Sub-Nyquist Additive Random Sampling (sNARS)

• Sub-Nyquist Additive Random Sampling (sNARS) is a technique that samples analog signals below the Nyquist rate by exploiting inherent signal sparsity through structured randomness.
• It employs multiple parallel analog channels that mix signals using Gabor frame-based window functions and randomized coefficients to capture sparse signal information.
• The method integrates compressed sensing recovery via convex optimization to robustly reconstruct multipulse or multiband signals with reduced measurement rates and hardware complexity.

Sub-Nyquist Additive Random Sampling (sNARS) refers to a class of analog-to-digital conversion strategies that acquire continuous-time signals at rates significantly below the classical Nyquist threshold by leveraging inherent signal sparsity and structured randomness in the acquisition process. sNARS generalizes and rigorously extends traditional uniform and non-uniform sub-Nyquist methods by combining analog mixing with randomized modulation and compressed sensing (CS) recovery. This class of techniques enables stable, low-rate sampling architectures for multiplex, spectrally sparse, and otherwise compressible analog signals, even when critical parameters such as pulse shapes or locations are unknown.

1. The sNARS Sampling Model and Gabor Frame Mixing

sNARS exploits the structural sparsity of analog signals (e.g., sparse pulses in time) by basing the measurement process on inner products with a family of shifted and modulated window functions derived from a Gabor frame: $$G(g, a, b) = \{ M_{bl}\,T_{ak}\,g(t) = e^{2\pi i\,b l t}\,g(t - a k) : k,l \in \mathbb{Z} \},$$ where $a = \mu W$, $b = 1/W$, and the window $g(t)$ is compactly supported on $[ -W/2, W/2 ]$ and well-localized in frequency for a chosen $\mu \in (0,1)$. Arbitrary continuous-time signals $f(t)$ with finite support are decomposed as

$$f(t) = \sum_k \sum_l z_{k,l}\; M_{b l} T_{a k} \tilde{g}(t),$$

with “Gabor coefficients” $z_{k,l} = \langle f, M_{b l} T_{a k} g \rangle$ and a dual synthesis window $\tilde{g}(t)$.

sNARS systems use $JM$ parallel analog channels, each mixing $f(t)$ with a unique function $q_{j,m}(t) = w_j(t)\,s_m(t)$, where

$$w_j(t) = \sum_{l=-L_0}^{L_0} d_{j,l} e^{-2\pi i b l t},\qquad s_m(t) = \sum_{k=-K_0}^{K_0} c_{m,k} \overline{g(t-ak)}.$$

Here, $d_{j,l}$ and $c_{m,k}$ are mixing coefficients, typically random or designed as sign sequences. The output of each channel,

$$y_{j,m} = \int_{-T/2}^{T/2} f(t)\,q_{j,m}(t)\,dt = \sum_{k,l} c_{m,k}\,d_{j,l}\,z_{k,l},$$

is thus a linear mixture of Gabor coefficients. Collecting all outputs leads to a system

$Y = D\,X^T,\quad X = C\,Z,$

with $Z$ the matrix of Gabor coefficients, $C$ and $D$ the mixing matrices constructed from $\{c_{m,k}\}$ and $\{d_{j,l}\}$, and $Y$ the observed samples. This architecture realizes additive random sampling in both time and frequency, multiplexing sparse information into a compressed measurement ensemble.

2. Signal Sparsity and Model-Agnostic Reconstruction

The primary target class for sNARS is signals that are “multipulse”: finite unions of short-duration pulses occurring at unknown times with unknown shapes,

$$f(t) = \sum_{n=1}^N h_n(t), \quad \mathrm{supp}(h_n) \leq W.$$

Given the large ambient support $T \gg W$, the vast majority of Gabor coefficients $z_{k,l}$ vanish for non-overlapping shifts—only coefficients corresponding to time shifts near pulse locations (indexed by $k$) are nonzero. Thus, the Gabor coefficient matrix $Z$ is row-sparse, corresponding to the sparse time structure, even in the extreme “blind” case where neither pulse shape nor locations are known. The only required prior is an upper bound on the number $N$ of pulses and their duration $W$, enabling near-universal sparse recovery.

This row-sparsity generalizes directly to multiband or sparse-in-frequency settings, where the coefficient matrix becomes doubly sparse (row and, for multiband signals, column sparse), reducing the requisite measurement count further.

3. Compressed Sensing Recovery and Convex Programming

Since the direct measurements are linear mixtures of sparse Gabor coefficients, sNARS naturally fits the compressed sensing (CS) paradigm. The recovery challenge is to reconstruct the sparse matrix $Z$ from the undersampled mixtures $Y$ via the mixing matrices $C$ and $D$.

