Spectral Bandwidth

📐 Definition

For Fourier coefficients $\hat{u}(k)$, define

$$B = \left(\frac{\sum_{k} |k|^{2} |\hat{u}(k)|^{2}}{\sum_{k} |\hat{u}(k)|^{2}}\right)^{1/2}$$

Copied

the square root of the energy-weighted second moment of wavenumbers.

Domain and Codomain

Applicable to spectra with finite energy $\sum_k |\hat{u}(k)|^{2} < \infty$ and finite second moment. Output $B \ge 0$ has units of inverse length (or frequency).

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⚙️ Key Properties

Invariant to uniform scaling of $\hat{u}$. Sensitive to high-wavenumber content; spectral filtering that removes large $|k|$ decreases $B$.

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For a single wavenumber $k_0$, $B = |k_0|$. For isotropic Gaussian spectra with variance $\sigma_k^2$, $B$ scales like $\sigma_k$.

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🎯 Special Cases and Limits

• Single mode at $k_0$: $B=|k_0|$.
• Narrowband spectra have small $B$; broadband spectra have larger $B$.

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🔗 Related Functions

Spectral entropy captures distribution uniformity; filtering modifies $B$ by attenuating selected bands.

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Usage in Oakfield

Oakfield computes spectral bandwidth as part of runtime field diagnostics:

• SimFieldStats.spectral_bandwidth is computed from the FFT power distribution as an RMS width over frequency-bin coordinates.
• Used in the UI/diagnostics path to monitor whether dynamics (or operators like dispersion/linear_dissipative) are concentrating or spreading energy across modes.

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Historical Foundations

📜 Moments of Spectra

Bandwidth measures arise by treating normalized spectral energy as a distribution over wavenumber and computing its moments, analogous to variance and standard deviation.

🌍 Modern Perspective

Spectral bandwidth is widely used as a compact diagnostic for resolution and regularity in spectral simulations.

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📚 References

• Bracewell, The Fourier Transform and Its Applications
• Stein & Shakarchi, Fourier Analysis