Information in waves
Signals carry information by changing a measurable property such as amplitude, frequency, phase, or timing. A receiver recovers symbols from noisy sampled data using an explicit decision rule. The information is not the drawn sine curve; it is the agreed mapping between symbols, measurable states, and receiver decisions.
Think like a programmer
Model checklist
- Inputs
- Symbol stream, amplitude or phase mapping, channel noise, sample timing, receiver threshold, seed, and message length.
- State
- Sent symbols, transmitted samples, received samples, decoded symbols, and error count.
- Rule
- Encode, perturb only the transmitted state, sample at documented times, then classify using the receiver rule.
- Output
- Recovered symbols, error positions, bit-error rate, and all configuration values needed to replay the run.
- Check
- With zero noise, decoding recovers the exact input; the same seed and inputs reproduce the same received samples; bit-error rate is errors divided by sent bits.
For a deliberately small binary amplitude model, encode 0 as amplitude 0 and 1 as amplitude 1. The channel adds bounded noise, then the receiver chooses 1 at or above threshold 0.5.
Seeded binary-channel experiment
Each sent bit is encoded as amplitude 0 or 1. A repeatable bounded-noise channel changes it, and the receiver decides 1 when the received amplitude is at least 0.5.
- Correctly decoded bit
- Incorrectly decoded bit
Noise amplitude 0.4; 0 errors in 8 bits; bit error rate 0.0%.
The lab uses a fixed short message and a seeded noise sequence. At zero noise, its decoded row must exactly match the sent row; that is a test of the complete pipeline, not merely a visual expectation. As noise rises, some amplitudes cross the threshold and produce errors. One short run may happen to show no errors even when a channel is risky, so compare multiple documented seeds or a longer stream before estimating a rate.
Try this experiment
Prediction: With the threshold fixed, larger bounded noise makes threshold crossings and decoding errors possible; zero noise produces no errors.
Set noise to zero and verify every decoded bit. Raise it in steps, predict which received amplitudes are closest to failure, and record error count and BER. Then replace amplitude mapping in pseudocode with phase or frequency mapping, while keeping the receiver's available data explicit.Where this model breaks
Summary
Model communication as a reproducible pipeline from symbols to samples and back. Keep sender knowledge separate from receiver knowledge, test the noiseless identity case, and report error rate with the channel and decoding assumptions that produced it.
Glossary
- Modulation: mapping information onto a carrier property.
- Channel: model of transmission and noise.
- Bit error rate: fraction of decoded bits that differ from input.
- Threshold decoder: receiver that classifies a sample by comparing it with one boundary value.
- Symbol: one encoded message value from an agreed alphabet.
- Channel model: explicit rule that transforms transmitted into received data.
Self-check
- Which stages must a communication model separate?
- Which full-pipeline test should pass without noise?
- What information may the decoder use?
- How is BER calculated?
Sources
- C. E. Shannon, A Mathematical Theory of Communication.
- OpenStax, University Physics Volume 3, Communication Systems.
Model contract
Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.
\[\text{observable output} = f(\text{inputs},\,\text{state})\]- Inputs
- Quantities you set or measure, with units and useful bounds.
- State
- Values the program must retain to reproduce the next result.
- Rule
- The relationship or update that turns inputs and state into a result.
- Check
- A known limit, unit check, invariant, or measured result that can expose a bad model.
Implement the idea as a model
For Information in Waves, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Waves II — Analysis and Transformations, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.
Guided experiment
Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.
Where this model breaks
This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.
Summary
Treat Information in Waves as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.
Glossary
- Input: a measured value or chosen parameter supplied to a model.
- State: the smallest set of values needed to continue or reproduce a model.
- Validation: comparing an output with a known result, limit, invariant, or measurement.
Self-check
- Which values are inputs, and which values must remain state?
- What observable result would tell you the model is behaving as expected?
- Which assumption would you test first before applying the model to a real system?
Model review: turn Information in Waves into a test
Model encoding, a noisy channel, sampling, and decoding as separate testable signal-processing steps.
- Name the inputs and units that the waves ii — analysis and transformations model needs.
- Separate the state you must keep from values you can calculate when needed.
- Write one rule that maps the current state and inputs to an observable result.
- Choose a limiting case, unit check, invariant, or known result before trusting an output.
- State one assumption you would change before using this simplified model for a real decision.