The coefficient recovery is formulated as a multiple measurement vector (MMV) $\ell_{2,1}$ minimization: $$\min_{Z} \|Z\|_{2,1} \quad \mathrm{subject~to} \quad X = C Z,$$ where $$\|Z\|_{2,1} = \sum_k \left(\sum_l |z_{k,l}|^2\right)^{1/2}$$. Under standard restricted isometry property (RIP) conditions—specifically, if $C$ has RIP of order $\mathcal{O}(N)$ and corresponding constants—the solution $\hat{Z}$ yields a stable approximation of the true Gabor coefficients and hence the original signal $f(t)$. Reassembly follows via Gabor synthesis: $$\hat{f}(t) = \sum_{k,l} \hat{z}_{k,l}\, M_{b l} T_{a k} \tilde{g}(t).$$

Rigorous a priori error bounds are established: $$\|f - \hat{f}\|_2 \leq \mathcal{C}_0 (\varepsilon_\Omega + \varepsilon_B) \|f\|_2 + \mathcal{C}_1 \|n_1\|_2 + \mathcal{C}_2 \|n_2\|_2,$$ with constants $\mathcal{C}_i$ dependent on the system design, and $\varepsilon_\Omega, \varepsilon_B$ quantifying out-of-band energy and window approximation error.

4. Performance for Multiband and Simultaneously Sparse Signals

For signals sparse in both time and frequency (essentially multiband multipulse), sNARS has an even more favorable scaling. If, in addition to at most $N$ pulses of duration $W$, the signal’s spectrum occupies no more than $S$ bands (each of bandwidth $\Delta_W$), then per row of $Z$, only $\mathcal{O}(S (\Delta_W + B) W)$ columns may be significant. The total sample complexity reduces accordingly, with bounds such as

$$\text{Number of measurements} \sim 8\,(\Delta_W + B) \, W N S,$$

which approaches the theoretical Landau rate up to logarithmic factors and modest oversampling in practical systems. The design of the mixing coefficients in both “time” ($C$) and “frequency” ($D$) domains enables effective measurement reduction; for instance, frequency-domain mixing may be “shrunken” when few columns are anticipated to be nonzero.

5. Robustness and Noise Analysis

sNARS explicitly addresses robustness to noise and model mismatch. In the presence of signal and measurement noise,

$Y = D\,X^T + \tilde{N}, \quad X = C\,Z + N,$

the $\ell_{2,1}$-minimization recovery yields a reconstruction whose error is bounded in terms of the energy outside the model, the “compressibility” of $Z$ (i.e., how sparse it is), and the norm of the noise matrices: $$\|f - \hat{f}\|_2 \leq \mathcal{C}_0 (\varepsilon_B + \varepsilon_{\Omega})\,\|f\|_2 + \mathcal{C}_1\,\|Z - Z^S\|_{2,1} + \mathcal{C}_2\,\|N\|_2.$$ Here $Z^S$ denotes the best $S$-row sparse approximation to $Z$. This proves that sNARS systems are inherently stable under moderate to severe corruption, provided the mixing matrices maintain suitable incoherence or RIP properties.

6. Applications and System Implications

The sNARS paradigm is well-suited for applications where information is naturally sparse in time, frequency, or both:

• Radar and Sonar: Echoes as pulse trains (delays unknown), sparse target scenes.
• Medical and Bio-Imaging (Ultrasound, Neuronal): Multipulse signals from scattering, firing, or reflectivity events.
• Communications: Bursty channels, random access protocols.
• Wideband Sensing: Cognitive radio, spectral occupancy where high frequency bands are mostly empty.

The multichannel, additive random mixing design is highly advantageous for hardware: analog front-ends can integrate and modulate at low rates, dramatically reducing requirements for ADC bandwidth and throughput. sNARS also aligns with the broader “Xampling” architecture, wherein analog acquisition is statistically compressed before digitization.

7. Summary of Key Mathematical Formulations

Formula	Description
$$f(t) = \sum_{k,l} z_{k,l} M_{bl}T_{ak}\, \tilde{g}(t)$$	Gabor synthesis of signal
$z_{k,l} = \langle f, M_{bl}T_{ak}\, g\rangle$	Gabor coefficient definition
$q_{j,m}(t) = w_j(t)\,s_m(t)$	Analog mixing function per channel
$$y_{j,m} = \int f(t)\, q_{j,m}(t)\,dt = \sum_{k,l} c_{m,k}\, d_{j,l}\, z_{k,l}$$	Channel output as mixed sum of coefficients
$Y = D\,X^T$, $X = C\,Z$	Matrix form of measured samples
$\min \|Z\|_{2,1}$ s.t. $X = C Z$	Compressed sensing recovery objective

8. Theoretical and Practical Impact

sNARS provides a mathematically rigorous and practically validated approach for sampling analog signals at rates matching their intrinsic information content, rather than their ambient Nyquist bandwidth. By combining randomized mixing aligned with Gabor frame theory and robust convex optimization, sNARS systems are provably stable, adaptable to signal, model, and noise uncertainties, and deployable in low-power, hardware-limited environments. The transition from theory to implementation is facilitated by the modular design—mixing, integrating, digitizing, and CS-based reconstructing—which directly translates to modular analog front-ends and scalable digital back-ends in real-world measurement and communication systems.