nyquist: Frequency Domain Signal Reconstruction

Mathematical Signal Processing through Adaptive Bandwidth Control

The Nyquist function implements a frequency domain filter that prevents aliasing artifacts through adaptive bandwidth limiting. Named after Harry Nyquist's pioneering work in signal reconstruction, this function creates smooth transitions between original and filtered signals based on spectral width analysis.

The mathematical foundation operates on the Nyquist-Shannon sampling theorem, where signals must be filtered below half the sampling rate to prevent aliasing distortion:

$$f_{nyquist}(x, width) = \text{mix}(0.5, x, \text{smoothstep}(0.75, 0.25, width))$$

The filter establishes critical frequency bounds:

• Lower cutoff: $f_c - \Delta f = 0.5 - 0.25 = 0.25$
• Upper cutoff: $f_c + \Delta f = 0.5 + 0.25 = 0.75$
• Center frequency: $f_c = 0.5$ (Nyquist critical point)
• Transition band: $\Delta f = 0.25$ (adaptive width control)

Dynamic Bandwidth Control

ライブエディター

const fragment = () => {
      const time = iTime.mul(1.2)
      const field = uv.sub(0.5).mul(10)

      const quantum1 = field.length().mul(3).add(time).sin().pow(3)
      const quantum2 = field.x.mul(field.y).mul(2).sub(time.mul(1.4)).cos()
      const entanglement = quantum1.add(quantum2).mul(0.5)

      const oscillatingWidth = field.length().mul(0.06).add(
              time.mul(0.15).sin().mul(time.mul(0.25).cos()).mul(0.01).add(0.04)
      )

      const filtered = nyquist(entanglement, oscillatingWidth)
      const energy = filtered.abs().pow(1.5)

      const spectrum = vec3(
              energy.mul(1),
              energy.mul(0.7).add(0.1),
              energy.mul(0.4).add(0.2)
      )

      return vec4(spectrum, 1.0)

}