Capstone & Appendix · Study deck
IoT Mathematics: Fusion and Sampling Decisions
GPS drifts slowly while an accelerometer reacts fast.
Test Tessa is your guide for this deck.
After studying this chapter
Learning objectives
You will be able to:
- Explain: Decision: In this ADC-only model, 2 kHz sampling provides a 2.8x margin over the Nyquist minimum while preserving 7+ years of battery life.
- Explain: In this simplified link budget, the link has 31 dB margin (1,259x power above minimum); verify fading and interference before claiming reliable field operation.
- Explain: The student's 10 dB answer is off by 10 dB (a 10× power-ratio difference, or about 3.16× in voltage ratio).
- Explain: Key Insight: Kalman filter prevents dead reckoning drift by periodically re-anchoring position with GPS, while smoothing GPS noise with prediction.
Major section
Worked Example: Designing a Kalman Filter for GPS-Accelerometer Fusion
Example: Current state is position (100m, 200m), velocity (15 m/s, 0 m/s), $\Delta t = 0.1s$.
- Step 3: Measurement Update (GPS Reading): GPS reports position (103m, 201m) with sigma = 10m uncertainty.
- Key Insight: Kalman filter prevents dead reckoning drift by periodically re-anchoring position with GPS, while smoothing GPS noise with prediction.
Major section
Decision Framework: Choosing Sampling Rates Using Nyquist Theorem
Where $f_{max}$ is the highest frequency component in your signal.
- Step 3: Add Safety Margin: For this example, use 2.5x the 720 Hz Nyquist minimum to allow filter roll-off.
- Decision: 2 kHz (exceeds requirement, standard IC availability).
- Decision: In this ADC-only model, 2 kHz sampling provides a 2.8x margin over the Nyquist minimum while preserving 7+ years of battery life.
Major section
Decision Framework: Choosing Sampling Rates Using Nyquist Theorem (continued)
Nyquist requires $f_{sample} > 120$ Hz, so 100 Hz causes aliasing.
- The Problem: Students often use the power formula $dB = 10\log_{10}(P_2/P_1)$ when comparing voltages, leading to 6 dB errors.
- The student's 10 dB answer is off by 10 dB (a 10× power-ratio difference, or about 3.16× in voltage ratio).
Major section
Common Mistake: Misapplying dB Calculations to Voltage vs. Power
Memory trick: Voltage uses 20 log because power depends on voltage SQUARED (the 2 becomes a multiplier in the log).
- Given: LoRa transmitter outputs 14 dBm (25 mW), path loss is 120 dB, receiver sensitivity is -137 dBm.
- In this simplified link budget, the link has 31 dB margin (1,259x power above minimum); verify fading and interference before claiming reliable field operation.
- Common mistake: Student adds voltage gain (in dBV) to power budget (in dBm) - units must match!
Deck summary
Key takeaways
Example: Current state is position (100m, 200m), velocity (15 m/s, 0 m/s), $\Delta t = 0.1s$.
- Where $f_{max}$ is the highest frequency component in your signal.
- Nyquist requires $f_{sample} > 120$ Hz, so 100 Hz causes aliasing.
- Memory trick: Voltage uses 20 log because power depends on voltage SQUARED (the 2 becomes a multiplier in the log).
Retrieval practice
Recall check 1 of 4
Test Tessa says: answer from memory, then check your reasoning.
Q11. Decibels (dB) are primarily a:
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Answer: B dB scales compress large ratios using logarithms (e.g., 10*log10(P2/P1) for power ratios), which is useful in RF and signal analysis.
Q22. In an RC circuit, the time constant is defined as:
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Answer: C The RC time constant tau = R*C sets the exponential charging/discharging rate (e.g., V(t) = V0*e^(-t/tau) for discharge).
Retrieval practice
Recall check 2 of 4
Test Tessa says: answer from memory, then check your reasoning.
Q34. Modular arithmetic is most directly relevant to:
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Answer: D Many cryptographic schemes rely on arithmetic over finite fields or modular rings (e.g., operations mod a prime), making modular arithmetic foundational for crypto.
Q46. The Fourier transform converts a signal from:
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Answer: C The Fourier transform decomposes a time-domain signal into its frequency components.
Retrieval practice
Recall check 3 of 4
Test Tessa says: answer from memory, then check your reasoning.
Q57. Shannon's channel capacity formula C = B*log2(1 + SNR) shows that doubling bandwidth:
Show answer
Answer: A Shannon's formula shows capacity is linear with bandwidth (B) but logarithmic with SNR.
Retrieval practice
Recall check 4 of 4
Test Tessa says: answer from memory, then check your reasoning.
Q68. An IoT sensor samples at 1000 Hz and applies a moving average filter. After 5 samples, what is the maximum frequency that can be accurately represented without aliasing?
Show answer
Answer: D The Nyquist limit is half the sampling rate: 1000/2 = 500 Hz.
Print reference
Answers 1 of 2
Answer key.
- B · dB scales compress large ratios using logarithms (e.g., 10*log10(P2/P1) for power ratios), which is useful in RF and signal analysis.
- C · The RC time constant tau = R*C sets the exponential charging/discharging rate (e.g., V(t) = V0*e^(-t/tau) for discharge).
- D · Many cryptographic schemes rely on arithmetic over finite fields or modular rings (e.g., operations mod a prime), making modular arithmetic foundational for crypto.
Print reference
Answers 2 of 2
Answer key.
- C · The Fourier transform decomposes a time-domain signal into its frequency components.
- A · Shannon's formula shows capacity is linear with bandwidth (B) but logarithmic with SNR.
- D · The Nyquist limit is half the sampling rate: 1000/2 = 500 Hz.