29TH ESA ANTENNA WORKSHOP ON MULTIPLE BEAMS AND RECONFIGURABLE ANTENNAS,
18-20 APRIL 2007
Experience with sparse arrays at the University of Oslo
Sverre Holm, Andreas Austeng, Jan Egil Kirkebø,
Department of Informatics, University of Oslo, P.O. Box 1080, NO-0316 Oslo, Norway
Email: Sverre.holm( at) ifi.uio.no
ABSTRACT
This paper gives an overview over work on layout and weight optimization of sparse 1D and 2D arrays which has been
done over the last decade. The optimization methods have been linear programming, genetic methods and simulated
annealing. The latter is our preferred method. The paper shows examples of optimized arrays on Cartesian and nonCartesian grids such as polar and hex. Non-planar arrays (curved and cylindrical) have also been optimized.
INTRODUCTION
We have been working for more than a decade on optimizing the layout of thinned arrays. We have been optimizing
symmetric 1D and 2D planar arrays and curved and cylindrical arrays. Most of our problems have involved binary
weights, i.e. elements that are either in or out, but in some cases weighting functions have been found also.
Our applications have primarily been monostatic systems for use in medical ultrasound and sonar. In imaging systems,
it is often been possible to take advantage of different array layouts for transmission and reception. These methods have
also been tested experimentally.
Our work now is concerned with geometries where elements are not aligned on a regular grid, and on optimization
using broadband excitation.
RANDOM, BINNED, AND PERIODIC ARRAYS
Some basic properties of sparse arrays will be given in this section which builds on [6]. Assume an array with M
elements where only K elements are kept after random thinning. The average array pattern is equal to that of an aperture
weighted with the probability density function of the thinning. This also determines the shape of the main lobe. For a
uniform thinning it turns out that the ratio of average sidelobe power to main lobe power is 1/K. But the variance is
about K for |u| = |sin φ| > λ/L, where L is the aperture. The relative peak level of a 1D random array is √K ln K [1]
which gives in our experience a fairly good estimate of the peak level.
In the binned array, the aperture is divided into K equal size bins and one element is chosen at random in each bin. This
resembles a nearest neighbor restriction as there can be no more than two neighbor elements in a 1D binned array. With
a uniform distribution in each bin, it can be shown that the binned array has the same properties as the random array
except that the variance does not reach the value of K until the angle reaches |u| = |sin φ| > K lambda/L. Thus the binned
array has close-in random sidelobes that are much lower than the random array [1].
Periodic arrays are thinned with a periodicity, e.g. 1001001001, 101010101, or 11001100 etc. This means that grating
lobes are formed whose position and size are easily predicted. Periodic arrays have turned out to be particularly useful
in imaging systems where one can use different periodicities for the transmitter and the receiver. The two-way
beampattern is the product of the transmitter and receiver beampatterns, so by proper design, grating lobes formed by
the transmitter are suppressed by the receiver and vice versa. A special case is the vernier array which has periodicities
p and p-1 [1].
OPTIMIZATION OF STEERED ARRAYS
For optimization of the unsteered beam pattern of a 2D array, the sidelobe level should be minimized over all visible
angles. This is the region defined by all elevation angles and with the azimuth angle in the range [φ1,π], where φ1 is the
29TH ESA ANTENNA WORKSHOP ON MULTIPLE BEAMS AND RECONFIGURABLE ANTENNAS,
18-20 APRIL 2007
boundary between the mainlobe and sidelobe regions. Because (kx, ky) = 2π/λ*(sinφ cosθ, sinφ sinθ), this is an annular
region in k-space of radius 2π/λ, except the small mainlobe region in the center, as shown in Fig. 23 of [6].
Due to the sampling of the aperture, the beampattern will be repeated for argument of kx and ky larger than 2π/λ. This
means that the annulus will repeat along the kx-axis and the ky-axis. When the element distance is λ/2, the circles will
exactly touch along the direction of the axis. If the element distance is larger than λ/2, there is undersampling and the
circles will partly overlap. Grating lobes may be explained in this way.
When steering is applied to the array, the visible region will shift to have its center at the steering direction, while the
optimized region from the array is still centered at the origin. There is, therefore, no longer full overlap between the
optimized region and the visible region. In order to deal properly with steering one must, therefore, optimize a larger
region. For an array with element distance λ/2, and for all possible steering angles, one must optimize over the area not
covered by the pattern of repeating circles, i.e. a square region. These results can easily be applied to the 1D case also
[6], and this has been utilized in [9].
