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Lab 9: Sampling Rate and Aliasing

Time: ~50 minutes | Prerequisites: Lab 8 | Hardware: Pico 2, INMP441, OLED, a tone source

This lab is going to lie to you

Echo waving welcome On purpose. You'll play a tone that keeps rising, and your Pico will confidently report a frequency that's falling. Not noisy. Not approximate. Precisely, sincerely wrong. Once you've seen it happen you'll never forget why sample rate matters.

What You'll Build

A frequency meter that folds. You'll sweep a tone upward and watch the reading climb, stall, and then run backwards — discovering the Nyquist limit by walking off the edge of it.

Learning Objectives

  • Define sampling rate, sample period and Nyquist frequency
  • Predict what frequency a sampler will report for a tone above Nyquist
  • Demonstrate aliasing with real audio
  • Explain why an anti-alias filter is necessary
  • Justify a sample rate choice for a given application

Concepts Introduced

ID Concept
280 Sample Period
281 Sampling Theorem
282 Nyquist Frequency
283 Aliasing Artifact
284 Frequency Folding
285 Anti Aliasing Filter
286 Undersampling
287 Tone Generator
288 Sample Rate Selection
289 Sample Rate Selection Tradeoff
290 Productive Failure

Background

Sampling is snapshots

A microphone signal is continuous. We look at it only at intervals — 12,800 times a second in this course. Between snapshots, we have no idea what happened.

The Nyquist limit

To capture a wave faithfully you need at least two samples per cycle — one for the top, one for the bottom. So the highest frequency you can represent is half the sampling rate:

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Nyquist frequency = sample rate / 2
Sample rate Highest honest frequency
4,000 Hz 2,000 Hz
8,000 Hz 4,000 Hz
12,800 Hz 6,400 Hz
44,100 Hz (CD) 22,050 Hz

CD audio uses 44.1 kHz because human hearing tops out near 20 kHz. That's not a coincidence.

What happens above the limit

The sampler doesn't fail, complain, or return an error. It reports a different frequency — one that fits. The true frequency folds back like light off a mirror:

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        reported
           ▲
    Nyq ───┼──────╮
           │     ╱ ╲
           │    ╱   ╲
           │   ╱     ╲
         0 └──┴───────┴────► true frequency
              Nyq    rate

At an 8,000 Hz sample rate:

True tone Reported as
3,000 Hz 3,000 Hz ✓
5,000 Hz 3,000 Hz ✗
6,000 Hz 2,000 Hz ✗
7,000 Hz 1,000 Hz ✗

A 7 kHz tone and a 1 kHz tone become genuinely indistinguishable. The information is gone, not merely degraded.

You've seen this before

Echo thinking Wagon wheels rolling backwards in films. A camera samples 24 times a second; if the spokes move slightly less than one full gap between frames, your brain reads it as slow backwards motion. Same phenomenon, different sensor. Aliasing isn't an audio quirk — it's what sampling does.

The fix: filter before you sample

You can't undo aliasing afterwards, so you prevent it: an anti-alias filter removes everything above Nyquist before the sampler sees it. Better a missing high frequency than a fake low one.

Setup

You need a tone source with adjustable frequency:

  • a tone generator app on a phone, or
  • an online tone generator in a browser, or
  • a musical instrument if you're feeling fancy

Hold it near the microphone. Loud and steady works best.

Procedure

Step 1 — Predict first

Before running anything, fill this in. The program samples at 4,000 Hz, so Nyquist is 2,000 Hz.

You play You predict it reads
500 Hz
1,500 Hz
2,500 Hz
3,500 Hz

Step 2 — Run it

Open 09-aliasing.py and run it:

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# Lab 9: Sampling Rate and Aliasing
#
# This lab is designed to LIE TO YOU, and then explain why.
#
# Play a steady tone at the microphone and this program reports the frequency
# it thinks it hears. Start low and walk upward. Somewhere past half the
# sampling rate, the reported frequency stops rising and starts FALLING --
# even though the real tone keeps going up.
#
# That is aliasing. The reading is not noisy or approximate. It is confidently,
# precisely wrong.
#
# We measure frequency by counting zero crossings, because we have not built
# an FFT yet. A clean tone crosses zero exactly twice per cycle.

import config
import math
import struct
import time

SAMPLES = 512
FULL_SCALE = 8388608
NOISE_GATE = 20000        # ignore anything quieter than this (raw RMS)

# Try each of these. Nyquist -- the highest frequency each can represent --
# is half the sample rate.
RATES = [4000, 8000, 12800]


def capture(mic, raw):
    n = mic.readinto(raw)
    words = struct.unpack("<%di" % (n // 4), raw[:n])
    samples = [w >> 8 for w in words]
    dc = sum(samples) / len(samples)
    return [s - dc for s in samples]


def rms_of(ac):
    total = 0.0
    for v in ac:
        total += v * v
    return math.sqrt(total / len(ac))


def estimate_frequency(ac, rate):
    """Count zero crossings to estimate the dominant frequency.

    A sine wave crosses zero twice per cycle, so:
        cycles = crossings / 2
        frequency = cycles / duration
    """
    crossings = 0
    for i in range(1, len(ac)):
        # A crossing is a change of sign between consecutive samples.
        if (ac[i - 1] < 0) != (ac[i] < 0):
            crossings += 1
    duration = len(ac) / rate
    return (crossings / 2) / duration


def what_it_should_read(true_hz, rate):
    """What an ideal sampler reports for a tone at true_hz.

