Spectrum blind reconstruction and direction of arrival estimation of multi-band signals at sub-Nyquist sampling rates

Abstract

In this paper we consider the problem of spectrum blind reconstruction (SBR) and direction of arrival (DOA) estimation of constituent sources of a disjoint multi-band signal (MBS) at sub-Nyquist sampling rates. Transformation of the problem into frequency domain indicates that the steering vector is a function of both the carrier frequency and its corresponding DOA. Employing the existing two dimensional frequency-DOA search algorithms suffers from the drawbacks of increased computational complexity and ambiguity issues. To overcome these drawbacks, in this paper we propose a simple modification to the receiver architecture by introducing an additional delay channel at every sensor. Estimation algorithms based on ESPRIT is then employed to estimate the carrier frequencies, while MUSIC algorithm is employed to estimate their corresponding DOAs. Using the knowledge of both these parameters, the MBS spectrum is then reconstructed. A two-dimensional iterative grid refinement algorithm is also described to further improve the estimation accuracy in the presence of noise. Identifiability issues are addressed and the conditions for unique identifiability are discussed. Furthermore, by assuming a two dimensional uniform array the advantages of the proposed approach in terms of identifiability is also provided. We further show that an $M \ge N+1$ sensors and an overall sampling rate of at least $2(N+1)B$ would be sufficient to achieve SBR and DOA estimation of an MBS comprising of N disjoint bands each of maximal bandwidth B. Numerical simulations are finally presented which verifies the validity of the proposed approach and compares the performance against appropriate bounds.

Appendix

1.1 Proof of Proposition 1

To properly reconstruct X(f), for any $m \in \mathbb {Z}$ and $i = \{1,2,...,N\}$, $S_i(f-mf_s-f_i) \bigcap S_i(f-(m+1)f_s-f_i) = \emptyset$ (see Fig. 2). Since the maximum bandwidth of the information band is B, this can be satisfied if and only if $f_s \ge B$, which also implies $L \le 1/BT$, thus proving the first condition. Now, if $\text {krank}(\mathbf{A }_c(\tilde{\varvec{f}},\tilde{\varvec{\theta }})) > N$, then by Lee et al. (2012, Proposition 4.2)  it is guaranteed that the MUSIC spectrum shows null for steering vectors corresponding to $\{f_i,\theta _i\}_{i = 1}^N$. We now prove the “only if” by contradiction. Assume this condition is satisfied and in addition to N nulls, MUSIC shows null corresponding to some $(f_k,\theta _k) \notin \{f_i,\theta _i\}_{i = 1}^N$. This implies that $(f_k,\theta _k)$ resides in the range space of $\mathbf{A }_c(\tilde{\varvec{f}},\tilde{\varvec{\theta }})$. By the definition of the Kruskal rank, this is possible only if $\text {krank}(\mathbf{A }_c(\tilde{\varvec{f}},\tilde{\varvec{\theta }})) \le N$ which is a contradiction. Hence by satisfying $\text {krank}(\mathbf{A }_c(\tilde{\varvec{f}},\tilde{\varvec{\theta }})) > N$, MUSIC shows only N nulls corresponding to $\{f_i,\theta _i\}_{i = 1}^N$, which completes the proof.$\square$

1.2 Proof of Theorem 1

The condition (26) ensures that the generators of the Vandermonde matrices $\mathbf{A }_c^{l_{1}}({\varvec{f}},\tilde{\varvec{\theta }})$ and $\mathbf{A }_c^{l_{2}}({\varvec{f}},\tilde{\varvec{\theta }})$ are unique, hence by the well known property of Vandermonde matrices, $\mathbf{A }_c^{l_{1}}({\varvec{f}},\tilde{\varvec{\theta }})$ and $\mathbf{A }_c^{l_{2}}({\varvec{f}},\tilde{\varvec{\theta }})$ are full rank as well as full Kruskal rank matrices (Sidiropoulos and Liu 2001, Lemma 2). Since $NN_\theta > M_{1}$ and $NN_\theta > M_{2}$, $\text {Krank}(\mathbf{A }_c^{l_{1}}({\varvec{f}},\tilde{\varvec{\theta }})) = M_{1}$ and $\text {Krank}(\mathbf{A }_c^{l_{1}}({\varvec{f}},\tilde{\varvec{\theta }})) = M_{2}$. Now, using the bound on the Krukal rank (Sidiropoulos and Liu 2001, Lemma 1), $\text {Krank}(\mathbf{A }_c({\varvec{f}},\tilde{\varvec{\theta }})) = \text {Krank}(\mathbf{A }_c^{l_{1}}({\varvec{f}},\tilde{\varvec{\theta }}) \odot \mathbf{A }_c^{l_{2}}({\varvec{f}},\tilde{\varvec{\theta }})) \ge \text {min}(M_{1}+M_{2}-1,NN_\theta )$. Hence if $M_{1}+M_{2}-1 > N$, $\text {Krank}(\mathbf{A }_c({\varvec{f}},\tilde{\varvec{\theta }})) > N$.$\square$