Abstract
The problem is to detect a multi-dimensional source transmitting an unknown sequence of complex-valued symbols to a multi-sensor array. In some cases the channel subspace is known, and in others only its dimension is known. Should the unknown transmissions be treated as unknowns in a first-order statistical model, or should they be assigned a prior distribution that is then used to marginalize a first-order model for a second-order statistical model? This question motivates the derivation of subspace detectors for cases where the subspace is known, and for cases where only the dimension of the subspace is known. For three of these four models the GLR detectors are known, and they have been reported in the literature. But the GLR detector for the case of a known subspace and a second-order model for the measurements is derived for the first time in this paper. When the subspace is known, second-order generalized likelihood ratio (GLR) tests outperform first-order GLR tests when the spread of subspace eigenvalues is large, while first-order GLR tests outperform second-order GLR tests when the spread is small. When only the dimension of the subspace is known, second-order GLR tests outperform first-order GLR tests, regardless of the spread of signal subspace eigenvalues. For a dimension-1 source, first-order and second-order statistical models lead to equivalent GLR tests. This is a new finding.
Highlights
Begin with a multisensor array consisting of L elements, and a measurement plan that records N independent measurements in time
Generalized likelihood ratio (GLR) tests based on firstorder statistical models, where an unknown signal is received through a basis for a known subspace have been derived for the past two decades for various signal and noise model assumptions [6]–[10]
For a dimension-1 source, we prove that firstorder and second-order statistical models lead to equivalent generalized likelihood ratio (GLR) tests under both cases, known subspace and unknown subspace of known dimension
Summary
Begin with a multisensor array consisting of L elements, and a measurement plan that records N independent measurements in time. Generalized likelihood ratio (GLR) tests based on firstorder statistical models, where an unknown signal is received through a basis for a known subspace have been derived for the past two decades for various signal and noise model assumptions [6]–[10] These GLR tests have found numerous applications, such as radar and sonar [11], passive radar and source localization [12], and eavesdropper detection [13]. When only the dimension of the channel subspace is known, numerical experiments indicate that the GLR derived from a second-order model outperforms the GLR derived from a first-order model This is consistent with findings reported in [9] for a detector derived from a non-Gaussian prior distribution on the source signals. This column vector x is a stack of columns of the matrix X
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