The Analog Edge, Part 13: Coherence, Not dBFS — Scale-Invariant Health

By Matt Cheramie · August 30, 2026

Part 13 of The Analog Edge, a 14-part field guide to the analog half of a GopherTrunk system. Part 12 classified front ends by watching branch_phase_deg — a log field standing in for test equipment. This part is about the number that made the whole diversity stack honest, and the incident that forced it into existence. Back in Part 3 we promised the full story of the operator who raised gain 65 → 82 dB to clear a software constant by 0.2 dB. Here it is — and the scale-invariant replacement is the most transferable idea this series has to offer.

TL;DR: MRC calibration used to be gated on the reference branch clearing −40 dBFS — so an operator raised front-end gain from 65.0 to 82.0 dB purely to push a meter past a constant (landing at −39.8 dBFS, clearing it by 0.2 dB). Any absolute-power gate is a gain-staging trap that re-fires on the next front end. The replacement is the normalised cross-correlation |rho| = |Σ x1·conj(x0)| / sqrt(Σ|x0|²·Σ|x1|²) (diversity.CrossStats), which is scale-invariant and answers the question the calibrator actually has: are these two receivers seeing the same signal? The mapping |rho| = γ/(1+γ) makes thresholds interpretable — 0.50 ≈ 0 dB per-branch SNR, 0.35 ≈ −2.7 dB — against a noise-only floor near sqrt(π/4N). Inside the correlator, DC removal is load-bearing, not hygiene; absolute power survives only as a −100 dBFS digitally-dead-branch reject.

Update (18 Aug): a fixed |rho| threshold turned out to be a staging trap of its own — a bandwidth-staging trap. Wideband |rho| is diluted by every hertz of noise-only bandwidth around the coherent carrier (rho_wb ≈ rho_ch·sqrt(f0·f1) for in-channel power fractions), and an X310 operator raised RF gain 5 dB purely to push wideband coherence past the 0.50 constant while their phase estimate had been accurate to ~4° all along. The gates now bound the estimate’s projected phase error sqrt((1−ρ²)/(2Nρ²)) instead (diversity.TrackingOptions.LockPhaseSigmaRad / TrackPhaseSigmaRad, defaults 0.10/0.16 rad): the minimum |rho| falls as 1/sqrt(N) yet stays ~8×/~5× above the sqrt(π/4N) noise floor, so noise still cannot lock. The principle of this post stands — gate on evidence, not level — the evidence was just one derivative sharper than |rho| itself.

Key takeaways

• An absolute dBFS gate makes operators tune the radio to the software. The 17 dB gain raise bought nothing but intermod risk — the number it chased measured level, and the decision needed evidence. Every absolute threshold quietly re-fires when the hardware changes.
• Coherence is the question, so measure the question. “Are these branches the same signal through a constant complex gain?” has a direct statistic, and it reads the same at any gain setting — which is why the not-coherent WARN can honestly say raising gain will not help.
• The γ/(1+γ) mapping turns a unitless number into dB. A threshold of 0.5 isn’t folklore; it’s “0 dB per-branch SNR,” chosen on a scale where the noise floor for N samples is computable.
• An uncentred correlator lies confidently. Shared LO leakage is common-mode DC on both branches; without DC removal, two branches of pure independent noise read |rho| → 1 — a calibrator would lock onto nothing, in exactly the weak-signal regime the gate exists to protect.

Cheat sheet

Concern	What it does	Where it lives
The statistic	DC-removed normalised cross-correlation over a window	internal/dsp/diversity/crossstats.go (CrossStats.Coherence)
The gain beside it	least-squares branch gain from the same sums	CrossStats.Gain (h = cov(x,ref)/var(ref))
Thresholds	phase-error bounds 0.10/0.16 rad ⇒ |rho| ≥ 1/sqrt(1+2Nσ²) (see the Update above)	internal/sdr/soapyremote/mrc.go (mrcLockPhaseSigmaRad, mrcTrackPhaseSigmaRad)
DC-removal proof	common DC must not read as correlation	TestCrossStatsRejectsCommonDCOffset (crossstats_test.go)
The power floor that remains	reject a digitally dead branch only	MinBranchPower 1e-10 ≈ −100 dBFS (tracking.go)
Operator surface	coherence in the 30 s MRC health line	diversityReporter (mrc.go), Part 11
The deep dive	full calibration design + measurements	WSE Part 10

In this post

• The 0.2 dB incident — how a threshold turns into a tuning target.
• The scale-invariant question — what coherence measures.

