28 Sampling and Aliasing: Nyquist and Filters

28.1 Start With the Decision

A sensor can report a slow false wave when it samples a fast real signal. Sampling rate and filtering must protect the measurement before analysis begins.

28.2 Route Overview

This is part 1 of 3. Continue with Sampling Design: Product Scenarios.

28.3 Part Objectives

• Apply the Nyquist condition to a bounded sensor signal.
• Explain how aliasing and anti-alias filters change sampled evidence.

28.4 Start Simple

Prove a Fast Signal Does Not Become a Slow Lie

Picture a vibration monitor that reports a gentle wobble while the machine is shaking much faster. The stored trace may be neat and still be false.

Sampling rate means how many measurements are taken each second. State the fastest signal of interest and what the input filter removes before choosing that rate.

Apply a known slow tone, a near-limit tone, and one above the limit. Keep the source frequency, filter, sampling rate, raw samples, spectrum, and false component so another person can repeat the check.

This check covers chosen tones and one signal path, not every transient. The deeper sections develop aliasing, limits, filtering, oversampling, and reconstruction.

Imagine measuring vibration but sampling so slowly that a fast wobble appears as a slow false pattern. Nyquist sampling is the rule that prevents a changing signal from pretending to be something else. Start with the highest frequency you need to trust, then choose sample rate and filtering before collecting data.

28.5 Learning Objectives

By the end of this section, you will be able to:

• Apply Nyquist Theorem: Calculate minimum sampling rates for different signals
• Diagnose Aliasing: Explain why undersampling causes incorrect frequency representation and predict aliased frequencies
• Design Anti-Aliasing Filters: Specify hardware low-pass filter parameters to prevent high-frequency contamination before ADC
• Evaluate Sampling-Power Trade-offs: Select optimal sampling rates that satisfy Nyquist while minimizing energy consumption in battery-powered IoT devices
• Implement Oversampling: Use extra samples to improve effective resolution and reduce anti-aliasing filter complexity

28.6 Prerequisites

Before diving into this chapter, you should be familiar with:

• ADC Fundamentals: Understanding how ADCs convert analog to digital
• Binary Fundamentals: Understanding of resolution and bit depth
• Basic Signal Concepts: Understanding of frequency (Hz) and waveforms

28.7 Why This Chapter Matters

Follow these connections in order: Sampling theory sounds mathematical, but it answers a brutally practical question: how often do I need to read a sensor so the data still represents reality. In real IoT systems, bad sampling choices create fake signals, missed events, wasted battery, and misleading analytics. The device may still produce numbers, but the numbers no longer describe the real world. This chapter gives students a way to choose sampling rates intentionally instead of guessing.

28.8 How To Use This Chapter

• Start with the Nyquist rule and the aliasing examples. Those are the ideas you must keep.
• Read the tables with real signal types slowly and compare them to devices you already know: temperature sensors, microphones, ECG signals, accelerometers, and vibration monitors.
• When you finish a section, ask yourself what would happen in a real product if the sampling rate were cut in half.

28.9 Nyquist-Shannon Sampling Theorem

The first design question is simple: what is the fastest part of the signal you must preserve. Nyquist turns that question into the minimum sampling rate.

Phoebe’s Field Notes: Why Aliases Fold Back

The mathematical gist. Sampling at $f_s$ gives a Nyquist ceiling $f_s/2$. A band ending at $f_{max}$ stays distinct only when $f_s\geq2f_{max}$; otherwise a tone folds to $|f_{signal}-nf_s|$. Thus the chapter’s 800 Hz harmonic sampled at 1000 Hz appears falsely at 200 Hz, while 2500 Hz covers the named 1 kHz precursor with a 1250 Hz Nyquist ceiling.

Math Bridge · guided foundationsWhy can an 800 Hz vibration masquerade as 200 Hz?Let Eddie connect sample rate, Nyquist ceiling, and alias folding. Learn the maths

Inspect Figure 28.1 at this point because its depiction of nyquist-Shannon sampling theorem summary supplies the physical context for the prose.

Figure 28.1 gives separate roles to Nyquist-Shannon Sampling Theorem, pn = period of the “fastest” sine wave, and fn = frequency of the “fastest” sine wave. Following those roles reveals nyquist-Shannon sampling theorem summary. That reading provides the evidence needed for the section’s next calculation, comparison, or field check.

