wavyr

wavyr is an R package designed for studying traveling waves in space and time.

Signal Processing of Traveling Waves

Rational Signals and Rational Spectra

When we transform a wave from the physical domain to the spectral domain, uncertainty is introduced. Gabor established the uncertainty limit for the time-frequency dimension, a principle that also applies to space-wavenumber dimensions. Rational fraction representations of waves have proven helpful for feature extraction and fundamental frequency detection. In our novel method, we combine Gabor uncertainty with rational fraction approximations of complex waves. We use the Stern-Brocot tree to locate rational fractions within the Gabor uncertainty limit, allowing us to create signals and spectra that are both rational and bounded by the uncertainty product of the entire waveform. As long as the uncertainty of the sensor system is smaller than the Gabor uncertainty product, the uncertainty principle enables us to reliably generate rationalized waveforms, without needing to account for the intricacies of those sensors or even the uncertainty trade-offs between the physical and spectral domains. This uncertainty-bounded, rational-approximation is a parameter-free kernel for Fourier transformations. This new method, the Stern-Brocot-Gabor Fourier Transform, offers a novel approach for fundamental wave discovery and feature extraction. By comparing our model to neural models and experimental studies, we propose that for signals and spectra, the ambiguity that arises from the Gabor uncertainty principle could be an enabler of pattern perception.

The Stern-Brocot-Gabor Fourier Transform:

Time-Frequency Dimension

$$ \phi(\omega) = Q{SB}(\Delta t \Delta \omega, \ \omega) \int{-\infty}^\infty \psi(t) \ e^{-i \omega t} \ dt $$

Space-Wavenumber Dimension

$$ \phi(k) = Q{SB}(\Delta x \Delta k, \ k) \int{-\infty}^\infty \psi(x) \ e^{-i k x} \ dx $$

The Inverse Stern-Brocot-Gabor Fourier Transform:

Time-Frequency Dimension

$$ \psi(t) = \frac{1}{2\pi} \int{-\infty}^\infty Q{SB}(\Delta t \Delta \omega, \ \omega) \ \phi(\omega) \ e^{i \omega t} \ d\omega $$

Space-Wavenumber Dimension

$$ \psi(x) = \frac{1}{2\pi} \int{-\infty}^\infty Q{SB}(\Delta x \Delta k, \ k) \ \phi(k) \ e^{i k x} \ dk $$

The Stern-Brocot-Gabor Quantizer:

The Stern-Brocot-Gabor quantizer $Q_{SB}(\Delta_G, r)$ traverses the Stern-Brocot tree of all rational numbers $\mathbb{Q}$ in order, where fractions with smaller denominators appear earlier in the traversal, and returns the first rational number that approximates $r$ within the Gabor uncertainty limit $\Delta_G$.

$$ Q_{SB}(\Delta_G, r) = \frac{p}{q}, \quad \text{where } \left| r - \frac{p}{q} \right| \le \Delta_G \quad \text{and } \gcd(p, q) = 1 $$

The Stern-Brocot-Gabor quantizer ensures that $\frac{p}{q}$ is a co-prime fraction in simplest form, providing a rational approximation of $r$ within the specified uncertainty.

Uncertainty Principles

Time-Frequency Uncertainty:

$$ \Delta t \Delta \omega \geq \frac{1}{2} $$

Space-Wavenumber Uncertainty:

$$ \Delta x \Delta k \geq \frac{1}{2} $$

Combined Time-Frequency and Space-Wavenumber Uncertainty:

$$ \Delta t \Delta \omega \Delta x \Delta k \geq \frac{1}{4} $$

Relationships among Wave Properties