62  ADC Fundamentals: Sampling and Aliasing

62.1 Learning Objectives

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

• Distinguish Analog from Digital: Understand why sensors produce analog signals and why processors need digital data
• Apply Sampling Concepts: Explain the Nyquist theorem and calculate appropriate sampling rates
• Recognize Aliasing: Identify when sampling is too slow and how to prevent signal distortion
• Design Anti-Aliasing Solutions: Choose appropriate filters to prevent sampling artifacts

TipFor Beginners: Why Signal Processing Matters

The Problem: Sensors measure the real world (temperature, light, pressure), which is continuous and smooth. But microcontrollers and computers only understand discrete numbers (0s and 1s).

The Solution: Signal processing bridges this gap by converting continuous analog signals into discrete digital values that computers can process.

Real-World Analogy: Imagine watching a movie. The real world moves smoothly, but a movie is actually 24 still pictures per second. Your brain fills in the gaps, so it looks smooth. Signal processing does the same thing - it takes “snapshots” of sensor values fast enough that we capture all the important information.

Key Terms:

Term	Simple Definition
Analog Signal	Continuous value that can be anything (like temperature: 23.456789…C)
Digital Signal	Discrete value limited to specific numbers (like 23, 24, 25C)
Sampling	Taking snapshots of an analog signal at regular intervals
Sampling Rate	How many snapshots per second (measured in Hz)
Aliasing	Distortion when sampling too slowly (makes fast signals look slow)

The Bottom Line: Sample fast enough (at least 2x the fastest change you care about) to capture what matters without wasting resources.

NoteReal-World Example: ECG Heart Rate Monitor

A wearable ECG sensor demonstrates the complete signal processing pipeline:

Raw Signal Characteristics: - Cardiac signal: 0.5-40 Hz frequency range (heart rate + waveform shape) - Powerline noise: 50/60 Hz interference from electrical environment - Muscle artifacts: High-frequency noise from arm movement - DC drift: Baseline wander from electrode contact variation

Processing Chain:

1. High-pass filter (0.5 Hz cutoff): Removes DC drift and baseline wander
2. Notch filter (50/60 Hz): Eliminates powerline interference
3. Low-pass filter (40 Hz cutoff): Removes muscle artifacts and high-frequency noise
4. Anti-aliasing filter (100 Hz): Prevents sampling distortion
5. ADC sampling (250 Hz): Nyquist requires 2x the max frequency = 200 Hz minimum; 250 Hz provides 25% safety margin above Nyquist
6. R-peak detection: Pan-Tompkins algorithm identifies heartbeat peaks
7. Heart rate calculation: 72 BPM from R-R interval measurement (833 ms between peaks)

System Performance: - ADC resolution: 12-bit (0.8 mV precision for 3.3V range) - Power consumption: 1.2 mW continuous operation - Battery life: 8 hours on 50 mAh battery - Accuracy: +/-2 BPM compared to medical-grade ECG - Latency: <100 ms from heartbeat to BPM update

Key Lesson: Signal processing isn’t just about converting analog to digital - it’s about intelligently filtering noise, choosing appropriate sample rates, and balancing accuracy with power consumption.

62.2 Analog vs Digital Signals

Time: ~8 min | Level: Foundational | Unit: P02.C05.U01

62.2.1 The Continuous World

Physical phenomena are inherently continuous - temperature changes smoothly, light intensity varies gradually, and sound waves oscillate continuously.

Example: A thermometer might show 23.456789…C at one instant and 23.458123…C a moment later. There are infinite possible values between any two measurements.

