PUMA criterion = MODE criterion

0.06 MB·

by  Dave Zachariah

1
PUMA criterion = MODE criterion
Dave Zachariah, Petre Stoica and Magnus Jansson
Abstract—We show that the recently proposed (enhanced) is a Toeplitz matrix with coefficients c = [c0 c1··· cr]⊤.
PUMAestimator for array processing minimizesthesame crite- These coefficientsalso define a polynomialwith roots that lie
rionfunctionasthewell-establishedMODEestimator.(PUMA=
on the unit circle,
principal-singular-vector utilization for modal analysis, MODE
= method of direction estimation.) r
c +c z+···+c zr =c (1−e−jφkz), c 6=0.
7 0 1 r 0 0
1 k=1
Y
0 I. PROBLEM FORMULATION
Therefore there is a direct correspondence between φ and c
2
The standard signal model in array processing is [1], [2]. As a consequenceof (4) the orthogonalprojectorcan

an y(t)=A(φ)s(t)+n(t)∈Cm (1) be written as
J whereφ=[φ ···φ ]⊤ parameterizestheunknowndirections Π⊥A =ΠT =T∗(TT∗)−1T
7 1 r
1 of arrival from r < m far-field sources, s(t) is a vector of which yields an equivalent problem to (3) in terms of c:
unknown source signals, n(t) is a noise term, and A(·) is

] a known function describing the array response [1], [2]. The c=argmin VML(c), (5)
T c
covariance matrix of the received signals is
O
where
. R=APA∗+σ2Im, (2) b
at VML(c)=tr ΠTR =tr (TT∗)−1TRT∗ . (6)
t where P and σ2Im are the signal and noise covariances,
s n o n o
respectively. The data is assumed to be circular Gaussian. Usingthisalternativeparameterization,tractableminimization
[ b b
Given T independent snapshots {y(t)}T , the maximum algorithms can be formulated. Next, we consider two alterna-
t=1
1
likelihood (ML) estimate of φ is given by tive estimation criteria and prove that they are equivalent.
v
3 ⊥
φ=argmin tr Π R , (3)
8 A II. PUMA CRITERION EQUALS MODE CRITERION
φ
5 n o
4 where b b Usingtheeigendecomposition,thecovariancematrixcanbe
0 1 T written as
1. R= T y(t)y∗(t) R=UsΛU∗s +σ2UnU∗n
t=1
0 X
7 denotes the sample cbovariance matrix and Π⊥ is the orthog- where R(Us)=R(A) and Λ=diag(λ1,...,λr)≻0 is the
A matrixofeigenvaluesthatarelargerthanσ2.Insteadoffitting
1 onal projector onto R(A)⊥ and is a nonlinear function of
: the subspace to the sample covariance R, as in (6), consider
v φ. The nonconvex problem in (3) can be viewed as fitting
fitting to a weighted estimate of the signal subspace [3], [4]:
i the signal subspace spanned by A to the data, and it can be
X b
tackled using numerical search techniques. U ΓU∗,
r s s
a When consideringuniform linear arrays, the columnsof A
where
have a Vandermonde structure: b bb
1 1 ··· 1 (λˆ −σˆ2)2 (λˆ −σˆ2)2
Γ,diag 1 ,..., r
A= ej.φ1 ej.φ2 ··· ej.φr . λˆ1 λˆr !
. . .
ej(m−.1)φ1 ej(m−. 1)φ2 ··· ej(m−.1)φr and whereb{λˆi} and σˆ2 are obtained from the eigendecompo-
  sition of R. Then the cost function in (5) is replaced by
 
In this case we have the following orthogonal relation
V (c)=tr (TT∗)−1TU ΓU∗T∗ .
bMODE s s
TA=0 (4)
n o
This leads to the asymptoticallyefficient ‘methodof direction
b bb
where
estimation’ (MODE) [3] [2, ch. 8.5]. A simple two-step algo-
c c ··· c rithmwasproposedin[3] toapproximatetheminimumofthe
0 1 r
T=0 ... ... ... 0∈C(m−r)×m above estimation criterion.
Another approach for array processing, called ‘principal-
c c ··· c
 0 1 r singular-vector utilization for modal analysis’ (PUMA), has
 
