Robust forward backward based beamformer for a general-rank signal model with real-valued implementation
Highlights
► A robust beamformer for an incoherently scattered general-rank model is proposed. ► It is based on the uniform linear array structure with forward backward processing. ► A real-valued close-form solution is provided by introducing a preprocessing transformation. ► Computational complexity of the system is reduced significantly. ► It has achieved a higher output signal-to-interference-plus-noise ratio.
Introduction
Beamforming is a technique for receiving the signal of interest from some specific directions while suppressing the interfering signals from other directions using an array of sensors (Note that the same technique can also be used for signal transmission) [1], [2]. One of the most commonly used beamformers is the linearly constrained minimum variance (LCMV) beamformer [3]. However, in practice, the performance of an LCMV beamformer degrades significantly when the sample size of the received signal is small or the steering vector of the desired signal has a mismatch. Various robust algorithms have been proposed in the past decades, such as the diagonal-loading-based beamformer and the eigenvector-based beamformer [4], [5]. A drawback of the diagonal-loading-based beamformer is that it is not clear how to choose the optimum diagonal-loading factor, while the eigenvector-based beamformer suffers from the subspace swap phenomenon in low signal-to-noise ratio (SNR) environments. In [6], the worst-case optimization-based robust beamformer was proposed. By estimating the real steering vector while maximizing the beamformer output power, a robust Capon beamformer was proposed in [7] and then implemented in a recursive form in [8]. By generalizing the signal covariance matrix into a higher-rank (non-point source) one [9], a robust approach for general-rank signal models was derived, which was further developed in [10] with positive semidefinite constraints.
One key issue in many of the beamforming algorithms is the estimation of data covariance matrix based on finite samples. Based on the specific structure of uniform linear arrays (ULAs) and the resultant persymmetric structure of their covariance matrices, a forward backward (FB) averaging method was proposed for covariance matrix estimation with a significant performance improvement [11], [12]. It was then exploited in different applications such as directional of arrival (DOA) estimation [13], [14], spectrum estimation [15], and adaptive beamforming [16]. It is further shown in [17] that the optimum weight vector of a ULA (or the more general case: symmetrically distributed arrays) has a generalized conjugate symmetric property, which is employed to form a constrained beamforming problem and leads to improved performance. However, the conjugate symmetric property of the optimum weight vector will be destroyed in the presence of model perturbations, such as array mutual coupling, sensor position errors, discrepancies in sensor responses, etc., which leads to a degraded performance. Therefore, there is a need to develop a robust FB algorithm against all kinds of model errors.
In this paper, we will first propose a robust FB beamformer, based on a general-rank signal model and a ULA structure; then by introducing a transformation matrix to preprocess the received array data, the original complex-valued optimum weight vector will be reduced to a real-valued one, which reduces computational complexity of the system significantly.
Section snippets
Beamforming based on general-rank signal models
Consider a linear array with sensors, the nth snapshot vector of the received array signals can be expressed aswhere , and are the desired signal, interference and noise vectors, respectively. By applying a set of coefficients wi, to the received array signals , we obtain the beamformer outputwhere denotes the Hermitian transpose operation and is the weight vector.
In the absence of model errors, the optimum solution of
Robust forward backward beamformer
The general-rank signal model in Section 2 is based on an arbitrary array structure. Now we focus on the ULA structure and suppose there are M omnidirectional sensors with an adjacent sensor spacing d and a signal wavelength . We can see that DOA angle mismatch and incoherent scattering of the desired signal do not destroy the centrohermitian structure of , i.e.where is the transpose operation,
Simulations and results
In this part we will examine the performance of the proposed algorithms and the original one in terms of output SINR. The optimum output SINR is given bywith being the correlation matrix of interference plus noise, and being the optimum weight vector.
Our simulations are based on a ULA with 10 elements and an adjacent sensor spacing . There is an interfering signal arriving from the DOA angle and the desired signal is an incoherently
Conclusion
A forward backward-based robust beamformer with real-valued implementation has been proposed for the incoherently scattered general-rank signal model, by introducing a unitary transformation matrix for preprocessing. As a result, computational complexity of the original general-rank robust beamformer is reduced by at least 50%. Moreover, simulation results have shown that the proposed beamformer is less sensitive to the involved parameters and has achieved a higher output SINR at low input SNR
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Multi-channel post-filtering based on spatial coherence measure
2014, Signal ProcessingCitation Excerpt :Further, the proposed spatial coherence measure can be easily extended to multi-rank signal models encompassing incoherently scattered source, etc. Multi-rank signal models or rank relaxation has been widely used in sensor array localization [18–21], beamforming [22–25], or quadratic optimization problems [25,26]. It is more convenient to consider various design requirements than previous methods using microphone array.
Robust Capon beamforming exploiting the second-order noncircularity of signals
2014, Signal ProcessingCitation Excerpt :Compared to the early DL methods, the main advantage of these methods [15–18] is that the corresponding DL factor can be appropriately calculated based on the uncertainty set of the array steering vector. There are many extensions of these methods, for example, the doubly constrained robust Capon beamformer [19], the fully automatic DL beamformer [20], and the beamformers considering interference nonstationarity [21], general-rank signal models [22,23] and partly calibrated sparse sensor arrays [24]. To exploit the SO noncircularity of complex signals, widely linear (WL) beamformers have been developed in [26–29].
Robust beamforming for coherent signals based on the spatial-smoothing technique
2012, Signal ProcessingCitation Excerpt :As a result, the same improved performance is achieved as in the case of (43) and (46), but with much lower computational complexity. As shown in [22], a robust beamformer based on a ULA structure can be implemented with real-valued coefficients for the reception of incoherently-scattered general-rank signals. Following the same approach, a real-valued implementation can also be developed for our proposed beamformers.
Robust Beamforming Based on Complex-Valued Convolutional Neural Networks for Sensor Arrays
2022, IEEE Signal Processing LettersMaximum Entropy-Based Interference-Plus-Noise Covariance Matrix Reconstruction for Robust Adaptive Beamforming
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