Spectral Filtering

📐 Definition

For a spectral multiplier $M(k)$ and Fourier coefficients $\hat{u}(k)$,

$$\hat{u}_{\text{filt}}(k) = M(k)\,\hat{u}(k)$$

Copied

defines filtering in the frequency domain. Common choices include sharp cutoff ($M(k) = \mathbf{1}_{|k|\le k_c}$), exponential filters $M(k) = e^{-\alpha (|k|/k_c)^p}$, or Gaussian tapers.

Domain and Codomain

Applicable to Fourier-transformed signals $\hat{u}(k)$ over $\mathbb{Z}^d$ or $\mathbb{R}^d$. Output resides in the same spectral space with modified amplitudes.

---

⚙️ Key Properties

Linear and diagonal in the spectral basis; preserves phases when $M(k)$ is real and nonnegative. Spatially, filtering corresponds to convolution with the inverse transform of $M(k)$.

---

Sharp truncation corresponds to ideal low-pass filtering. As $\alpha \to 0$ in exponential filters, $M(k) \to 1$ and filtering vanishes; increasing $\alpha$ strengthens attenuation of high-$|k|$ modes. A true hard cutoff is given by the indicator filter $M(k)=\mathbf{1}_{|k|\le k_c}$ (or can be approached by increasing the filter order $p$ while keeping $M(0)=1$).

---

🎯 Special Cases and Limits

• Sharp cutoff: $M(k)=\mathbf{1}_{|k|\le k_c}$ (ideal low-pass).
• Exponential filter: $M(k)=e^{-\alpha(|k|/k_c)^p}$ with tunable strength $\alpha$ and order $p$.

---

🔗 Related Functions

Spectral bandwidth quantifies the spread remaining after filtering; spectral entropy measures the concentration of energy after applying $M(k)$.

---

Usage in Oakfield

Oakfield provides both physical-space and spectral-space filtering mechanisms:

• Physical-space filtering: sieve performs a discrete Gaussian-like low-pass convolution (or complementary high-pass residual) on real or complex fields (complex is component-wise).
• Spectral-space filtering: linear_dissipative acts as a fractional-Laplacian spectral damper by multiplying FFT bins by an exponential decay factor.
• Spectral conversion: fft_convert provides explicit physical↔spectral transforms when a workflow needs to operate directly in the spectral domain.

---

Historical Foundations

📜 Filtering in Fourier Space

Fourier methods make linear, shift-invariant filtering diagonal: multiplication by $M(k)$ in frequency corresponds to convolution in physical space.

🌍 Modern Perspective

Spectral filtering is a standard stabilization tool in pseudo-spectral methods, controlling aliasing and suppressing unresolved high-frequency energy.

---

📚 References

• Canuto et al., Spectral Methods
• Trefethen, Spectral Methods in MATLAB