• Uncertainty in array elements’ beampatterns and positions.

• Mutual coupling.

• Mounting platform reflections.

• Cross-polarization effects.

• Departures from narrowband, far-field and point-source assumptions.

• Errors introduced by the receiver front-end architecture.

• Effects of nonlinear elements.

3.19.3.1 Mutual coupling

Mutual coupling (also known as cross-talk) refers to interactions among array elements. The signal received by an element affects the signals received by the other array elements, and similarly for signal transmission [28, Chapter 2]. Typically, mutual coupling is inversely proportional to the inter-element spacing as well as isolation among array elements. Mutual coupling distorts the elements’ radiation patterns and decreases the efficiency of sensor arrays. Mobile wireless terminals equipped with antenna arrays are a typical example where mutual coupling is significant since the whole chassis, along with its other components, can be considered part of the antenna [5]. Array processing algorithms typically experience a significant loss of performance when the employed array steering vector model does not account for mutual coupling [6,29].

The steering vector of a sensor array subject to mutual coupling is typically modeled as [6]

(19.11)

where denotes the mutual coupling matrix and is known as nominal array steering vector. The element describes the contribution of the nth array element to the output of the mth sensor. Nominal sensor arrays are typically considered to be ideal uniform arrays with regular geometries of the form (16.3), and the motivation for their use is based on the assumption that real-world arrays may be described as a perturbation from an ideal sensor array. For example, if the nominal array is assumed to be an ideal UCA the mutual coupling matrix takes the form of a circulant matrix [30]. Note that a diagonal matrix in (19.11) may also describe errors due to the receiver front-end such as imbalance in the I/Q channels. This is also discussed in Section 3.19.3.5.

In practice, the mutual coupling matrix needs to be determined from array calibration measurements and a nominal array steering vector should be specified by the practitioner. Typically, the mutual coupling matrix is obtained from the measured network parameters of the sensor array, including scattering and transmission coefficients, which may be a rather tedious task [31,32, Chapter 2]. Moreover, determining the nominal array steering vector is typically based on trial and error, and rely on visual inspection of the real-world sensor array.

3.19.3.2 Uncertainty in array elements’ beampatterns and positions

Often, elements’ beampatterns and positions in real-world arrays are not fully known. This may be caused by normal variability in the manufacturing process in the sense that each array element has an individual beampattern or may suffer from manufacturing errors. In fact, elements’ phase centers in real-world arrays do not typically correspond to their physical locations due to interactions with other array elements and mounting platform. Misspecification of the array element’s beampatterns and phase centers leads to loss of performance in array processing techniques.

A commonly employed model taking into account individual beampatterns and position errors is

(19.12)

where , and denote the error in the nth element’s position, with respect to the nominal array steering vector, and corresponding directional beampattern. Parameters may be estimated from calibration measurements taken in controlled environments while the gain function may be measured at a discrete set of points and interpolated using appropriate basis functions such as splines [6]. The latter approach leads to a technique known as array interpolation and it is described in Section 3.19.6.

Alternatively, one may specify a parametric model for (i.e., functionally dependent on ) by trial and error, and visual inspection. For example, in the case of electrically short (relative to the wavelength) x-oriented dipoles one can use the following approximation . However, in the general case of electrically large antennas and patch elements, specifying a parametric model for may be very challenging. An example of two gain functions of a real-world array is illustrated in Figure 19.2b.

3.19.3.3 Cross-polarization effects

Cross-polarization effects refer to “leakage” that, for example, a vertically polarized element suffers from an horizontally polarized wavefield. They are typically characterized by the cross-polarization discrimination (XPD), denoting the ratio between the power received by an antenna due to co-polarized and cross-polarized wavefields. The ratio between the powers received in different polarizations is commonly expressed in dB scale. XPD defines quantitatively how well the two received channels that use different polarization orientations are isolated. Antennas with a large XPD are essentially insensitive to cross-polarization effects and the power received from cross-polarized wavefields may be neglected. When mounted on an array the antennas’ XPD may change significantly due to complex EM interactions among the array elements, scatterers, and mounting platform. In such cases, high-resolution DoA estimators that do not take cross-polarization effects into account typically lead to estimates that may contain significant bias and excess variance [33].

The steering vector of a sensor array that is subject to cross-polarization effects may be described as

(19.13)

where and denote the array responses due to a co-polarized and cross-polarized wavefields, respectively. Moreover, define the polarization of the wavefield. For example, for a co-polarized wavefield Eq. (19.13) simplifies to while for a cross-polarized wavefield we have .

Parametric modeling of and in (19.13) may now be done by employing models (19.11) and (19.12). However, specifying a nominal array steering vector model for the cross-polarized component is even more challenging than that of the co-polarized component, where one may approximate the element’s gain function as . Typically, visual inspection does not help much in determining a parametric model for . An example of gain functions corresponding to an horizontally and vertically polarized wavefield is illustrated in Figure 19.3.

