Diss / (Springer Series in Information Sciences 25) S. Haykin, J. Litva, T. J. Shepherd (auth.), Professor Simon Haykin, Dr. John Litva, Dr. Terence J. Shepherd (eds.)-Radar Array Processing-Springer-Verlag
.pdf5. Systolic Adaptive Beamforming |
241 |
Here 0' has been chosen such that the iteration of 0 remains unchanged in the square-root-free algorithm. Making the simple scale transformation
(S.C.3S)
It becomes clear that the factor g can be absorbed completely into the scale parameters 5 and 5' and therefore need never appear explicitly in cell computations: it appears only as an initialization of 5 at the first boundary cell, where it is entered with the associated input vector. Note again that only g, and not g1/2, need ever be used in computations.
Acknowledgements. The authors are indebted to their colleagues at STL-Dr. P. Hargrave, Dr. C. Ward, Dr. J. Hudson, and Dr. P. Gosling-for their essential contribution to the collaboration which has resulted in much of the material presented iiI this chapter. The authors should also like to thank STL for providing Figs. 5.6 and 5.19-21.
Appendix S.D Principal Symbols Used in this Chapter
A
A(n)
a(/)(n), a(l)(n)
B
c
C(J) |
(I) |
A |
|
C, Ci'C, C |
|
C, Ci' C, C
D(n), Do
e(n), e(l)(n) e(t)
e(n, n) e(n, n - 1) ej(n, n)
F
F{n)
(f(x», f(x)
G(n), G' g(q), g
Blocking matrix for constrained least squares Submatrix of unitary update matrix Q(n) Auxiliary vectors employed in MVDR processing Exponential deweighting matrix
Matrix of constraint vectors Condition number of given matrix J Constraint vector
ith element of constraint vector C
Cosine (or cosine-related) rotation parameter Diagonal matrix containing squares of R-matrix diagonal elements
Element of matrix D(n)
Cost function (otherwise known as "signal power" or "metric")
Residual error vector at time tn residual error at time t
a posteriori residual a priori residual
jth a posteriori sub-residual Submatrix of weighting matrix G' Update matrix for the matrix R(n) (Estimated) vector function of vector x
Weighting matrix for weighted least-squares Real-valued scalar weighting coefficient in weighted least-squares
242 |
T.J. Shepherd and J.G. McWhirter |
|
gjj _ |
A |
Coefficient for Gram-Schmidt orthogonaIization |
H, H(n), H(n) |
General metric-preserving transformation matrix |
|
H (superscript) |
Matrix Hermitian conjugation |
|
In |
|
n x n identity matrix |
L |
|
Submatrix of weighting matrix G' |
L |
|
Number of independent constraints in MVDR |
|
|
beamformer |
M(n) |
|
Estimated data covariance matrix at time tn |
M |
|
MVDR control bit |
Mo |
|
Number of taps on broad-band beamformer |
m,m' |
|
Vector of linear constraint gains |
N |
|
Number of simultaneous constraints in constraint |
|
|
pre-processor |
Nc |
|
Number of radial basis function centre vectors |
n |
|
Index of tn - time epoch of most recently received data |
P(a, b) |
Vector projection operator, orthogonalizing given |
|
|
|
vectors a and b |
p |
eo, Q'Jn), |
Total number of input data channels |
e(n), |
Unitary matrix |
|
Q(n), Q(n), Q(n) |
||
qj(n), (/Mn» |
Orthogonalized (orthonormalized) vector |
|
(R(n», R(n) |
(Unit) upper triangular matrix |
|
rj, rij' (rj) |
Element of matrix R, (R) |
|
S(n) |
|
Submatrix of unitary matrix Q(n) |
S |
|
Time-invariant upper trapezoidal matrix |
S, Si> S, S |
Sine (or sine-related) rotation parameter |
|
(f), T |
Time-invariant (unit) upper triangular matrix |
|
T (superscript) |
Matrix transposition |
|
tn |
|
nth time epoch |
U |
|
Time-independent rectangular matrix |
u(n), uj(n), |
|
|
uG(n), u(n) |
Part of data vector y(n) after rotation |
|
V, jJ |
|
Time-independent rectangular matrix |
v(n), vj(n), vG(n) |
Part of data vector y(n) after rotation |
|
w, w(n) |
Weight vector |
|
X(n) |
|
Data matrix containing n data vector "snapshots" |
x(t), (x(tn ), x(n» |
Vector "snapshot" of data at time t (tn ) |
|
Xj(tn), xj(n) |
Vector of input data, from time t1 to tno in channel i |
|
xi,xI |
|
Centre and training data vectors, respectively, in radial |
|
|
basis function algorithm |
Xi> Xj, Xnj, Xj(tn) |
ith channel input datum |
|
y(tn), y(n) |
Vector of primary channel input data, up to time tn |
|
y(t) |
|
Primary channel data value at time t |
Z, i, Z, i(l)(n) |
General vector output from time-independent or |
|
"frozen" network
a(n), iX(n), ain)
p
rp(n)
Y, Yj
(j,{l
8m
1Ii
A (g)
A,l,(A,N)
A,
fl, fli, fl{l) l!(n)
tp(n)
q>(r)
tpi(n)
'P(g)
lJI(n)
5. Systolic Adaptive Beamforming |
243 |
Rationalized residual
Exponential data deweighting factor
Matrix of Gram-Schmidt orthogonalization coefficients Specific element of Qmatrix. (Multiplier in square-root algorithm)
Multiplier in square-root-free algorithm
