Capstone & Appendix · Study deck

IoT Mathematics: Linear Algebra and Probability

Several sensor values form a vector, while noise makes each value unsure.

Sammy is your guide for this deck.

mathematicalfoundations

$Test Tessa, the module guide, in a scene from this chapter.$

After studying this chapter

Learning objectives

You will be able to:

Explain: Key Property of modular exponentiation in suitable large groups: Easy to compute forward; the discrete logarithm can be hard to recover.
Explain: Diffie-Hellman relies on forward calculations such as 5^7 mod 23 = 17 being easy while the reverse problem is hard.
Explain: The update shifts the state and cuts that doubt based on how much each source can be trusted.
Explain: The FFT sorts those same samples by frequency; it cannot bring back a tone lost before the ADC.

Major section

The Core Idea

Modular arithmetic is "clock arithmetic." After reaching a maximum value, numbers wrap around to zero.

Notation: $a \mod n$ = remainder when a is divided by n.
Key Property of modular exponentiation in suitable large groups: Easy to compute forward; the discrete logarithm can be hard to recover.
One device mixes in private red and sends orange.

Diffie-Hellman key exchange between two IoT devices

Major section

The Core Idea (continued)

This one-way property supports Diffie-Hellman key agreement and related public-key systems.

Two IoT devices can agree on a shared secret over an insecure channel.
The other mixes in private blue and sends green.
Each device adds its own secret colour to the mix it got, so both reach brown.

Public paint mixtures let both devices reach the same private colour without exposing either secret contribution.

Major section

The Core Idea

Entropy measures information content or uncertainty.

More unpredictable data requires more bits to represent.
Where $p_i$ is the probability of each outcome.
The alarm provides only 0.08 bits of information per reading (very predictable).
Key Insight: To double capacity, you need to roughly quadruple SNR.

Major section

Checkpoint: Security and Signals

That wraparound supports the forward operations used in the key-exchange example.

Diffie-Hellman relies on forward calculations such as 5^7 mod 23 = 17 being easy while the reverse problem is hard.
Authentication and key handling remain separate requirements.
Complex numbers track amplitude and phase, while information theory connects entropy, SNR, bandwidth, and Shannon capacity.

Major section

Kalman Filter Visualization

Its range of doubt will often grow.

The new measure then gives the filter a separate fact.
The update shifts the state and cuts that doubt based on how much each source can be trusted.
The loop starts again from this fixed state.

Major section

Signal Processing Pipeline

A digital filter shapes the time record that remains.

The FFT sorts those same samples by frequency; it cannot bring back a tone lost before the ADC.
The Nyquist test gives the key limit.
If the highest kept tone is 2 kHz, the sample rate must be at least 4 kHz.

Deck summary

Key takeaways

Modular arithmetic is "clock arithmetic." After reaching a maximum value, numbers wrap around to zero.

This one-way property supports Diffie-Hellman key agreement and related public-key systems.
Entropy measures information content or uncertainty.
That wraparound supports the forward operations used in the key-exchange example.
Its range of doubt will often grow.

Retrieval practice

Recall check 1 of 2

Test Tessa says: answer from memory, then check your reasoning.

Q1An accelerometer produces an ordered set of axis readings. What does the vector magnitude describe?

AThe value of the first axis alone

BThe order in which readings were transmitted

CThe combined acceleration magnitude

DThe covariance between successive samples

Show answer

Answer: C The chapter uses the square root of summed squared components to describe total magnitude.

Q2A sensor-fusion design stores a covariance matrix. What do its off-diagonal entries describe?

AThe variance of each variable by itself

BThe vector’s total acceleration magnitude

CHow variables change together

DThe next state without a transition model

Show answer

Answer: C The chapter distinguishes cross-variable correlation from diagonal variance.

Retrieval practice

Recall check 2 of 2

Test Tessa says: answer from memory, then check your reasoning.

Q3What checks whether sampled evidence fits timing, loss, capacity and link-budget constraints?

ADevice Measurements and Uncertainty

BApplication Models and Decisions

CSignal and Network Calculations

Show answer

Answer: C Signal and network calculations test whether sampled evidence can be represented and delivered within timing, loss, capacity, and link-budget constraints.

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Answers

Answer key.

C · The chapter uses the square root of summed squared components to describe total magnitude.
C · The chapter distinguishes cross-variable correlation from diagonal variance.
C · Signal and network calculations test whether sampled evidence can be represented and delivered within timing, loss, capacity, and link-budget constraints.