    Below Nyquist it reports the truth. Above it, the frequency FOLDS back
    down like light off a mirror.
    """
    nyq = rate / 2
    f = true_hz % rate
    return f if f <= nyq else rate - f


oled = config.init_display()

print("=== Aliasing demo ===")
print("Play a steady tone near the mic and slowly raise its pitch.")
print("A tone generator app or an online tone generator works well.")
print()
for rate in RATES:
    print("  at %5d Hz sampling, anything above %4d Hz will be a LIE"
          % (rate, rate // 2))
print()
print("Ctrl-C to stop.")
print()

try:
    while True:
        for rate in RATES:
            mic = config.init_microphone(rate=rate)
            raw = bytearray(SAMPLES * 4)
            for _ in range(3):                 # let the new rate settle
                mic.readinto(raw)
                time.sleep_ms(30)

            ac = capture(mic, raw)
            level = rms_of(ac)
            mic.deinit()

            if level < NOISE_GATE:
                reading = "-- too quiet --"
                hz = 0
            else:
                hz = estimate_frequency(ac, rate)
                reading = "%6.0f Hz" % hz

            nyq = rate // 2
            flag = "  <-- ABOVE NYQUIST!" if hz > nyq * 0.95 and hz > 0 else ""
            print("rate %5d Hz (Nyquist %4d) reads %s%s"
                  % (rate, nyq, reading, flag))

            oled.fill(config.BLACK)
            oled.text("Aliasing", 0, 0, config.WHITE)
            oled.hline(0, 10, config.WIDTH, config.WHITE)
            oled.text("rate  %5d" % rate, 0, 16, config.WHITE)
            oled.text("nyq   %5d" % nyq, 0, 28, config.WHITE)
            oled.text("reads %5d" % int(hz), 0, 40, config.WHITE)
            if hz > nyq * 0.95 and hz > 0:
                oled.text("SUSPECT!", 0, 52, config.WHITE)
            oled.show()
            time.sleep_ms(600)
        print()

except KeyboardInterrupt:
    oled.fill(config.BLACK)
    oled.text("Stopped.", 32, 28, config.WHITE)
    oled.show()
    print("Stopped.")

It cycles through three sample rates, reporting what it hears at each.

Step 3 — Sweep upward

Start your tone at about 300 Hz and raise it slowly, watching the 4,000 Hz row.

Somewhere near 2,000 Hz the reading stops climbing. Keep going up. The reported frequency starts coming down. You are now in alias territory.

Step 4 — Find a collision

Find a tone that reads the same at 4,000 Hz sampling as some much lower tone does. Two completely different sounds, one identical reading. That's what "information is lost" means in practice.

If this feels unsettling, good

Echo encouraging Most measurement errors are noise — a bit off, obviously imperfect. Aliasing isn't like that. Your instrument reports a clean, confident, completely fictional number. Sitting with that discomfort is the point of this lab.

Step 5 — Compare rates

Play a 3,000 Hz tone and watch all three rows at once:

Sampling at Nyquist Reads
4,000 Hz 2,000 1,000 Hz — wrong
8,000 Hz 4,000 3,000 Hz — correct
12,800 Hz 6,400 3,000 Hz — correct

Same sound, same room, same microphone. Only the sample rate differs.

Expected Output

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  at  4000 Hz sampling, anything above 2000 Hz will be a LIE
  at  8000 Hz sampling, anything above 4000 Hz will be a LIE
  at 12800 Hz sampling, anything above 6400 Hz will be a LIE

rate  4000 Hz (Nyquist 2000) reads   1500 Hz
rate  8000 Hz (Nyquist 4000) reads   1500 Hz
rate 12800 Hz (Nyquist 6400) reads   1500 Hz

…until you cross a limit, at which point the rows disagree with each other. When two sample rates disagree about the same sound, at least one of them is aliasing. That's a genuinely useful diagnostic.

Troubleshooting

Symptom Likely cause Fix
"too quiet" always Tone not loud enough Move the source closer, or lower NOISE_GATE
Readings jump around Zero-crossing counting is fooled by noise Use a pure tone; avoid background sound
Reads about half the expected Noise adding extra crossings Increase the tone volume
Every rate agrees, no folding Tone still below the lowest Nyquist Go above 2,000 Hz to break the 4 kHz row first
Reading drifts while tone is steady Room echoes Try a less reverberant spot

Challenges

  1. Map the fold. At 4,000 Hz sampling, record the reported frequency for true tones at 500 Hz intervals from 500 to 7,500 Hz. Plot it. You should get a triangle wave.
  2. Predict exactly. Derive a formula for the reported frequency given the true frequency and sample rate, then check it against your measurements.
  3. Pick a rate. You need to capture speech (up to ~3,400 Hz). What's the lowest sample rate that works, and why wouldn't you just use the highest available?

Check Your Understanding

  1. State the Nyquist criterion in one sentence.
  2. At a 10,000 Hz sample rate, what does a 7,000 Hz tone report as?
  3. Why can't aliasing be corrected after sampling?
  4. What does an anti-alias filter do, and where in the chain does it sit?
  5. Why don't we simply sample everything as fast as possible?

You broke it on purpose and learned why

Echo celebrating Every spectrum you build from Lab 21 onward stops at 6,400 Hz — and now you know that isn't a limitation someone chose to annoy you, it's arithmetic.

Next: Lab 10: Bit Depth, Headroom and Clipping | Previous: Lab 8