17-03-2014, 10:23 PM
Abstract
A new computationally simple MDL approach is addressed
in this paper. Unlike the eigenvalue-based MDL
methods, the proposed method suggests to use the minimum
mean square errors (MMSEs) of the multi-stage Wiener
lter (MSWF) to calculate the description length required
to encode the observed data. As a result, the proposed
method is more robust to the nonuniform noise than the
eigenvalue-based MDL methods. On the other hand, since
the proposed method does not involve the estimation of
the observed covariance matrix and its eigedecomposition,
its computational complexity can be signicantly reduced.
Numerical results are presented to illustrate the consistency
and robustness of the proposed method.
1. Introduction
Computationally efcient and robust methods for source
detection are of signicant interest in practical applications
of array processing [1]-[10]. This is due to the fact that,
on one hand, when a large array is employed to localize
the signals of interest (SOI) in a real-time manner, the
required computational load of the classical methods is quite
heavy. On the other hand, the assumption of spatially and
temporally white noise across such a large array might not
be true because the unknown noise environment may change
slowly with time [11], and the sensor noises thereby become
correlated from sensor to sensor and unequal in power level.
Although the sensor noises may be uncorrelated among all
sensors in many practical applications, their power levels are
in general unequal due to the nonidealities of the practical
arrays, such as the nonideality of the receiving channel,
the nonuniformity of the sensor response and the mutual
coupling between sensors. As a consequence, the sensor
noise becomes a spatially inhomogeneous white process,
i:e:; of unequal power level and uncorrelated from sensor
to sensor. When implemented in such an environment, the
classical model-dependent methods, such as the classical
MDL methods [1], may fail to yield the reliable estimate
of the number of sources in a real-time manner.
While there have been some papers, such as [4]-[8],
dealing with the robust estimation of the number of sources,
these methods need to be further improved in computational
complexity and/or detection performance. The eigenvectorbased
methods, such as [8], can yield the reliable estimate
of the number of sources in the nonuniform noise
environment. Similar to the eigenvalue-based methods [1],
however, the eigenvector-based methods necessarily involve
the estimation of the observed covariance matrix and its
EVD calculation, making them to be quite computationally
intensive. Although the MDL method addressed by
Fishler and Poor [7] is robust against the deviations from
the assumption of spatially and temporally white noise, it
involves N iterations and each iteration needs the EVD
computation, thereby requiring around O(N4) ops besides
the calculation of the covariance matrix, which is rather
computationally burdensome especially when N becomes
large. Recently, we addressed a computationally efcient
Gerschgorin disk estimator for source number without eigendecomposition
(GDEWE) in [6]. The GDEWE method is
more robust and computationally efcient than the classical
methods for source enumeration. Nevertheless, like the
GDE estimator, the detection performance of the GDEWE
estimator also relies on a non-increasing function that needs
to be carefully designed in the practical applications.
In this paper, a novel computationally efcient MDL
method is addressed for the detection of source number.
In contrast to the eigenvalue-based MDL methods, the
proposed mMDL method only involves the MMSEs of the
MSWF [14] to calculate the code length of the observed
data, independent of the eigenvalues of the observed covariance
matrix. As a result, the mMDL method is more
robust to nonuniform noise than the eigenvalue-based MDL
methods. Meanwhile, the proposed method does not involve
the estimation of the observed covariance matrix and its
EVD computation, requiring lower computational cost than
the EVD-based methods, particularly for a large array.
A new computationally simple MDL approach is addressed
in this paper. Unlike the eigenvalue-based MDL
methods, the proposed method suggests to use the minimum
mean square errors (MMSEs) of the multi-stage Wiener
lter (MSWF) to calculate the description length required
to encode the observed data. As a result, the proposed
method is more robust to the nonuniform noise than the
eigenvalue-based MDL methods. On the other hand, since
the proposed method does not involve the estimation of
the observed covariance matrix and its eigedecomposition,
its computational complexity can be signicantly reduced.
Numerical results are presented to illustrate the consistency
and robustness of the proposed method.
1. Introduction
Computationally efcient and robust methods for source
detection are of signicant interest in practical applications
of array processing [1]-[10]. This is due to the fact that,
on one hand, when a large array is employed to localize
the signals of interest (SOI) in a real-time manner, the
required computational load of the classical methods is quite
heavy. On the other hand, the assumption of spatially and
temporally white noise across such a large array might not
be true because the unknown noise environment may change
slowly with time [11], and the sensor noises thereby become
correlated from sensor to sensor and unequal in power level.
Although the sensor noises may be uncorrelated among all
sensors in many practical applications, their power levels are
in general unequal due to the nonidealities of the practical
arrays, such as the nonideality of the receiving channel,
the nonuniformity of the sensor response and the mutual
coupling between sensors. As a consequence, the sensor
noise becomes a spatially inhomogeneous white process,
i:e:; of unequal power level and uncorrelated from sensor
to sensor. When implemented in such an environment, the
classical model-dependent methods, such as the classical
MDL methods [1], may fail to yield the reliable estimate
of the number of sources in a real-time manner.
While there have been some papers, such as [4]-[8],
dealing with the robust estimation of the number of sources,
these methods need to be further improved in computational
complexity and/or detection performance. The eigenvectorbased
methods, such as [8], can yield the reliable estimate
of the number of sources in the nonuniform noise
environment. Similar to the eigenvalue-based methods [1],
however, the eigenvector-based methods necessarily involve
the estimation of the observed covariance matrix and its
EVD calculation, making them to be quite computationally
intensive. Although the MDL method addressed by
Fishler and Poor [7] is robust against the deviations from
the assumption of spatially and temporally white noise, it
involves N iterations and each iteration needs the EVD
computation, thereby requiring around O(N4) ops besides
the calculation of the covariance matrix, which is rather
computationally burdensome especially when N becomes
large. Recently, we addressed a computationally efcient
Gerschgorin disk estimator for source number without eigendecomposition
(GDEWE) in [6]. The GDEWE method is
more robust and computationally efcient than the classical
methods for source enumeration. Nevertheless, like the
GDE estimator, the detection performance of the GDEWE
estimator also relies on a non-increasing function that needs
to be carefully designed in the practical applications.
In this paper, a novel computationally efcient MDL
method is addressed for the detection of source number.
In contrast to the eigenvalue-based MDL methods, the
proposed mMDL method only involves the MMSEs of the
MSWF [14] to calculate the code length of the observed
data, independent of the eigenvalues of the observed covariance
matrix. As a result, the mMDL method is more
robust to nonuniform noise than the eigenvalue-based MDL
methods. Meanwhile, the proposed method does not involve
the estimation of the observed covariance matrix and its
EVD computation, requiring lower computational cost than
the EVD-based methods, particularly for a large array.