For verification, record the fastest sine-wave period pn, derive fn = 1/pn, and compare the sampling interval with half that period.

The theorem has direct implications for how we choose sampling rates in practice. Bare Nyquist is the mathematical minimum, but real devices usually need extra margin for noise, timing uncertainty, and anti-aliasing filter roll-off.

Trace Figure 28.2 before continuing. It turns how sample rate changes the reconstructed signal from a verbal claim into labelled evidence.

Figure 28.2 gives separate roles to How Sample Rate Changes, Teal is the true signal, dots are the samples,, and 1. Undersampling. Following those roles reveals how sample rate changes the reconstructed signal. That reading provides the evidence needed for the section’s next calculation, comparison, or field check.

28.9.1 Why This Matters

Undersampling (too slow) causes aliasing - high frequencies appear as low frequencies!

Inspect Figure 28.3 before why this matters; its labelled elements make aliasing in a real sampling scenario concrete enough to test against the circuit narrative.

Anchor Figure 28.3 on True waveform: period = 20 ms, which corresponds to the stated Shaft speed = 3000 RPM. Under Signal Aliasing and Sampling, compare that real motion with the more slowly changing sample sequence. The mismatch shows how sparse observations can imply a false shaft speed or direction, connecting the sampling theorem to a concrete monitoring failure rather than an abstract waveform exercise.

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28.10 Practical Sampling Rate Examples

Arduino Uno ADC:

Compare the examples by moving from the stated input to the resulting electrical behavior: Maximum sample rate: ~9,600 Hz (9.6 kHz). Practical rate: ~1-5 kHz for stable readings.

ESP32 ADC:

Compare the examples by moving from the stated input to the resulting electrical behavior: Maximum sample rate: ~83 kHz. Practical audio sampling: 16-48 kHz.

28.11 Understanding Aliasing

Meeting the minimum is only half the story. Out-of-band frequencies can still make sampled data look clean but wrong.

28.12 Sampling Rate vs Power Consumption

Once the signal is sampled honestly, every extra sample must justify its current, storage, and transmission cost.

Checkpoint: Aliasing and Power

You now know:

• Sampling 500 Hz at 500 Hz fails: the Nyquist frequency is only 250 Hz, and the defect can fold to 0 Hz.
• A 1000 Hz design has no margin for an 800 Hz harmonic, which can appear as 200 Hz.
• Cost scales quickly: 1 Hz creates 86,400 samples/day and about 345 kB/day; 1000 Hz creates 86.4 million samples/day and about 345 MB/day.

28.13 Anti-Aliasing Filters

Anti-Aliasing Filter Design Rule:

Before the ADC, use a low-pass filter to attenuate frequencies above the Nyquist frequency ($f_{sample}/2$). In practice, set the filter cutoff below Nyquist to allow for the filter’s gradual roll-off:

$f_{cutoff} \leq \frac{f_{sample}}{2}$

A common practical guideline is $f_{cutoff} = f_{sample}/4$, which provides sufficient transition bandwidth for the filter to reach adequate attenuation by $f_{sample}/2$.

Why? If high-frequency noise (>f_sample/2) reaches the ADC, it aliases to lower frequencies, corrupting data.

Example: 1 kHz accelerometer sampling without anti-aliasing filter

• Motor emits 1.2 kHz vibration harmonic
• Nyquist frequency: 1000 Hz / 2 = 500 Hz
• 1.2 kHz signal aliases to: |1200 Hz - 1000 Hz| = 200 Hz
• False reading: 200 Hz vibration appears in data (doesn’t exist!)
• Solution: Add RC low-pass filter with f_c = 400 Hz before ADC

28.14 ADC Sampling and Aliasing Demo

// Waveform visualization
{
const width = 700;
const height = 250;
const margin = {top: 20, right: 20, bottom: 35, left: 50};
const plotW = width - margin.left - margin.right;
const plotH = height - margin.top - margin.bottom;

// Time span: show 2-3 full cycles of the signal
const cycles = 3;
const duration = cycles / signalFrequency;
const numPoints = 500;

// Generate continuous signal
const continuous = Array.from({length: numPoints}, (_, i) => {
const t = (i / (numPoints - 1)) * duration;
return {t, v: signalAmplitude * Math.sin(2 * Math.PI * signalFrequency * t)};
});

