PannerBlock

Multi-format spatial panning supporting stereo, surround, and ambisonics.

Overview

PannerBlock positions a mono signal in 2D or 3D space using three different algorithms:

• Stereo: Constant-power panning between left and right
• Surround: VBAP (Vector Base Amplitude Panning) for 5.1/7.1 layouts
• Ambisonic: Spherical harmonic encoding for immersive audio

Mathematical Foundation

Coordinate System

For surround and ambisonic modes, positions are specified in spherical coordinates:

• Azimuth $\theta$: Horizontal angle from front
  • $0°$ = front
  • $90°$ = left
  • $-90°$ = right
  • $±180°$ = rear
• Elevation $\phi$: Vertical angle from horizon
  • $0°$ = horizon
  • $90°$ = directly above
  • $-90°$ = directly below

Stereo: Constant-Power Pan Law

Simple left/right panning uses a constant-power pan law based on sine and cosine functions. This preserves perceived loudness as the sound moves across the stereo field.

For a pan position $p \in [-100, 100]$:

1. Normalize to $[0, 1]$: $$ t = \frac{p + 100}{200} $$

2. Convert to angle: $$ \alpha = t \cdot \frac{\pi}{2} $$

3. Calculate gains: $$ g_L = \cos(\alpha), \quad g_R = \sin(\alpha) $$

Why Constant-Power?

The key insight is that perceived loudness relates to power, not amplitude. When a sound is centered, it plays equally from both speakers, and the listener perceives both contributions combined.

Linear panning (using $g_L = 1-t$ and $g_R = t$) causes a loudness dip in the center because: $$ g_L^2 + g_R^2 = (1-t)^2 + t^2 \neq 1 \quad \text{(varies from 0.5 to 1)} $$

Constant-power panning maintains consistent loudness because: $$ g_L^2 + g_R^2 = \cos^2(\alpha) + \sin^2(\alpha) = 1 \quad \text{(always)} $$

Surround: Vector Base Amplitude Panning (VBAP)

VBAP generalizes panning to arbitrary speaker layouts by computing gains based on angular distance from each speaker.

Speaker Positions

Standard speaker layouts define specific azimuths:

5.1 (ITU-R BS.775-1):

7.1 (ITU-R BS.2051):

Gain Calculation

For a source at azimuth $\theta_s$ and speaker $i$ at azimuth $\theta_i$:

1. Calculate angular difference (handling wrap-around): $$ \Delta\theta_i = \min(|\theta_s - \theta_i|, 2\pi - |\theta_s - \theta_i|) $$

2. Apply gain based on a spread angle $\sigma$ (typically $\frac{\pi}{2}$): $$ g_i = \begin{cases} \cos\left(\frac{\sigma - \Delta\theta_i}{\sigma} \cdot \frac{\pi}{2}\right) & \text{if } \Delta\theta_i < \sigma \ 0 & \text{otherwise} \end{cases} $$

3. Normalize for constant energy: $$ g_i' = \frac{g_i}{\sqrt{\sum_j g_j^2}} $$

This ensures that the total power remains constant regardless of source position.

Ambisonics: Spherical Harmonic Encoding

Ambisonics represents sound fields using spherical harmonics—mathematical functions that describe how sound varies with direction. Unlike channel-based formats, ambisonics is speaker-independent: the encoded signal can be decoded to any speaker layout.

Spherical Harmonics

Spherical harmonics $Y_l^m(\theta, \phi)$ form an orthonormal basis for functions on the sphere, where:

• $l$ is the order (0, 1, 2, 3...)
• $m$ is the degree ($-l \leq m \leq l$)

Higher orders capture finer spatial detail but require more channels:

• Order 0: 1 channel (omnidirectional)
• Order 1: 4 channels (directional)
• Order 2: 9 channels (improved localization)
• Order 3: 16 channels (high spatial resolution)

The channel count for order $L$ is $(L+1)^2$.

