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.
| Date of Award | 2 Oct 2013 |
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| Original language | English |
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| Awarding Institution | - City University of Hong Kong
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| Supervisor | Hing Cheung SO (Supervisor) |
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