Discrete angular sampling and Nyquist structure#

PETAL2D samples

\[ \theta_j=\frac{2\pi j}{N_\theta},\qquad j=0,\ldots,N_\theta-1, \]

and computes a discrete Fourier transform.

Representable FFT indices#

For odd \(N_\theta\), the canonical integer indices are

\[ -\frac{N_\theta-1}{2},\ldots,0,\ldots,+\frac{N_\theta-1}{2}. \]

For even \(N_\theta\), NumPy’s FFT convention contains

\[ 0,1,\ldots,\frac{N_\theta}{2}-1, -\frac{N_\theta}{2},\ldots,-1. \]

The Nyquist harmonic is represented by the single index \(-N_\theta/2\). The index \(+N_\theta/2\) gives the same sampled sequence and is not a second independent bin.

Aliasing#

For sampled angles,

\[ e^{i(m+kN_\theta)\theta_j}=e^{im\theta_j} \]

for any integer \(k\). Thus the data determine \(m\) only modulo \(N_\theta\). PETAL2D’s m_all reports the canonical FFT representative of each discrete class.

Real fields#

For real angular samples the DFT satisfies discrete conjugate symmetry. Away from \(m=0\) and the even-grid Nyquist singleton, bins occur in \((+m,-m)\) pairs. PETAL2D’s adaptive selector preserves those pairs automatically.

Choosing Ntheta#

The strict Nyquist condition is only the beginning. In practical sampled-data problems, interpolation and finite Cartesian resolution can generate weak higher harmonics. Use enough angular samples that the physically relevant range is comfortably below Nyquist, then verify convergence by increasing Ntheta.