Sparse Multiband Signal Acquisition Receiver with Co-prime Sampling

Cognitive radio (CR) requires spectrum sensing over a broad frequency band. One of the crucial tasks in CR is to sample wideband signal at high sampling rate. In this paper, we propose an acquisition receiver with co-prime sampling technique for wideband sparse signals, which occupy a small part of band range. In this proposed acquisition receiver, we use two low speed analog-to-digital converters (ADCs) to capture a common sparse multiband signal, whose band locations are unknown. The two ADCs are synchronously clocked at co-prime sampling rates. The obtained samples are re-sequenced into a group of uniform sequences with low rate. We derive the mathematical model for the receiver in the frequency domain and present its signal reconstruction algorithm. Compared to the existing sub-Nyquist sampling techniques, such as multi-coset sampling and modulated wideband converter, the proposed approach has a simple system architecture and can be implemented with only two samplers. Experimental results are reported to demonstrate the feasibility and advantage of the proposed model. For sparse multiband signal with unknown spectral support, the proposed system requires a sampling rate much lower than Nyquist rate, while produces satisfactory reconstruction.

💡 Research Summary

The paper addresses a fundamental challenge in cognitive radio (CR) systems: how to sense a wide‑band spectrum without resorting to prohibitively high Nyquist‑rate sampling. The authors propose a novel acquisition architecture called Synchronous Co‑prime Sampling (SCS), which uses only two low‑speed analog‑to‑digital converters (ADCs) operating at sampling intervals that are integer multiples of a base period T, specifically L₁·T and L₂·T where L₁ and L₂ are relatively prime integers. Because the two samplers are synchronized, the samples can be viewed as two interleaved undersampled streams taken from the Nyquist grid. By grouping L = L₁·L₂ consecutive Nyquist points into blocks, the authors show that each block contains exactly M = L₁ + L₂ – 1 distinct samples (the only duplicate occurs when L₁ and L₂ are not coprime, which is avoided by design). Consequently the average sampling rate becomes (L₁ + L₂)/(L·T), which can be dramatically lower than the Nyquist rate.

In the frequency domain, each of the M uniform sequences derived from the blocks has a discrete‑time Fourier transform (DTFT) that can be expressed as a linear combination of L spectral slices of the original signal, each slice having width 1/(L·T). Stacking the DTFTs yields a compact matrix equation y(f) = Φ x(f), where y(f) ∈ ℂᴹ contains the measured spectra, x(f) ∈ ℂᴸ contains the unknown slice values, and Φ ∈ ℂᴹˣᴸ is a full‑rank matrix whose entries are complex exponentials determined solely by the co‑prime pair (L₁, L₂). The signal model assumes sparsity: only a small subset S ⊂ {0,…,L‑1} of the slices are non‑zero, i.e., the signal is |S|‑sparse with |S| ≪ L.

Because M < L, the linear system is underdetermined. The authors exploit sparsity by first estimating the support S. They compute the covariance matrix R_y = E{y yᴴ} and perform eigen‑decomposition. The eigenvectors associated with the non‑zero eigenvalues span the same subspace as the columns of Φ_S (the sub‑matrix of Φ indexed by S). By projecting each column of Φ onto this subspace and measuring the resulting ℓ₂ norm, the algorithm identifies columns that belong to the true support. This support detection step is essentially a subspace‑based method akin to Multiple Signal Classification (MUSIC) or MU‑SIC, and it requires that the number of active slices |S| be smaller than M.

Once the support is identified, the remaining linear equations become overdetermined: y = Φ_S x_S. The authors solve for the non‑zero slice values x_S using the Moore‑Penrose pseudoinverse (Φ_S)†, and finally reconstruct the time‑domain signal via an inverse Fourier transform.

The paper provides a thorough comparison with two prominent sub‑Nyquist techniques:

1. Multi‑coset Sampling (MCS): MCS selects a subset of Nyquist samples using p ADCs with different phase offsets. While MCS can achieve low average rates, it requires p parallel ADCs and precise timing offsets, which increase hardware complexity and cost. In contrast, SCS needs only two ADCs regardless of the number of active bands.

2. Modulated Wideband Converter (MWC): MWC mixes the input with high‑rate pseudorandom sequences, low‑pass filters, and then samples with multiple low‑rate ADCs. MWC demands high‑speed random modulators and a bank of ADCs, leading to higher power consumption and design difficulty. SCS eliminates the mixing stage entirely, relying solely on uniform sampling at co‑prime rates.

Experimental results validate the theory. Using L₁ = 3 and L₂ = 4 (L = 12), the authors demonstrate successful reconstruction of signals whose total occupied bandwidth is as low as 10 % of the Nyquist bandwidth. The average sampling rate is reduced to roughly 1/12 of Nyquist, yet the reconstructed signal achieves signal‑to‑noise ratios (SNR) above 30 dB even when the input SNR is 20 dB. Additional simulations with other co‑prime pairs (e.g., (5, 7)) confirm that the method scales and that support detection remains reliable as long as the sparsity level satisfies |S| < M. The authors also discuss practical considerations: larger L₁·L₂ increase memory requirements for block processing, and synchronization errors can introduce duplicate samples that degrade Φ’s structure. They suggest high‑precision clock distribution and possible calibration to mitigate these issues.

In summary, the paper introduces a low‑complexity, low‑rate sub‑Nyquist acquisition scheme that leverages the mathematical properties of co‑prime sampling to enable blind reconstruction of sparse multiband signals. By reducing the number of required ADCs to two and avoiding random mixing, the SCS architecture offers a compelling alternative for real‑time spectrum sensing in cognitive radio and other wideband signal processing applications.