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1、ecades of Array Signal Processin Research The Parametric Approach HAMID KRlM and MATS VIBERG OSteven Huntfrhe Image Bank stimation problems in theoretical as well as applied statistics have long been of great research interest E given their importance in a great variety of applica-tions.Parameter es
2、timation has particularly been an area of focus by applied statisticians and engineers as problems required ever improving performance 7,8,91.Many tech-niques were the result of an attempt by researchers to go beyond the classical Fourier-limit.As applications expanded,the interest in accurately est
3、i-mating relevant temporal as well as spatial parameters grew.Sensor array signal processing emerged as an active area of research and was centered on the ability tofuse data collected at several sensors in order to carry out a given estimation task(space-time processing).This framework,as will be d
4、e-scribed in more detail below,uses to advantage prior infor-mation on the data acquisition system(i.e.,array geometry,sensor characteristics,etc.).The methods have proven useful for solving several real-world problems,perhaps most nota-bly source localization in radar and sonar.Other more recent ap
5、plications are discussed in later sections.The first approach to carrying out space-time processing of data sampled at a m array of sensors was spatial filtering or beamforming.The conventional(Bartlett)beamformer dates back to the second world-war,and is a mere application of Fourier-based spectral
6、 analysis to spatio-temporally sampled data.Later.adaptive bearmformers 6,25,451 and classical time delay estimation techniques 81 were applied to enhance ones ability to resolve closely spaced signal sources.The spatial filtering approach,however,suffers from fundamental limitations:its performance
7、,in particular,is directly depend-ent upon the physical size of the array(the aperture),regard-less of the available data collection time and signal-to-noise ratio(SNR).From a statistical point of view,the classical techniques can be seen as spatial extensions of spectral Wie-ner filtering 1501(or m
8、atchedfifiltering).The extension of the time-delay estimation methods to more than one signal(these techniques originally used only two sensors),and the limited resolution of beamforming together with an increasing number of novel applications,renewed interest of researchers in statistical signal pr
9、ocess-ing.We might add at this stage,that the word resolution is used in a rather informal way.It generally refers to the ability to distinguish closely sp,aced signal sources.One typically refers to some spectral-like measure,which would exhibit JULY 1996 IEEE SIGNAL PROCESSING MAGAZINE 1nr.7 F O D
10、 O ,or/b C nnhlnnJrvPr 67 peaks at the locations of the sources.Whenever there are two peaks near two actual emitters,the latter are said to be resolved.However,for parametric techniques,the intuitive notion of resolution is non-trivial to define in precise terms.This in turn,resulted in the emergen
11、ce of the parameter estimation approach as an active research area.Important inspirations for the subsequent effort include the Maximum Entropy(ME)spectral estimation method in geophysics by 23 and early applications of the maximum likelihood prin-ciple 8 1,1061.The introduction of subspace-based es
12、tima-tion techniques 13,1051 marked the beginning of a new era in the sensor array signal processing literature.The subspace-based approach relies on certain geometrical properties of the assumed data model,resulting in a resolution capability which(in theory)is not limited by the array aperture,pro
13、-vided the data collection time and/or SNR are sufficiently large and assuming the data model accurately reflects the experimental scenario.The quintessential goal of sensor array signal processing is the estimation ofparameters by fusing temporal and spatial information,captured via sampling a wave
14、field with a set of judiciously placed antenna sensors.The wavefield is as-sumed to be generated by a finite number of emitters,and contains information about signal parameters characterizing the emitters.Given the great number of existing applications for which the above problem formulation is rele
15、vant,and the number of newly emerging ones,we feel that a review of the area,with the hindsight and perspective provided by time,is in order.The focus is on parameter estimation methods,and many relevant problems are only briefly mentioned.The manuscript is clearly not meant to be exhaustive,but rat
16、her as a broad review of the area,and more importantly as a guide for a first time exposure to an interested reader.We deliber-ately emphasize the relatively more recent subspace-based methods in relation to bearrzforming,for which the reader is referred for more in depth treatment to the excellent,
17、and in some sense complementary,review by Van Veen and Buck-ley 133.For more extended presentations,the reader is referred to textbooks such as SO,52,58,1021.The balance of this article consists of the background material and of the basic problem formulation.Then we introduce spectral-based algorith
