Source enumeration and prewhitening techniques for high-resolution and robust multidimensional array processing
Student thesis: Doctoral Thesis
Related Research Unit(s)
Estimating the number of sources impinging on an array of sensors, termed as source enumeration, is required prior to the use of parametric methods, e.g., estimation of signal parameters via rotational invariance techniques (ESPRIT) and multiple signal classification (MUSIC), for extraction of the signal parameters in sensor array applications. For source enumeration from a matrix of measurements, whose rows and columns typically correspond to the spatial and temporal dimensions, respectively, is termed as one-dimensional (1-D) source enumeration. Numerous 1-D source enumerators have been developed in the literature, including the classical information theoretic criterion (ITC) based methods, which are optimal when the number of snapshots is much larger than the number of sensors, and the random matrix theory (RMT) algorithm that is designed for relatively small snapshot scenarios. As the first contribution, we investigate the optimal choice of the number of source signals in the threshold region of nonlinear parameter estimators. Due to the low signal-to-noise ratio (SNR) and/or presence of closely-spaced sources, some signal parameters can be accurately estimated while others cannot. By introducing the concept of the effective source number (ESN), which is the number of available accurate parameter estimates, it is proposed to combine a radical source enumerator that tends to overestimate the number of signals with a conservative source enumerator that tends to underestimation for parameter estimation. Such a scheme retains the benefit of the under-enumerators with only accurate estimates while remarkably improves the estimation accuracy. For source enumeration from a multidimensional matrix of measurements, whose dimensions can correspond to the spatial dimensions such as 1-D linear or 2-D planar arrays at the transmitter and/or receiver, as well as time, propagation delay and polarization, much less attention has been paid in the literature compared to 1-D source enumeration. In sensor array applications, often the noise-free signals follows the parallel factor (PARAFAC) model, in which case two approaches to R-D source enumeration can be applied: matrix-based and tensor-based. In matrix-based solutions to R-D source enumeration, the measurement tensor is unfolded into a matrix along individual (e.g., temporal) dimensions, and then the 1-D eigenvalue- or eigenvector/subspace-based detection methods are applied. For eigenvalue-based detection, we generalize the r-mode unfolding of a tensor so that the unfolding along merged dimensions is included. Using the generalized unfolding of the measurement tensor, more unfolded matrices and hence mode eigenvalues are available for use. By optimally applying one or combining more sets of mode eigenvalues in 1-D source enumerators, we devise R-D extensions of the 1-D RMT, 1-D eigenvalue fluctuation information criterion (EFIC) and 1-D minimum description length (MDL). Moreover, using the sequential source enumeration scheme, the identifiable number of signals in R-D RMT/EFIC/MDL reaches up to the size of the most squared" unfolded matrix minus one, which is a significant improvement for R ≥ 3. For eigenvector-based detection, we devise R-D extensions of the subspacebased estimation error (ESTER), which is used for uniform R-D HR in colored noise environments. The matrix-based R-D source enumerators have limited identifiability (identifiable number of signals) due to the unfolding operation. Instead, the CORe CONsistency DIAgnostic (CORCONDIA), which is a tensor-based R-D detection method relying on the computationally expensive alternating least squares (ALS) PARAFAC decomposition, is able to identify more signals. The core consistency, defined as the relative distance of the estimated core and an identity tensor, is compared with a pre-defined threshold for determining the number of signals. One drawback of the CORCONDIA is its low detection probability even at sufficiently high SNRs. To tackle this, it is proposed to utilize the reconstruction error to assist in detecting the number of components. The resultant scheme presents accurate detection at both low and high SNRs. In subspace based R-D parameter estimation, the signal and noise subspaces are utilized for parameters estimation. In the presence of colored noise or interference, estimation of the signal subspace may be seriously affected due to the overlap between the signal and noise subspaces. To improve the performance of signal subspace/ parameter estimation, prewhitening is required prior to the use of subspace based R-D parameter estimators. In electroencephalogram/magnetoencephalogram and multiple-input multiple-output (MIMO) applications, the multidimensional colored noise has a Kronecker correlation structure which for the 2-D case means that the noise covariance matrix in joint spatio-temporal dimensions is equal to the Kronecker product of the spatial and temporal covariance matrices. By exploiting the Kronecker correlation structure of the multidimensional colored noise, the multidimensional prewhitening (MD-PWT) algorithm is developed by removing the noise correlation sequentially along individual dimensions, using the corresponding correlation factors estimated from the noise-only measurements. The MD-PWT employing only a few noise-only snapshots significantly improves the performance of the closedform PARAFAC based parameter estimator (CFP-PE). When noise-only measurements are unavailable, an algorithm for joint estimation of noise and signal parameters and prewhitening is proposed by iteratively alternating MD-PWT and CFP-PE. Moreover, to reduce the algorithm complexity, adaptive convergence thresholds are designed as the stopping conditions such that the iterative algorithm automatically stops at an optimal number of iterations. Simulation results show that the iterative prewhitening scheme performs nearly the same as the MD-PWT utilizing an equal number of noise-only and signal-bearing snapshots, in all scenarios except for a special one of intermediate SNRs and high noise correlation levels.
- Signal processing, Digital techniques, Array processors