Residual vector in radial basis function algorithm Unit basis vector
Metric tensor
Maximum (minimum) singular value of matrix Accumulation parameter in MVDR processor Linear constraint gain
Estimated primary/auxiliary channel cross-correlation vector at time tn
Error criterion for constrained system Extended data matrix in Gram-Schmidt orthogonalization
Specific vector from column of Qmatrix Radial basis function
General data column of matrix 4>p(n) Generalized plane rotation matrix Specific vector from row of Qmatrix
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6.Two-Dimensional Adaptive Beamforming: Algorithms and Their Implementation
T.V. Ho and J. Litva
With 19 Figures
Adaptive beamforming technology has been actively discussed in the literature for at least two decades, and is now, increasingly, finding applications in radar, sonar, and communications systems [6.1,2]. The reason for all of this interest lies in the ability of adaptive arrays to automatically steer nulls in the direction of interfering sources. Recently, with the rapid growth of VLSI technology, and particularly with the advent of systolic arrays, the use of VLSI array processors in adaptive beamforming has become a subject of considerable interest [6.3-5]. However, most of the work that has been carried out in the past has been concentrated in the area of adaptive beamforming with linear array antennas, i.e., the one-dimensional (1D) case. Since most antenna arrays are, in practice, planar arrays, the focus of future developments in adaptive beamforming must start to shift to the two-dimensional (20) case.
Two-dimensional adaptive beamforming is rarely discussed in the literature. It is thought that there are two reasons for the lack of 20 results. First, there is the general impression among workers in the field that the principles underlying the 20 case are a simple extension of the 10 case [6.6,7]. Secondly, it is felt that the only way around the complexity that is inherent to the 20 case is by means of subarraying, i.e., by reducing the degrees-of-freedom [6.8]. As a result, very little work has been carried out to optimize 2D adaptive beamforming techniques.
In the case of conventional adaptive beamforming, the computational overhead is proportional to the number of degrees-of-freedom. For a 2D array with L rows and M columns, the number of degrees-of-freedom is given by (LM - 1). A fully adaptive array is one in which every element of the array is individually controlled adaptively [6.9]. A partially adaptive array is one in which elements are controlled in groups (the subarray approach), or in which only certain elements, called auxiliary elements, are made controllable.
The reason for reducing an array's Degrees-Of-Freedom (OOF) usually revolves around the cost. If one uses conventional processors, it is not possible to achieve full dimensionality with arrays consisting of thousands of elements. First, one has the problem of designing a processor that has sufficient speed and accuracy for meeting the computational requirements of a fully functioning 2D array. Next, one has to ensure that the processor is not prohibitively expensive. If conventional techniques are used, these two requirements are usually mutually exclusive.
Springer Series in Information Sciences, Vol. 25 |
Radar Array Processing |
Eds.: S. Haykin J. Litva T. J. Shepherd |
© Springer-Verlag Berlin, Heidelberg 1993 |
250 T.V. Ho and J. Litva
A number of configurations have been considered for reducing the dimensionality of the array while maintaining as much control as possible over the size of a given aperture. Two that come to mind immediately are: (1) grouping the physically contiguous elements to form what are termed "simple arrays", and (2) grouping larger numbers of elements together to form "super arrays". In the latter case, the elements in each group may not necessarily be consistent with the array's natural lattice geometry. An example of a simple array is one consisting of the rows and columns of a 20 array. If 10 processing is first carried out on the L rows, the number of OOF is (L - 1), and if it is carried out on the M columns, as well, there is an additional (M - 1) OOF. In the case of a super array, the subarrays are formed after phase-shifting at the element level takes place. When the outputs ofthe arrays are digitized, they constitute a super array consisting of elements with patterns corresponding to the subarray patterns which are all steered to the same direction. Both of these arrangements preserve the homogeneous antenna front-end hardware, which is very desirable for the application of highly integrated modules.
