Sensors & Measurement · Study deck

Sensor Fusion and Kalman Filtering

Picture a strain sensor on a bridge.

Physics Phoebe is your guide for this deck.

sensortypes

Physics Phoebe, the module guide, in a scene from this chapter.

After studying this chapter

Learning objectives

You will be able to:

Identify 1/f noise in sensor data and explain why it limits long-term averaging effectiveness
Design multi-sensor fusion architectures that outperform individual sensors
Implement Kalman filtering algorithms for optimal state estimation
Construct robust sensing systems with error handling, redundancy, and watchdog recovery for production deployment

Major section

Start With the Measurement Story

Its reading changes with load, but also with heat, age, mounting stress, and small random effects.

A clean number on a screen can hide those very different causes.
Combining sensors can expose faults, yet it can also combine shared errors.
An advanced method is useful only when its limit is visible.

Major section

Field Sensors Fail Differently

Maria, an agricultural IoT engineer, was frustrated.

Her soil moisture sensors worked perfectly in the lab---readings were stable, accurate, and repeatable.
The solution wasn't more averaging---it was smarter signal processing and combining multiple sensors together.
This chapter takes you on the journey from lab-perfect sensors to field-reliable systems.

Major section

In 60 Seconds

Advanced sensor topics include 1/f noise (which limits long-term averaging effectiveness), multi-sensor fusion (combining GPS, Wi-Fi, barometer, and IMU for better positioning), and Kalman filtering (optimally blending predictions with measurements).

Production sensor systems need robust error handling, redundancy, watchdog timers, and traceable calibration.
But an averaging window has bandwidth near $1/(2T)$.

Major section

Part 1: The 1/f Noise Problem

You've probably learned that averaging reduces noise.

This works beautifully for "white noise"---the random fluctuations that are equally likely at any frequency.
But here's what the textbooks often skip: not all noise is white noise.
Below this frequency, 1/f noise dominates and averaging becomes ineffective.

Major section

Part 1: The 1/f Noise Problem (continued)

Its power increases at lower frequencies, which means slow, wandering drift dominates over long time periods.

The impact is counterintuitive: short-term averaging works exactly as expected (10 samples gives you ~3× improvement), but long-term averaging hits a wall.
At some point, taking more samples stops helping---and can even make things worse as you capture slow baseline drift.
Every sensor has a "1/f corner frequency"---the frequency where 1/f noise equals the white noise floor.
Know when to stop averaging.: Check your sensor's datasheet for the 1/f corner frequency.

Major section

Optional High-Pass Filter Pattern

This moves your signal above the 1/f corner frequency where noise is well-behaved.

The 1/f noise, being correlated between samples, largely cancels out.
The plot separates frequency-independent white noise from the low-frequency rise associated with flicker noise.

1/f Noise Spectrum: The frequency domain view shows white noise (flat) and 1/f noise (rising at low frequencies) intersecting at the corner frequency. Below the corner, averaging becomes ineffective.

Major section

When One Sensor Is Not Enough

Each sensor has complementary weaknesses.

GPS drifts indoors where Wi-Fi works.
Accelerometers drift over time where GPS provides corrections.
The insight:: Instead of trying to build a perfect single sensor (impossible), combine imperfect sensors that fail in different ways.
Each witness sees a different angle of the same event.

Major section

Fusion Rule of Thumb

The route connects redundancy to improved observability while preserving the need to reject correlated faults and biased inputs.

The key is figuring out the right weights---how much to trust each sensor at each moment.

Sensor Fusion Architecture: Multiple sensors with complementary strengths feed into a Kalman filter fusion engine, producing a best estimate more accurate than any individual sensor.

Major section

MEMS Sensor Implementation Chain

Miniaturisation does not remove the measurement chain; it packs the chain into fewer packages and makes power-domain timing more important.

The wake sequence is causal.
For a simple 1 ms active, 1 s sleep schedule, use charge rather than averaging currents by eye.

A miniature sensor node converts a transducer signal through bias, analog conditioning, ADC, MCU, and radio under a one-second sleep and one-millisecond active schedule.

Major section

MEMS Sensor Implementation Chain (continued)

If $I_a$ is active current, $I_s$ is sleep current, $t_a=1\ \text{ms}$, and $t_s=1\ \text{s}$, then.

