Chapter 4 · Audio-Centric Recovery

An Audio-Centric Guarantee

Theorem 2 certifies recovery using the worst-case bandwidth $\Omega$: universal, but conservative. Audio rarely uses its full band. Speech and heart sounds concentrate their energy far below $\Omega$. Replace $\Omega$ by a signal-adaptive effective bandwidth, the order-$h$ spectral moment, and the admissible sampling step grows, sometimes by orders of magnitude.

Definition 1 · Order-$h$ Spectral Moment

$$\Oeff{h}=\left(\frac{\int_{-\Omega}^{\Omega}|\omega|^{2h}|\widehat g(\omega)|^2\,d\omega}{\int_{-\Omega}^{\Omega}|\widehat g(\omega)|^2\,d\omega}\right)^{\frac1{2h}}=\left(\frac{\norm{g^{(h)}}_{L^2}}{\norm{g}_{L^2}}\right)^{\frac1h}.$$

For a Gaussian $|\widehat g(\omega)|^2=e^{-\sigma^2\omega^2}$: $\Oeff{h}=\bigl(\pi^{-1/2}\Gamma(h+\tfrac12)\bigr)^{1/(2h)}/\sigma$, and $\Oeff{1}=1/(\sqrt2\sigma)$.

Theorem 3 · Audio-Centric Recovery Guarantee

Under the setting of Theorem 2, $\tilde g$ is recovered with error at most $\eta_\lambda^B$ (up to $2\lambda\Z$) provided

$$T\le\max_{h\in\mathcal H_B}\frac{1}{\Oeff{h}}\left(\frac{\lambda}{2\sqrt{\Omega/\pi}\,\norm{g}_{L^2}}\right)^{1/h},\quad \mathcal H_B=\Bigl\{h\in\Z_+:h<\Bigl\lfloor\log_2\tfrac{2^B-1}{2}\Bigr\rfloor\Bigr\}.$$

Spectrum

Concentration $\omega_0/\Omega$

Dynamic-range expansion $\DRX$

Bit depth $B$

Spectrum and Effective Bandwidths

$|\widehat g(\omega)|^2$$\Oeff{h}$$\Omega$

Required Oversampling by Order

worst case, $T^{(2)}_{\max}(h)$audio-centric, $T^{(3)}_{\max}(h)$

Where the Worst Case Enters

Theorem 2 bounds $\norm{\partial_t^hg}_{L^\infty}\le\Omega^h\norm{g}_{L^\infty}$ (Bernstein): every frequency up to $\Omega$ is treated as fully occupied. Its maximal step is

$$T^{(2)}_{\max}(h)=\frac1\Omega\Bigl(\frac{\lambda}{2\norm{g}_{L^\infty}}\Bigr)^{1/h}.$$

Bernstein–Nikol'skii and Parseval

Instead, $\norm{\partial_t^hg}_{L^\infty}\le\sqrt{\Omega/\pi}\,\norm{\partial_t^hg}_{L^2}=\sqrt{\Omega/\pi}\,\Oeff{h}^h\norm{g}_{L^2}$. The derivative is weighed by where the energy actually lies:

$$T^{(3)}_{\max}(h)=\frac1{\Oeff{h}}\Bigl(\sqrt{\tfrac\pi\Omega}\,\frac{\lambda}{2\norm{g}_{L^2}}\Bigr)^{1/h}.$$

When Theorem 3 Wins

The relaxation factor is $T^{(3)}_{\max}/T^{(2)}_{\max}=(\Omega/\Oeff{h})\bigl(\sqrt{\pi/\Omega}\,\norm{g}_{L^\infty}/\norm{g}_{L^2}\bigr)^{1/h}$. It exceeds one whenever $(\Oeff{h}/\Omega)^h<\sqrt{\pi/\Omega}\,\norm{g}_{L^\infty}/\norm{g}_{L^2}$: concentrated spectra and a moderate-to-large peak-to-energy ratio.

In the paper Section III: Definition 1 (eq. (9)), Theorem 3 (eq. (10)) and its proof, the relaxation factor; Table I of the Supplementary Material lists $\Oeff{1}/\Omega$ for the audio corpus (speech and heart sounds: $10^{-3}$ to $10^{-2}$).

Spectra are real and non-negative on $[-\Omega,\Omega]$ (zero phase), so $\norm{g}_{L^\infty}=g(0)=\frac1{2\pi}\int\widehat g$, scaled to $\DRX\cdot\lambda$ with $\lambda=1$; $\norm{g}_{L^2}^2=\frac1{2\pi}\int|\widehat g|^2$. Moments are computed by quadrature on 20,000 log-spaced frequencies. $\Omega=2\pi\cdot20$ kHz. Oversampling factor $\OF=\pi/(\Omega T)$; $\OF=1$ is Nyquist. Admissible orders are restricted to $\mathcal H_B$ for both theorems.