// Generate sample points
const sampleInterval = 1 / samplingRate;
const numSamples = Math.floor(duration / sampleInterval) + 1;
const samples = Array.from({length: numSamples}, (_, i) => {
const t = i * sampleInterval;
const v = signalAmplitude * Math.sin(2 * Math.PI * signalFrequency * t);
// Quantize
const levels = Math.pow(2, adcBits);
const step = (2 * signalAmplitude) / levels;
const quantized = Math.round((v + signalAmplitude) / step) * step - signalAmplitude;
return {t, v: quantized};
});

// Generate aliased (reconstructed) waveform if aliasing
const aliased = isAliasing ? Array.from({length: numPoints}, (_, i) => {
const t = (i / (numPoints - 1)) * duration;
return {t, v: signalAmplitude * Math.sin(2 * Math.PI * aliasedFrequency * t)};
}) : null;

const svg = d3.create("svg")
.attr("viewBox", [0, 0, width, height])
.attr("width", "100%")
.style("max-width", width + "px")
.style("background", "#fafafa")
.style("border-radius", "8px");

const xScale = d3.scaleLinear().domain([0, duration]).range([0, plotW]);
const yScale = d3.scaleLinear().domain([-1.1 * signalAmplitude, 1.1 * signalAmplitude]).range([plotH, 0]);

const g = svg.append("g").attr("transform", `translate(${margin.left},${margin.top})`);

// Axes
g.append("g").attr("transform", `translate(0,${plotH})`).call(d3.axisBottom(xScale).ticks(6).tickFormat(d => (d * 1000).toFixed(1) + " ms"));
g.append("g").call(d3.axisLeft(yScale).ticks(5).tickFormat(d => d.toFixed(1) + " V"));

// Original signal (blue, thin)
const line = d3.line().x(d => xScale(d.t)).y(d => yScale(d.v));
g.append("path").datum(continuous).attr("d", line).attr("fill", "none").attr("stroke", "#3498DB").attr("stroke-width", 1.5).attr("opacity", 0.6);

// Aliased reconstruction (red dashed) if aliasing
if (aliased) {
g.append("path").datum(aliased).attr("d", line).attr("fill", "none").attr("stroke", "#9b1c1c").attr("stroke-width", 2).attr("stroke-dasharray", "6,3");
}

// Reconstructed from samples (green line connecting sample points)
const sampleLine = d3.line().x(d => xScale(d.t)).y(d => yScale(d.v));
g.append("path").datum(samples).attr("d", sampleLine).attr("fill", "none").attr("stroke", "#0b6b50").attr("stroke-width", 1).attr("opacity", 0.5);

// Sample points (orange dots)
g.selectAll("circle").data(samples).join("circle")
.attr("cx", d => xScale(d.t)).attr("cy", d => yScale(d.v))
.attr("r", 3.5).attr("fill", "#E67E22").attr("stroke", "#2C3E50").attr("stroke-width", 0.5);

// Legend
const legend = g.append("g").attr("transform", `translate(${plotW - 200}, 5)`);
legend.append("line").attr("x1", 0).attr("x2", 20).attr("y1", 0).attr("y2", 0).attr("stroke", "#3498DB").attr("stroke-width", 1.5).attr("opacity", 0.6);
legend.append("text").attr("x", 25).attr("y", 4).text("Original signal").attr("font-size", "10px").attr("fill", "#2C3E50");
legend.append("circle").attr("cx", 10).attr("cy", 15).attr("r", 3).attr("fill", "#E67E22");
legend.append("text").attr("x", 25).attr("y", 19).text("ADC samples").attr("font-size", "10px").attr("fill", "#2C3E50");
if (isAliasing) {
legend.append("line").attr("x1", 0).attr("x2", 20).attr("y1", 30).attr("y2", 30).attr("stroke", "#9b1c1c").attr("stroke-width", 2).attr("stroke-dasharray", "6,3");
legend.append("text").attr("x", 25).attr("y", 34).text("Aliased (" + aliasedFrequency.toFixed(1) + " Hz)").attr("font-size", "10px").attr("fill", "#9b1c1c");
}

return svg.node();
}

28.15 Sampling Rate Design Calculator

28.16 Continue to the Next Part

Carry this evidence into Sampling Design: Product Scenarios, which begins with For Kids: Meet the Sensor Squad!.