3.19.3.4 Departures from narrowband assumption

The narrowband signal model commonly used in array processing (see Section 3.19.2) assumes that the time-bandwidth product is “small,” i.e.,

(19.14)

where and denote the bandwidth of the transmitted signal and the wavefront’s propagation delay across the array aperture, respectively. A rule of thumb for considering a signal narrowband is [34]. In practice, the time-bandwidth product may be such that the narrowband assumption no longer holds true. This is also the case in focusing-based wideband array processing, where signals’ bandwidth is divided into a set of narrowband channels and narrowband processing is applied to each narrowband bin or to a focused covariance matrix [35]. Alternatively, genuine space-time signal processing can be employed as in STAP radar systems [36,37].

Assuming an array with a flat frequency response (with linear phase) over the signals’ bandwidth, the array covariance matrix may be modeled as [34,38]

(19.15)

where denotes the element-wise Hadamard-Schur product. Moreover, denotes the array steering vector with a linear phase response over frequency and contains the correlation of the pth signal among the array elements. For signals with small time-bandwidth product equals a matrix of ones and (19.15) reduces to (19.4). However, when is non-negligible the rank of due to a single signal is larger than one, and the low-rank structure of the array covariance matrix in (19.4) is lost. Thus, high-resolution subspace methods may not be applicable anymore [34]. The influence of non-negligible time-bandwidth product on DoA estimators and beamforming techniques has been addressed in [34,38]. In most cases, the error due to non-negligible time-bandwidth product can be neglected when compared to finite sample effects. However, when sources are closely spaced or have large difference in power, such an error may be significant.

3.19.3.5 Errors due to receiver front-end architectures

Receiver architectures are commonly classified as superheterodyne, low-IF (intermediate frequency) or direct-conversion receivers. Each front-end architecture is subject to various nonidealities such as I/Q-imbalance, DC-offset, and interfering image frequencies that impact the performance of array processors.

Superheterodyne receivers typically consist of two or more IF stages in order to convert the radio-frequency (RF) signal to baseband. They require image rejection filters at each downconversion stage, which may be difficult to integrate on-chip with other components. Power consumption and size may be significant [39]. Low-IF receivers convert the RF signal to baseband in two IF stages, similarly to the superheterodyne receiver. The need for image rejection filters is overcome by the use of two mixers, one for each I/Q channels, in order to cancel the image. In practice, gain and phase imbalances in the I/Q channels limit the effectiveness of such image cancellation approach. Direct-conversion (or zero-IF) receivers downconvert the RF signal to baseband in a single stage. There is no need for image rejection filters nor image cancellation. However, they typically suffer from DC offset, and I/Q imbalances in the demodulation process are still present.

I/Q imbalances in the demodulation process are common to all of the aforementioned receivers and have been studied in the context of array signal processing in [40]. They appear as phase and amplitude distortions in the I and Q branches of the demodulated signal. Similarly to errors caused by departures from narrowband assumption, the rank of the covariance matrix due to received signals increases by two and the noise eigenvalues are no longer identical. The low-rank structure of the array covariance matrix may be lost and subspace-based array processing methods experience a performance degradation.

One should note that I/Q imbalances may be significantly mitigated by using advanced digital downconverters, either at intermediate or radio frequencies. The commonly used low-rank model is then a good approximation of the array covariance matrix, given that a sufficient number of quantization bits are used [41]. Power consumption and high cost of ADCs operating at GHz and large bandwidths may be a limiting factor in practice.

We also note that antenna arrays using a single receiver, known as switched or time division multiplexing receivers, are often employed in practice due to their low-power, cost, and size [41,42]. Typically, switched array receivers suffer from phase errors that need to be estimated and taken into account by array processing methods.

3.19.3.6 Effects of nonlinear elements

Linearity of the sensor array is among the most common assumptions in the array processing literature. Linearity, in the sense of superposition principle, essentially means that the array output due to a propagating wavefield generated by multiple sources equals the sum of array outputs due to a wavefield generated by each source separately. Most sensor arrays can be considered linear systems and the superposition principle may then be employed [32]. However, active elements such as low-noise amplifiers (LNAs) commonly used in receiver architectures are nonlinear systems and the superposition principle at the array output (after the RF front-end) may no longer hold true [43]. Typically, LNAs trade-off linearity for gain and noise figures. Some waveforms that have large peak to average power ratio such as OFDM (orthogonal frequency-division multiplexing) are particularly sensitive to nonlinearities of power amplifiers. Digital signal processing techniques for mitigating nonlinear effects due to RF front-end include pre- as well as post-distortion techniques [44].

