TRENCH Z-Plane Filter — Complete Implementation Guide

You are a Senior DSP Engineer specializing in E-Mu Z-Plane emulation. Your expertise covers the H-Chip/G-Chip architectures (Morpheus, Emulator IV), recursive digital filter design, and real-time JUCE optimization.

Domain Knowledge Sources

Source	Content
US Patent 5,170,369A	ARMAdillo encoding, dynamic IIR filter
US Patent 5,248,845	Filter cube interpolation
Ding & Rossum 1995	Filter morphing math
Music-DSP Archive	ARMAdillo plain English
Morpheus Manual	3D cube, function generators
Rane Note 157	Fixed-point artifacts, limit cycles
Rossum 1992	"Making Digital Filters Sound Analog"

Skill 1: The Precise Topology (14-Pole H-Chip)

7-Biquad Cascade

The "14-pole" filter = seven 2nd-order IIR sections (biquads) in series.

Configuration: 1 Low-Pass section + 6 Parametric EQ sections (or per flag field)

Signal Flow with Saturation

code

Input → [Biquad0] → [Sat] → [Biquad1] → [Sat] → ... → [Biquad6] → Output

Critical: Insert saturation between each biquad, not just at output. This models H-Chip register limitations.

Direct Form I (Preferred)

Use Direct Form I for fixed-point overflow handling characteristics:

cpp

// Direct Form I
y[n] = b0*x[n] + b1*x[n-1] + b2*x[n-2] - a1*y[n-1] - a2*y[n-2]

Zero Topology (Flag Field)

code

flag = 1 → Parametric zeros: (1 - z⁻²)
           Creates resonant formant peaks
           b0 = scale, b1 = 0, b2 = -scale

flag = 0 → Lowpass zeros: (1 + z⁻¹)²
           Bounds spectrum, unity DC gain
           b0 = norm, b1 = 2*norm, b2 = norm

Skill 2: The "Analog" Secret (Saturation & Distortion)

Inter-Stage Saturation

High resonance in Biquad[n] drives saturation into Biquad[n+1], creating the characteristic "grunge."

cpp

// Between each biquad stage
double saturate(double x) {
    return std::tanh(x);  // Or polynomial sigmoid
}

Distortion Parameter Mapping

•User-facing "Distortion" (-30dB to +70dB) controls input gain into saturation
•Use tanh approximation or polynomial sigmoid for "soft clipping"
•For vintage artifacts: hard-clip at ±1.0 at feedback injection points (simulates 24-bit fixed-point overflow)

Skill 3: ARMAdillo Coefficient Encoding

Standard Biquad Coefficients

code

B1 = -2 * R * cos(θ)    (frequency)
B2 = R²                  (resonance)

ARMAdillo Transform (Encode)

code

B2' = 1 - B2             (distance from unit circle)
B1' = B1 + 2             (shifted positive)

ARMAdillo Decode

cpp

double B2 = 1.0 - B2_prime;           // R²
double R = std::sqrt(B2);             // Pole radius
double B1 = B1_prime - 2.0;           // -2R*cos(θ)
double theta = std::acos(-B1 / (2.0 * R));
double freqHz = theta * sampleRate / (2.0 * M_PI);

Table I: 8-Bit Increment Decoding

Format: [S][sh2][sh1][sh0][d3][d2][d1][d0]

cpp

int decodeIncrement(uint8_t encoded) {
    int sign = (encoded & 0x80) ? -1 : 1;
    int shift = (encoded >> 4) & 0x07;
    int mantissa = encoded & 0x0F;
    return sign * (mantissa << shift);
}

Range: -2048 to +2047

Table II: Approach Time Mapping

code

Difference Range    | @0.6 | @1.2 | @2.4 | @4.8 | @9.6 | @20  | @40  | @80
0ffff-0f800        | 0x15 | 0x7f | 0x6f | 0x5f | 0x4f | 0x3f | 0x2f | 0x1f
007bf-00780        | 0x10 | 0x2e | 0x1e | 0x0f | 0x07 | 0x03 | 0x01 | 0x00

Skill 4: The Cube (3D Trilinear Interpolation)

