Chirplet Transform And Controlled Chirp Codebook Deep Dive

Core Distinction

There are two different machines that share similar math:

1. A general chirplet transform asks: “what chirp-like atoms exist in this signal?”
2. Mimir’s active receiver asks: “which known emitted atom arrived when, through which distorted path?”

The second machine is much cheaper because the codebook, slopes, timing plan, and symbol grammar are known.

Uses For Mimir

Timing Synchronization

Each decoded code-valid chirp tuple can become a canonical timeline anchor. With enough anchors, fit:

• offset;
• sample-rate offset;
• confidence;
• jitter/residual.

Frequency Response Normalization

Each classified chirp yields observed energy across bins. Aggregated over time:

• reliable bands;
• dead bands;
• alias/confusion bands;
• path coloration.

Phase / Group Delay

If phase is coherent enough, dechirped complex bin responses can estimate:

• per-bin phase;
• phase slope across frequency;
• group delay correction;
• reflection/path stability.

Localization

Known emitted time plus observed receiver time gives a distance constraint when the source and receiver clocks are modeled. Multiple receivers turn this into a position estimate. Multiple emitters/receivers turn it into room calibration.

Watermarking

Hybrid mode can embed low-gain coded chirps under program audio. The watermark should be:

• sparse enough not to annoy;
• deterministic enough to identify time;
• adaptive enough to avoid dead acoustic bands;
• continuous enough to track drift.

Decoder Pipeline

flowchart TD
    PCM["PCM stream"] --> Energy["streaming energy/onset proposal"]
    Energy --> Cand["candidate event windows"]
    Cand --> Dechirp["dechirp with known slope"]
    Dechirp --> Bins["bin / Goertzel / FFT score"]
    Bins --> Likelihood["calibration-weighted likelihood"]
    Likelihood --> Grammar["codebook grammar / de Bruijn tuples"]
    Grammar --> Clock["clock fit"]
    Clock --> Model["response + delay model"]

What The Current Code Has

• Fixed-slope chirp-bin renderer.
• De Bruijn codebook plan.
• Energy/onset proposals.
• Dechirp/bin scoring.
• Ambiguity candidates.
• Global bin-shift hypotheses.
• Clock fit and source-offset refinement.
• Calibration model with response/confusion/delay/phase/adaptive codebook.

What Is Missing

• Streaming candidate state.
• Calibration-weighted likelihood before path selection as the default physical path, not just optional model support.
• Better group-delay model.
• Reduced reliable-bin physical proof.
• Native/SIMD/GPU scoring path after correctness is proven.

Algorithm Options

Option A: Goertzel Bank

Best when:

• bin count is small;
• sample window is short;
• CPU SIMD is available;
• frequencies are fixed.

Pros:

• No FFT plan overhead.
• Easy to weight per bin.
• Simple to port to AVX2.

Cons:

• Repeats work per bin.
• Less attractive if bin count/window count grows.

Option B: Dechirp + FFT

Best when:

• many bins;
• many candidates;
• batching is possible.

Pros:

• Canonical CSS receiver shape.
• Good fit for GPU/native FFT batches.

Cons:

• FFT overhead and memory motion can dominate small cases.

Option C: Chirp Z Transform

Best when:

• only a narrow frequency band is needed at high resolution;
• arbitrary frequency grid matters.

Pros:

• Efficient zoomed spectral analysis.

Cons:

• More complex than current fixed bins.
• Might be premature unless physical data shows bin boundaries are the problem.

Option D: Matched Chirplet Atoms

Best when:

• emitted atoms are varied and not reducible to one common slope.

Pros:

• General.

Cons:

• Too expensive and too unconstrained for the current controlled beacon.

Implementation Rule

Build the controlled CSS-like receiver first. Pull in full chirplet machinery only when the codebook stops being controlled or when a research proof shows a specific invariant the controlled receiver cannot protect.