Chapter 3 · Recovery Guarantee

Why Recovery Works

The folded samples hide an unknown multiple of $2\lambda$ at every instant. Two facts recover it. Folding commutes with integer differences, and differences of an oversampled smooth signal are small. With finite bits there is a catch: differencing also amplifies quantization noise. Theorem 2 of the paper balances the two and turns them into a law linking oversampling, bit depth and dynamic range.

Theorem 2 · Bit–Sampling Law

Let $g\in\BL$ and $\yq[n]=\Qeta\bigl(\Mlam(g(nT))\bigr)$ with $\eta_\lambda^B=\lambda/(2^B-1)$. Then $\tilde g[n]$ can be recovered from $\yq[n]$ with $\min_{\gamma\in\Z}\norm{\tilde g[n]-g[n]+2\gamma\lambda}_{\ell^\infty}\le\eta_\lambda^B$, provided that

$$T\le\frac{1}{2^\alpha\Omega},\ \alpha>0,\qquad B>\log_2\Bigl(1+4\sqrt[\alpha]{2\DRX}\Bigr),\qquad \DRX=\frac{\norm{g}_{L^\infty}}{\lambda}.$$

Equivalently, the oversampling factor is $\OF=\pi/(\Omega T)\ge\pi2^\alpha$.

Start from

Dynamic-range expansion $\DRX$

Oversampling exponent $\alpha$

Bit depth $B$

Difference order $h$

Quantized Folded Samples and Recovery

$g(t)$ $\yq[n]$ $\tilde g[n]$

Shrink Versus Amplify

$\max|\Delta^kg|$$(\Omega T)^k\norm{g}_\infty$$\max|\Delta^kq|$$2^k\eta_\lambda^B$

The Bit–Sampling Law

$B_{\min}(\DRX)$ at your $\alpha$$\alpha=1,2,3$PCM: $\log_2\DRX+B$

Differences, Folded Again

$\Delta^k\yq$ $\Delta^k(g+q)$ $\Mlam(\Delta^k\yq)$ $[-\lambda,\lambda)$

Signal Contraction

By the fundamental theorem of calculus and Bernstein's inequality, $|\Delta g[n]|\le T\norm{\partial_tg}_\infty\le\Omega T\norm{g}_\infty$. Induction on $h$ gives

$$\norm{\Delta^hg}_{\ell^\infty}\le(\Omega T)^h\norm{g}_{L^\infty}.$$

This explicit induction removes the legacy $2e$ factor of Theorem 1: $T<1/\Omega$ suffices in the ideal case.

Quantization-Noise Amplification

Each difference can at most double the noise: $\norm{\Delta^hq}_{\ell^\infty}\le2^h\eta_\lambda^B$. If both terms stay below $\lambda/2$, then $|\Delta^h(g+q)|<\lambda$, and because $\Delta^h\res\in2\lambda\Z$,

$$\Mlam\bigl(\Delta^h\yq\bigr)=\Delta^hg+\Delta^hq.$$

Residue Estimation

Now $\Delta^h\res=\Mlam(\Delta^h\yq)-\Delta^h\yq$ is known. Anti-difference $h$ times; each constant of summation lies in $2\lambda\Z$ and is fixed because $\res=g-y$ stays bounded over a long window. Then $\tilde g=\yq+\res=g+q$, up to $2\lambda\Z$: the error is the quantization noise, $\le\eta_\lambda^B$.

Consequences

Remarks 1–4 of the Paper

R1 · Oversampling Lowers the Bit Budget

For $\DRX=16$: $\alpha=1$ needs $B>7.01$; $\alpha=2$ needs $B>4.56$. Doubling the oversampling saves 2.45 bits. A conventional ADC would need about $\log_2\DRX=4$ extra bits for the same range and resolution.

R2 · Few Bits Support a Large Range

Rearranged, $\DRX\le\tfrac12\bigl((2^B-1)/4\bigr)^{\alpha}$: at fixed $B$ the achievable range grows exponentially in $\alpha$, whenever $B\ge\log_29\approx3.17$.

R3 · Accuracy Improves by $20\log_{10}\DRX$ dB

Under the uniform-quantization model, $\mathrm{MSE}_{\lambda,B}=(\DRX)^{-2}\,\mathrm{MSE}_B$. A 15-fold $\DRX$ gives about 23 dB, consistent with the hardware results in Chapter 6.

R4 · Oversampling Only at Acquisition

Oversampling is needed only at the front end. After recovery, standard decimation returns the audio to conventional PCM rates such as 44.1–192 kHz.

In the paper Section III: eqs. (4)–(7), Theorem 2 and its proof (signal contraction, quantization-noise amplification, residue estimation), Remarks R1–R4, eq. (12).

Here $\lambda=1$, $\Omega=\pi$ rad/s and $T=1/(2^\alpha\Omega)$. The input is a fixed sum of fourteen shifted sincs on $0\le t\le40$, scaled to the chosen $\DRX$. Order $h$ “auto” uses the theorem's $H=\lceil\ln(2\DRX)/\ln((\Omega T)^{-1})\rceil$. Recovery follows the proof: fold the $h$-th difference, snap $\Delta^h\res$ to $2\lambda\Z$, anti-difference with constants fixed by boundedness over the whole record. Recovered samples are drawn with the unrecoverable global offset $2\gamma\lambda$ taken from the truth; the decoder never sees it. The theorem's conditions are sufficient, not necessary, so recovery can succeed outside them.