…scaled Bessel functions

…led variants to eliminate NR2 Gamma crackling (#1507) (#3275)

## Summary

- Replaces `bessI0()`/`bessI1()` in `SpectralNR` with
exponentially-scaled `bessI0e()`/`bessI1e()` that return `exp(-|x|) *
Iₙ(x)` instead of `Iₙ(x)` directly
- Removes the now-redundant `std::exp(-0.5 * v)` factor from
`computeGainGamma()` — the cancellation happens inside the scaled
functions, no overflow possible
- Adds `spectral_nr_test` (112 assertions) covering Bessel finiteness,
mathematical equivalence, integration, and gain formula at extreme v
- OTA validated TC-1 through TC-7 on FLEX-8400 with no crackling at any
tested signal level, GainMax setting, or NPE method

Fixes #1507.

## Root cause

The large-x branch of `bessI0()`/`bessI1()` (A&S 9.8.1/9.8.2) computed
`std::exp(ax) / std::sqrt(ax) * poly`. In `computeGainGamma()`, the
Ephraim-Malah gain expression is:

```
gain = Γ(3/2) · √v / γ · exp(-v/2) · [(1+v)·I₀(v/2) + v·I₁(v/2)]
```

The `exp(-v/2)` factor is intended to cancel the `exp(ax)` inside
`bessI0`/`bessI1`, but because they are computed as separate
intermediate values, the Bessel calls overflow to `+inf` when `v/2 >
710` (i.e. `v > 1420`). The product `0 · ∞ = NaN` per IEEE 754. The NaN
guard clamps gain to `0.01` (~−40 dB) for that hop, producing a
suppression burst every 5.33 ms — audible as crackling.

`v = ehr · γ` where `γ` is capped at `GammaMax = 1e4`, so the overflow
threshold (`v > 1420`) is crossed for any signal with SNR above ~31.5 dB
under the Gamma gain method. This is routine for a strong HF signal or
any in-band carrier.

PR #1696 (merged April 2026) addressed a distinct mechanism — `gainMax =
1.5` causing output amplification above ±1.0 float32, clipped at the
QAudioSink. That fix is orthogonal; the Bessel overflow is still present
in the current tree.

## Investigation note re: PR #1508

PR #1508 (the earlier automated Bessel fix) was closed on 2026-04-18
with the rationale that Opus compression artifacts were "the real root
cause." After further investigation we believe both issues are real but
independent:

- **Opus/SmartLink path**: NR2's spectral gain estimator amplifies Opus
codec artifacts — valid, should be tracked separately
- **Bessel overflow path**: triggered by any signal with SNR > 31.5 dB,
regardless of audio codec — present on direct DAX connections with no
Opus in the chain

The original reporter used a FLEX-8600 on a direct connection ("inject a
strong signal — a −10 dBm test tone, or tune to a strong local
broadcast"), not a SmartLink session. Disabling NR2 under Opus would not
have fixed their report. This PR addresses only the Bessel path; the
Opus interaction is left for a separate issue and fix.

## Changes

**`src/core/SpectralNR.h`** — rename two private static declarations:
```cpp
// Before
static double bessI0(double x);
static double bessI1(double x);

// After
static double bessI0e(double x);   // returns exp(-|x|) * I0(x)
static double bessI1e(double x);   // returns exp(-|x|) * I1(x)
```

**`src/core/SpectralNR.cpp`** — rewrite both Bessel implementations and
update the calling site:

- *Small-x branch*: existing polynomial multiplied by `std::exp(-ax)` —
trivial, no overflow possible
- *Large-x branch*: `exp(-ax) · exp(ax) / √ax · poly` reduces
analytically to `1 / √ax · poly` — **no exponential computed at all**;
the overflow path is eliminated by construction
- `computeGainGamma()`: remove `std::exp(-0.5 * v)` from the gain
expression; rename Bessel calls

```cpp
// Before
double gain = gf1p5 * std::sqrt(v) / std::max(gamma, EpsFloor)
            * std::exp(-0.5 * v)
            * ((1.0 + v) * bessI0(0.5 * v) + v * bessI1(0.5 * v));

// After
double gain = gf1p5 * std::sqrt(v) / std::max(gamma, EpsFloor)
            * ((1.0 + v) * bessI0e(0.5 * v) + v * bessI1e(0.5 * v));
```

Result is mathematically identical to the original for all `v < 1420`;
numerically stable for `v` up to `GammaMax = 1e4` and beyond.

