Covariance beamforming,covariance matrix tapers and matrix beamforming are related J.Li,P.Stoica and T.Yardibi
It is shown that the covariance beamforming,covariance matrix tapers
and matrix beamforming approaches,which were considered par-
ately from one another in the previous array processing literature,are
in fact related.The relationships between them in terms of both gener-
ality and design procedures are clarified.
Introduction:Matrix beamforming(MB)approaches have been of interest recently becau they offer certain design conveniences[1–4]. In this Letter,we comment on the relationship between MB and the earlier approaches of covariance beamforming(CB)and of covariance matrix tapers(CMT).Despite the fact that the beamforming approaches have been considered parately in the previous literature, we show here that they are all cloly related.
Covariance beamforming(CB):The idea of CB may be traced back to the approach propod in[5]and it
s references.Bad on equations(17) and(30)of[5],we can obtain the following generalid result.Let R denote an estimate of the array covariance matrix.Then the signal power estimate of the conventional delay-and-sum(DAS)beamformer is
s2¼aÃRað1Þwhere(.)Ãdenotes the conjugate transpo(or the complex conjugate for scalars)and a denotes the MÂ1array steering vector with M being the number of nsors in an array.Let R m,m˜denote the(m,m˜)th element of R.Then(1)can be rewritten as:
s2¼
P M
m¼1P M
~m¼1
R m;~m aÃm a~mð2Þ
where a m denotes the m th element of a.CB extends the conventional DAS beamforming approach as follows:
s2 CB ¼
P M
m¼1
P M
~m¼1
U m;~m R m;~m aÃ
m
a~mð3Þ
Here f U m,m˜g denote weighting factors that are to be designed by the ur; however,there is little discussion in[5]or in subquent works as to how to choo f U m,m˜g.
Let denote the Hadamard(element-wi)matrix product operator. Then
U m;~m R m;~m¼½U R m;~mð4Þwhere[.]m,m˜denotes the(m,m˜)th element of the matrix.Therefore,(3) can be rewritten as:
s2
CB
¼aÃðU RÞað5Þ
This expression of s CB2makes the connection with CMT more trans-parent(e following Section).
Covariance matrix taper(CMT):Interestingly,even though CMT was introduced independently of CB,CMT us the tapered covariance matrix U R,instead of R,like CB does[6–11].However,there are also differences.In particular,the works on CMT,unlike the pre-vious ones on CB,contain more preci guidelines for choosing the taper matrix U:for instance,U can be lected to broaden the nulls or peaks of the beampattern in adaptive array processing[6–11];further-more,U must be a positive midefinite matrix to guarantee that U R is so(this constraint on U,which follows from well-known properties of the Hadamard matrix product,was mentioned for thefirst time in the context of CMT in[9]).Also,unlike CB which us a DAS(data-inde-pendent)beamformer for power estimation,CMT makes u of a Capon (data-dependent)beamformer[12,13]:
w¼
ðU RÞÀ1a
aÃðU RÞÀ1a
ð6Þ
which is ud to calculate the signal power estimate as follows:
s2
CMT
¼wÃRwð7Þ
Matrix beamforming(MB):In MB[1–4],the signal power is estimated
as follows:
s2
MB
¼trðWRÞ¼
P M
m¼1
P Mproblem
~m¼1
W m;~m R m;~mð8Þ
where tr(.)denotes the trace.Becau for(3),
s2
CB
¼ðs2
CB
Þü
P M
m¼1
51voa
P M
~m¼1
UÃm;~m RÃm;~m a m aÃ~m
¼
P M
m¼1
P M
~
m¼1
ðU m;~m aÃ
m
a~mÞÃR~m;mð9Þ
where we have ud the fact that R is R m,m˜¼RÃm,m˜)it is
obvious that we can choo W m,m˜in(8)to get(3)and U m,m˜in(3)to get
(8).It follows that the CB class of power estimates is neither smaller nor
larger than that of MB estimates.However,while there are few clear
guidelines for choosing U in CB;the cited works on MB propo a
host of methods for lecting W in(8)(either in a data-independent or
a data-dependent manner).Regarding CMT,it is apparent from(7)
and(8)that CMT is a special ca of MB(and hence of CB)correspond-
ing to the choice W¼wwÃ.Additionally,as already indicated in the
preceding Section,CMT us the same type of tapered covariance
matrix as CB.
Concluding remarks:Recent rearch on MB([1–4])has
failed to recogni the connection of MB to the earlier approaches of
[5])and of CMT([6–11]),and even the latter approaches
were considered parately from one another in the previous literature.
In this Letter we have tried to correct this situation by putting all
the approaches in the right perspective.We have shown that,regarding
generality,MB and CB are equivalent in the n that the class of
totalrecallpower estimates that can be obtained by varying the ur parameters
of CB and,respectively,MB coincide with one another,whereas
letalone
CMT leads to a more restricted class of power estimates.We have
also noted the interesting fact that CMT makes u of the same type
武汉商务英语培训班of tapered covariance matrix as CB.In what concerns guidelines for
choosing the ur parameters of the three approaches to beamforming,
we have explained that the earlier work on CB([5])has only
touched upon this topic,whereas there are clearer ideas about how to
perform this choice for CMT(e[6–11])and very clear(optimal,in
a specified n)lection procedures for MB(e[1–4]).In line
autumn怎么读with the comments,therefore,our preferred approach is MB which
posss both rality orflexibility)and
optimal design procedures),as neither power without control nor
control without power is appealing.
Note:Opinions,interpretations,conclusions,and recommendations in
this Letter are tho of the authors and are not necessarily endord
cansitive
by the United States Government.
Acknowledgments:This work was supported in part by the National
Science Foundation under grant no.ECCS-0729727,grant no.
CCF-0634786,and the Swedish Science Council(VR).
#The Institution of Engineering and Technology2008
28October2007
Electronics Letters online no:20083103
doi:10.1049/el:20083103
directions是什么意思
J.Li and T.Yardibi(Department of Electrical and Computer
Engineering,University of Florida,P.O.Box116130,Gainesville,
FL32611,USA)
E-mail:ytarik@dsp.ufl.edu
P.Stoica(Department of Information Technology,Uppsala University,
Uppsala,Sweden)
ELECTRONICS LETTERS28th February2008Vol.44No.5
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bleeding love什么意思
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英语拼写
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ELECTRONICS LETTERS28th February2008Vol.44No.5