One of the more attractive techniques or configurations studied is that of beam-space adaptive beamforming. The technique involves transforming a large array of N elements into an equivalent small array of J + 1 elements, where J is the number of jammers present [6.10]. In this technique, J auxiliary beams are formed using the whole array. The auxiliary beams are pointed at the unwanted signals, one beam for each signal. The outputs of the auxiliary beams, together with the main beam signal, form the adaptive transformed array. This can result in a considerable reduction in the computational overhead.
One of the major problems with the super array approach for reducing an array's dimensionality is that of grating lobes. Grating lobes are generated due to the spacing of the super arrays, thereby creating spurious notches which result in blind directions for antennas. For multiple jammers the number of blind directions very soon becomes intolerably high. Grating lobes can be avoided by irregular spacing of the super array, which results in irregular subarrays. This reduces the homogeneity of the antenna front-end hardware.
One of the disadvantages of applying 10 beamforming to a 20 array is that the auxiliary beams (eigenbeams) that are used to cancel interferers are fan beams. Ifthe interferers are located on either side ofthe main beam, 10 adaptive beamforming can be carried out successfully. On the other hand, when the interferers are located above or below the main beam in the plane of the fan shaped eigenbeams, main beam cancellation can take place. This may lead to a degradation in the &ignal-to-Interference Ratio (SIR) rather than to its enhancement, thereby defeating the purpose for applying adaptive beamforming to an array antenna in the first place. One way of overcoming this problem is by carrying out 10 adaptive beamforming in both planes, and then choosing the result that gives the best SIR. In this latter case, the processing overhead is proportional to (L + M - 2).
The problem of main beam cancellation can be totally circumvented, in the majority of instances, by employing 20 adaptive beamforming. One of the
6. Two-Dimensional Adaptive Beamforming |
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advantages of 20 beamforming lies in the fact that the eigenbeams are now pencil beams, which have higher gain than the fan beams that are used for 10 adaptive beamforming. Therefore, adaptive nulling of interferers in the antenna sidelobe region can take place without any cancellation of the main beam. Also, 2D adaptive nulling will result in the formation of deeper nulls for cancellation of interferers than in the case of 10 adaptive beamforming. The deeper nulls in the 20 adaptive beamforming case come about as a result of lowering the noise floor due to the higher gain of the pencil beams. Ultimately, it is the noise floor that sets the limit on the null depth that can be achieved.
It should be pointed out that the null depth that is achieved during beamforming does not depend, per se, on the algorithm used, i.e., whether it is an extended 10 algorithm or a 20 algorithm. What is important is that the full dimensionality of the array is preserved. In the case of the extended 10 algorithm, one does not have the advantage of being able to visualize the eigenbeams as in the case of the 20 algorithm which will be described in the chapter. As well, it is expected that the computational overhead for the extended 10 processor will be considerably greater than that for the 20 processor.
The optimum configuration, then, for adaptive nulling is the fully adaptive array, which, by definition, suppresses the interference by applying some matrix operation to all array element outputs. In theory, this provides the necessary OOF to lower all of the deterministic sidelobes to any arbitrary level, as well as nulling out unwanted signals. This is the approach that is being followed here. An adaptive beamformer based o~ a three-dimensional systolic array will be introduced, which has the potential for processing data from a fully adaptive 2D array.
6.1 Arrangement of the Chapter
The chapter is presented in four sections. The first section, which is now almost concluded, gave a short introduction to adaptive beamforming and has indicated a persistent need to derive a solution to the 20 problem. Section 6.2 reviews some of the key contributions in adaptive beamforming, such as the Howells-Apple~aums algorithm, the LMS (least mean square error) algorithm, the SMI (sample matrix inversion) algorithm, and others which are referred to as classical adaptive beaniforming techniques. The QRD-LS (QR decompositionleast squares) algorithm, a modem adaptive beamforming approach, and its systolic array implementation are introduced. Two-dimensional adaptive beamforming techniques are derived in Sect. 6.3. The 20 versions of the LMS algorithm and the Howells-Applebaum algorithm are developed in this section. The QRO-LS algorithm and its systolic array implementation for 2D adaptive beamforming are then presented. The concept of 20 eigenbeams is used to interpret the performance of the 20 adaptive nulling algorithm. Finally, simulation studies are given in Sect. 6.4 to demonstrate the performance of the adaptive beamforming algorithms developed in Sect. 6.3.