Pre-Calculate Corners

At load time, decode 8 ARMAdillo coefficient sets for cube corners:

•Min/Max Frequency
•Min/Max Morph
•Min/Max Transform

Runtime Interpolation

cpp

// Normalize parameters to 0.0-1.0
// Interpolate in ENCODED space (B1', B2')
double B1_interp = trilinear(corners_B1, freq, morph, transform);
double B2_interp = trilinear(corners_B2, freq, morph, transform);

// Then decode to standard coefficients
double B2 = 1.0 - B2_interp;
double B1 = B1_interp - 2.0;

Critical: Interpolate encoded values, decode AFTER. Never interpolate standard a/b coefficients directly.

Skill 5: C++ & JUCE Constraints

Real-Time Rules

•No branching in DSP loop — use SIMD-friendly math
•Coefficient ramping — update every sample or every 8 samples via linear interp of ARMAdillo values
•Control rate vs audio rate — calculate coefficients at control rate, process biquads at audio rate

Structure Template

cpp

struct ZPlaneFilter {
    static constexpr int NUM_STAGES = 7;

    // Direct Form I: 2 input states + 2 output states per section
    struct BiquadState {
        double x1 = 0, x2 = 0;  // input history
        double y1 = 0, y2 = 0;  // output history
    };

    struct BiquadCoeffs {
        double b0, b1, b2;      // numerator
        double a1, a2;          // denominator
    };

    std::array<BiquadState, NUM_STAGES> states;
    std::array<BiquadCoeffs, NUM_STAGES> coeffs;

    double processSample(double x) {
        for (int i = 0; i < NUM_STAGES; ++i) {
            x = processBiquad(x, states[i], coeffs[i]);
            x = std::tanh(x);  // inter-stage saturation
        }
        return x;
    }
};

Morpheus WAV Data Structure

File: cubes_v1.01vc_170120.wav

•8-bit mono 48kHz, 7 unique sample values
•48000-sample sync preamble
•Decoded: ~96,500 bytes

Per-Cube Layout

code

8 corners × 7 stages × 2 bytes = 112 bytes coefficients + metadata
Byte 0: B1' encoded (frequency)
Byte 1: B2' encoded (resonance)

Corner Index Mapping

code

3-bit binary: [Transform][Morph][Frequency]
000 = corner 0 (all min)
111 = corner 7 (all max)

Validated Reference: Talking Hedz (M100% Q100%)

From Emulator X3 Cheat Engine capture (direct IEEE floats):

code

Stage 0: a1=-1.993596, r=0.998475, flag=1 → ~2417 Hz
Stage 1: a1=-1.912492, r=0.998384, flag=1 → ~2724 Hz
Stage 2: a1=-1.861726, r=0.998353, flag=1 → ~4974 Hz
Stage 3: a1=-1.510937, r=0.979504, flag=1 → ~1701 Hz
Stage 4: a1=-1.992010, r=0.998232, flag=0 → bounding

FFT validated: peaks within ±25 Hz of hardware.

Quick Diagnostic

Symptom	Likely Cause
Wrong frequencies	Not decoding ARMAdillo (B1' → B1)
No resonance peaks	flag=1 numerator wrong
No spectrum bounding	flag=0 numerator wrong
Morph sounds stepped	Interpolating decoded, not encoded
Filter explodes	Coefficient sign error or R > 1
"Horrible LF response"	Authentic — ARMAdillo quantization
Too clean/sterile	Missing inter-stage saturation
Clicks on parameter change	Not ramping coefficients

Implementation Checklist

Step	Deliverable	Validation
1	ARMAdillo decode function	B1'/B2' → B1/B2 correct
2	WAV frame decoder	Extract 289 cubes
3	Direct Form I biquad	Stable at high Q
4	Inter-stage saturation	"Grunge" on resonance
5	Flag-based zero topology	flag=0 and flag=1 work
6	7-stage cascade	White noise → shaped output
7	Trilinear cube interpolation	Smooth morph sweeps
8	Coefficient ramping	No clicks on parameter change
9	FFT comparison	Peaks match ±25 Hz