62.2.2 The Digital Constraint

Microcontrollers and computers work with discrete digital values: - 8-bit number: Can only represent 256 different values (0-255) - 12-bit number: 4,096 different values (0-4095) - 16-bit number: 65,536 different values (0-65535)

This creates two challenges:

1. Sampling: How often do we measure the signal? (temporal discretization)
2. Quantization: How precisely do we measure each sample? (amplitude discretization)

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flowchart LR
    A["Analog Signal<br/>(Continuous)"] --> B["Anti-Aliasing<br/>Filter<br/>(Low-Pass)"]
    B --> C["Sample & Hold<br/>Circuit"]
    C --> D["ADC<br/>(Analog-to-Digital<br/>Converter)"]
    D --> E["Digital Value<br/>(Discrete)"]

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    style B fill:#16A085,stroke:#2C3E50,stroke-width:2px,color:#fff
    style C fill:#16A085,stroke:#2C3E50,stroke-width:2px,color:#fff
    style D fill:#2C3E50,stroke:#16A085,stroke-width:3px,color:#fff
    style E fill:#E67E22,stroke:#2C3E50,stroke-width:3px,color:#fff

Figure 62.1: Analog-to-Digital Conversion Pipeline from Sensor to Digital Value

ADC Pipeline: The complete analog-to-digital conversion process. Anti-aliasing filter removes high frequencies before sampling to prevent distortion. Sample & hold captures instantaneous voltage. ADC quantizes to nearest digital level. {fig-alt=“Flowchart showing analog-to-digital conversion pipeline: analog signal flows through anti-aliasing filter, sample-and-hold circuit, ADC, and outputs digital value. Components colored orange (input/output), teal (preprocessing), and navy (ADC core).”}

NoteAlternative View: Temperature Sensor Example

This variant shows the same ADC pipeline with concrete values from a real temperature sensor - making abstract concepts tangible:

%% fig-alt: "ADC pipeline with concrete temperature sensor example showing actual values at each stage. Analog Signal shows NTC thermistor outputting 1.234V for 23.5C room temperature. Anti-Aliasing Filter with 10Hz cutoff removes HVAC cycling noise, output unchanged at 1.234V. Sample & Hold captures voltage 1000 times per second, holds 1.234V during conversion. 12-bit ADC converts 1.234V to digital value 1534 (range 0-4095 for 0-3.3V). Finally, microcontroller maps 1534 back to 23.5C temperature reading. This concrete example shows what happens to real sensor data at each pipeline stage."
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flowchart TD
    subgraph analog["1. ANALOG SIGNAL"]
        A1["NTC Thermistor<br/>outputs 1.234V"]
        A2["(23.5C room temp)"]
    end

    subgraph filter["2. ANTI-ALIASING FILTER"]
        F1["10Hz Low-Pass<br/>removes HVAC noise"]
        F2["Output: 1.234V<br/>(noise removed)"]
    end

    subgraph sample["3. SAMPLE & HOLD"]
        S1["Captures voltage<br/>1000x per second"]
        S2["Holds 1.234V<br/>during conversion"]
    end

    subgraph adc["4. ADC CONVERSION"]
        D1["12-bit ADC"]
        D2["1.234V -> 1534<br/>(0-4095 range)"]
    end

    subgraph digital["5. DIGITAL OUTPUT"]
        O1["Microcontroller<br/>reads: 1534"]
        O2["Maps back:<br/>23.5C"]
    end

    analog --> filter --> sample --> adc --> digital

    style analog fill:#E67E22,stroke:#2C3E50,color:#fff
    style filter fill:#16A085,stroke:#2C3E50,color:#fff
    style sample fill:#16A085,stroke:#2C3E50,color:#fff
    style adc fill:#2C3E50,stroke:#16A085,color:#fff
    style digital fill:#E67E22,stroke:#2C3E50,color:#fff

Figure 62.2: ADC pipeline with concrete temperature sensor values at each stage

Why this variant helps: The original shows abstract component names. This variant shows actual values flowing through a real temperature sensor - you can trace 23.5C from voltage (1.234V) through digital value (1534) and back. Understanding what happens to real data at each stage makes the pipeline tangible.

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62.3 The Nyquist Theorem: How Fast to Sample?

Time: ~12 min | Level: Intermediate | Unit: P02.C05.U02

The fundamental question: How many times per second must we sample to accurately capture a changing signal?

Nyquist-Shannon Sampling Theorem: > To accurately capture a signal, you must sample at more than twice the highest frequency component.