OPTIMIZATION OF ARRAY LAYOUTS FOR ONE-WAY BEAMPATTERNS
Given that a random sparse array has such a large variation in peak sidelobe level, it is natural to look for optimization
methods in order to find arrays with good sidelobe behavior. This can be done in 1D and 2D, and also with elements on
an underlying regular grid or with freely chosen positions in the aperture (avoiding overlapping elements). It also
makes a difference whether the array is flat or curved. The optimization criterion we have mainly used is to minimize
the peak sidelobe in the beampattern with a condition on the maximum mainlobe width.
The variables to optimize may either be just element positions or only element weights (for e.g. a random thinned
array), or joint optimization of positions and weights. There may also be several conditions imposed on the problem to
reduce the search space such as symmetry or binning.
In [4] we used linear programming to create a uniform sidelobe pattern for a randomly thinned array. The linear
programming method used there required a symmetric array since it could only deal with a real object function. It
should be noted that the weight functions have seemingly random properties and are quite different from just sampled
versions of the weighting that would apply to a full array. A nice feature of linear programming is that it guarantees that
a global minimum is reached. We also used linear programming for finding element layout. In our experience linear
programming can be used for finding weights for large 2D array problems. It can, however, only be used for solving
rather small combinatorial problems if a global optimum is to be guaranteed. In our experience it is only useful for 1D
layout optimizations.
Therefore, we turned our attention to heuristic methods. In [2], simulated annealing was used to minimize the maximum
sidelobe level for an array with aperture L=50λ (M=101) with N=25 active elements lying on a grid with λ/2 element
distance. This is a problem that has been optimized since the sixties, and some of the best solutions that have been
found when the mainlobe is allowed to be 50% wider than in the full array are:
•
•
•
With condition that the end-elements are always active so that the aperture size is kept: -12.36 dB [5]
With no no condition on the end-elements: -12.42 dB [9]
When the steering range is limited and the elements are allowed on arbitrary positions: -14.02 dB [9]
In [5] this problem was solved with a simulated annealing method. It has a speed improvement since an incremental
procedure is used for finding the array pattern. The evaluation of the beam pattern requires a discrete Fourier transform
that can be implemented by a FFT algorithm in order to speed it up. A further speed increase can be obtained by the
observation that the simulated annealing algorithm consists in perturbing just a single element at a time. Therefore, the
array pattern of the perturbed array can be found by subtracting the contribution of the element that was moved and
adding the contribution of that which was added. When all contributions from all the elements at all the angles are
precomputed and stored in memory, this results in a speed increase. It was shown that for an N=256 point evaluation,
this results in 6.7 times faster execution than when the FFT algorithm is used. For larger N’s, the speed increase is even
larger. The faster evaluation of the array pattern means that one can evaluate more configurations and allow
perturbation at an arbitrary location in the array. An applet for this problem that can be found on the web-site
http://heim.ifi.uio.no/~sverre/anneal/ where there is a high score list for the 10 best results obtained.
29TH ESA ANTENNA WORKSHOP ON MULTIPLE BEAMS AND RECONFIGURABLE ANTENNAS,
18-20 APRIL 2007
Fig. 1: Low element number, symmetric, sparse hex-grid array thinned from 112 to 80 elements, a square
element array thinned from 121 to 77 elements, and a ring array thinned from 112 to 80 elements.
Recently we have also enhanced the simulated annealing algorithm. The latest version is able to handle arbitrary layouts
where the elements are not locked to a Cartesian grid and where each individual element can have different directivity.
It can also easily be extended to handle wideband excitations, and optimization of the response in the near field
(focused arrays) [10]. This was also done in [12] using a least angle regression method.
The simple examples of Fig. 1 illustrate the geometries that the algorithm can handle and shows a sparse hex-grid
optimized array with hexagonally shaped elements, a linear grid design with square elements, and a ring array having
increasing element sizes for increasing ring radii.
The genetic algorithm is also well suited for searching for solutions to large 2D array layout problems as we have done
in [6]. Each gene is coded with 1s and 0s indicating whether an element is included or not. An improved initialization
method is then used. Genetic algorithms are usually initialized with uniform probability distributions over the array and
according to theory of random arrays the first sidelobes will be in the -13 dB range. The main operation in the genetic
algorithm is the cross-over. However, this operation does not significantly alter the probability distribution, so the
probability density distribution is still close uniform. The randomness introduced by the mutation operation is often not
large enough to significantly alter the probability density distribution. Therefore, one should initialize the search with
density functions that already have desirable sidelobe properties. This improves convergence time and more importantly
makes convergence to a good solution possible. In [6] one can also find examples of large optimized arrays such as
selection of 500 elements from a 2D array with 50 x 50 elements. It also shows a solution with a weighted optimization
criterion so that one for instance can let the sidelobe level increase with angle away from the mainlobe.
