polaris-audio

polaris-audio is a research-oriented DSP library for sound synthesis and analysis based on
poles, transfer functions, and state-space representations.

The library focuses on explicit z-plane design, discrete-time dynamical systems,
and control-theoretic approaches to sound synthesis rather than fixed filter recipes.

polaris-audio is designed for experimentation, analysis, and algorithmic sound construction,
not for real-time audio I/O or plugin development.

---

Minimal example

1. Pole-based resonator

import polaris as pa

fs = 48000
n = 4096

a0, a1, a2 = pa.ar2_from_parameters(fs, f0=440.0, T60=1.2)
b0, b1, b2, _ = pa.ma2_from_attack_exp3(fs, attack_s=0.01)

A, B, C, D = pa.arma22_controllable_canonical_ss(
    b0, b1, b2,
    a0, a1, a2,
)

u = pa.make_excitation(n, fs, kind="pluck", seed=0)
y = pa.apply_ss_filter(u, A, B, C, D)
y = pa.normalize(y)

This designs a resonant system directly from frequency and decay time,
excites it with a short noise burst, and simulates the resulting sound.

---

2. Sweeping pole decay

import polaris as pa
import numpy as np

fs = 48000
n = 2048

for T60 in [0.2, 0.5, 1.0, 2.0]:
    a0, a1, a2 = pa.ar2_from_parameters(fs, 440.0, T60)
    b0, b1, b2, _ = pa.ma2_from_attack_exp3(fs, 0.01)

    A, B, C, D = pa.arma22_controllable_canonical_ss(
        b0, b1, b2,
        a0, a1, a2,
    )

    u = pa.make_excitation(n, fs, "impulse")
    y = pa.apply_ss_filter(u, A, B, C, D)

    print(T60, y[-1])

This illustrates how pole radius (via T60) affects stability and decay.

---

3. Modal synthesis

import polaris as pa
import numpy as np

fs = 48000
n = 4096

modes = [
    (220.0, 2.0),
    (440.0, 1.5),
    (660.0, 1.0),
]

u = pa.make_excitation(n, fs, "pluck", seed=1)
y_total = np.zeros(n)

for f0, T60 in modes:
    a0, a1, a2 = pa.ar2_from_parameters(fs, f0, T60)
    A, B, C, D = pa.arma22_controllable_canonical_ss(
        1.0, 0.0, 0.0,
        a0, a1, a2,
    )
    y_total += pa.apply_ss_filter(u, A, B, C, D)

y_total = pa.normalize(y_total)

This combines multiple resonant modes into a single signal.

---

Core components

Pole utilities

• omega_from_f0
  Convert frequency to normalized angular frequency.

• r_from_T60
  Convert decay time (T60) to pole radius.

• pole_from_f0_T60
  Construct a complex pole from frequency and decay.

---

Transfer function design

• ar2_from_poles
  Second-order AR denominator from complex poles.

• ma2_from_zeros
  Second-order MA numerator from complex zeros.

• ma2_from_attack_exp3
  Exponential attack excitation as MA coefficients.

These functions explicitly bridge the z-plane and difference equations.

---

State-space models

• arma22_controllable_canonical_ss
  ARMA(2,2) system in controllable canonical form.

• ss_step
  One-sample state update.

• apply_ss_filter
  Apply a state-space system to a signal.

All synthesis in polaris-audio is ultimately expressed as:

x[n+1] = A x[n] + B u[n]
y[n]   = C x[n] + D u[n]

---

Excitation signals

• make_excitation
  Generate impulse, noise, or pluck excitation signals.

Excitation is treated as an external force applied to a system,
not as sound itself.

---

Signal utilities

• normalize
  Simple peak normalization with headroom for numerical safety.

This is not a perceptual loudness normalizer.

---

RBJ filters (reference)

RBJ Audio EQ Cookbook formulas are provided under:

polaris.reference.rbj

They are included for comparison and interoperability only.
RBJ filters are not the primary design method in polaris-audio.

---

Design principles

• Explicit z-plane and state-space representations
• Clear separation of excitation and system dynamics
• Deterministic, sample-by-sample simulation
• Minimal abstractions
• Emphasis on inspectability and research use

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Non-goals

polaris-audio intentionally does not provide:

• Real-time audio I/O
• Plugin formats (VST, AU, AAX)
• Preset-based synthesizers
• DAW-style workflows
• Perceptual loudness models

These concerns are expected to live in higher-level systems.

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Project status

• Python >= 3.10
• Offline / non-realtime oriented
• Research-focused
• Experimental API
• v0.x series may evolve

polaris-audio prioritizes conceptual clarity over convenience.