Real-Time State-Space Parameterization and Lock-Free Semantic Analysis in Digital Equalization Update:

Real-Time State-Space Parameterization and Lock-Free Semantic Analysis in Digital Equalization

Gary Doman (GareBear99 / TizWildin)

FreeEQ8 / ProEQ8 Open-Source DSP Project

https://github.com/GareBear99/FreeEQ8

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Abstract

This paper presents the architecture of a production-grade 8-band parametric
equalizer designed to eliminate high-frequency magnitude cramping without the
computational overhead of brute-force oversampling. By utilizing a 64-bit
double-precision implementation of the Simper State Variable Filter (SVF)
topology via trapezoidal integration, the system achieves a de-cramped
frequency response near the Nyquist limit while consuming only 0.62% of a
single-core real-time CPU budget at 44.1 kHz (161× headroom). To ensure
absolute real-time safety within modern Digital Audio Workstations (DAWs), a
lock-free Single-Producer Single-Consumer (SPSC) triple-buffering swap-chain
isolates the audio hot-path from UI rendering. We further introduce a
variable-cadence coefficient engine that switches between 4-sample batching
during sustained signals and per-sample accuracy on transients, reducing Dynamic
EQ CPU cost by up to 75% under stable envelope conditions. Finally, an
allocation-free, log-frequency resonance detection array (ResonanceDetector.h)
maps live spectral coefficients to localized semantic labels—providing a
framework for zero-latency, explainable mix-assist workflows.

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1. Introduction

Traditional parametric EQ implementations use the Robert Bristow-Johnson (RBJ)
Audio EQ Cookbook biquad formulas [6] in Transposed Direct Form II (TDF-II).
While mathematically correct, the bilinear transform introduces frequency
cramping near Nyquist: a Bell filter at 16 kHz with Q=1.0 at 44.1 kHz exhibits
an effective bandwidth 199% narrower than intended. Industry solutions
include brute-force oversampling (adding latency and CPU cost) or proprietary
polynomial analog-matching curves (FabFilter Pro-Q family).

This paper documents a third path: the Simper SVF topology, which pre-warps
the cutoff frequency via g = tan(π·fc/fs), achieving exact cutoff placement
at any frequency up to Nyquist without oversampling. All eight required filter
types (Bell, LowShelf, HighShelf, LP, HP, Bandpass, Notch, AllPass) emerge from
a single two-integrator core, simplifying maintenance and enabling structural
modulation safety that TDF-II cannot provide.

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2. Filter Topology

2.1 RBJ TDF-II (Legacy Path — FreeEQ8)

H(z) = (b₀ + b₁z⁻¹ + b₂z⁻²) / (1 + a₁z⁻¹ + a₂z⁻²)

Coefficients computed per the RBJ cookbook [6]. 64-bit double internal state,
float I/O. Parameter smoothing: 20 ms linear ramp, coefficients refreshed every
16 samples (smoothing path) or every sample (Dynamic EQ path).

Measured Q distortion (Bell, Q=1.0, 44.1 kHz):

Frequency	RBJ effective Q	Error
1 kHz	1.005	+0.5%
8 kHz	1.337	+33.7%
16 kHz	2.990	+199%

2.2 Simper SVF (Modern Path — ProEQ8)

Reference: Andrew Simper, "Solving the continuous SVF equations using
trapezoidal integration and equivalent currents," Cytomic, 2013 [1].

Pre-warped cutoff:

g  = tan(π · fc / fs)          // exact pre-warp
k  = 1/Q
a1 = 1 / (1 + g·(g + k))      // shared across all filter types
a2 = g · a1
a3 = g · a2

Per-sample processing (optimised bounded form):

v3  = v0 - ic2eq
t   = a2 * ic1eq               // cached — eliminates redundant mul
v1  = a1 * ic1eq + a2 * v3    // bandpass
v2  = ic2eq + t + a3 * v3     // lowpass
ic1eq = v1 + v1 - ic1eq       // v+v avoids 2.0* mul
ic2eq = v2 + v2 - ic2eq
out = m0·v0 + m1·v1 + m2·v2  // filter-type-specific mix

Output mix coefficients per filter type (from Simper paper §§3–8):

Type	m0	m1	m2
LP	0	0	1
HP	1	−k	−1
BP	0	1	0
Bell	1	kA·(A²−1)	0
LowShelf	1	k·(A−1)	A²−1
HighShelf	A²	k·(1−A)·A	1−A²

where A = 10^(gainDb/40) and kA = k/A (Bell uses modified denominator).

Measured performance (g++ -O3, 44.1 kHz, 512-sample block, 8-band stereo):

Path	ns/sample	CPU headroom
RBJ 8-band	40.7	277×
SVF 8-band	70.4	161×
SVF overhead	1.73×	—
CPU budget used	0.62%	—

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3. Real-Time Safety Architecture

3.1 SPSC Triple-Buffer (SpectrumFIFO)

Three buffer slots indexed by {writeSlot, midSlot, readSlot} (a permutation
of {0, 1, 2}). Audio thread writes into writeSlot; when a frame is complete,
atomically swaps writeSlot ↔ midSlot with memory_order_release. UI thread
reads by atomically swapping midSlot ↔ readSlot with memory_order_acquire.
No mutex, no lock, no blocking on either thread.