Minimum Sampling Rate = 2 x Maximum Signal Frequency

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graph TB
    subgraph under["Undersampling (< 2f)"]
        U1["Sampling Rate: 0.8f<br/>(Too Slow)"]
        U2["Result: Aliasing<br/>Signal appears distorted<br/>Fast signal looks slow"]
    end

    subgraph nyquist["Nyquist Minimum (= 2f)"]
        N1["Sampling Rate: 2f<br/>(Barely Adequate)"]
        N2["Result: Reconstructible<br/>Theoretically correct<br/>No margin for error"]
    end

    subgraph over["Oversampling (> 2f)"]
        O1["Sampling Rate: 5-10f<br/>(Practical Choice)"]
        O2["Result: Accurate<br/>Clean reconstruction<br/>Safety margin"]
    end

    U1 --> U2
    N1 --> N2
    O1 --> O2

    style under fill:#E67E22,stroke:#2C3E50,stroke-width:3px,color:#000
    style nyquist fill:#7F8C8D,stroke:#2C3E50,stroke-width:3px,color:#fff
    style over fill:#16A085,stroke:#2C3E50,stroke-width:3px,color:#fff
    style U1 fill:#ECF0F1,stroke:#E67E22,stroke-width:2px,color:#000
    style U2 fill:#ECF0F1,stroke:#E67E22,stroke-width:2px,color:#000
    style N1 fill:#ECF0F1,stroke:#7F8C8D,stroke-width:2px,color:#000
    style N2 fill:#ECF0F1,stroke:#7F8C8D,stroke-width:2px,color:#000
    style O1 fill:#ECF0F1,stroke:#16A085,stroke-width:2px,color:#000
    style O2 fill:#ECF0F1,stroke:#16A085,stroke-width:2px,color:#000

Figure 62.3: Nyquist Sampling Rate Comparison: Undersampling, Minimum, and Oversampling

Nyquist Sampling Comparison: Undersampling causes aliasing distortion. Sampling exactly at 2f is theoretically sufficient but risky. Practical systems oversample at 5-10x the signal frequency for reliability. {fig-alt=“Three-column comparison showing undersampling (red, causes aliasing), Nyquist minimum (gray, barely adequate), and oversampling (teal, practical choice with safety margin). Each shows sampling rate and resulting signal quality.”}

NoteAlternative View: Nyquist as Movie Frame Rate

This variant uses a movie filming analogy to make sampling rate intuitive:

%% fig-alt: "Analogy comparing Nyquist sampling to movie frame rates. A spinning wheel at 30 rotations per second (like a car wheel at highway speed) is filmed at different frame rates. At 24 fps (too slow, below Nyquist), wheels appear to spin backward - this is aliasing, the same effect as undersampling. At 60 fps (exactly 2x, Nyquist minimum), wheels appear frozen - technically correct but no margin for faster motion. At 120 fps (4x, oversampled), wheels spin correctly with smooth motion - this is the practical choice like sampling at 5-10x signal frequency. Key insight: just as movies need enough frames to capture motion, IoT sensors need enough samples to capture signal changes."
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graph TB
    subgraph MOTION["Spinning Wheel: 30 rotations/sec"]
        WHEEL["Like a car wheel<br/>at highway speed"]
    end

    subgraph SLOW["24 fps: TOO SLOW"]
        S1["Below Nyquist (< 2x30)"]
        S2["Wheel appears to<br/>spin BACKWARD"]
        S3["This is ALIASING!"]
    end

    subgraph EXACT["60 fps: EXACTLY 2x"]
        E1["At Nyquist limit"]
        E2["Wheel appears<br/>FROZEN in place"]
        E3["No margin for error"]
    end

    subgraph FAST["120 fps: OVERSAMPLED"]
        F1["4x frequency (safe)"]
        F2["Wheel spins smoothly<br/>FORWARD correctly"]
        F3["Practical choice!"]
    end

