A Multifamily GLRT for CFAR Detection of Signals in a Union of Subspaces

Testing whether a signal lies within a known subspace is a well-studied problem in the framework of matched-subspace detectors [1]–[2][3][4][5][6]. However, in practice, signals are often generated by multimodal processes so that there is not one but several possible subspace models. Among the set of possible models, the active subspace that generated the observed signal may be unknown a priori. This kind of signals complies with what is referred to as the UoS model. More precisely, x∈RN is an unknown signal belonging to some union of M known subspaces, i.e., x∈⋃Mi=1Si, if and only if there exists i0 such that x∈Si0 [7]. In other words, x belongs to one of the subspace Si, but we do not know a priori to which one. UoS examples include signals with unknown spectral support [7], spectral signatures of radar targets [8] or sparse representations [9].

Signal detection under the UoS model has been partly addressed in the literature in the specific case where all subspaces are distinct, i. e., there are no subspaces in the union such that Si⊂Sℓ, for i≠ℓ [8]–[9][10]. All these works resort to the GLRT and estimate the active subspace index i0 as the one that maximizes the likelihood functions. By assuming distinct subspaces, these methods fail to handle nested models properly. In our UoS context, nesting occurs when the signal can be expressed as the sum of an unknown number of basis vectors so that Si⊂Sℓ, for 1≤i<ℓ≤M. Examples include signals of unknown duration, periodic signal with an unknown number of harmonics, signals transmitted through an unknown multipath channel, etc. In that case, the GLRT always chooses the subspace model with the largest dimension leading to possible detection losses. More generally, problems arise with the GLRT when subspaces are of different dimensions and are not pairwise disjoint.

In [11], the MFLRT has been introduced as a general solution to accommodate nested signal models. The idea is to add a penalty term to the GLR statistic to counteract its tendency to increase with the model order. More recently, a new normalizing transformation has been introduced in [12]. It generalizes the results of [11] and relies on the Legendre transform of the cumulant generating function (LT-CGF) of the test statistic. Although effective for various applications [13], [14], these results are not directly applicable to our context since the GLR statistic involved in our problem does not satisfy the required properties.