3.2 Off-Thread FIR Reconstruction (LinearPhaseEngine)

Linear phase mode latency: 2048 samples (4096-tap Hann-windowed FIR / 2).
Parameter changes set an atomic linPhaseDirty flag. A dedicated background
thread (LinPhaseRebuildThread) parks via wait(-1), wakes on notify, rebuilds
the FIR kernel (magnitude → IFFT → circular shift → Hann window → forward FFT),
and publishes via the same triple-buffer atomic swap protocol. Audio thread reads
with a single acquire load—never blocks.

3.3 Allocation-Free Hot Path

All oversamplers (1×/2×/4×/8× via JUCE polyphase IIR) are pre-constructed in
prepareToPlay() into std::array<std::unique_ptr<Oversampling<float>>, 3>.
Mid-playback order changes call Oversampling::reset() (non-allocating) and
trigger a 128-sample linear crossfade to eliminate the transient pop (v2.2.3).

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4. Variable-Cadence Dynamic EQ (v2.2.3)

Dynamic EQ recomputes biquad coefficients per-sample when active, matching the
one-pole envelope follower's cadence. With all 8 bands in dynamic mode the
bq.set() call dominates (22 ns/call for SVF Bell). We observe that during
held/sustained notes the envelope dynGainMod is near-static: changes of less
than 0.1 dB between samples are inaudible within the 4-sample batch window
(0.09 ms at 44.1 kHz).

Variable-cadence algorithm:

δ = |dynGainMod − lastDynGainMod|
if δ > 0.1 dB:
    update coefficients (per-sample — zero transient lag)
    intervalCounter = 0
else:
    if intervalCounter++ >= 4:
        update coefficients (batched)

Measured savings at 8 bands all-dynamic, sustained note:
up to 75% reduction in bq.set() calls with no audible difference.
On a transient attack the first sample with δ > 0.1 dB immediately restores
per-sample accuracy.

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5. Allocation-Free Semantic Analysis (ResonanceDetector)

Traditional "smart EQ" products apply machine-learning inference models (Soothe2,
iZotope Neutron) — opaque, CPU-heavy, non-deterministic. We introduce a
deterministic alternative that adds zero latency and allocates no memory:

1. Take the 2048-bin log-magnitude spectrum from SpectrumFIFO (UI thread rate).
2. Re-sample into a 96-bin log-frequency grid (20 Hz → Nyquist).
3. Estimate local baseline via ±0.5-octave moving average.
4. Flag peaks where (magnitude − baseline) ≥ 3 dB and peak is local max in ±3-bin neighbourhood.
5. Score each peak: score = deviation × intentWeight(hz, mode) where intentWeight is a log-frequency Gaussian bump per IntentMode profile.
6. Return top-4 suggestions: {freqHz, gainDb, q, confidence, label}.

The intentWeight function encodes instrument-specific problem zones:

Mode	Primary Bump	Zone
VocalClean	×1.6 at 300 Hz	mud
DrumPunch	×1.5 at 300 Hz	boxiness
GuitarSpace	×1.5 at 250 Hz	mud
MasterPolish	×1.3 at 250 Hz	low-end buildup

Semantic labels from FrequencyExplainer.h map frequency ranges to strings
("mud", "harshness", "sibilance", "air") enabling explain-on-hover UX.

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7. Benchmarks (Measured — Reproducible)

All benchmarks run from Tests/FeatureBench.cpp — standalone, no JUCE, no DAW,
no mock. Build: g++ -std=c++17 -O3 -DNDEBUG -pthread Tests/FeatureBench.cpp -o FeatureBench -ISource.
Platform: Linux x86-64, g++ 13.3.0. Median of 16 trials, 4 warmup runs discarded.

7.1 Single-Instance Filter Cost

Path	ns/sample	MB/s	CPU% (44.1kHz/512/50%)	Headroom
RBJ 8-band stereo	41.0	98	0.36%	277×
SVF 8-band stereo	72.7	55	0.63%	161×
SVF overhead vs RBJ	1.61×	—	—	—
SVF DynEQ per-sample	68.8	58	0.61%	165×

7.2 Instance Scaling (Real DAW Load Simulation)

The critical gap identified in post-release review: "not yet crossed into industrial
benchmark validation under real DAW stress matrices." This table fills that gap.
Each row simulates N simultaneous independent 8-band stereo plugin instances.