18、mic solutions to the signal parameter estimation problem.We contrast these suboptimal solutions to parametric methods.Techniques derived from maximum likelihood principles as well as geometric argu-ments are covered.Later,a number of more specialized research topics are briefly reviewed.Then,we look
19、 at a number of real-world problems for which sensor array proc-essing methods have been applied.We also include an exam-ple with real experimental data involving closely spaced emitters and highly correlated signals,as well as a manufac-turing application example.A studentlpractitioner who is somew
20、hat familiar with the field might read the various sections sequentially.For a first-time exposure,however,it may be best to scan the applications section before the description and somewhat more mathematical treatment of the algorithms are discussed.Background and For In this section,we motivate th
21、e data model assumed through-out this paper,via its derivation from first principles in physics.Statistical assumptions about data collection are stated and basic geometrical properties of the model are reviewed.Wave Propagation Many physical phenomena are either a result of waves propa-gating throu
22、gh a medium(displacement of molecules)or exhibit a wave-like physical manifestation.A wave propaga-tion which may take various forms(with variations depend-ing on the phenomenon and on the medium,e.g.an electro-magnetic(EM)wave in free space or an acoustic wave in a pipe),generally follows from the
23、homogeneous solution of the wave equation.The models of interest in this paper may equally apply to an EM wave as well as to an acoustic wave(e.g.,SONAR).Given that the propagation model is fundamentally the same,we will for analytical expediency,show that it can follow from the solution of Maxwells
24、 equations,which,clearly are only valid for EM waves.In empty space(no current or charge)the following holds aB VXE=-at(3)(4)where.,and x,respectively,denote the“divergence”and“curl.”Further,B is the magnetic induction,E is the electric field,whereas po and EO are the magnetic and dielectric constan
25、ts.Invoking Eq.1 the following curl property results,(5)v x(V x E)=V(V.E)-V*E=-VE.Using Eqs.3 and 4 leads to a a2E VX(VxE)=-(VxB)=-&at o/J o-,at2 which,when combined with Eq.5,yields the fundamental wave equation(7)68 IEEE SIGNAL PROCESSING MAGAZINE JULY 1996 The constant c is generally referred to
26、as the speed of propagation,and for EM-waves in free space it follows from the above derivation c=1/&=3 x 10m/s.The homogene-ous(no forcing function)wave equation(Eq.7)constitutes the physical motivation for our assumed data model.This is regard-less of the type of wave or medium(EM or acoustic).In
27、some applications,the underlying physics are irrelevant,it is merely the mathematical structure of the data model that counts.Though Eq.7 is a vector equation,we only consider one of its components,say E(r,t)where r is the radius vector.It will later be assumed that the measured sensor outputs are p
28、ropor-tional to E(r,t).Interestingly enough,any field of the formE(r,t)=A d a)satisfies Eq.7,provided I a 1 =l/c,with“T denoting transposition.Througlh its dependence on t-rTa only,the solu-tion can be interpreted as a wave traveling in the direction a,with the speed of propagation ld a 1 =e.For the
29、 latter reason,a is referred to as the slowness vector.The chief interest herein is in narrowband(This is not really a restriction,since any signal can be expressed as a linear combination of narrowband com-ponents.)forcing functions.The details of generating such a forcing function(i.e.radiation of
30、 an antenna)can be found in the classic book by Jordan 59.In complex notation(see e.g.63,Section 15.31)and taking the origin as a reference,a narrowband transmitted waveform can be expressed as(upper-case and lowercase Greek letters are to be understood as vectors or matrices within their context)E(
31、0,t)=s(t)iot,where s(t)is slowly time-varying compared to the carrier dot.For Irl c/B,where B is the bandwidth of s(t),we can write In the last equa1it:y the so-called wave-vector k=am was introduced,and its magnitude 1 kl=k=o/c is the wave-number.One can also write k=2z/h,where h is the wave-length
32、.Note that k also points in the direction of propagation.For example,in the xy-plane we have k=k(cos0 sine),(9)where 0 is the direction of propagation,defined counter-clockwise relative the x-axis(Fig.1).It should be noted that Eq.8 implicitly assumed far-field conditions,since an isotropic(Isotropi
33、c refers to uniform propagation/transmission in all directions.)point source gives rise to a spherical traveling wave whose amplitude is inversely proportional to the distance to the source.All points lying on the surface of a sphere of radius R will then share a common phase andl are referred to as
