Capstone & Appendix · Study deck
IoT Mathematics: Fusion and Sampling Decisions
GPS drifts slowly while an accelerometer reacts fast.
Sammy is your guide for this deck.
After studying this chapter
Learning objectives
You will be able to:
- Work through a Kalman filter for GPS and motion data.
- Choose a sample rate with the Nyquist rule.
- Explain: Example: Current state is position (100m, 200m), velocity (15 m/s, 0 m/s), Δt=0.1s.
- Explain: Where fmax is the highest frequency component in your signal.
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), Δt=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 fmax 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 fsample>120 Hz, so 100 Hz causes aliasing.
- The Problem: Students often use the power formula dB=10log10(P2/P1) 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), Δt=0.1s.
- Where fmax is the highest frequency component in your signal.
- Nyquist requires fsample>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:
Show answer
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:
Show answer
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:
Show answer
Answer: C 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:
Show answer
Answer: B 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.
Q5In Shannon's formula C = B log2(1 + SNR), if SNR stays fixed, what does doubling bandwidth do to the theoretical capacity?
Show answer
Answer: D 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.
Q6A sensor samples at 1000 Hz and then applies a five-sample moving average without decimation. What is the nominal Nyquist frequency of the sampled sequence?
Show answer
Answer: A At a 1000 Hz sample rate the nominal Nyquist frequency is 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).
- C · 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.
- B · The Fourier transform decomposes a time-domain signal into its frequency components.
- D · Shannon's formula shows capacity is linear with bandwidth (B) but logarithmic with SNR.
- A · At a 1000 Hz sample rate the nominal Nyquist frequency is 500 Hz.