Sparse multiband signal spectrum sensing with asynchronous coprime sampling

Abstract

Cognitive radio requires spectrum sensing over a broad frequency band and leads to a high sampling rate. In this paper, we propose an asynchronous coprime sampling technique for capturing and reconstructing of sparse multiband signals that occupy a small part of a given broad frequency band. The band locations of signal are not known a priori. In this proposed approach, we use a sub-Nyquist sampling rate by exploiting a low-dimensional representation of the original high-dimensional signal. A common input sparse multiband signal is digitized using a pair of uniform samplers, which are (not necessarily synchronously) clocked at coprime sampling rates. The captured samples are then re-sequenced and the multi-coset signal processing algorithm is employed. We derive the system model in the frequency domain, where the phase mismatch is compensated. Compared to the conventional multi-coset sampling, the proposed approach needs fewer samplers and does not require synchronous clock phase. Simulation results are provided to demonstrate the feasibility and effectiveness of the proposed asynchronous coprime sampling for sparse multiband signal.

Appendix 1

1.1 Real-valued signals

In this Appendix, we present the modifications to (12) for real-valued signal reconstruction. The sparse multiband signal is required to be bandlimited to ℱ ⊂ [−1/(2T), 1/(2T)]. The FT is conjugate symmetric for a real-valued signal x(t), its FT X(f) contains positive and negative frequencies. To form x(f), ℱ is still equally partitioned into L frequency intervals. Then, for odd and even values of L, ℱ₀ is redefined by

$$\mathcal{F}_{0} = \left\{ {\begin{array}{*{20}c} {\left[ { - \frac{1}{2LT},\text{ }\frac{1}{2LT}} \right],\text{ }odd\text{ }L\text{ }} \\ {\left[ {0,\text{ }\frac{1}{LT}} \right],\text{ }even\text{ }L} \\ \end{array} } \right.$$

(31)

Define the set of L consecutive integers:

$$K = \left\{ {\begin{array}{*{20}c} {\left\{ { - \frac{L - 1}{2}, \ldots ,\frac{L - 1}{2}} \right\},\text{ }odd\text{ }L\text{ }} \\ {\left\{ { - \frac{L}{2} + 1, \ldots ,\frac{L}{2}} \right\},\text{ }even\text{ }L} \\ \end{array} } \right.$$

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Consequently, x(f) is rewritten as

$${\mathbf{x}}_{l} \left( f \right) = X\left( {f + \frac{{K_{l} }}{LT}} \right),\text{ }f \in \mathcal{F}_{0} ,$$

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where 0 ≤ l < L. The (i, l)th entry of Φ is given by

$${\varvec{\Phi}}_{{\text{ }i,l}} = \left\{ {\begin{array}{*{20}c} {\exp \left( {j2\pi c_{i}^{1} \frac{{K_{l} }}{L}} \right),\text{ }0 \le i < L_{2} \text{ }} \\ {\exp \left( {j2\pi \left( {c_{{i - L_{2} }}^{2} + \frac{\tau }{T}} \right)\frac{{K_{l} }}{L}} \right),\text{ }L_{2} \le i < M} \\ \end{array} } \right. .$$

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