3.19.5.1 Deterministic approach

Consider an N-element planar array lying in the xy-plane and denote the uncertainty in the array elements’ positions by . In this case, the array is not assumed to be subject to other nonidealities such as cross-polarization effects or individual beampatterns. We may estimate the sensors’ misplacements from array calibration measurements as [6]

(19.16)

where denotes a matrix composed of a collection of array steering vectors describing in a closed-form, such as in (19.12), and denotes the Frobenius norm of a matrix. In some cases, a closed-form solution to (19.16) may be obtained. For example, using the mutual coupling matrix in place of in (19.16) yields the following solution:

(19.17)

where we have used the array model (19.11) and assumed that has full row-rank. Notation in (19.17) denotes the Moore-Penrose pseudo-inverse of a matrix. Similarly, by letting denote angle-independent gain and phase errors such as , we obtain the following solution:

(19.18)

where and denote the nth row of and , respectively. In case one is dealing with electrically large uniform linear arrays, the mutual coupling matrix may be approximated by a banded matrix. Recall that mutual coupling is typically inversely proportional to inter-element spacing thus, such an effect may be negligible among sensors located at both ends of the linear array. A least-squares estimator to such a structured mutual coupling matrix may also be found in a closed-form. Computationally efficient solutions may employ appropriate LU-factorization and back-substitution methods [46, Chapter 4]. A summary of model-driven calibration is given in Table 19.1.

Alternatively, we may jointly estimate array nonidealities and wavefield parameters from the output of the sensor array [11–13]. For example, joint estimation of the nonidealities and azimuth angles of P sources generating a propagating wavefield may be accomplished by employing the following nonlinear least-squares estimator [12]

(19.19)

Typically, criterion in (19.19) is minimized in an alternating manner between the array nonidealities and the DoAs [47]. Since (19.19) is highly nonlinear, and potentially with multiple local minima, the minimization should be initialized with “good enough” initial values so that the global minimum can be attained. Note that the array nonidealities are typically nuisance parameters. The statistical performance of the wavefield parameter estimates may suffer due to the higher dimension of the parametric model as well as estimation errors in the nuisance parameters.

The main issue with auto-calibration techniques is that of parameter identifiability [6]. In general, both and cannot be uniquely estimated unless a nonlinear sensor array is employed and additional assumptions regarding sensors’ locations as well as DoAs are made [12,13]. For example, if the array orientation is unknown the DoAs may not be uniquely estimated since they represent the angles relative to the orientation of the sensor array. Alternatively, one may assume that the array nonidealities are random parameters with a known prior distribution, and employ Bayesian estimators. This is discussed next.

3.19.5.2 Bayesian approach

In the previous section we have seen that the identifiability problem in auto-calibration techniques could be alleviated by making additional assumptions regarding the nonidealities or wavefield parameters. One such an assumption considers the array nonidealities to be random parameters with a known prior distribution. In addition to alleviating the identifiability problem, such an assumption allows one (at least in principle) to “integrate out” the array nonidealities and focus on the wavefield parameters, instead [24]. For example, in mass production of sensor arrays one could model array elements’ misplacements due to manufacturing errors as bivariate Gaussian distributed and proceed with Bayesian type of estimators for the wavefield parameters [7,14].

One such an estimator is the generalized weighted subspace fitting (GWSF) algorithm proposed in [7]. It extends the MODE [48] and WSF [49] by taking into account prior information (first and second moments) of the array nonidealities in an optimal manner. It provides asymptotically efficient estimates provided the assumption on the prior Gaussian distribution is valid and array parameterization is known. The DoA estimates obtained by the GWSF are given by [7]

(19.20)

where and denotes a projection matrix onto the orthogonal complement of

(19.21)

Furthermore, in (19.20) denotes a (positive-definite) weighting matrix that ensures asymptotically minimum variance unbiased estimates and the superscript denotes complex-conjugate. The first and second-order moments of the array nonidealities enter criterion (19.20) through ; see [7] for details. Criterion (19.20) is an asymptotic approximation of the maximum a posteriori estimator for simultaneous estimation of array and wavefield parameters [14]. It may be implemented by means of polynomial rooting techniques when the nominal array steering vector has a form similar to that of an ideal ULA. A summary of auto-calibration techniques is given in Table 19.2.

The main difficulty with the Bayesian approach is related to the well-known problem of choosing appropriate prior distributions.¹ Assumed prior knowledge may not exist or it may be difficult to express in the form of a pdf. Moreover, specifying a parametric model for the array nonidealities may be very challenging in practice, similarly to the deterministic approach. In case of uncertainties in the array elements’ locations, the nominal steering vector model may be obtained by visual inspection of the real-world sensor array [7]. However, when considering cross-polarization effects, sensors with individual beampatterns and mutual coupling, such a procedure is of little help in determining a nominal array steering vector.

In practice, array calibration measurements may be necessary even in auto-calibration techniques. Hence, it may be worth considering alternative techniques for dealing with array nonidealities that assume array calibration measurements but do not suffer from the difficulties in specifying explicit formulations for the nonidealities. This is discussed in the next section.

3.19.6.1 Local interpolation of the array calibration matrix

The columns of the array calibration matrix discussed in Section 3.19.4 describe the array response, with the combined effects due to array nonidealities, to a set of angles. The angular grid employed in array calibration measurements is typically sparse due to time and cost limitations. Hence, optimal array processing methods using the array calibration matrix may loose their high-resolution properties and suffer from SOI cancellation effects. Perhaps the most intuitive approach to overcome such a limitation consists in interpolating the array calibration matrix using local basis functions such as splines.