SN3D Normalization and ACN Ordering

This implementation uses:

• SN3D (Semi-Normalized 3D): Schmidt semi-normalization for consistent gain
• ACN (Ambisonic Channel Number): Standard channel ordering

Encoding Coefficients

For a source at azimuth $\theta$ and elevation $\phi$:

Order 0 (omnidirectional): $$ Y_0^0 = 1 \quad \text{(W channel)} $$

Order 1 (first-order, directional): $$ \begin{aligned} Y_1^{-1} &= \cos\phi \sin\theta \quad \text{(Y channel)} \ Y_1^0 &= \sin\phi \quad \text{(Z channel)} \ Y_1^1 &= \cos\phi \cos\theta \quad \text{(X channel)} \end{aligned} $$

Order 2: $$ \begin{aligned} Y_2^{-2} &= \sqrt{\frac{3}{4}} \cos^2\phi \sin(2\theta) \quad \text{(V channel)} \ Y_2^{-1} &= \sqrt{\frac{3}{4}} \sin(2\phi) \sin\theta \quad \text{(T channel)} \ Y_2^0 &= \frac{3\sin^2\phi - 1}{2} \quad \text{(R channel)} \ Y_2^1 &= \sqrt{\frac{3}{4}} \sin(2\phi) \cos\theta \quad \text{(S channel)} \ Y_2^2 &= \sqrt{\frac{3}{4}} \cos^2\phi \cos(2\theta) \quad \text{(U channel)} \end{aligned} $$

Order 3 follows similar patterns with $\cos(3\theta)$, $\sin(3\theta)$, and higher powers of $\cos\phi$ and $\sin\phi$.

Intuition for Spherical Harmonics

• W (order 0): Captures overall level—same in all directions
• X, Y, Z (order 1): Capture front-back, left-right, and up-down gradients
• Higher orders: Capture increasingly fine directional detail

A source at the front ($\theta=0, \phi=0$) encodes as:

• $W = 1$ (present everywhere)
• $Y = 0$ (no left-right component)
• $Z = 0$ (on horizon)
• $X = 1$ (fully in front)

Panning Modes

Stereo Mode

Traditional left-right panning using constant-power pan law.

#![allow(unused)]
fn main() {
use bbx_dsp::{blocks::PannerBlock, graph::GraphBuilder};

let mut builder = GraphBuilder::<f32>::new(44100.0, 512, 2);

// Position: -100 (left) to +100 (right)
let pan = builder.add(PannerBlock::new(0.0));  // Center
let pan = builder.add(PannerBlock::new_stereo(-50.0));  // Half left
}

Surround Mode (VBAP)

Uses Vector Base Amplitude Panning for 5.1 and 7.1 speaker layouts.

#![allow(unused)]
fn main() {
use bbx_dsp::{blocks::PannerBlock, channel::ChannelLayout, graph::GraphBuilder};

let mut builder = GraphBuilder::<f32>::new(44100.0, 512, 6);

// 5.1 surround panner
let pan = builder.add(PannerBlock::new_surround(ChannelLayout::Surround51));

// 7.1 surround panner
let pan = builder.add(PannerBlock::new_surround(ChannelLayout::Surround71));
}

Control source position with azimuth and elevation parameters.

Ambisonic Mode

Encodes mono input to SN3D normalized, ACN ordered B-format for immersive audio.

#![allow(unused)]
fn main() {
use bbx_dsp::{blocks::PannerBlock, graph::GraphBuilder};

let mut builder = GraphBuilder::<f32>::new(44100.0, 512, 4);

// First-order ambisonics (4 channels)
let pan = builder.add(PannerBlock::new_ambisonic(1));

// Second-order ambisonics (9 channels)
let pan = builder.add(PannerBlock::new_ambisonic(2));

// Third-order ambisonics (16 channels)
let pan = builder.add(PannerBlock::new_ambisonic(3));
}

Port Layout

Port counts vary by mode:

Parameters

Position Values (Stereo)

Azimuth/Elevation (Surround/Ambisonic)

