Multirate Synchronous Sampling (SMRS)

• Multirate Synchronous Sampling (SMRS) is a technique that exploits spectral sparsity by synchronizing multiple sampling channels at distinct rates to reduce the overall required rate.
• It utilizes structured sensing matrices and aliasing principles to formulate overdetermined systems for efficient sparse recovery, often surpassing traditional multicoset methods in stability and noise robustness.
• The approach underpins practical applications in radar, communications, spectral estimation, and control by achieving near-Landau rate sampling while offering improved computational efficiency and resilience to noise.

Multirate Synchronous Sampling (SMRS) is a paradigm for sub-Nyquist acquisition, reconstruction, and estimation of sparse signals in both continuous- and discrete-time domains, with core applications in radar, communications, spectral estimation, and robust networked control. SMRS leverages synchronized sampling at multiple distinct rates, exploiting spectral or temporal sparsity to reduce the total sampling rate—often to a modest multiple of the Landau limit—while maintaining robustness, simplicity of reconstruction, and architectural efficiency. The approach has inspired significant theoretical development in signal processing, compressive sensing, and optimal control.

1. Mathematical Signal Model and Synchronous Multirate Sampling

The canonical SMRS signal model considers a continuous-time signal $x(t)$ with Fourier transform $$X(f)=\int_{-\infty}^\infty x(t)e^{-j2\pi ft}\mathrm{d}t$$, bandlimited to $[0,F_{max}]$ but occupying only a sparse union $S \subset [0, F_{max}]$ of $N$ disjoint intervals:

$$S = \bigcup_{n=1}^N (a_n, b_n]\,, \qquad B = \sum_{n=1}^N(b_n - a_n) \ll F_{max}\,.$$

The minimum sampling rate for perfect reconstruction of all signals supported on $S$ is the Landau rate $R_L = |S| = B$.

In the SMRS framework, $x(t)$ is sampled in $M$ parallel channels, each operating at a rate $f_m$:

$$x_m(t) = x(t)\cdot \sum_{k\in\mathbb{Z}} \delta\left(t - \frac{k}{f_m}\right)\,,$$

with all channels initiated synchronously (zero delay). Each $f_m$ is typically an integer multiple of a frequency step $\Delta f$, facilitating a uniform frequency resolution across all channels:

$$f_m = M_m\cdot \Delta f\,, \quad M_m \in \mathbb{N}\,.$$

The aggregate sampling rate $\sum_{m=1}^M f_m$ is designed to be substantially less than the Nyquist rate, provided the underlying support $S$ is sufficiently sparse (0806.0579).

2. SMRS Matrix Formulation and Reconstruction Strategies

Sampling at rate $f_m$ aliases $X(f)$ into $[0, f_m]$ in channel $m$. Discretizing $f = k\Delta f$, one defines the vectorized spectrum $x[k] = X(k\Delta f)$ and samples $y_m[\ell] = X_m(\ell\Delta f)$. Aggregating all output channels yields the matrix system:

$\mathbf{y} = \mathbf{A}\,\mathbf{x},$

where $\mathbf{A}$ is a structured sensing matrix with entries

$$A_{(m,\ell),k} = \begin{cases} f_m, & k \equiv \ell \ (\mathrm{mod}\ M_m),\ 0, & \text{otherwise}. \end{cases}$$

If the spectral support is known, columns of $\mathbf{A}$ corresponding to unoccupied frequencies are pruned, yielding an (often overdetermined) system for the band content. For unknown support but sparse spectrum, a zero-support reduction is first applied—indices $k$ with zero output in all channels are deemed unoccupied, further sparsifying the system. Robust recovery employs least-squares or sparse-block pursuit (e.g., block-OMP) depending on problem conditioning (0806.0579).

3. Noise Robustness, Condition Number, and Architectural Comparison

The condition number $$\kappa(\mathbf{A}) = \sigma_{max}(\mathbf{A}) / \sigma_{min}(\mathbf{A})$$ governs noise stability:

$$\|\delta\mathbf{x}\| \leq \kappa(\mathbf{A}) \|\mathbf{n}\|,$$

with empirical ranges $\kappa(\mathbf{A})\approx 1$–$6$ for typical SMRS configurations (channels at pairwise coprime or mutually prime rates). As the per-channel rate $f_m$ increases, quantization noise per alias diminishes, and SNR in digitized samples scales as $10\log_{10}(f_m / \Delta f)$. Notably, SMRS requires far fewer channels than conventional multicoset architectures for comparable signal recovery and yields superior numerical stability (e.g., $\kappa\sim 5$ versus $20$–$50$ in coset sampling for large systems) (0806.0579).