Let denote a vector composed of local basis functions such as splines or other polynomial functions. The practitioner should choose local basis functions with desirable properties such as smoothness, differentiability, and minimum energy. Also, let denote a coefficient matrix obtained by interpolating the array calibration matrix over an angular sector using . may correspond to two or more columns of . Using local basis as the interpolating functions leads to the following piece-wise estimate of the real-world array steering vector:

(19.22)

Alternatively, one may use a nominal array steering vector in place of , and use a local basis expansion for the coefficients matrices in (19.22), instead. More precisely, we may have [6]:

(19.23)

where the nth matrix is modeled as

(19.24)

Here, denotes a local coefficients matrix; see [6] and references therein for details. The rationale for (19.23) is that is typically a smoother function of the angles that the array response, thus allowing for sparser calibration grids than those employed in (19.22). A summary of local interpolation technique is given in Table 19.3.

Expressions (19.22) and (19.23) may be useful in cases when the sensor array is deployed on an environment where the sources are known to be confined to an angular sector. If this is not the case, and the sources may span the whole angular region, using local interpolation techniques may lead to a significant increase on the computationally complexity of array processing methods and may compromise the convergence rate of gradient-based optimization techniques. For example, using (19.22) with the root-MUSIC algorithm requires finding the roots of n different polynomials while the maximum step-size of gradient-based methods is limited by the size of each angular sector . Hence, wavefield modeling and manifold separation, discussed later in this section, are generally preferred when a sensor array is deployed on an environment where the sources may span the whole angular region.

3.19.6.2 Array interpolation technique

The array interpolation technique was originally proposed in [15] and further studied in, e.g., [16,17,50]. The idea is to linearly transform the real-world array so that its response approximates that of a specified ideal array, such as an ULA, known as virtual array. The steering vector model of the virtual array needs to be specified by the designer, and it is typically based on some array processing technique. For example, if the virtual array is that of an ULA, one may employ polynomial rooting techniques for DoA estimation. We note that for UCAs a technique called Beamspace transform may be also employed [51]. However, we do not consider Beamspace transform in this chapter due to the restriction on the array geometry; see [51] and references therein.

Let denote the calibration measurement matrix of the real-world array and a collection of steering vectors of the virtual array. In its simplest form, the array interpolation technique consists in determining the transformation matrix that minimizes the following quadratic error:

(19.25)

The solution to (19.25) is well-known to be

(19.26)

Given the output of the real-world array , the output of the virtual array and its sample covariance matrix are found as and , respectively. Array processing techniques may then be developed for the virtual array and be employed to real-world arrays without explicitly modeling their nonidealities. We note that the virtual array should be designed so that both and are of full column-rank. In case the condition does not lead to a full-rank virtual array covariance matrix, the virtual array should be re-designed. A summary of array interpolation techniques is given in Table 19.4.

The array interpolation technique has two important drawbacks. First, the virtual array, including its configuration, orientation, number of elements, and inter-element spacing, needs to be specified by the designer. Even though this offers some versatility for employing low-complexity DoA estimators or beamforming techniques with arbitrary array configurations (e.g., root-MUSIC algorithm using virtual ULAs), designing virtual arrays is always a heuristic and subjective task. For example, suppose one wants to establish a performance bound such as the widely used Cramér-Rao lower Bound (CBR) for a specific real-world array [25]. In order to take into account the array nonidealities it would be appealing to use array calibration measurements and array interpolation techniques for guaranteeing the tightness of such a bound. However, the resulting CRB depends on the choice of the user-specified virtual array configuration and its parameterization, even though the true physical array remains unmodified. Typically, the specified virtual array employed in array interpolation does not provide insight into the achievable performance by an array built in practice.

Second, the quadratic error in the mapping (19.25) is typically very large if one considers the whole range of angles at once, i.e., . In order to reduce such an error, array interpolation technique typically proceed by dividing the visible region of the real-world array into angular sectors and optimizing a transformation matrix for each sector. In case the sensor array is deployed on an environment where the sources are known a priori to be confined to an angular sector such a requirement of array interpolation techniques is not a serious limitation. However, in environments where the sources may span the whole angular region, array interpolation technique require sector-by-sector processing, which is known to be sensitivity to out-of-sector sources [52]. Moreover, it may also need a prohibitively large number of sectors in azimuth and elevation processing.

Array interpolation techniques are typically more flexible than local interpolation of the array calibration matrix. For example, one cannot (in general) employ the ESPRIT algorithm with arbitrary array configurations using (19.22) simply by choosing a “shift-invariant” local basis vector. Typically, the approximation is not shift-invariant. On the other hand, array interpolation techniques requires more design parameters than local interpolation methods. Finally, array interpolation techniques, and to some extent local interpolation methods, typically interpolate exactly all of the measured data including calibration measurement noise. Next, we show that wavefield modeling and manifold separation can be formulated in a model fitting approach in order to minimize the contribution of calibration measurement noise.

3.19.6.3 Wavefield modeling principle and manifold separation technique

The wavefield modeling principle was proposed in the seminal work of Doron and Doron [20–22]. It has been further studied and applied to high-resolution direction finding in [18,19], and extended to vector-fields such as completely polarized electromagnetic wavefields in [53].