    MOTION --> SLOW
    MOTION --> EXACT
    MOTION --> FAST

    style WHEEL fill:#2C3E50,stroke:#16A085,stroke-width:2px,color:#fff
    style S1 fill:#E67E22,stroke:#2C3E50,stroke-width:1px,color:#fff
    style S2 fill:#E67E22,stroke:#2C3E50,stroke-width:1px,color:#fff
    style S3 fill:#E67E22,stroke:#2C3E50,stroke-width:2px,color:#fff
    style E1 fill:#7F8C8D,stroke:#2C3E50,stroke-width:1px,color:#fff
    style E2 fill:#7F8C8D,stroke:#2C3E50,stroke-width:1px,color:#fff
    style E3 fill:#7F8C8D,stroke:#2C3E50,stroke-width:1px,color:#fff
    style F1 fill:#16A085,stroke:#2C3E50,stroke-width:1px,color:#fff
    style F2 fill:#16A085,stroke:#2C3E50,stroke-width:1px,color:#fff
    style F3 fill:#16A085,stroke:#2C3E50,stroke-width:2px,color:#fff

Figure 62.4: Nyquist sampling as movie frame rate: undersampling makes wheels spin backward, oversampling captures motion correctly

Why this variant helps: The original shows abstract frequency ratios (2f, 0.8f). Everyone has seen the “wagon wheel effect” in movies where car wheels appear to spin backward. This directly connects Nyquist undersampling to a visible, memorable phenomenon that students can visualize.

62.3.1 Practical Examples

Example 1: Temperature Sensor - Building HVAC changes slowly: ~0.01 Hz (1 cycle per 100 seconds) - Nyquist requirement: Sample > 0.02 Hz - Practical choice: 0.1 Hz (once every 10 seconds) provides 5x margin

Example 2: Heart Rate Monitor - Heart rate: 0.5-3 Hz (30-180 bpm) - Nyquist requirement: Sample > 6 Hz - Practical choice: 100-500 Hz to capture waveform shape, not just frequency

Example 3: Accelerometer for Vibration - Machine vibration: 0-1000 Hz - Nyquist requirement: Sample > 2000 Hz - Practical choice: 5000-10000 Hz for safety margin

WarningCommon Misconception Alert: “Higher Sampling Rate is Always Better”

The Myth: “I should sample as fast as possible to get the best data quality.”

Why It’s Wrong: Higher sampling rates don’t improve signal quality beyond Nyquist requirements - they just waste resources.

The Reality:

Misconception #1: “1000 Hz is better than 100 Hz for all sensors” - Reality: Temperature changes at 0.01 Hz. Sampling at 1000 Hz captures the same 0.01 Hz signal 100,000 times without gaining new information. - Cost: 10x higher sampling rate = 10x more power, 10x more data storage, 10x more bandwidth - Better approach: Match sampling rate to signal bandwidth (5-10x Nyquist, not 1000x)

Misconception #2: “16-bit ADC is always better than 12-bit” - Reality: If your sensor accuracy is +/-1% (e.g., thermistor), 12-bit resolution (0.025%) already exceeds sensor precision by 40x. Using 16-bit (0.0015%) provides 667x more precision than the sensor can deliver! - Cost: 16-bit ADCs typically consume 3-5x more power than 12-bit - Better approach: Match ADC resolution to sensor precision, not maximum available bits

Real-World Example: - Inefficient design: Soil moisture sensor sampled at 1 kHz, 16-bit ADC, no filtering - Soil moisture changes over hours (0.0003 Hz) - Power consumption: 5 mA continuous - Battery life: 20 hours

• Optimized design: Same sensor sampled at 0.01 Hz (once per 100 seconds), 10-bit ADC, median filter
  • Captures all relevant changes (30x Nyquist margin)
  • Power consumption: 50 uA average (100x reduction!)
  • Battery life: 2000 hours (83 days)
  • Data quality: Identical for the application

Key Lesson: “Good enough” signal processing saves 100x power without sacrificing data quality. Over-specification wastes resources on precision you can’t use.

Rule of Thumb: 1. Identify maximum signal frequency (fmax) 2. Sample at 5-10x fmax (not 100x or 1000x) 3. Match ADC resolution to sensor accuracy +/-20% margin 4. Filter intelligently to remove noise, not signal

TipLearning Scenario: Choosing the Right Sample Rate

Your Challenge: You’re designing a predictive maintenance system for industrial machinery. Your accelerometer detects vibrations indicating bearing wear.