Scheme	Channels	Total Rate	$\kappa$	Example (8 bands)
SMRS	3	$8R_L$	$\sim 5$	$3\times(3.8,4,4.2)$GHz
Multicoset	6	$13R_L$	$20$–$50$	$6\times 1$GHz

4. Generalizations: Co-Prime and Multirate Support, Gram Structures

Variations of SMRS include synchronous co-prime sampling, where two low-speed ADCs (rates $f_1, f_2$) with $\gcd(M, N)=1$ sample jointly. After re-sequencing, $L=M N$ original Nyquist intervals yield $M+N-1$ uniform, low-rate output streams. The key is a Vandermonde-structured sensing matrix $\Phi$, enabling blind spectral support estimation and robust sparse recovery through subspace methods like MUSIC (Zhao et al., 2018).

In general, any synchronized multirate system with rates admitting a common supporting grid can be represented via polynomial Gram structures. This enables efficient semidefinite programming (SDP) solutions for sparse recovery, leveraging compact Gram parametrizations to structure the SDPs at minimal dimension relative to the number of acquired samples (rather than the virtual grid length) (Costa et al., 2016).

5. Regularized SMRS and Approaching the Landau Limit

SMRS can be hybridized with regularized sampling, notably by band-limited windowing of the input and trigonometric polynomial approximation. Sampled data under $K$ synchronous grids yield a block-sparse linear system with “ones-and-zeros” structure, permitting accurate solution by pseudoinverse. Oversampling offers a trade-off between robustness to noise (quantified via amplification factor $\gamma$) and sampling rate efficiency, while the use of MUSIC-type subspace detection enables blind estimation of the spectral support (Selva, 2010).

When carefully designed, SMRS-based systems approach the Landau limit, with the minimal average sampling rate for perfect sparse recovery given by $R_{\text{avg}} \gtrsim |\mathcal S_z|$, where $|\mathcal S_z|$ is the spectral measure of the union of bands. Windowing and regularization introduce only modest excess above this limit, primarily for noise robustness.

6. SMRS in Optimal and Robust Control: Synchronous Multirate Self-Triggering

In sampled-data control, SMRS extends to the generation of aperiodic, multirate sampling sequences for LTI systems under state-feedback. At each sampling instant $t_h$, a finite-horizon sequence $\sigma_h = (T^1, ..., T^{\ell_h})$ of future sampling intervals is chosen to maximize average inter-sample time while guaranteeing Lyapunov-stability (for unperturbed) or uniform ultimate boundedness (for perturbed systems) via explicit linear matrix inequalities. Both online and offline implementations systematically select admissible sequences through quadratic or S-procedure-based tests, trading computation for memory as required (Tariverdi, 2 Nov 2025).

7. Algorithmic Complexity and Theoretical Guarantees

In SMRS-based estimation, the core bottleneck is solving structured linear or polynomial systems (or SDPs for spectrally continuous/finite-rate-of-innovation signals). With block or Gram structure, the computational demands reduce to solving systems sized by the actual number of samples ($N=\sum_j n_j$) rather than the hypothetical uniform grid length. Under the conditions of a common supporting grid and minimal spectral separation (e.g., $|\nu_\ell - \nu_{\ell'}| > 1/n_\sharp$), TV-minimization and its SDP duals guarantee unique, exact recovery of all spectrally sparse signals (Costa et al., 2016).

• "Multirate Synchronous Sampling of Sparse Multiband Signals" (0806.0579)
• "Sparse Multiband Signal Acquisition Receiver with Co-prime Sampling" (Zhao et al., 2018)
• "Regularized sampling of multiband signals" (Selva, 2010)
• "Achieving Super-Resolution in Multi-Rate Sampling Systems via Efficient Semidefinite Programming" (Costa et al., 2016)
• "Robust Self-Triggered Control Approaches Optimizing Sampling Sequences with Synchronous Measurements" (Tariverdi, 2 Nov 2025)