Instances	RBJ ns/samp	RBJ CPU%	SVF ns/samp	SVF CPU%	SVF/RBJ
1	44.4	0.39%	71.9	0.63%	1.62×
8	46.6	0.41%	73.4	0.65%	1.58×
32	47.5	0.42%	75.1	0.66%	1.58×
64	46.4	0.41%	75.9	0.67%	1.64×
128	46.8	0.41%	75.5	0.67%	1.61×

Key finding: Per-instance cost rises only 5% from 1→128 instances (cache
pressure from larger working set). The scaling is sub-linear — each instance
benefits from the previous instance's cache warmup on shared coefficient tables.

At 128 SVF instances total CPU = 128 × 0.67% = 85.8% of one core at 44.1 kHz
with 512-sample blocks. A modern 8-core CPU can host ~900 SVF instances.

7.3 Worst-Case Dynamic EQ

Document 11 review identified: "the actual limit is NOT filter math — it becomes
dynamic coefficient churn." This benchmark quantifies exactly that ceiling.

Configuration: 8 bands simultaneously in dynamic mode, white noise input
(maximum envelope follower excitation — all transients, all samples active),
variable-cadence engine active (v2.2.3 optimization).

Configuration	ns/sample	CPU%	Headroom
8-band DynEQ, white noise, all active	370.9	3.27%	30.6×

Finding: Even at absolute worst-case (8 active dynamic bands tracking white
noise), the variable-cadence engine keeps CPU below 3.3%. The 30.6× headroom
means a 50% CPU budget can host ~9 simultaneous worst-case dynamic EQ instances.

7.4 SvfBandArray — Packed SIMD Scaffold

The SvfBandArray<8> template (v2.2.4) packs all 8 band states into aligned
arrays for SIMD dispatch. On this test machine (SSE2, no AVX2 available at test
time), the scalar fallback runs:

Path	ns/sample (mono)	CPU%	vs SVF scalar stereo
SvfBandArray<8> scalar (SSE2 host)	23.5	0.21%	3.1× faster

The mono vs stereo difference accounts for half the gap. With AVX2 active
(8-wide float32), projected improvement is an additional 2–4× over scalar,
targeting < 10 ns/sample for all 8 bands mono — approaching 0.09% CPU.

7.5 MatchEQ Hot-Path Optimization

Path	ns/sample equivalent	Speedup
Naive pow(10) per bin (old)	7.4	—
Pre-computed correctionGain	2.8	3.0×

7.6 Reproducing These Results

git clone --recursive https://github.com/GareBear99/FreeEQ8.git
cd FreeEQ8
g++ -std=c++17 -O3 -DNDEBUG -pthread Tests/FeatureBench.cpp -o FeatureBench -ISource
./FeatureBench          # human-readable table
./FeatureBench --csv    # machine-readable CSV

# For ARC-AudioBench integration (JSON output):
g++ -std=c++17 -O3 -DNDEBUG Tests/ArcBenchIntegration.cpp -o ArcBench -ISource
./ArcBench --json > arc_results.json

Numbers will vary by CPU and compiler. The headroom ratios should remain
comfortably above 10× on any modern x86-64 or Apple Silicon machine.

Inspired by Ableton Live's EQ Eight compact device view. Design constraint: the
coordinate mapping (freqToX, dbToY), drag sensitivity (pixel delta → parameter
delta), Q drag acceleration, and node hit-test radius (as proportion of view
height) must be identical between full and compact views. Only visual density
changes: FFT resolution, grid label density, node text size.

This is enforced architecturally: setCompactMode(bool) sets a flag but never
modifies the mapping functions. The APVTS remains the single source of truth;
both renderers read the same parameter values.

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7. Future Work (v2.2.5+)

• Explicit SIMD vectorisation: group 8 bands into juce::dsp::SIMDRegister<float>, processing 4 bands per SSE instruction or 8 via AVX2.
• Cross-instance masking negotiation: via ARC-Core local IPC spine, multiple plugin instances communicate energy peaks and negotiate inverse dynamic notches.
• Spectral dynamics mode: per-bin FFT threshold clamping (Soothe2 territory) using the existing overlap-add Match EQ infrastructure.
• Dolby Atmos 9.1.6: expand isBusesLayoutSupported for discrete immersive channel arrays with spatial zone linking.

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References

[1] A. Simper, "Solving the continuous SVF equations using trapezoidal integration
and equivalent currents," Cytomic, 2013.
https://cytomic.com/files/dsp/SvfLinearTrapOptimised2.pdf

[2] W. Pirkle, "Designing Audio Effect Plugins in C++," Focal Press, 2019.

[3] J. Reiss and A. McPherson, "Audio Effects: Theory, Implementation and
Application," CRC Press, 2014.

[4] F. Renn-Giles and D. Rowland, "Real-time 101," ADC 2019.
https://github.com/hogliux/farbot

[5] JUCE, "juce::dsp::Oversampling," Articy, 2023.
https://docs.juce.com/master/classjuce_1_1dsp_1_1Oversampling.html

[6] R. Bristow-Johnson, "Audio EQ Cookbook," musicdsp.org, 1994.
https://www.musicdsp.org/files/Audio-EQ-Cookbook.txt