spharray

A self-contained Python toolkit for spherical microphone array processing. This repository is the intended open-source root; migration workspaces, local virtual environments, generated reports, and large reference assets should stay outside this directory.

Features

• Spherical harmonics: complex and real (tesseral) SH basis matrices, ACN/SN3D normalisation, forward and inverse SHTs, coefficient conversions
• Spatial sampling: Fibonacci, equiangle, and t-design grids with integration weights
• Array simulation: free-field plane-wave transfer functions
• Fixed beamforming: cardioid, hypercardioid (max DI), supercardioid (max F/B ratio), MaxEV (perceptual) weight vectors and pattern evaluation
• Adaptive beamforming: MVDR and LCMV SH-domain beamformers
• DOA estimation: plane-wave decomposition (PWD) and MUSIC spatial spectra with peak picking
• Diffuseness estimation: IE, TV, SV, and CMD estimators from FOA/SH covariance
• Diffuse-field coherence: sinc-based models for omnidirectional sensor arrays
• Acoustics: spherical Bessel/Hankel functions, modal coefficients for open/rigid arrays
• Plotting: 3-D array geometry, 2-D spatial maps, MATLAB-like figure style

Installation

Install the current public release from PyPI:

pip install spharray

For development from a local checkout:

python3 -m venv .venv
source .venv/bin/activate
python -m pip install -U pip
pip install -e ".[dev]"

Python >= 3.11 is required. Core dependencies: numpy, scipy, matplotlib.

Optional extras:

Extra	Adds
audio	soundfile for WAV I/O
image	scikit-image for image processing utilities
dev	pytest, build
notebook	jupyterlab, ipykernel

Quick start

import numpy as np
import spharray as sap

# ── Spatial sampling ──────────────────────────────────────────────────────────
grid = sap.array.fibonacci_grid(100)   # SphericalGrid, 100 directions
print(f"weights sum: {grid.weights.sum():.4f}")  # ≈ 4π

# ── SH basis and transforms ───────────────────────────────────────────────────
spec = sap.SHBasisSpec(max_order=3, basis="complex", angle_convention="az_colat")
Y  = sap.sh.matrix(spec, grid)          # (100, 16) complex SH matrix
f  = np.random.randn(grid.size)
nm = sap.sh.direct_sht(f, Y, grid)     # (16,) SH coefficients
f_rec = sap.sh.inverse_sht(nm, Y).real # reconstruct

# ── Fixed beamforming ─────────────────────────────────────────────────────────
b       = sap.beamforming.beam_weights_hypercardioid(3)   # max-DI weights, N=3
thetas  = np.linspace(0, np.pi, 181)
pattern = sap.beamforming.axisymmetric_pattern(thetas, b) # B(0) = 1

# ── Array simulation ──────────────────────────────────────────────────────────
sensor_grid = sap.array.fibonacci_grid(32)
geometry    = sap.ArrayGeometry(radius_m=0.042, sensor_grid=sensor_grid)
src_grid    = sap.array.fibonacci_grid(4)
freqs, H    = sap.array.simulate_plane_wave_array_response(
    fft_len=256, fs=16000.0, geometry=geometry, source_grid=src_grid
)
# H.shape = (129, 32, 4)  — (n_bins, n_mics, n_sources)

# ── DOA estimation ────────────────────────────────────────────────────────────
search = sap.array.fibonacci_grid(300)
Q      = spec.n_coeffs                        # (N+1)^2 = 16
R_eye  = np.eye(Q, dtype=complex)             # diffuse covariance → flat spectrum
result = sap.doa.pwd_spectrum(R_eye, search, spec, n_peaks=1)
print("Peak direction (az, el) rad:", result.peak_dirs_rad[0])

# ── MUSIC spectrum ────────────────────────────────────────────────────────────
y0  = sap.sh.matrix(spec, search)[42]         # steering vector at grid index 42
R1  = np.outer(y0, y0.conj()) + 0.01 * np.eye(Q)
res = sap.doa.music_spectrum(R1, search, spec, n_sources=1)
print("MUSIC peak index:", res.peak_indices[0])  # → 42

