Linear Constraint Minimum Variance Beamforming

A major aspect of digital beamforming in the context of SAR applications has not been considered yet, that is the suppression of spatial interference. While the purpose of MVDR beamformers is to maximize the sensitivity of a SAR for a specific direction, or wavenumber, respectively, another class of beamformers allows to optimize the sensitivity for the direction of interest and to suppress certain other directions, where interference is expected, at the same time. This

method is called linear constraint minimum variance (LCMV) [123] beamform-

ing. The optimization problem can be formulated in analogy to equations (4.51) and (4.52) according to

minimize wTR_vw∗ (4.58)

subject to ATw = c, (4.59)

with the cost function (4.58) being identical to the one of the MVDR beam- former. Here, a set of linear constraints is defined via equations (4.59), where the matrix

A =[a(k, φ1, ϑ1) a(k, φ2, ϑ2) . . . a(k, φNdir, ϑNdir)

]

∈ CNact×Ndir _{, (4.60)}

consists of the array manifold vectors with Nact active feed elements for Ndir

different directions. Here, the first column could correspond to the signal of in- terest, given for instance in the denominator of equation (4.46), while all further columns might correspond to them, lth ambiguity collected in the sum term in

the numerator. Of course the same consideration applies to the AASR (4.44)

and the RASR (4.45) individually. A closed form solution can be derived (see

appendix D) resulting in

w∗= R−1_v A(AHR−1_v A)−1c∗. (4.61)

The constraint vector c can in principle be chosen freely. A meaningful con- straint vector could for example take the form

c =[1 0 . . . 0]T , (4.62)

where the ’1’ corresponds to the direction of the signal of interest and a zero is placed in directions to be suppressed 2. Inserting equations (4.62) and (4.61) into (4.42) gives SNR(ky, t, k, r) = σo2g · wTa 2 cT_(AH_R−1 v A)−1c∗ . (4.63)

Under certain circumstances the term wTa in the numerator is equal to one.

This is the case if

AHR−1_v a = 0, (4.64)

2_{Wide-swath SAR acquisitions may require multiple signal directions simultaneously. In those}

cases simply another array manifold vector is added to the matrix A in equation (4.60) and in the constraint vector c another ’1’ is placed at the corresponding position.

4.3 Optimal Beamforming Techniques

with the definition

a = a(k, φ1, ϑ1) and A =

[

a(k, φ2, ϑ2) . . . a(k, φNdir, ϑNdir)

]

. (4.65)

This means, the array manifold with the wavenumber of interest must be or- thogonal to the array manifold for all other wavenumbers, under consideration of the noise channel covariance matrix R_v.

The key step in the numerical evaluation of the LCMV beamformer (4.61) is the computation of the inverse of AH_R−1

v A. Even if the channels are well

balanced R_v ∼ I, with I the identity matrix, this matrix can become ill posed quickly [123]. The reason for this is again the focusing ability, characteristic for reflector antennas. If for instance a single or multiple wavenumbers to be suppressed are outside the swath to be illuminated, the corresponding array manifold vectors in (4.60) will contain only small magnitude numbers com- pared to the signal vector. This means, the condition number, which is the ratio of the largest to the smallest eigenvalue, of the matrix AH_R−1

v A will be large.

In the worst case this matrix will become rank deficient and no solution can be obtained. The same problem arises from very close wavenumbers. In this case the corresponding array manifold vectors become linearly dependent, with the same consequence of rank deficiency. In that sense, from a purely beamforming perspective, two requirements for the design of an antenna can be formulated, that are mutually independent array manifold vectors and a small condition number.

A dedicated method to improve the robustness of such beamformers iseigen- value thresholding [75], or diagonal loading [72, 127]. Assuming that R−1_v exists,

AHR−1_v A = VΛVH (4.66)

can be decomposed into an eigenvalue matrix

Λ =     λ1 . .. λNact     , (4.67)

and the corresponding eigenvector matrices V. A modified eigenvalue matrix can now be constructed of the form

Λ =     max{µλ_max, λ1} . ..

max{µλmax, λNact}

  

 , (4.68)

with λmax = max{λ1, . . . , λNact}, µ ∈ [0,1] and replace the former eigenvalue ma-

beamformer. In the worst case this can result, with an improper selection of the parameterµ, in an insufficient suppression of spatial interference or even a

deterioration of the mainlobe.

The main motivation for the introduction of the LCMV beamformer in SAR applications is to minimize spatial interference in the ambiguity-to-signal ratio expression (4.46). If the array response matrix (4.60) is well posed, zeros can be placed in the directions to be suppressed. A fundamental limit of this con- cept to work is of course when spatial interference enters the main beam. An expression of the half-power beamwidth for center fed reflector antennas was given in equation (3.12). However, in practice the array response matrix is usu- ally not a square matrix, so that the LCMV beamformer must be understood as least square solution of a quadratic program. This means, some directions might not be suppressed as efficiently as others. The LCMV beamformer shows its best performance when interference needs to be damped in the region of a few sidelobes next to the mainlobe. This will be a major motivation to use the LCMV concept with azimuth processing techniques.

In summary, LCMV beamforming is a technique which optimizes the SNR

and the ambiguity-to-signal ratio at the same time. The signal of interest is preserved via the constraint ’1’ with the best possible SNR, while ambiguities

are suppressed by placingNdir− 1 zeros3 in the constraint vector.