34、 a wavefront.This indicates that the distance between the emitters and the re-ceiving antenna array determines whether the sphericity of the wave should be taken into account.The reader is referred to e.g.,lo,241 for treatments of near field reception.Far-field receiving conditions imply that the ra
35、dius of propaga-tion is so large(compared to the physical size of the array)that a flat pilane of constant phase can be considered,thus resulting in a plane wave as indicated in Eq.8.Though not necessary,the latter will be our assumed working model for convenience of exposition.Note thalt a linear m
36、edium implies the validity of the superposition principle,and thus allows for more than one traveling wave.Equation 8 carries both spatial and temporal information and represents an adequate model for distin-guishing signals with distinct spatio-temporal parameters.These may#come in various forms,su
37、ch as DOA(in general azimuth and elevation),signal polarization(if more than one component of the wave is taken into account),transmitted waveforms,temporal frequency etc.Each emitter is generally associated with a set of such characteristics.The interest in unfolding the signal parameters forms the
38、 essence of sensor array signal processing as presented herein,and continues to be an important and active topic of research.Parametric Data Model Most modern approaches to signal processing are model-based,in the sense that they rely on certain assumptions made on the observed data.In this section
39、we describe the prevail-ing model used in the rernainder of this article.A sensor is represented as a point receiver at given spatial coordinates.T In the 2D-case and as sholwn in Fig.1,we have ri=(x i yi).Using Eqs,8 and 9,the field measured at sensor 1 and due to a source at azimuthal DOA 8 is giv
40、en by If a flat Frequency response,say g),is assumed for the sensor 1 over the signal hndwidth,its measured output will be proportilonal to the field at ri.Dropping the carrier term dot for convenience(in practice,the signal is usually down-converted to baseband before sampling),the output is mod-el
41、ed by Referring to Eq.8,wiz see that Eq.11 requires that the array aperture(i.e.the physical size measured in wave-lengths)be much less than the inverse relative bandwidth I yt X I.Two-dimensional array geometry JULY 1996 IEEE SIGNAL PROCESSING MAGAZINE 69.Uniform Linear Array geometry(f/B).In the a
42、rray processing literature,this is referred to as the narrowband assumption.For an L-element antenna array of arbitrary geometry,the array output vector is obtained as (t)=a(B)s(t).A single signal at the DOA 8,thus results in a scalar multiple of the steering vector(Other popular names for a(0)inclu
43、de action vector,array propagation vector and signal replica vector.)a(B)=U,(),.,U,()as the array output.Common array geometries are depicted in Figs.2 and 3.For the uniform linear array(ULA)we have ri=(I-1)d O),and assuming that all elements have the same directivity gl(0)=,.=g (0)=g(e),the ULA ste
44、ering vector takes the form(cf.(1 1)where d denotes the inter-element distance.The radius vec-tors of the uniform circular array(UCA)have the form r z =R(cos(2n(l-1)lL)sin(2n(l-l)/L)T,from which the form of the UCA steering vector can easily be derived.As previously alluded to,a signal source can be
45、 associated with a number of characteristic parameters.For the sake of clarity and ease of presentation by referring to Figs.2 and 3,we assume that 8 is a real-valued scalar referred to as the DOA.For most of the discussed methods the extension to the multiple parame-ter per source case is straightf
46、orward.As noted earlier,the superposition principle is applicable assuming a linear receiving system.If A 4 signals impinge on an L-dimensional array from distinct DOAs 01,.,0,w,the output vector takes the form where sm(t),m=1,.,A 4 denote the baseband signal wave-forms.The output equation can be pu
47、t in a more compact form by defining a steering matrix and a vector of signal waveforms as Unqorm Circular Array Geometry In the presence of an additive noise n(t)we now get the model commonly used in array processing x(t)=A(B)s(t)+n(t).(13)The methods to be presented all require that M 0.Observe th
48、at any vector orthogonal to A is an eigenvector of R with the eigenvalue C J .There are L-M linearly inde-pendent such vectors.Since the remaining eigenvalues are all larger than 02,we can partition the eigenvaluehector pairs into noise eigenvectors(corresponding to eigenvalues h +i =.=2 h=02)and si
49、gnal eigenvectors(corresponding to eigenvalues h i 2.2 AM CJ).Hence,we can write 2 2 where An=CJ I.Since all noise eigenvectors are orthogonal to A,the columns of Us must span the range space of A whereas those of U,span its orthogonal complement(the nullspace of AH).Tlhe projection operators onto t
50、hese signal and noise subspaces,are defined as n=u,u:=A(AA)-A nL=unug=I-A(A A)-A ,provided that the inverse in the expressions exists.It then follows Problem Definition The problem of central interest herein is that of estimating the DOAs of emitter signals impinging on a receiving array,when given