Real-Time State-Space Parameterization and Lock-Free Semantic Analysis in Digital Equalization
Gary Doman (GareBear99 / TizWildin)
FreeEQ8 / ProEQ8 Open-Source DSP Project
GareBear99
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FreeEQ8
FreeEQ8 is a professional-grade, free and open-source 8-band parametric EQ plugin for macOS, Linux, and Windows. Linear phase, dynamic EQ, match EQ, per-band drive, band linking, M/S processing, oversampling, and a real-time spectrum analyzer. FabFilter Pro-Q alternative for $0–$20
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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.
- 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.
- 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% —
- 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::arraystd::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).
- 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.
- 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:
Take the 2048-bin log-magnitude spectrum from SpectrumFIFO (UI thread rate).
Re-sample into a 96-bin log-frequency grid (20 Hz → Nyquist).
Estimate local baseline via ±0.5-octave moving average.
Flag peaks where (magnitude − baseline) ≥ 3 dB and peak is local max in ±3-bin neighbourhood.
Score each peak: score = deviation × intentWeight(hz, mode) where intentWeight is a log-frequency Gaussian bump per IntentMode profile.
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.
- Compact View / Mini-Window (v2.2.3) 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.
- Future Work (v2.5+) Explicit SIMD vectorisation: group 8 bands into juce::dsp::SIMDRegister, 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. 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
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