnitude and phase
FINDpitch, beats, resonances, and spectral features
CHANGEedit selected bins, then transform back to PCM
MEASUREbandwidth, distortion, and energy by frequency
FFT = Fast Fourier Transform It efficiently computes the same frequency coefficients as a discrete Fourier transform. It changes the view—not the underlying sound.
ONE PCM BLOCK · A 3D FOURIER VIEWLIVE MODEL 08
HOLD TO ORBIT · 360°← → TURN · ↑ ↓ TILT
INPUT · 92.9 ms4,096 PCM SAMPLESthe section 07 composite
FFT →OUTPUT2,049 UNIQUE BINSmagnitude + phase; mirrored half omitted
CLICK THE VISUAL ONCEplanes show exact notes · bars show FFT bins
STRONG BINSC4 · E4 · G4
09 INSIDE ONE FFT BIN
Wrap the wave.
Measure the lean.
For one clean look at the math, keep only the chord’s three fundamentals. To test one frequency, the transform wraps every sample around a circle rotating at that frequency. A mismatch balances around the origin; a match leaves a large vector.
X[k] = Σ x[n]e−j2πkn/N
e−jθrotate around the complex plane
x[n] · e−jθturn each sample into a little vector
Σadd every vector tip to tail
|X[k]| ∠ X[k]distance is magnitude; angle is phase
One test becomes a spectrum Repeat this measurement for every candidate frequency. Place each resulting magnitude side by side and you get the frequency face of the cube.
THREE FUNDAMENTALS · ONE CANDIDATE FREQUENCYLIVE MODEL 09
TEST FREQUENCY · k262 Hz
→VECTOR MAGNITUDE · |X[k]|0%
∠PHASE · arg X[k]0°
HOLD A / D TO SCANlisten for the test tone and watch the vector
WHY IS IT “FAST”?Open the 8-sample butterfly →
THE DFT ANSWER, WITH REUSED WORK8 SAMPLES · 3 STAGES · 4 BUTTERFLIES PER STAGE
DIRECT 8-POINT DFT64 SAMPLE/BIN PAIRS
→RADIX-2 FFT12 REUSABLE BUTTERFLIES
09 SPECTRAL EDITING
Find it.
Change it.
Put it back.
An FFT is useful because we can edit the frequency coefficients. Multiply the spectrum by a mask, then run the inverse FFT to turn those edited bins back into PCM samples.
y[n] = IFFT{ M[k]X[k] }
FFTtime samples → frequency bins
×MASKkeep, reduce, or remove bins
→IFFTedited bins → time samples
A clean target A steady 60 Hz electrical hum occupies a narrow band, so a notch can remove it with little damage to the rest of the mix.
THE SAME COMPOSITE · EDITED IN FREQUENCY SPACELIVE MODEL 09
INPUTMIX + 60 HZ HUM
FFTMASKCUT 45–75 HZ
IFFTOUTPUTCLEANER MIX