That attractive number survives only if start-up, sensor warm-up, retries, leakage through GPIOs, and regulator quiescent current fit the model.
A 20 ms warm-up at the same active current would dominate the 1 ms radio burst.
The: ACTIVE · 1 ms overlay is credible only if reference settling, sensor warm-up, radio startup, acknowledgement, and retry work fit inside it; otherwise the apparent sleep ratio hides real energy.

Major section

Miniaturisation Limits Before Actuation

Smaller force sensors gain integration and may gain resonant frequency, but the signal and every noise mechanism do not scale together.

A spring mode stores mean energy on the order of $k_BT$, giving an RMS displacement.
The mechanical transfer function turns that force into displacement, and the readout adds roughly size-independent amplifier, reference, and quantisation noise.

Large and small force sensors compare thermal displacement, surface effects, and fixed electronics noise before a measured command drives an actuator.

Major section

Miniaturisation Limits Before Actuation (continued)

Consequently, shrinking geometry does not guarantee a quieter force estimate.

If a characteristic length is $L$, volume and mass scale approximately as $L^3$, while exposed area scales as $L^2$; therefore surface-to-volume ratio scales as $1/L$.
Surface charge traps, adsorption, contamination, package stress, and contact-related 1/f processes can occupy more of the error budget as the structure shrinks.
Electrical noise may stay almost fixed while the mechanical signal falls, so the interface circuit can set the resolution even when fabrication improves.

Deck summary

Key takeaways

Its reading changes with load, but also with heat, age, mounting stress, and small random effects.

Maria, an agricultural IoT engineer, was frustrated.
Advanced sensor topics include 1/f noise (which limits long-term averaging effectiveness), multi-sensor fusion (combining GPS, Wi-Fi, barometer, and IMU for better positioning), and Kalman filtering (optimally blending predictions with measurements).
You've probably learned that averaging reduces noise.
Its power increases at lower frequencies, which means slow, wandering drift dominates over long time periods.

Retrieval practice

Recall check 1 of 3

Physics Phoebe says: answer from memory, then check your reasoning.

Q1Per this chapter's 1/f Corner Frequency table, which sensor type has the highest typical corner frequency, meaning it can tolerate the least averaging time before hitting 1/f-dominated noise?

APhotodiode (100-1000 Hz)

BGas sensor (0.01 Hz)

CThermistor (0.1-1 Hz)

DMEMS accelerometer (1-10 Hz)

Show answer

Answer: A

Q2This chapter lists four mitigation strategies for 1/f noise. Which one works by periodically reversing the sensor's polarity or bias, then subtracting alternate readings to move the signal above the 1/f corner frequency?

AChopping/modulation

BHigh-pass filtering

CCorrelated double sampling

DKnow when to stop averaging

Show answer

Answer: A see answers page

Retrieval practice

Recall check 2 of 3

Physics Phoebe says: answer from memory, then check your reasoning.

Q3Your soil moisture sensor has 1/f noise corner at 0.5 Hz. You're averaging 100 samples over 10 seconds. What's the problem?

ANot enough samples - need 1000 for good averaging

BToo many samples - noise increases with sample count

CThe 10 s average falls below the 1/f corner

DThis is optimal - 10 seconds is perfect for noise reduction

Show answer

Answer: C

Retrieval practice

Recall check 3 of 3

Physics Phoebe says: answer from memory, then check your reasoning.

Q4Per this chapter's Single-Sensor Problem table, which limitation does GPS have that Wi-Fi RSSI specifically compensates for?

AGPS doesn't work indoors or in urban canyons with multipath, while Wi-Fi RSSI works indoors (though with lower +/-5-8m accuracy)

BGPS drifts over time due to integration error, while Wi-Fi never drifts

CGPS is distorted by nearby metal and electronics, while Wi-Fi is immune to interference

DGPS cannot detect altitude changes, while Wi-Fi provides precise floor-level altitude

Show answer

Answer: A The table pairs GPS's weakness ('Doesn't work indoors; multipath in urban canyons') against Wi-Fi RSSI's strength ('Works indoors'), illustrating the chapter's complementary-weakness pattern that motivates sensor fusion.

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Answers

Answer key.

A
A · The chapter's four mitigation strategies are: know the corner frequency and stop averaging in time, high-pass filter out low frequencies, chop/modulate by reversing polarity and subtracting alternate readings, and correlated double sampling by subtracting two differently-excited measurements.
C
A · The table pairs GPS's weakness ('Doesn't work indoors; multipath in urban canyons') against Wi-Fi RSSI's strength ('Works indoors'), illustrating the chapter's complementary-weakness pattern that motivates sensor fusion.