• **Reading

  rho

  in dB** — the γ/(1+γ) mapping and the noise floor.

• DC removal is load-bearing — the trap inside the correlator.
• The general moral — gating decisions on evidence, not level.

The 0.2 dB incident

The old MRC calibration logic refused to estimate branch gains until the reference branch’s level cleared −40 dBFS. Reasonable-sounding — “don’t calibrate on noise” — and wrong in a way that only shows up in the field. An operator with a healthy but conservatively-gained front end saw diversity refuse to engage, read the number it wanted, and did the only thing the gate rewarded: raised RF gain from 65.0 dB to 82.0 dB. They landed at −39.8 dBFS — clearing the constant by 0.2 dB — and diversity engaged. Nothing about their signal had improved. They had spent 17 dB of front-end headroom — the same headroom Part 4 showed protecting you from intermod — to move a meter past a line in the source code.

That is the anatomy of every absolute-power gate: it couples a decision (can I trust this estimate?) to a level (how hot is the ADC running?), and operators respond to what’s rewarded. Part 2 made the case that dBFS is a headroom meter, not a quality meter — the #764 captures both peaked ≈−48 dBFS and differed by 10 dB of usable SNR. A gate on dBFS is that category error, compiled in. And it re-fires on every new front end: the constant that a USRP clears at gain 65 needs 82 on the next radio, and whatever an RTL-SDR needs is nobody’s guess. The fix was to delete the question the gate was asking and ask the real one.

The scale-invariant question

What the calibrator actually needs to know is: are these two receivers seeing the same signal through a constant complex gain? That question has a direct statistic — the normalised cross-correlation of the two branches over an accumulation window:

// internal/dsp/diversity/crossstats.go (shape)
// Gain()      the least-squares complex gain  h = cov(x,ref)/var(ref)
// Coherence() the normalised cross-correlation |rho| in [0,1]
//
// Coherence is the quality gate. It answers "are these two receivers seeing
// the same signal through a constant complex gain?" … and — unlike an
// absolute power threshold — it is SCALE-INVARIANT.
func (s *CrossStats) Coherence() float64

Scale-invariance is the whole point: multiply either branch by any constant — turn any gain knob anywhere — and |rho| does not move, because the denominator normalises both branches’ energy away. The statistic reads the relationship between the branches, not their level. This is why the not-coherent WARN from Part 11 can state flatly that raising RF gain will not help — mathematically, it can’t. The same accumulated sums also yield the branch gain h itself, so the gate and the estimate come from one pass over one window: if |rho| says the window is trustworthy, h from that window is the calibration.

Reading |rho| in dB

A unitless 0-to-1 number invites folklore thresholds. What keeps this one honest is that it maps to something physical: for two branches carrying a common signal in independent noise at equal per-branch SNR γ,

**

rho

= γ / (1 + γ)**

So 0.50 is γ=1 — 0 dB per-branch SNR. 0.35 is about −2.7 dB. And two branches of pure independent noise over N samples don’t read zero — they read near sqrt(π/4N), about 0.03 at N=1000: the floor any threshold must clear. GopherTrunk’s gates originally sat right on this scale — lock at 0.50, track at 0.35 — until an operator’s 18 Aug captures showed the fixed constants were a bandwidth-staging trap (see the Update in the TL;DR): the gates now bound the estimate’s phase error, which puts the minimum |rho| at ~8× (lock) and ~5× (track) the sqrt(π/4N) floor for whatever window the stream is running (mrcLockPhaseSigmaRad, mrcTrackPhaseSigmaRad).

The γ/(1+γ) curve makes coherence thresholds interpretable: 0.5 means 0 dB per-branch SNR, 0.35 means −2.7 dB, and the noise-only floor is computable.