Usage Examples

Basic Stereo Panning

#![allow(unused)]
fn main() {
use bbx_dsp::{
    blocks::{OscillatorBlock, PannerBlock},
    graph::GraphBuilder,
    waveform::Waveform,
};

let mut builder = GraphBuilder::<f32>::new(44100.0, 512, 2);

let osc = builder.add(OscillatorBlock::new(440.0, Waveform::Sine, None));
let pan = builder.add(PannerBlock::new(50.0));  // Slightly right

builder.connect(osc, 0, pan, 0);
}

Auto-Pan with LFO

#![allow(unused)]
fn main() {
use bbx_dsp::{
    blocks::{LfoBlock, OscillatorBlock, PannerBlock},
    graph::GraphBuilder,
    waveform::Waveform,
};

let mut builder = GraphBuilder::<f32>::new(44100.0, 512, 2);

let osc = builder.add(OscillatorBlock::new(440.0, Waveform::Sine, None));
let lfo = builder.add(LfoBlock::new(0.25, 1.0, Waveform::Sine, None));  // 0.25 Hz, full depth
let pan = builder.add(PannerBlock::new(0.0));

builder.connect(osc, 0, pan, 0);
builder.modulate(lfo, pan, "position");
}

Surround Panning with VBAP

#![allow(unused)]
fn main() {
use bbx_dsp::{blocks::{LfoBlock, OscillatorBlock, PannerBlock}, channel::ChannelLayout, graph::GraphBuilder, waveform::Waveform};

let mut builder = GraphBuilder::<f32>::new(44100.0, 512, 6);

let osc = builder.add(OscillatorBlock::new(440.0, Waveform::Sine, None));
let pan = builder.add(PannerBlock::new_surround(ChannelLayout::Surround51));

builder.connect(osc, 0, pan, 0);

// Modulate azimuth for circular motion
let lfo = builder.add(LfoBlock::new(0.1, 1.0, Waveform::Sine, None));
builder.modulate(lfo, pan, "azimuth");
}

Ambisonic Encoding with Rotating Source

#![allow(unused)]
fn main() {
use bbx_dsp::{blocks::{LfoBlock, OscillatorBlock, PannerBlock}, channel::ChannelLayout, graph::GraphBuilder, waveform::Waveform};

let mut builder = GraphBuilder::<f32>::new(44100.0, 512, 4);

let osc = builder.add(OscillatorBlock::new(440.0, Waveform::Sine, None));
let encoder = builder.add(PannerBlock::new_ambisonic(1));

builder.connect(osc, 0, encoder, 0);

// Rotate source around listener
let az_lfo = builder.add(LfoBlock::new(0.2, 1.0, Waveform::Sine, None));
builder.modulate(az_lfo, encoder, "azimuth");

// Output channels: W, Y, Z, X (ACN order)
}

Full Ambisonics Pipeline

#![allow(unused)]
fn main() {
use bbx_dsp::{blocks::{AmbisonicDecoderBlock, OscillatorBlock, PannerBlock}, channel::ChannelLayout, graph::GraphBuilder, waveform::Waveform};

let mut builder = GraphBuilder::<f32>::new(44100.0, 512, 2);

// Source
let osc = builder.add(OscillatorBlock::new(440.0, Waveform::Sine, None));

// Encode to FOA
let encoder = builder.add(PannerBlock::new_ambisonic(1));
builder.connect(osc, 0, encoder, 0);

// Decode to stereo for headphones
let decoder = builder.add(AmbisonicDecoderBlock::new(1, ChannelLayout::Stereo));
builder.connect(encoder, 0, decoder, 0);  // W
builder.connect(encoder, 1, decoder, 1);  // Y
builder.connect(encoder, 2, decoder, 2);  // Z
builder.connect(encoder, 3, decoder, 3);  // X
}

Implementation Notes

• Click-free panning via linear smoothing on all parameters
• Uses ChannelConfig::Explicit (handles routing internally)
• All modes accept mono input (1 channel)
• VBAP normalizes gains for energy preservation
• Ambisonic encoding uses SN3D normalization and ACN channel ordering

Further Reading

• Pulkki, V. (1997). "Virtual Sound Source Positioning Using Vector Base Amplitude Panning." JAES, 45(6), 456-466.
• Zotter, F. & Frank, M. (2019). Ambisonics: A Practical 3D Audio Theory for Recording, Studio Production, Sound Reinforcement, and Virtual Reality. Springer.
• Daniel, J. (2001). Représentation de champs acoustiques, application à la transmission et à la reproduction de scènes sonores complexes dans un contexte multimédia. PhD thesis.
• Chapman, M. et al. (2009). "A Standard for Interchange of Ambisonic Signal Sets." Ambisonics Symposium.