# ── Diffuseness ───────────────────────────────────────────────────────────────
frame = (np.random.randn(4, 1024) + 1j * np.random.randn(4, 1024))
psi   = sap.diffuseness.diffuseness_ie(frame)
print(f"IE diffuseness mean: {psi.mean():.3f}")

New user path

Start with docs/getting_started.md. It walks through installation, the first runnable example, and the smallest useful SH workflow. Read docs/concepts.md when terms such as ACN, colatitude, modal coefficient, SHT, or unit front gain are not yet clear.

The most useful runnable tutorials are:

python examples/tutorials/01_sht_and_beamforming.py
python examples/tutorials/02_simulated_doa_pipeline.py
python examples/tutorials/03_modal_equalization_pipeline.py

Notebook users can open examples/notebooks/getting_started.ipynb.

Module reference

Module	Key symbols
sap.sh	matrix, complex_matrix, real_matrix, direct_sht, inverse_sht, acn_index, complex_to_real_coeffs, real_to_complex_coeffs, replicate_per_order
sap.array	fibonacci_grid, equiangle_sampling, get_tdesign_fallback, simulate_plane_wave_array_response, spatial_aliasing_frequency, max_sh_order
sap.acoustics	besseljs, besseljsd, besselhs, besselhsd, plane_wave_radial_bn, bn_matrix, sph_modal_coeffs, equalize_modal_coeffs
sap.beamforming	beam_weights_cardioid, beam_weights_hypercardioid, beam_weights_supercardioid, beam_weights_maxev, axisymmetric_pattern, mvdr_weights, lcmv_weights, steer_sh_weights, beamform_sh
sap.doa	pwd_spectrum, music_spectrum, peak_pick_spectrum, spatial_spectrum_from_map, estimate_sh_cov, forward_backward_cov, diagonal_loading
sap.diffuseness	diffuseness_ie, diffuseness_tv, diffuseness_sv, diffuseness_cmd, intensity_vectors_from_foa
sap.coherence	diffuse_coherence_matrix_omni, diffuse_coherence_from_weights
sap.coords	cart_to_sph, sph_to_cart, az_colat_to_azel, azel_to_az_colat, angular_distance, angular_distance_deg
sap.plotting	plot_mic_array, plot_directional_map_from_grid, apply_matlab_like_style, figure_style_context

Beamformer normalisation

All beamformers satisfy B(0) = 1 (unit front gain) in the axisymmetric_pattern convention:

B(θ) = Σ_{n=0}^{N} b_n · (2n+1)/(4π) · P_n(cos θ)

The weight vectors are normalised so that Σ b_n · (2n+1)/(4π) = 1.

Beamformer	DI (linear)	Notes
Cardioid	varies	Widest main lobe, no sidelobes
Hypercardioid	(N+1)²	Maximum directivity index
Supercardioid	varies	Maximum front-to-back energy ratio
MaxEV	varies	Von-Hann taper, good perceptual quality

SH conventions

Channel ordering follows ACN (Ambisonic Channel Numbering):

index q = n² + n + m,   n = 0…N,   m = -n…n

Complex SH are orthonormal (SN3D normalisation scaled to 4π). Real SH use tesseral form. Converting complex → real → complex is lossless only for conjugate-symmetric inputs (i.e., inputs that correspond to real-valued spatial functions).

Running tests

python -m pytest tests/ -q
# 248 passed

License

MIT. See LICENSE.

References

• Rafaely, B. (2015). Fundamentals of Spherical Array Processing. Springer.
• Zotter, F. & Frank, M. (2019). Ambisonics. Springer.
• Schmidt, R. O. (1986). Multiple emitter location and signal parameter estimation. IEEE Trans. Antennas Propag., 34(3), 276–280.