**`tests/spectral_nr_test.cpp`** (new) and **`CMakeLists.txt`** —
standalone test target `spectral_nr_test`, Qt6::Core only, no FFTW3
dependency:

| Group | What it tests | Assertions |
|---|---|---|
| Bessel finiteness | `bessI0e`/`bessI1e` finite and non-negative at
`v/2` for v ∈ {0…10000}, explicitly straddling v=1420 | 52 |
| Mathematical equivalence | Matches `exp(-x)·Iₙ_ref(x)` to 1e-9
relative error at x ≤ 50; symmetry / antisymmetry | 34 |
| Gain finiteness | `SpectralNR::process()`, Gamma method, 1000 hops
full-scale 1 kHz sine: zero NaN/Inf samples | 1 |
| Gain formula extreme v | Direct formula at v ∈ {1419, 1420, 1421,
2000, 5000, 10000}: finite, positive, not the 0.01 NaN-clamp sentinel |
24 |

## What this does NOT change

- `processNr2()` output clamp and NaN guard — retained as
belt-and-suspenders
- `GammaMax = 1e4` — no change needed; the full range is now numerically
safe
- All other gain methods (Linear, Log, Trained) — unaffected; they do
not call Bessel functions
- No behaviour change for signals below the overflow threshold (`v <
1420`, SNR < 31.5 dB)

## Test plan

Automated: `ctest -R spectral_nr_test` — 112/112 pass.

OTA (FLEX-8400, Gamma method, FFTW3 enabled production build):

- [x] TC-1: Strong broadcast/SSB S9+10, 60 s — no crackling
- [x] TC-2: Steady carrier S9+20–S9+40 (maximum-gamma stress) — clean
pass-through
- [x] TC-3: GainMax 0.5 / 1.0 / 1.5 — no crackling at any value
- [x] TC-4: Enable NR2 mid-signal (cold OSMS ramp) — clean transition
- [x] TC-5: All four gain methods compared — no regressions
- [x] TC-6: Weak-signal NR effectiveness — unchanged
- [x] TC-7: MMSE and NSTAT NPE methods — no crackling

🤖 Generated with [Claude Code](https://claude.com/claude-code)

---------

Co-authored-by: Claude Sonnet 4.6 <noreply@anthropic.com>

…led variants to eliminate NR2 Gamma crackling (aethersdr#1507) (aethersdr#3275)

## Summary

- Replaces `bessI0()`/`bessI1()` in `SpectralNR` with
exponentially-scaled `bessI0e()`/`bessI1e()` that return `exp(-|x|) *
Iₙ(x)` instead of `Iₙ(x)` directly
- Removes the now-redundant `std::exp(-0.5 * v)` factor from
`computeGainGamma()` — the cancellation happens inside the scaled
functions, no overflow possible
- Adds `spectral_nr_test` (112 assertions) covering Bessel finiteness,
mathematical equivalence, integration, and gain formula at extreme v
- OTA validated TC-1 through TC-7 on FLEX-8400 with no crackling at any
tested signal level, GainMax setting, or NPE method

Fixes aethersdr#1507.

## Root cause

The large-x branch of `bessI0()`/`bessI1()` (A&S 9.8.1/9.8.2) computed
`std::exp(ax) / std::sqrt(ax) * poly`. In `computeGainGamma()`, the
Ephraim-Malah gain expression is:

```
gain = Γ(3/2) · √v / γ · exp(-v/2) · [(1+v)·I₀(v/2) + v·I₁(v/2)]
```

The `exp(-v/2)` factor is intended to cancel the `exp(ax)` inside
`bessI0`/`bessI1`, but because they are computed as separate
intermediate values, the Bessel calls overflow to `+inf` when `v/2 >
710` (i.e. `v > 1420`). The product `0 · ∞ = NaN` per IEEE 754. The NaN
guard clamps gain to `0.01` (~−40 dB) for that hop, producing a
suppression burst every 5.33 ms — audible as crackling.

`v = ehr · γ` where `γ` is capped at `GammaMax = 1e4`, so the overflow
threshold (`v > 1420`) is crossed for any signal with SNR above ~31.5 dB
under the Gamma gain method. This is routine for a strong HF signal or
any in-band carrier.

PR aethersdr#1696 (merged April 2026) addressed a distinct mechanism — `gainMax =
1.5` causing output amplification above ±1.0 float32, clipped at the
QAudioSink. That fix is orthogonal; the Bessel overflow is still present
in the current tree.