Absolute power didn’t vanish entirely — it survives as exactly one check, at a level no gain-staging decision will ever brush against: a branch below MinBranchPower (1e-10 linear, ≈−100 dBFS, where one CS16 LSB is −90.3 dBFS) is digitally dead — all zeros, a receiver that never digitised — and there |rho| is 0/0. That’s not a signal-presence test; it’s a divide-by-zero guard, which is all absolute power was ever qualified to be.

DC removal is load-bearing

One trap hides inside the statistic itself, and it’s worth knowing because it generalizes. Every sum in CrossStats is a DC-removed covariance, not a raw correlation — and that’s not numerical hygiene. Both receivers of a shared front end carry LO leakage and converter DC offset, and that offset is perfectly common-mode: identical on both branches, always. Feed two branches of pure independent noise plus a shared DC into an uncentred correlator and it reports |rho| → 1 — mathematically correct, the DC really is perfectly correlated — and hands back h = dc1/dc0. A calibrator gated on that would confidently lock onto nothing, in exactly the weak-signal regime the gate exists to protect, and the combined output would be two noise floors aligned by their LO leakage. TestCrossStatsRejectsCommonDCOffset pins the trap shut: the centred statistic reads that same input as the noise it is. The cost of correctness is one extra running sum per branch. The cost of the bug would have been invisible.

The general moral

The series keeps meeting the same lesson wearing different clothes, so here it is stated once, plainly: never gate a DSP decision on an absolute level; gate it on scale-invariant evidence. Gain sweeps are scored by decode error rate, not power (The Hunt Part 5). Equalizers are judged by CRC yield, not EVM (the metrics-that-lie problem). Calibration is gated on coherence, not dBFS. In every case the absolute number was easier to compute and already on hand — and in every case it measured the radio’s configuration when the decision needed the signal’s testimony. When you find yourself adjusting hardware to satisfy a software number, stop and ask what question the number was standing in for. The full design that came out of this incident — window sizing, the hold- don’t-fallback rule, and what the gates measured on real captures — is Weak-Signal Engineering Part 10.

Where this goes next

That’s the last new instrument this series needed. Everything is now on the bench: dBFS read correctly, gain staged by decode quality, overload and phase noise recognized on sight, feedline and LNA budgets done in dB, captures that prove things, diversity with honest health numbers. Part 14 folds all of it into one field checklist — the is-it-RF-or-is-it-software triage tree — and closes the story of the reader whose hardware scanner used to win.

FAQ

Isn’t a coherence gate just a threshold too? Why is 0.5 better than −40 dBFS? Both are thresholds; they differ in what they measure. |rho|=0.5 is a statement about the signal (0 dB per-branch SNR) that reads identically on any hardware at any gain. −40 dBFS is a statement about the ADC’s operating point, which the operator controls — so it selects for operators who turn the knob, not for trustworthy estimates.

What coherence should I see on a healthy system? Track the mapping: branches at 5 dB SNR cohere at ~0.76, at 10 dB ~0.91. A co-located pair on a live trunking band typically reads 0.6–0.9. The two diagnostic regimes are near-floor (~0.03–0.1: branches see different signals — band, polarization, clocking) and stuck mid-range (~0.3–0.5: the wideband-scalar limit from Part 11).

Could the DC trap really happen, or is it theoretical? It’s the default outcome, not an edge case: every zero-IF front end leaks some LO, so the common-mode DC is always present, and the trap fires strongest precisely when there’s the least real signal to out-correlate it. That’s why the test pins it and why “hygiene” is the wrong word — remove the centring and the diversity stack fails quietly on weak signals only.

Where else does GopherTrunk apply this rule? Anywhere a decision used to want a power gate: autogain optimizes decode error rate across the sweep; the MRC dead-branch check sits at −100 dBFS (existence, not quality); and the equalizer work is validated exclusively by CRC-clean decode counts on captures. The diagnostic playbook collects the field-facing versions.

I’m not running diversity. What do I take from this part? The trap generalizes to your own debugging: any time you’re pushing gain, rate, or thresholds to make a dashboard number cross a line, you’re gating on level. Ask instead for scale-invariant evidence — decode rate, CRC yield, sync margin — the numbers that can’t be flattered by a knob.

Series navigation

Part 13 of 14 · ← Part 12: Front-End Classes — Shared LO vs Independent PLLs · Next → Part 14: The Field Checklist — Is It RF or Is It Software?