PUMA criterion = MODE criterion
aa r X i v : . [ s t a t . O T ] J a n PUMA criterion = MODE criterion
Dave Zachariah, Petre Stoica and Magnus Jansson
Abstract —We show that the recently proposed (enhanced)PUMA estimator for array processing minimizes the same crite-rion function as the well-established MODE estimator. (PUMA =principal-singular-vector utilization for modal analysis, MODE= method of direction estimation.)
I. P
ROBLEM FORMULATION
The standard signal model in array processing is y ( t ) = A ( φ ) s ( t ) + n ( t ) ∈ C m (1)where φ = [ φ · · · φ r ] ⊤ parameterizes the unknown directionsof arrival from r < m far-field sources, s ( t ) is a vector ofunknown source signals, n ( t ) is a noise term, and A ( · ) isa known function describing the array response [1], [2]. Thecovariance matrix of the received signals is R = APA ∗ + σ I m , (2)where P and σ I m are the signal and noise covariances,respectively. The data is assumed to be circular Gaussian.Given T independent snapshots { y ( t ) } Tt =1 , the maximumlikelihood (ML) estimate of φ is given by b φ = arg min φ tr n Π ⊥ A b R o , (3)where b R = 1 T T X t =1 y ( t ) y ∗ ( t ) denotes the sample covariance matrix and Π ⊥ A is the orthog-onal projector onto R ( A ) ⊥ and is a nonlinear function of φ . The nonconvex problem in (3) can be viewed as fittingthe signal subspace spanned by A to the data, and it can betackled using numerical search techniques.When considering uniform linear arrays, the columns of A have a Vandermonde structure: A = · · · e jφ e jφ · · · e jφ r ... ... ... e j ( m − φ e j ( m − φ · · · e j ( m − φ r . In this case we have the following orthogonal relation TA = (4)where T = c c · · · c r . . . . . . . . . c c · · · c r ∈ C ( m − r ) × m This work has been partly supported by the Swedish Research Council(VR) under contracts 621-2014-5874 and 2015-05484. is a Toeplitz matrix with coefficients c = [ c c · · · c r ] ⊤ .These coefficients also define a polynomial with roots that lieon the unit circle, c + c z + · · · + c r z r = c r Y k =1 (1 − e − jφ k z ) , c = 0 . Therefore there is a direct correspondence between φ and c [1], [2]. As a consequence of (4) the orthogonal projector canbe written as Π ⊥ A = Π T = T ∗ ( TT ∗ ) − T which yields an equivalent problem to (3) in terms of c : b c = arg min c V ML ( c ) , (5)where V ML ( c ) = tr n Π T b R o = tr n ( TT ∗ ) − T b RT ∗ o . (6)Using this alternative parameterization, tractable minimizationalgorithms can be formulated. Next, we consider two alterna-tive estimation criteria and prove that they are equivalent.II. PUMA CRITERION EQUALS
MODE
CRITERION
Using the eigendecomposition, the covariance matrix can bewritten as R = U s ΛU ∗ s + σ U n U ∗ n where R ( U s ) = R ( A ) and Λ = diag ( λ , . . . , λ r ) ≻ is thematrix of eigenvalues that are larger than σ . Instead of fittingthe subspace to the sample covariance b R , as in (6), considerfitting to a weighted estimate of the signal subspace [3], [4]: b U s b Γ b U ∗ s , where b Γ , diag (ˆ λ − ˆ σ ) ˆ λ , . . . , (ˆ λ r − ˆ σ ) ˆ λ r ! and where { ˆ λ i } and ˆ σ are obtained from the eigendecompo-sition of b R . Then the cost function in (5) is replaced by V MODE ( c ) = tr n ( TT ∗ ) − T b U s b Γ b U ∗ s T ∗ o . This leads to the asymptotically efficient ‘ m ethod o f d irection e stimation’ (M ODE ) [3] [2, ch. 8.5]. A simple two-step algo-rithm was proposed in [3] to approximate the minimum of theabove estimation criterion.Another approach for array processing, called ‘ p rincipal-singular-vector u tilization for m odal a nalysis’ (P UMA ), hasbeen recently proposed in [5] (see also references therein forpredecessors of that approach). It is motivated by properties of a related linear prediction problem and based on the followingfitting criterion V PUMA ( c ) = e ∗ c We , where c W , ( b Γ ⊗ ( TT ∗ ) − ) is a weighting matrix and e is a function of c and theeigenvectors in b U s . As shown in [5], e can be written as e = vec ( T b U s ) . It follows immediately that V PUMA ( c ) = e ∗ c We = vec( T b U s ) ∗ (cid:16)b Γ ⊗ ( TT ∗ ) − (cid:17) vec( T b U s )= vec( T b U s ) ∗ vec(( TT ∗ ) − T b U s b Γ )= tr n b U ∗ s T ∗ ( TT ∗ ) − T b U s b Γ o = tr n ( TT ∗ ) − T b U s b Γ b U ∗ s T ∗ o = V MODE ( c ) , where we made use of the following results vec( XYZ ) = ( Z ⊤ ⊗ X ) vec( Y ) tr { X ∗ Y } = vec( X ) ∗ vec( Y ) . Therefore the P
UMA criterion is exactly equivalent to theM
ODE criterion. The algorithm proposed in [5] is thus analternative technique for minimizing V MODE ( c ) .III. O THER VARIANTS
A fitting criterion on a similar form as V PUMA ( c ) wasproposed in [6] and shown to reduce to V MODE ( c ) in a specialcase. Alternative minimization techniques are also discussedtherein, see also [2, ch. 8]. See e.g. [7], [8] for additionalvariations of V MODE ( c ) .In scenarios with low signal-to-noise ratio or small samplesize T , subspace-fitting methods such as M ODE may sufferfrom a threshold breakdown effect due to ‘subspace swaps’[9], [10]. To reduce the risk that the signal subspace isfitted to noise in these cases, a modification was proposedin [11] consisting of using p < m − r extra coefficients in c .Then after computing the corresponding directions of arrival,all possible subsets of r directions are compared using themaximum likelihood criterion and the best subset is chosenas the estimate. This method is called the M ODE
X in [11]and its principle is exactly what is used in [5] to propose theEnhanced-P
UMA .Interestingly, while both papers [3] and [11] are referencedin [5], the equivalence (as shown above) of the P
UMA esti-mation criterion proposed there to M
ODE [3] and M
ODE
Xestimation criteria [11] was missed in [5].R
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