## Investigation note re: PR aethersdr#1508

PR aethersdr#1508 (the earlier automated Bessel fix) was closed on 2026-04-18
with the rationale that Opus compression artifacts were "the real root
cause." After further investigation we believe both issues are real but
independent:

- **Opus/SmartLink path**: NR2's spectral gain estimator amplifies Opus
codec artifacts — valid, should be tracked separately
- **Bessel overflow path**: triggered by any signal with SNR > 31.5 dB,
regardless of audio codec — present on direct DAX connections with no
Opus in the chain

The original reporter used a FLEX-8600 on a direct connection ("inject a
strong signal — a −10 dBm test tone, or tune to a strong local
broadcast"), not a SmartLink session. Disabling NR2 under Opus would not
have fixed their report. This PR addresses only the Bessel path; the
Opus interaction is left for a separate issue and fix.

## Changes

**`src/core/SpectralNR.h`** — rename two private static declarations:
```cpp
// Before
static double bessI0(double x);
static double bessI1(double x);

// After
static double bessI0e(double x);   // returns exp(-|x|) * I0(x)
static double bessI1e(double x);   // returns exp(-|x|) * I1(x)
```

**`src/core/SpectralNR.cpp`** — rewrite both Bessel implementations and
update the calling site:

- *Small-x branch*: existing polynomial multiplied by `std::exp(-ax)` —
trivial, no overflow possible
- *Large-x branch*: `exp(-ax) · exp(ax) / √ax · poly` reduces
analytically to `1 / √ax · poly` — **no exponential computed at all**;
the overflow path is eliminated by construction
- `computeGainGamma()`: remove `std::exp(-0.5 * v)` from the gain
expression; rename Bessel calls

```cpp
// Before
double gain = gf1p5 * std::sqrt(v) / std::max(gamma, EpsFloor)
            * std::exp(-0.5 * v)
            * ((1.0 + v) * bessI0(0.5 * v) + v * bessI1(0.5 * v));

// After
double gain = gf1p5 * std::sqrt(v) / std::max(gamma, EpsFloor)
            * ((1.0 + v) * bessI0e(0.5 * v) + v * bessI1e(0.5 * v));
```

Result is mathematically identical to the original for all `v < 1420`;
numerically stable for `v` up to `GammaMax = 1e4` and beyond.

**`tests/spectral_nr_test.cpp`** (new) and **`CMakeLists.txt`** —
standalone test target `spectral_nr_test`, Qt6::Core only, no FFTW3
dependency:

| Group | What it tests | Assertions |
|---|---|---|
| Bessel finiteness | `bessI0e`/`bessI1e` finite and non-negative at
`v/2` for v ∈ {0…10000}, explicitly straddling v=1420 | 52 |
| Mathematical equivalence | Matches `exp(-x)·Iₙ_ref(x)` to 1e-9
relative error at x ≤ 50; symmetry / antisymmetry | 34 |
| Gain finiteness | `SpectralNR::process()`, Gamma method, 1000 hops
full-scale 1 kHz sine: zero NaN/Inf samples | 1 |
| Gain formula extreme v | Direct formula at v ∈ {1419, 1420, 1421,
2000, 5000, 10000}: finite, positive, not the 0.01 NaN-clamp sentinel |
24 |

## What this does NOT change

- `processNr2()` output clamp and NaN guard — retained as
belt-and-suspenders
- `GammaMax = 1e4` — no change needed; the full range is now numerically
safe
- All other gain methods (Linear, Log, Trained) — unaffected; they do
not call Bessel functions
- No behaviour change for signals below the overflow threshold (`v <
1420`, SNR < 31.5 dB)

## Test plan

Automated: `ctest -R spectral_nr_test` — 112/112 pass.

OTA (FLEX-8400, Gamma method, FFTW3 enabled production build):

- [x] TC-1: Strong broadcast/SSB S9+10, 60 s — no crackling
- [x] TC-2: Steady carrier S9+20–S9+40 (maximum-gamma stress) — clean
pass-through
- [x] TC-3: GainMax 0.5 / 1.0 / 1.5 — no crackling at any value
- [x] TC-4: Enable NR2 mid-signal (cold OSMS ramp) — clean transition
- [x] TC-5: All four gain methods compared — no regressions
- [x] TC-6: Weak-signal NR effectiveness — unchanged
- [x] TC-7: MMSE and NSTAT NPE methods — no crackling

🤖 Generated with [Claude Code](https://claude.com/claude-code)

---------

Co-authored-by: Claude Sonnet 4.6 <noreply@anthropic.com>