USF-Audio · A Tutorial for Graduate Students
Re-thinking Digital Audio, One Explorer at a Time
This tutorial walks through the paper “Re-thinking Digital Audio via Unlimited Sensing: Theory, Algorithms, and Hardware Validation” by R. Guo, Y. Zhu and A. Bhandari. Each chapter states one claim, lets you test it in a live explorer whose plots are computed in your browser, then gives the argument in three steps, with pointers to the exact equations, theorems and figures of the paper.
Who It Is For
Graduate students and researchers in signal processing, audio and sensing. You should be comfortable with the Shannon–Nyquist sampling theorem, the Fourier transform, and uniform quantization. Everything about modulo sampling is built up from scratch.
Read the chapters in order: each uses the notation and results of the previous one. Chapters 3 and 5 are the core; Chapter 6 checks every theoretical claim against the paper's measurements.
How the Explorers Work
Presets load the settings discussed in the paper. Sliders change the parameters; every plot is recomputed from the paper's model. Chips report whether each condition of a theorem holds. Hover any plot for exact values. Buttons marked ▶ animate a process, such as a fold, a proof or an algorithm step.
No data leave your browser. Mathematics renders offline.
The Limits of PCM
Clipping, quantization noise and the range–resolution trade-off that no choice of input range escapes.
02Folding Before Digitizing
The centered modulo, the residue staircase on \(2\lambda\mathbb Z\), and why the same bits give finer steps.
03Why Recovery Works
Theorem 2: differences shrink the signal and amplify quantization noise; the bit–sampling law balances them.
04An Audio-Centric Guarantee
Spectral moments as effective bandwidths, Theorem 3, and how concentrated audio relaxes the sampling rate.
05The USF-ARA Algorithm
Frames, differences, DCT support separation and spike estimation, animated step by step.
06From Theory to Hardware
ENOB and SINAD at 6 bits, 14 recordings, severe clipping, and 52 listeners, drawn from the paper's tables.
Notation
| Symbol | Meaning |
|---|---|
| \(g\in\mathcal B_\Omega\) | finite-energy signal bandlimited to angular frequency \(\Omega\) |
| \(\mathscr M_\lambda\), \(\lambda\) | centered modulo with folding threshold \(\lambda\): values in \([-\lambda,\lambda)\) |
| \(y[n]=\mathscr M_\lambda(g(nT))\), \(T\) | folded samples, sampling step \(T\) |
| \(B\), \(\eta_\lambda^B=\lambda/(2^B-1)\) | bit depth and the modulo ADC's quantization half-step; \(y_q=\mathscr Q_\eta(y)\) |
| \(\varepsilon_g=g-y\) | residue: a staircase with jumps \(c_m\in2\lambda\mathbb Z\) at folding instants \(\tau_m\) |
| \(q\) | quantization noise, \(|q|\le\eta_\lambda^B\) |
| \(\mathsf{DR}_\mathsf{X}=\|g\|_{L^\infty}/\lambda\) | dynamic-range expansion |
| \(\Delta^h\) | \(h\)-th order forward difference, \((\Delta g)[n]=g[n+1]-g[n]\) |
| \(\mathsf{OF}=\pi/(\Omega T)\) | oversampling factor (1 at Nyquist) |
| \(\Omega_{\mathsf{eff},h}\) | order-\(h\) spectral moment (effective bandwidth) |
| \(N_g\), \(K\) | frame length and DCT band edge in USF-ARA |
Read and Listen
The paper (PDF) · USF Audio home, with a listening lab and playable spectrograms · Research note: hearing Für Elise in quantization noise · hardware audio demos.
Cite
R. Guo, Y. Zhu and A. Bhandari, “Re-thinking Digital Audio via Unlimited Sensing: Theory, Algorithms, and Hardware Validation.”