Trace new components from a quadratic two-tone model

Separate the visitor’s experience from the proposed cause

A fictional exhibition has two short sound cues: one acknowledges a visitor’s selection, while another introduces the next display. Each was reviewed in isolation. During a later discussion, someone reports a new tonal element when both events occur close together and asks the composer to remove it from the source. The producer should first establish what was actually heard, which version was used and how the two events were presented.

Write the intended communication role of each cue. The acknowledgement should make the selection understandable, and the introduction should move attention toward the next display. The creative owner may ultimately prefer a different overlap policy, but that is a separate decision from locating an unexpected production effect. Do not let a convenient technical explanation decide the exhibition’s sound design by default.

Let a=cos(2π1000t), b=cos(2π1200t), with t in seconds. Stipulate x=a+b and y=x+0.1x², with dimensionless amplitudes. This is a memoryless quadratic model, not an ADC specification.

Keep the synthetic exercise apart from the actual event report. The supplied tones do not represent a recording of either cue, and the chosen operation is not a claim about the installed equipment. It is a controlled explanation to discuss with the sound owner. No speaker, amplifier or software setting is changed by constructing the teaching record.

Account for the complete stipulated output

Expanding gives y=a+b+0.1+0.05cos(2π2000t)+0.05cos(2π2400t)+0.1cos(2π200t)+0.1cos(2π2200t). The cross-term creates sum and difference components; the squared individual terms create harmonics and DC. Kester’s tutorial identifies second-order intermodulation products.

The linear control y=x contains only the 1000 and 1200 Hz tones. Evaluating t=n/8000 for n=0…7999 and checking the Fourier coefficients reproduces every stated amplitude within 1e−11. No audio playback occurs.

Attach the input definition and complete output record to the investigation note. A reviewer should be able to tell whether a displayed result belongs to the proposed model or to an actual capture. Keep those identities visible when the note is copied into a presentation. Otherwise an illustrative calculation can acquire undeserved authority simply because its chart looks precise.

Ask another technical reader to reproduce the record before using it in the production discussion. If they obtain a different answer, resolve the arithmetic and representation first. Do not choose the plot that looks most like the reported symptom and call that agreement. The purpose of a controlled exercise is to expose its assumptions, not to decorate a preferred diagnosis.

Keep the investigation question narrow: which evidence from the actual route would distinguish this explanation from other possibilities? The owner may need the source assets, event timing and processing history. The lesson does not prescribe a universal measurement procedure or establish which part of the installed route caused the report.

Request evidence without rewriting the composition prematurely

Preserve the approved cue versions before asking for another creative edit. If the composer receives only the instruction to remove a tone, they may alter intentional content that was never responsible for the reported discrepancy. A good request identifies the disputed event and says what remains unknown. It does not accuse a supplier or imply a defect that has not been established.

Have the sound owner identify the actual comparison they can perform under the production team’s procedures. Keep the conditions of the original report available: which events occurred, what was playing beforehand and where the result was observed. A clean isolated preview may answer the wrong question if the concern occurred only during a particular interaction sequence.

At the same time, ask the creative owner whether overlapping cues serve the visitor. Even if the technical route is working as intended, two competing messages can be the wrong experience. Treat a proposed change to the interaction design as an explicit creative alternative, with its own reason, rather than calling it a repair proved by the numerical exercise.

Close the decision at the level the evidence supports

When the actual investigation returns, distinguish an established cause from an unresolved resemblance. If the route owner identifies a specific processing issue, ask for a separately authorized reversible correction and an appropriate comparison. If the cause remains unclear, preserve that uncertainty. The team can choose a conservative presentation alternative without pretending that it has solved the original technical question.

Review the selected experience with the intended sequence of visitor actions and the actual campaign content. Confirm which cue version and interaction behavior the creative owner accepted. The mathematical record does not establish comfort, intelligibility or preference. Those questions belong to the real experience and its established review process.

Keep the original report, the bounded hypothesis and the eventual production finding linked. A later editor should be able to tell whether the team changed an asset, corrected a route or deliberately revised the experience. That distinction protects the composition from repeated speculative edits and gives future production teams a useful explanation instead of an unexplained instruction to avoid a particular file.

Make the reasoning concrete.

A producer is asked to revise an approved cue after an unexpected tonal element is reported during overlapping events. The actual cause has not been established.

The producer requests the bounded investigation and keeps any new interaction concept separate from the technical finding.

Take the questions into your next brief.

Use these prompts in your own notes. Nothing is submitted from this page.

1. Report

   Which event sequence and asset versions produced the concern?

2. Intent

   What should each cue communicate to the visitor?

3. Identity

   Which displayed evidence is synthetic and which comes from production?

4. Discrimination

   What actual route evidence could reject the proposed explanation?

5. Decision

   Did the team change an asset, a route or the experience itself?

Watch for the wrong conclusion.

• Asking a composer to repair an unverified cause.
• Presenting an illustrative chart as a measured production result.
• Calling a new interaction concept a technically proved repair.

Sources & scope

All business examples and numerical fixtures are original and fictional. Sources support the identified mathematical or implementation distinctions. Qualified analysts own model specification and inference. No causal result, model fit, actual customer data or operational change is claimed.

• Kester: Intermodulation Distortion Considerations for ADCs, MT-012 →Primary technical tutorial · Supports: Supports second-order product locations and model limits.