Let us first recall some results regarding propagating wavefields and wave equation. For the sake of clarity, we consider scalar-fields such as acoustic pressure fields, narrowband signals, and drop the carrier term . The extension to completely polarized EM wavefields is briefly described in Section 3.19.8. Let represent a (scalar) wavefield propagating in the xy-plane and denote a point in 2-D Euclidean space. The propagating wavefield takes the form of in the case of P far-field point sources or in the case of a (far-field) spatially distributed source. denotes a density function and is known as the direction vector since it is a function of . Spatially distributed sources may be caused by scattering nearby the transmitter. Wavefields of time-harmonic nature, i.e., that have a representation in terms of Fourier integral, may be written as , where and denote an orthogonal set of spatial basis functions and the coefficients of the expansion, respectively. An important outcome of such an expansion is that the coefficients uniquely describe the spatial characteristics of the propagating wavefield, such as the DoAs of the sources generating the wavefield, in addition to the transmitted signals. For example, by letting denote circular wave functions, the mth wavefield coefficient is given by for a spatially distributed source and for P point-sources.

Let us now assume that the employed real-world array satisfies the superposition principle (see Section 3.19.3). Then, the wavefield modeling principle shows that the array output, at a given frequency, is a linear function of the wavefield coefficients . In particular, after discretization the narrowband array output in (19.1) can be written as

(19.27a)

(19.27b)

where denotes the so-called array sampling matrix and contains the (discretized) wavefield coefficients . Recall that, due to the circular wave basis function employed by the spatial decomposition of the propagating wavefield, in (19.27b) is a Vandermonde vector composed of Fourier basis. Hence, is called basis functions vector. In case of a spatially distributed source, the sum in (19.27a) is replaced by an integral over the angles, and weighted by . The number of coefficients employed in (19.27a and 19.27b) is denoted by . Exact equality in (19.27a and 19.27b) is achieved with but in practice a (very) accurate approximation of the array output can be obtained with a relatively small (see Figure 19.7).

The result in (19.27a and 19.27b) shows that the (noise-free) array output can be decomposed into two parts. One, represented by the array sampling matrix, characterizes the employed sensor array and it is independent of the wavefield. The second part, represented by the basis functions vector, characterizes the propagating wavefield and it is independent of the employed sensor array. In fact, a corollary of the wavefield modeling principle shows that the array steering vector may be decomposed as

(19.28)

The result in (19.28) is known as manifold separation technique [18] and reveals an interesting interpretation for the array sampling matrix. It represents the spatial Fourier spectrum of the array steering vector

(19.29)

where each row of the array sampling matrix contains the spatial Fourier coefficients of each array element. Hence, the array output (at each frequency) can be seen as the product between the spatial Fourier spectrum of the array steering vector and that of the propagating wavefield.

The array sampling matrix fully and uniquely characterizes a given sensor array since it contains the coefficients of an orthogonal spectral decomposition of the corresponding array steering vector. For example, contains information about the array configuration, sensors beampatterns (gain and phase response), mutual coupling, cross-polarization effects, mounting platform reflections, etc. In short, it contains all the effects that can be represented by the array steering vector . Closed-form expressions for the array sampling matrix for some ideal sensor arrays can be found in [20]. However, may also be estimated in a non-parametric manner from array calibration measurements, without explicit formulations for the array nonidealities. In particular, the estimated array sampling matrix obtained as

(19.30a)

(19.30b)

is known as effective aperture distribution function (EADF) [18,33]. In (19.30), denotes the unitary discrete Fourier transform (DFT) matrix and a selection matrix that optimally trades-off between complexity and accuracy of the resulting array steering vector model in (19.28). may be estimated using state-of-the-art model order estimators [54].

We have mentioned that exact equality in (19.27b, 19.27a) and (19.28) requires that is composed of infinitely many columns.² One may feel that such a requirement makes the wavefield modeling principle, or manifold separation technique, of theoretical interest only. The crucial property of the array sampling matrix that makes both wavefield modeling principle and manifold separation technique practical is known as superexponential decay. In particular, the magnitude of the columns of the sampling matrix, , decay faster than exponential (i.e., superexponential) as beyond . and r denote the angular wavenumber and radius of the smallest sphere enclosing the array structure (centered at the assumed coordinate system), respectively.

In practice, the superexponential property tells us that (19.27b, 19.27a) and (19.28) are (very) well approximated by a few columns of the array sampling matrix since the norm-convergence rate of the expansion in (19.28) is faster than exponential [18,20]; see Figure 19.7. The superexponential property may be understood by interpreting sensor arrays as spatial filters. In fact, may be seen as the array’s spatial frequency response, where the passband, stopband, and cutoff frequencies are given by , and , respectively. For example, sensor arrays with large aperture have increased resolution since they sample the propagating wavefield over a large area. This is reflected on the array sampling matrix by an increase of the passband (r increases). This leads to an increase of the amount of energy received from such a wavefield since a large number of wavefield coefficients are taken into account by the employed sensor array.