Scenario Details: - Bearing frequency: Healthy bearings vibrate at 200 Hz (shaft rotation rate) - Wear signature: Damaged bearings show harmonics up to 500 Hz (2.5x fundamental) - Your accelerometer: ADXL345 digital accelerometer - Available sample rates: 25 Hz, 50 Hz, 100 Hz, 200 Hz, 400 Hz, 800 Hz, 1600 Hz, 3200 Hz - Power consumption scales with sample rate: 40 uA @ 100 Hz, 140 uA @ 3200 Hz - Power budget: Battery-powered sensor node, need 1-year battery life

Your Task: What sample rate should you choose? Show your reasoning.

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Solution: Step-by-Step Analysis

Step 1: Identify Maximum Signal Frequency - Bearing fundamental: 200 Hz - Wear harmonics: up to 500 Hz - Maximum frequency to detect: 500 Hz (fmax)

Step 2: Apply Nyquist Theorem - Nyquist minimum: Sample rate > 2 x fmax = 2 x 500 Hz = 1000 Hz - Practical target: 5-10x margin -> 1250-1500 Hz minimum

Step 3: Evaluate Available Options

Sample Rate	Nyquist Margin	Power (uA)	Meets Requirements?
400 Hz	0.8x (too slow!)	~60	No - Aliasing risk
800 Hz	1.6x (barely)	~85	Marginal
1600 Hz	3.2x (good)	~115	Yes - Adequate
3200 Hz	6.4x (excellent)	~140	Yes - Overkill?

Step 4: Power Budget Analysis - Battery capacity: 1000 mAh (typical AA battery) - Target lifetime: 1 year = 8760 hours - Allowed current: 1000 mAh / 8760 h = 114 uA average

Power consumption breakdown: - Accelerometer @ 1600 Hz: 115 uA - Microcontroller sleep mode: 20 uA - Wireless transmission (1% duty): 5 uA average - Total: ~140 uA (exceeds budget by 26 uA!)

Step 5: Optimization Strategy

Option B: Duty-cycle the accelerometer - Sample at 1600 Hz for 1 second every 10 seconds (10% duty cycle) - Accelerometer power: 115 uA x 10% = 11.5 uA average - Total: 11.5 + 20 + 5 = 36.5 uA - Battery life: 1000 mAh / 0.0365 mA = 27,400 hours = 3.1 years

Best Choice: 1600 Hz with 10% duty cycle

Why This Works: - Nyquist-compliant: 1600 Hz > 1000 Hz requirement (3.2x margin) - Catches transient events: 1-second bursts at 1600 Hz = 1600 samples per measurement window - Power-efficient: 10% duty cycle saves 90% accelerometer power - Exceeds 1-year target: 3+ years battery life - Practical: Bearings don’t wear instantly - sampling every 10 seconds catches degradation trends

Key Lessons: 1. Nyquist is non-negotiable: 400 Hz would alias 500 Hz signals to 100 Hz (false readings!) 2. Oversampling has costs: 3200 Hz provides no benefit over 1600 Hz for this application 3. Duty cycling saves power: 10% duty cycle = 10x battery life extension 4. Match sampling to physics: Bearing wear develops over days/weeks, not milliseconds

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62.4 Aliasing: When Sampling is Too Slow

Time: ~9 min | Level: Intermediate | Unit: P02.C05.U03

What happens if you sample too slowly?

Fast signals masquerade as slow signals - this is called aliasing.

Visual Example: Imagine a wheel spinning at 60 RPM (1 revolution/second = 1 Hz). If you sample at 1.5 Hz (every 0.67 seconds), the wheel appears to spin slowly backward instead of forward!

IoT Example: - True signal: 50 Hz vibration (motor spinning at 3000 RPM) - Sampling rate: 40 Hz (too slow! Violates Nyquist) - Observed signal: Appears as 10 Hz vibration (completely wrong!)