been recently proposed in [5] (see also references therein for
This work has been partly supported by the Swedish Research Council
(VR)undercontracts 621-2014-5874 and2015-05484. predecessorsofthatapproach).Itismotivatedbypropertiesof
2
arelatedlinearpredictionproblemandbasedonthefollowing [4] M.VibergandB.Ottersten,“Sensorarrayprocessingbasedonsubspace
fitting criterion fitting,” IEEETrans.SignalProcessing,vol.39,no.5,pp.1110–1121,
∗ 1991.
V (c)=e We,
PUMA [5] C.Qian, L.Huang, N.Sidiropoulos, andH.C. So,“Enhanced PUMA
for direction-of-arrival estimation and its performance analysis,” IEEE
where
c TransactionsonSignalProcessing,vol.64,no.16,pp.4127–4137,2016.
[6] M.Jansson,A.L.Swindlehurst, andB.Ottersten, “Weighted subspace
W,(Γ⊗(TT∗)−1) fitting for general array error models,” IEEE Transactions on Signal
Processing,vol.46,no.9,pp.2484–2498, 1998.
is a weighting matrix and e is a function of c and the [7] P.StoicaandM.Jansson,“Onforward–backwardMODEforarraysignal
c b
eigenvectors in U . As shown in [5], e can be written as processing,”DigitalSignalProcessing,vol.7,no.4,pp.239–252,1997.
s [8] M. Kristensson, M. Jansson, and B. Ottersten, “Modified IQML and
e=vec(TUs). It follows immediately that weightedsubspacefittingwithouteigendecomposition,” Signalprocess-
b ing,vol.79,no.1,pp.29–44,1999.
∗
VPUMA(bc)=e We [9] M. Hawkes, A. Nehorai, and P. Stoica, “Performance breakdown of
subspace-based methods: prediction and cure,” in Proc. IEEEInterna-
=vecc(TUs)∗ Γ⊗(TT∗)−1 vec(TUs) tional Conference Acoustics, Speech, andSignal Processing(ICASSP),
vol.6,pp.4005–4008vol.6,2001.
=vec(TU )∗(cid:16)vec((TT∗)−1T(cid:17)U Γ) [10] B.A.Johnson,Y.I.Abramovich,andX.Mestre,“Theroleofsubspace
bs b s b
swap in MUSIC performance breakdown,” in Proc. IEEE Interna-
=tr U∗T∗(TT∗)−1TU Γ tional Conference Acoustics, Speech, andSignal Processing(ICASSP),
sb s b b pp.2473–2476, March2008.
=trn(TT∗)−1TU ΓU∗T∗o [11] A. B. Gershman and P. Stoica, “New MODE-based techniques for
b s bs b direction finding with an improved threshold performance,” Signal
=V n (c), o Processing,vol.76,no.3,pp.221–235, 1999.
MODE
b bb
where we made use of the following results
⊤
vec(XYZ)=(Z ⊗X)vec(Y)
tr{X∗Y}=vec(X)∗vec(Y).
Therefore the PUMA criterion is exactly equivalent to the
MODE criterion. The algorithm proposed in [5] is thus an
alternative technique for minimizing V (c).
MODE
III. OTHERVARIANTS
A fitting criterion on a similar form as V (c) was
PUMA
proposedin [6] andshownto reducetoV (c) in a special
MODE
case. Alternative minimization techniques are also discussed
therein, see also [2, ch. 8]. See e.g. [7], [8] for additional
variations of V (c).
MODE
In scenarios with low signal-to-noise ratio or small sample
size T, subspace-fitting methods such as MODE may suffer
from a threshold breakdown effect due to ‘subspace swaps’
[9], [10]. To reduce the risk that the signal subspace is
fitted to noise in these cases, a modification was proposed
in [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 the
maximum likelihood criterion and the best subset is chosen
as the estimate. This method is called the MODEX in [11]
and its principle is exactly what is used in [5] to propose the
Enhanced-PUMA.
Interestingly,while both papers [3] and [11] are referenced
in [5], the equivalence (as shown above) of the PUMA esti-
mation criterion proposed there to MODE [3] and MODEX
estimation criteria [11] was missed in [5].
REFERENCES
[1] P.StoicaandR.L.Moses,Spectralanalysisofsignals.Pearson/Prentice
Hall,2005.
[2] H.L.VanTrees,Detection,estimation,andmodulationtheory:Optimum
arrayprocessing. JohnWiley&Sons,2004.
[3] P. Stoica and K. C. Sharman, “Maximum likelihood methods for
direction-of-arrival estimation,” IEEE Trans. Acoustics, Speech and
SignalProcessing,vol.38,no.7,pp.1132–1143, 1990.