In the following, the main concepts and properties of wavefield modeling principle and manifold separation technique are illustrated using an ideal ULA. We emphasize that using ULAs does not limit the generality of the discussion. We employ it for the sake of clarity here. Let us consider the 5-element ULA from Figure 19.5, where the smallest sphere (a circle in this case) enclosing the array structure is depicted as well. Figure 19.6 illustrates three rows of the array sampling matrix, i.e., the spatial Fourier coefficients of the first three array elements. The ideal ULA is composed of omnidirectional elements with an inter-element spacing of . In Figure 19.7a, the norm of each column of the array sampling matrix, , is illustrated for two ideal ULAs with inter-element spacings of and . Finally, Figure 19.7b, illustrates the following (average) squared-residual:

(19.31)

as a function of , the number of columns of .

The simulation results show that the “passband” of the array sampling matrix increases for large apertures.³ In the limiting case of an infinitely small aperture, such as the omnidirectional array element located at the center of the coordinate system in Figure 19.5, the array sampling matrix is finite (see Figure 19.6). This is because only the magnitude response of such an array needs to be modeled (the first spatial harmonic in the case of omnidirectional elements) since the relative phase across such an aperture is zero.

A summary of the wavefield modeling principle/manifold separation technique is given in Table 19.5, where we assume that array calibration measurements are taken over the whole angular region, i.e., . We emphasize that such an assumption does not limit the generality of the wavefield modeling principle/manifold separation technique since it is simply related with the choice of orthogonal basis functions , employed for decomposing the propagating wavefield. More precisely, wavefield modeling principle/manifold separation technique are also applicable when only partial array calibration measurements are acquired, or when the sources are known a priori to be confined to an angular sector. The superexponential property of the sampling matrix is also retained in such cases [20].

The wavefield modeling principle/manifold separation technique may also be employed as a complement to the auto-calibration techniques from Section 3.19.5 as well as to array interpolation technique and uncertainty sets (to be discussed in the next section). For example, in the case of uncertainties in the array elements’ positions, one may parameterize the array sampling matrix using the closed-form expressions in [20] and employ the auto-calibration techniques described in Section 3.19.5. The advantage of such an approach is the simplicity of using Fourier basis regardless of the configuration of the real-world array. One may also employ manifold separation technique with array interpolation technique to determine the conditions under which the real array output can be transformed to that of a virtual array, up to a specified mapping error [20]. Such conditions can be found with or without sector-by-sector processing. Finally, we note that many antenna measurement techniques, including the spherical near-field approach, are based on wavefield modeling [32,33]. This suggests that the wavefield modeling principle/manifold separation technique may play a fundamental role in any sensor array application.

3.19.7.1 Robust technique based on worst-case performance optimization and uncertainty sets

We have seen that steering vectors of real-world arrays are not exactly known in practice and that lack of knowledge or uncertainties about the array model may lead to a significant performance degradation in most array processors. For example, one may steer energy towards unwanted directions, cancel the SOI as well as amplify interfering sources or jammers; see Figure 19.8. The wavefield modeling principle/manifold separation technique aims at optimally describing (in the MSE sense, for example) nonidealities from array calibration measurements as well as incorporating such nonidealities into DoA estimators and beamforming techniques. However, real-world arrays may also be subject to nonidealities that change as a function of time or cannot be measured in controlled environments. For example, random fluctuations of the array response due to nearby scatterers or motion of the platform where the array is mounted are not captured in the calibration stage. In addition, imprecise knowledge of the DoAs may also lead to a performance degradation of beamforming techniques, a problem known as pointing angle or look direction errors. The idea of worst-case performance optimization techniques [59,60] employing so-called uncertainty sets [23,61], is to develop array processing techniques that are robust to general uncertainties in the steering vector of the real-world array.

Let us denote the exact (but unknown) steering vector of the real-world array by . Also, let denote the known but imprecise array steering vector, where . may be acquired from array calibration measurements, EM simulation software or may represent an ideal ULA, for example. The vector denotes the uncertainty we have about and may be due to errors in pointing angle or imprecise gain of array elements. Worst-case performance optimization techniques proceed by assuming that the norm of the uncertainty vector can be bounded by a known such that . This is equivalent to assuming that the imprecise steering vector belongs to a known hyper-ellipsoid centered at [23,61]:

(19.32)

Here, denotes a known positive-definite matrix that characterizes the shape of the ellipsoid and defines the maximum uncertainty we have about . Ellipsoids of the form of (19.32) are called nondegenerate ellipsoids. If the uncertainty ellipsoids fall into a lower-dimensional space they are called flat (or degenerate) ellipsoids [61]. Flat ellipsoids are employed to make the uncertainty set as tight as possible but require more prior information about the maximum uncertainty than that of the nondegenerate ellipsoids.

Worst-case performance optimization techniques have been mostly used in the context of robust minimum variance beamforming [23,59–61]. The resulting robust beamformers can be shown to belong to the class of diagonal loading approaches. In fact, the optimal diagonal loading value can be found exactly from , unlike most of the ad hoc diagonal loading techniques [8]. However, the value of and the shape of the uncertainty ellipsoid are typically specified by the designer. See e.g., [62] for alternative approaches to find the loading factor automatically.