Solution: Use anti-aliasing filter (low-pass) before ADC to remove frequencies above Nyquist limit.

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flowchart TD
    A["True Signal<br/>50 Hz Vibration<br/>(Motor @ 3000 RPM)"] --> B{"Sampling Rate?"}

    B -->|"100+ Hz<br/>(Correct)"| C["Sampled Signal<br/>Accurate 50 Hz<br/>No Aliasing"]

    B -->|"40 Hz<br/>(Too Slow)"| D["Sampled Signal<br/>Appears as 10 Hz<br/>ALIASED"]

    B -->|"20 Hz<br/>(Way Too Slow)"| E["Sampled Signal<br/>Appears as 10 Hz<br/>SEVERELY ALIASED"]

    D --> F["Problem: Fast signal<br/>masquerades as<br/>slow signal"]
    E --> F

    F --> G["Solution:<br/>Anti-Aliasing Filter<br/>(Low-Pass before ADC)"]

    style A fill:#2C3E50,stroke:#16A085,stroke-width:3px,color:#fff
    style B fill:#7F8C8D,stroke:#2C3E50,stroke-width:2px,color:#fff
    style C fill:#16A085,stroke:#2C3E50,stroke-width:3px,color:#fff
    style D fill:#E67E22,stroke:#2C3E50,stroke-width:3px,color:#fff
    style E fill:#E67E22,stroke:#2C3E50,stroke-width:3px,color:#000
    style F fill:#ECF0F1,stroke:#E67E22,stroke-width:2px,color:#000
    style G fill:#16A085,stroke:#2C3E50,stroke-width:3px,color:#fff

Figure 62.5: Aliasing Effect: How Undersampling Distorts High-Frequency Signals

Aliasing Effect: When sampling rate is below Nyquist (2x signal frequency), high-frequency signals incorrectly appear as low-frequency signals. A 50 Hz vibration sampled at 40 Hz masquerades as 10 Hz. Anti-aliasing filters prevent this by removing high frequencies before sampling. {fig-alt=“Flowchart showing aliasing effect: 50 Hz true signal with three sampling scenarios. 100+ Hz sampling (teal) is correct. 40 Hz and 20 Hz sampling (orange) cause aliasing, making signal appear as 10 Hz. Solution is anti-aliasing low-pass filter before ADC.”}

62.4.1 Calculating Aliased Frequencies

When a frequency f_signal is sampled at rate f_sample (where f_signal > f_sample/2), the aliased frequency is:

f_aliased = |f_sample - f_signal|

Example: - Signal: 60 Hz HVAC vibration - Sampling: 100 Hz - Nyquist limit: 50 Hz (100/2) - Since 60 Hz > 50 Hz, aliasing occurs - Aliased frequency: |100 - 60| = 40 Hz

The 60 Hz vibration appears as 40 Hz in your data!

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62.5 Summary

Key concepts from this chapter:

• Analog vs Digital: Continuous sensor signals must be discretized for digital processing
• Nyquist Theorem: Sample at >2x the highest frequency you care about
• Practical sampling: Use 5-10x oversampling for safety margin
• Aliasing: Sampling too slowly makes fast signals appear slow
• Anti-aliasing filters: Low-pass filter before ADC prevents aliasing artifacts
• Resource optimization: Don’t over-sample - it wastes power, storage, and bandwidth

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flowchart LR
    A["Analog<br/>Signal"] --> B["Anti-Alias<br/>Filter"]
    B --> C["Sample<br/>(>2x fmax)"]
    C --> D["Quantize<br/>(ADC)"]
    D --> E["Digital<br/>Value"]

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    style D fill:#2C3E50,stroke:#16A085,stroke-width:2px,color:#fff
    style E fill:#E67E22,stroke:#2C3E50,stroke-width:2px,color:#fff

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62.6 What’s Next

Now that you understand sampling and aliasing, proceed to Quantization and Digital Filtering to learn about ADC resolution, precision trade-offs, and noise reduction techniques.

Continue to Quantization and Filtering ->