From Signals to Bits: Sampling and Quantization

Sampling means measuring a signal at regular moments, like taking snapshots, and quantization means rounding each measurement to the nearest rung on a fixed ladder of levels. The name quantization comes from "quantum," a fixed amount that can't be split: every reading has to be a whole number of steps. Together they're how a sensor's smoothly changing voltage becomes the list of numbers a processor can work with. The formulas look like a lot, but they're simple pieces: $f_s > 2f_{\max}$ (take snapshots more than twice as fast as the fastest wiggle), $\Delta = V_\text{ref}/2^N$ (the step is the full range cut into $2^N$ equal pieces) and $6.02N + 1.76$ dB (each extra bit makes the result about 6 dB cleaner). It's really just "how often" and "how finely"; bear with it.

An analog-to-digital converter (ADC) does both jobs: it samples the signal at regular instants, and it quantizes each sample to one of $2^N$ levels.

Sampling and aliasing. Here's the catch with snapshots: between them, the ADC sees nothing. Sampling at rate $f_s$ records $x[n] = x(n/f_s)$. A cosine at frequency $f$ and one at $f + kf_s$ give the same samples for every integer $k$:

So the samples only know $f$ mod $f_s$, the same modular arithmetic as before, and since cosine is even, $-f$ looks like $f$ too. Every frequency folds into the band from 0 to $f_s/2$, the Nyquist frequency, and a tone above it shows up as a false, lower alias. That's the wagon wheel that spins backward on film, which samples at 24 frames per second.

The sampling theorem (Nyquist–Shannon) is the good news: if the signal has no frequencies at or above $f_s/2$, the samples determine it completely. CD audio samples at 44.1 kHz to capture everything up to 20 kHz.

A common mistake. It's tempting to think you can clean up an alias in software after sampling, the way you'd filter any other noise. You can't: once sampled, an alias can't be told apart from a real signal at that lower frequency. That's why an analog low-pass anti-aliasing filter removes the high frequencies before the ADC ever sees them.

Quantization. An $N$-bit ADC with reference $V_\text{ref}$ splits its range into $2^N$ steps of $\Delta = V_\text{ref}/2^N$, one LSB. A 12-bit ADC at 3.3 V has $\Delta \approx 0.81$ mV, and firmware turns a code back into volts with $V \approx \text{code} \cdot \Delta$. Rounding makes an error between $-\Delta/2$ and $\Delta/2$ that behaves like noise spread evenly over that interval. Its mean square is the average of $e^2$:

Comparing a full-scale sine wave's RMS with the noise RMS $\Delta/\sqrt{12}$ gives $\text{SNR} \approx 6.02N + 1.76$ dB: each extra bit halves the noise and buys about 6 dB. 16-bit audio reaches about 98 dB.

uint16_t code = adc_read();          // 0 to 4095 on a 12-bit ADC
float volts = code * (3.3f / 4096);  // one step is about 0.81 mV

PWM goes the other way. A digital pin can only be high or low, but switching it fast, high for a fraction $D$ of each period (the duty cycle), gives an average of

A motor's inertia, your eye watching an LED, or an RC low-pass filter smooths the pulses into that average. A timer with an $N$-bit counter sets $D$ in steps of $1/2^N$ at a PWM frequency of $f_\text{clk}/2^N$, so more resolution means a lower frequency.

Every microphone, accelerometer and temperature sensor on a microcontroller goes through an ADC, so firmware engineers pick a sample rate (Nyquist) and a resolution (6 dB per bit) for each one. PWM runs the other way, dimming LEDs, driving motors and making sound from pins that can only be on or off.