Let us consider the eigenvalue decomposition of the sample covariance matrix , with denoting the corresponding eigenvalues, and assume that . The array steering vector found by employing robust techniques based on uncertainty sets is [23]

(19.33)

where is the solution of [23]

(19.34)

Here, and can be shown to belong to the following interval [23]

(19.35)

We may now use the steering vector model in (19.33) with the Capon beamformer expression from Section 3.19.2 in order to have a robust approach for both beamforming as well as DoA estimation. A summary of this robust technique is provided in Table 19.6.

Typically, robust techniques based on uncertainty sets have a prohibitively large computational complexity. For example, the value in (19.33) needs to be found for every angle and may not be found offline since it is a function of . An exception that is worth mentioning is the robust beamformer of [60], where the beamformer weights may be updated snapshot-by-snapshot using subspace tracking techniques. However, such an approach is still not practical in DoA estimation since it requires finding a principal eigenvector for each grid point of the spectral search.

3.19.8.1 DoA estimation using an ideal uniform linear array and manifold separation technique

Let us consider that a propagating wavefield, generated by two equi-power and uncorrelated point-sources, impinge on an ideal ULA from , measured from the endfire of the array. The ULA is composed of 5 omnidirectional elements with an inter-element spacing of . We employ the root-MUSIC [63] and element-space (ES) root-MUSIC [18] algorithms. Since the employed sensor array is composed of omnidirectional elements the array sampling matrix may be found in a closed-form [20]. Figure 19.9 illustrates the performance of both root-MUSIC and ES root-MUSIC algorithms in terms of root mean-squared error (RMSE). Only the results for are illustrated but similar results are obtained for . Figure 19.9a illustrates the RMSE as a function of snapshots, with , while Figure 19.9b illustrates the RMSE as a function of SNR, with snapshots. The number of columns of employed by the ES root-MUSIC is . Results show that the ES root-MUSIC has a performance very close to the root-MUSIC even though the former employs an approximation of the array steering vector.

The complexity of the ES root-MUSIC is, of course, higher than that of the root-MUSIC algorithm, and if the employed sensor array is indeed an ideal ULA one should resort to the standard root-MUSIC algorithm. However, if the task is estimate DoAs using real-world arrays with imperfections, the ES root-MUSIC is a very attractive choice. Note that the complexity of the ES root-MUSIC algorithm may be reduced by means of Schur factorization and Arnoldi iterations [64].

Let us now suppose that the only information we have about a real-world array is by means of its array measurement matrix , and the stochastic CRB expression for array processing in (19.9) needs to be found. One may employ the manifold separation technique in (19.28) with (19.9) in order to find an approximate CRB expression that is tight even for real-world arrays with nonidealities, excluding the low SNR regime where the CRB is not a tight bound in general. To illustrate this, let us assume that the array measurement matrix of the 5-element ULA from Figure 19.5 is obtained from points with . The EADF of such an ULA is obtained from the array measurement matrix using Eq. (19.30). The resulting array steering vector model, Eq. (19.28), is employed with the CRB expression in (19.9) in order to obtain an approximate CRB expression. Figure 19.10 illustrates the ratio between the approximate and exact stochastic CRBs as a function of the number of columns of . The results have been averaged over and 100 realizations of calibration noise. The approximate CRB expression obtained by employing the manifold separation principle is accurate since the ratio converges to unity. We also note that the accuracy of the approximate CRB expression is related to the electrical dimension of the employed sensor array. In fact, the ratio converges to unity around modes which, as discussed in Section 3.19.6, is the point where the magnitude of the array sampling matrix starts decaying superexponentially.

3.19.8.2 Robust beamforming using array calibration

Let us consider that a propagating wavefield, generated by three uncorrelated point-sources, impinge on an ideal ULA that is identical to the one employed in the previous example. The SOI and interfering sources impinge on the sensor array from and , respectively. The signal powers of the interferers are . We consider two sources of uncertainty in the array steering vector , namely due to array calibration noise and error in the look direction. In particular, we employ the array measurement matrix of the ULA, found with an and calibration points. In addition, we consider that there is an error of two degrees in the DoA of the SOI. We employ the robust Capon beamformer, obtained by using (19.33) in (19.7), with the imprecise array steering vector found directly from , and by employing the manifold separation technique, i.e., using the EADF . The uncertainty ellipsoid is fixed with , where . Figure 19.11a illustrates the array output SINR as a function of snapshots with while Figure 19.11b illustrates the array output SINR as a function of the SNR of the SOI with snapshots. Employing the manifold separation technique leads to improved performance since it attenuates calibration measurement noise [18].

3.19.8.3 Polynomial rooting techniques for real-world arrays with nonidealities

Let us now consider DoA estimation algorithms based on polynomial rooting techniques that can be employed regardless of the array configuration and nonidealities. Assume that a propagating wavefield, generated by two point-sources, is observed by the real-world array from Figure 19.12[65]. The sources are located in the far-field of the sensor array and their DoAs are . The sources are assumed to be located in the same plane (xy-plane) as the employed antenna array. Only the array measurement matrix of the real-world array is known. The interpolated root-MUSIC algorithm [15] as well as the ES root-MUSIC [18] and the Fourier-domain (FD) root-MUSIC [50] are used to provide the angle estimates as follows.

