Chapter 2 · The USF-Audio Acquisition Model

Folding Before Digitizing

A modulo ADC does not clip. It applies the centered modulo $\Mlam$ in the analog domain, so its output stays in $[-\lambda,\lambda]$ however large the input. What it digitizes is exactly what PCM discards: the fractional part, or analog quantization noise. Because $B$ bits now cover $[-\lambda,\lambda]$ rather than $[-\norm{g}_\infty,\norm{g}_\infty]$, every quantization step shrinks by the dynamic-range expansion $\DRX=\norm{g}_\infty/\lambda$.

Centered Modulo · Eq. (2)

$$\Mlam: g\mapsto 2\lambda\left(\left[\!\left[\frac{g}{2\lambda}+\frac12\right]\!\right]-\frac12\right),\qquad \lambda>0,$$

where $[\![\cdot]\!]$ is the fractional part. The folded signal $y(t)=\Mlam(g(t))$ is sampled, $y[n]=\Mlam(g(nT))$, then quantized with $B$ bits: $\yq[n]=\Qeta(y[n])$, with $\eta_\lambda^B=\lambda/(2^B-1)$.

Start from

Dynamic-range expansion $\DRX$

Bit depth $B$

Input and Folded Signal

$g(t)$ $y(t)=\Mlam(g(t))$ $[-\lambda,\lambda]$

The Residue $\res=g-y$

$\res(t)$$2\lambda\Z$

Same Bits, Finer Steps

PCM levelsmodulo-ADC levels

Integer and Fractional Parts

Write $g=\lfloor g\rfloor+[\![g]\!]$. PCM, $\Sigma\Delta$ and noise shaping encode the integer part, which grows with $\norm{g}_\infty$, and discard the fractional part. USF-Audio digitizes the bounded fractional part and recovers the integer part from smoothness.

Folding Subtracts a Staircase

Folding removes a piecewise-constant residue whose jumps lie on the lattice $2\lambda\Z$, eq. (4):

$$\Mlam(g)=g-\res,\qquad \res(t)=\sum_{m=0}^{M-1}c_m\mathbb{1}(t-\tau_m),\ c_m\in2\lambda\Z.$$

Quantization Noise Becomes Small

With $\yq=y+q$ and $|q|\le\eta_\lambda^B$, eq. (6) reads $\yq[n]=g[n]-\res[n]+q[n]$. The step is $\DRX$ times finer than PCM's, so, eq. (1),

$$\mathrm{SQNR}_{\lambda,B}=\mathrm{SQNR}_B+20\log_{10}\bigl(\norm{g}_\infty/\lambda\bigr).$$

Theorem 1 of the Paper (Unlimited Sampling, [Bhandari et al. 2017, 2020])

Let $g\in\BL$ and $y[n]=\Mlam(g(nT))$. Then $T\Omega\le(2e)^{-1}$ is a sufficient condition for recovering $g(t)$ from $y[n]$, up to an unknown constant in $2\lambda\Z$. No condition on $\norm{g}_\infty$ appears: the amplitude is unlimited. Chapter 3 shows why, sharpens $2e$ to a bit–sampling law, and adds quantization.

In the paper Section I-D (reframing quantization noise as the signal), Section II (acquisition model, eqs. (2)–(3), Theorem 1), Section III (eqs. (4)–(6)).

Test signal: a fixed sum of shifted sincs bandlimited to $\Omega=\pi$ rad/s, scaled to $\norm{g}_\infty=\DRX\cdot\lambda$ with $\lambda=1$, sampled at $T=1/8$ s. The right-hand plot compares the two quantizers over a 1.5 s window: PCM spreads $2^B$ levels over $[-\norm{g}_\infty,\norm{g}_\infty]$; the modulo ADC spreads them over $[-\lambda,\lambda]$ and its samples are shown after adding back the true residue.