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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 estimation 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 esti- 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 de- 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 applications are discussed in later sections. The first approach to carrying out space-time processing of data sampled at am 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 analysis to spatio-temporally sampled data. Later. adaptive bearmformers [6, 25, 451 and classical time delay estimation techniques [ 81 were applied to enhance one’s ability to resolve closely spaced signal sources. The spatial filtering approach, however, suffers from fundamental limitations: its performance, 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 matchedfifiltering). 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 process- 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 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 emergence 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 estima- 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- 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 wavefield 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 relevant, 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 rather 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, 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 algorithmic 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 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 somewhat 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 the 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 through 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 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 Maxwell’s equations, which, clearly are only valid for EM waves. In empty space (no current or charge) the following holds aB V X E = - - 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 constants. 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 as the speed of propagation, and for EM-waves in free space it follows from the above derivation c = 1 / & = 3 x 10’m / 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 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 propor- tional to E(r,t). Interestingly enough, any field of the formE(r,t) = A d a ) satisfies Eq. 7, , provided I a1 = 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 l d a1 = e. For the 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 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(0,t) = s ( t ) io t , 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 = a m 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. 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 (Isotropic 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 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 radius 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 medium 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, such 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 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 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 = (xi yi) . Using Eqs, 8 and 9, the field measured at sensor 1 and due to a source at azimuthal DOA 8 is given 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- eled 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 y t X I . Two-dimensional array geometry JULY 1996 IEEE SIGNAL PROCESSING MAGAZINE 69 . Uniform Linear Array geometry (f/B). In the array 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) include 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 steering 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 rz = 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 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 straightforward. As noted earlier, the superposition principle is applicable assuming a linear receiving system. If A4 signals impinge on an L-dimensional array from distinct DOAs 01, ..., 0,w, the output vector takes the form where sm(t), m = 1, ..., A4 denote the baseband signal wave- forms. The output equation can be put 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 < L, which is therefore assumed throughout the development. It is inter- esting to note that in the noiseless case, the array output is then confined to an M-dimensional subspace of the complex L-space, which is spanned by the steering vectors. This is the signal subspace, and this observation forms the basis of subspace-based methods . The sensor outputs are appropriately pre-processed and sampled at arbitrary time instances, labeled t = 1,2, ...,A7 for simplicity. Clearly, the process x(t) can be viewed as a multichannel random process, whose characteristics can be well understood from its first and second order statistics determined by the underlying signals and noise. The pre- processing of the signal is often done in such a way that x(t) can be regarded as temporally white. Assumptions The signal parameters which are of interest in this article are spatial in nature, and thus require the cross-covariance infor- mation among the various sensors, i.e. the spatial covariance matrix R = E{x(t)xH(t)} = AE(s(t)sH(t)}AH + E{n(t)nH(t)) (14) where E( ' } denotes statistical expectation, E { s ( t ) s H ( t ) ) = P is the source covariance matrix and E{n(t)nH(t)} = 021 (16) JULY 1996 70 IEEE SIGNAL PROCESSING MAGAZINE is the noise covariance matrix. The latter covariance structure is a reflection of the noise having a common variance CJ at all sensors and being uncorrelated among all sensors. Such noise is usually termed spatially white, and is a reasonable model, for example receiver noise. However, other man- made noise sources need not result in spatial whiteness, in which case the noisle must be pre-whitened in many of the methods to be described. More specifically, if the noise covariance matrix is Q, the sensor outputs are multiplied by (Q-"' denotes a Hermitian square-root factor of Q-') prior to further processing. The source covariance matrix, P, is often assumed to be nonsingular (a rank-deficient P, as in the case of coherent signals, is discussed later) or near-singu- lar for highly correlated signals. In the later development, the spectral factorization of R will be of central importance, and its positivity guarantees the following representation, 2 R = A P A " + d I =UAUH, (17) with U unitary and A. = diag{ hi, 12, . . ., h ~ ) a diagonal matrix of real eigenvalues ordered such that hi 2 h2 2 . . . 2 h ~ > 0. Observe that 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 signal eigenvectors (corresponding to eigenvalues hi 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 these signal and noise subspaces, are defined as n = u,u: = A ( A ~ A ) - ' 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 a finite data set { ~ ( t ) ] observed overt = 1,2, ...,N. As noted earlier, vie will primarily focus on reviewing a number of techniques based on second-order statistics of data. All of the earlier formulation assumed the existence of exact quantities, i.e. infinite observation time. It is clear that in practice only sample estimates which we denote by a hat , i.e., *, are ;available. A natural estimate of R is the sample covariance matrix 1 N * = I R = - Z x ( t ) x " ( t ) , for which (a spectral repiresentation similar to that of R is defined as A , . , . A A , . R = U,A ,U: +U,AnUf . This representation will1 be extensively used in the descrip- tion and implementation of the subspace-based estimation algorithms. Indeed, if xit) is a stationary white Gaussian process wifh unknown structure (i.e., the data model (13) is not used), then R and its eigen-elements are maximum likelihood estimates of the corresponding exact quantities Throughout the paper, the number of underlying signals, M , in the observed process is considered known. There are, however, good and consiistent techniques for estimating the M signals present [31, 68, 108, 1441 in the event that such information is not available (see also the "Additional Top- ics"). In the following two sections we discuss the best known parameter estimation techniques, respectively classified as Spectral-Based and Parametric methods. Due to space limi- tations, a number of good variations which address specific aspects of the underlying problem and which have appeared in t