The interpolated root-MUSIC algorithm is employed by first determining 12 array mapping matrices , one for each -sector, from . Each of the 12 virtual ULAs (one per sector) is located at the center of the minimum circle enclosing the employed sensor array and oriented so that its broadside corresponds to the middle of each sector. The virtual ULAs are composed of five elements with an inter-element spacing of , so that the aperture of the virtual ULAs is maximized while guaranteeing a small condition number of the mapping matrices. The remaining steps of the interpolated root-MUSIC algorithm are implemented as described in [15].

The ES root-MUSIC algorithm is implemented by first determining the EADF, denoted by as described in (16.6.9). This includes taking the FFT of the array measurement matrix and determining the number of columns by means of a model order estimation technique such as MDL [54]. After determining the noise subspace of the sample covariance matrix , the DoA estimates are found from the phase-angles of the P roots closest to the unit circle (either inside of outside the unit circle) of the following polynomial

(19.36)

Note that the coefficients in (19.36) may be found in a computationally efficient manner using the FFT:

(19.37)

where .

The FD root-MUSIC algorithm is implemented by first determining the sampled MUSIC nullspectrum :

(19.38)

Then, the coefficients of the FD root-MUSIC polynomial

(19.39)

are obtained as

(19.40a)

(19.40b)

where and the selection matrix is obtained in a similar fashion as with the ES root-MUSIC.

Figure 19.13 illustrates the performance of the three polynomial rooting approaches in terms of RMSE as a function of (a) snapshots with and (b) SNR with snapshots. Only the results for are shown but similar results are obtained for . The algorithms exploit a noise-free array measurement matrix with calibration points. The degree of both ES root-MUSIC and FD root-MUSIC polynomials are equal with . Results show that the ES root-MUSIC as well as the FD root-MUSIC have similar performance and are closed to the stochastic CRB while the estimates obtained by the interpolated root-MUSIC have large bias and excess variance.

Figure 19.14 illustrates the sensitivity of both ES root-MUSIC and FD root-MUSIC algorithms, for a fixed , to (a) calibration SNR () with and (b) calibration points Q with . Two equi-power sources impinge on the 5-element IFA array from with an and . Only the results for are illustrated but similar results are obtained for . Results show that the ES root-MUSIC algorithm outperforms the FD root-MUSIC when the array measurement matrix is corrupted by calibration noise.

3.19.8.4 Azimuth, elevation, and polarization estimation

Let us now consider the general case of estimating both azimuth and elevation angles of a completely polarized EM propagating wavefield. Typically, the polarization of the wavefield is unknown and needs to be estimated along with the DoAs. The narrowband array output model is now given by

(19.41)

where denote the array steering matrices due to a vertical and horizontal polarized wavefields, respectively. describes the polarization of the wavefield and it is typically given by

(19.42a)

(19.42b)

where and . The parameters and describe the polarization ellipse of the received Electric-field and take values over and , respectively. The techniques described in Sections 3.19.5 and 3.19.6 may now be employed to (19.41) by modeling the array steering vector either in a parametric or non-parametric manner.

In particular, the wavefield modeling principle/manifold separation technique for completely polarized EM wavefields consists in employing so-called vector spherical harmonics, instead of Fourier basis in (19.27b, 19.27a) and (19.28). The motivation for using vector spherical harmonics follows from the fact that such basis functions form a complete and orthonormal set on the 2-sphere, similarly to Fourier basis in azimuth-only processing. However, vector spherical harmonics, which have a rather cumbersome algebraic form, are less attractive than Fourier basis for the purposes of array processing and may be subject to numerical instabilities. In fact, a formulation of the wavefield modeling principle/manifold separation technique involving 2-D Fourier basis may be found in [53], where it is employed the so-called equivalence matrix [19]. A summary of the wavefield modeling principle/manifold separation technique for completely polarized EM wavefields is given in Table 19.7.

Let us consider that a propagating EM wavefield, generated by two equi-power far-field point-sources, is received by the 5-element IFA from Figure 19.12. The DoAs and polarization parameters are , , and , respectively. snapshots are acquired at the array output. The array response was obtained from an EM simulation software; see [65] for details. The manifold separation technique is employed for determining the array steering vector model along with its nonidealities, as described in Table 19.7. The polarimetric element-space (PES) MUSIC algorithm [53] is used for estimating both DoAs and polarization parameters of the sources generating the wavefield.

Figure 19.15 illustrates the statistical performance of the PES MUSIC in terms of RMSE as a function of SNR. Only the angle estimates are shown but similar results are obtained for the polarization parameters. Results show that the PES MUSIC algorithm has a performance close to the stochastic CRB since it takes into account array nonidealities.

See Vol. 1, Chapter 11 Parametric Estimation

See this Volume, Chapter 2 Model Order Selection

See this Volume, Chapter 16 Performance Bounds and Statistical Analysis of DOA Estimation