The genetic algorithm can also be speed enhanced as described for simulated annealing, but it will require much more
memory as the responses of the whole population needs to be stored in memory. We have concluded that it is preferable
to use simulated annealing for optimization.
OPTIMIZATION OF 2-WAY BEAMPATTERNS
In an imaging system, more degrees of freedom can be obtained in the optimization if one allows the layouts of the
transmitter and the receiver to be different. We will still assume that the receiver and transmitter arrays are located in
the same position (monostatic), and allow for partial or no overlap at all between the selected elements.
The simplest way to utilize this freedom is to use the properties of periodic arrays and variants where grating lobes in
the transmitter are used to cancel the receiver’s grating lobes and vice versa. References [7] and [8] describe simulations
and experiments with a 2D array for medical ultrasound of size 48 x 48 elements where several variations and
combinations are utilized for minimizing sidelobes. The different variants investigated were:
• Periodic 2D arrays with different periodicity on transmission and reception
• Arrays with diagonal periodicity combined with each other and with periodic arrays
• Arrays with periodicity along only a single axis and combinations with diagonal periodic arrays
• Arrays with periodicity along radius vectors rather than the x- and y-axes
29TH ESA ANTENNA WORKSHOP ON MULTIPLE BEAMS AND RECONFIGURABLE ANTENNAS,
18-20 APRIL 2007
Arrays which had no overlap between the transmitter and receiver elements were also considered, such as:
•
•
Binned arrays with different layouts for transmission and reception
Polar binned arrays where the bins are along rays emanating from the center of the array
All of the proposed configurations were also tested on experimental data as the array was actually built. Good
correspondence with simulations was obtained.
CURVED ARRAYS
More recently we have started work on optimizing curved arrays for e.g. medical ultrasound applications. Due to the
Fourier projection-slice theorem, the beampattern is equivalent to that of an array with unequal spacing between the
elements. This reduces grating lobes. We have applied some of the layouts of the previous section, usually with
different transmitter and receiver layouts. We have found that there is an optimal radius of curvature for the array which
minimizes peak sidelobe in the two-way response [13]. The optimal value varies with thinning method. Similar work,
involving optimization of the one-way response has also been applied to cylindrical arrays for sonar applications [11].
CONCLUSION
This paper has given an overview of random, random binned and periodic arrays and how the properties of these arrays
are used for improved optimization of 1D and 2D arrays. The optimization methods have been linear programming,
genetic methods and simulated annealing. We have found that the simulated annealing method is best suited for this
kind of problems due to its convergence properties, execution speed, and relatively low memory requirements. The
paper has also shown examples of optimized arrays on non-Cartesian grids such as polar and hex. Non-planar arrays
(curved and cylindrical) have also been optimized. At present we are working on establishing an empirical lower limit
on peak sidelobe level for a given array with a given percentage of thinning,
REFERENCES
[1] W. J. Hendricks, “The totally random versus the bin approach for random arrays,” IEEE Trans. Antennas
Propagat., vol. 39, pp. 1757–1761, Dec. 1991.
[2] G. R. Lookwood, P.-C. Li, M. O’Donnell, and F. S. Foster, “Optimizing the radiation pattern of sparse periodic
linear arrays,” IEEE Trans. Ultrason., Ferroelect.,Freq. Contr., vol. 43, pp. 7–14, Jan. 1996.
[3] V. Murino, A. Trucco, and C. S. Regazzoni, “Synthesis of unequally spaced arrays by simulated annealing,”
IEEE Trans. Signal Processing, vol. 44, pp. 119–123, Jan. 1996.
[4] S. Holm, B. Elgetun, G. Dahl, "Properties of the beampattern of weight- and layout-optimized sparse arrays,"
IEEE Trans. Ultrason., Ferroelec. Freq. Contr., vol. 44, no. 5, pp. 983-991, Sept. 1997.
[5] J. F. Hopperstad and S. Holm, "Optimization of sparse arrays by an improved simulated annealing algorithm",
Proc. Int. Workshop on Sampling Theory and Applications, pp. 91-95, Loen, Norway, August 1999.
[6] S. Holm, A. Austeng, K. Iranpour and J. F. Hopperstad, "Sparse sampling in array processing," ch. 19 in
"Sampling theory and practice" (F. Marvasti Ed.), Plenum, NY, 2001. Available from http://folk.uio.no/sverre
[7] A. Austeng and S. Holm, "Sparse 2D Arrays for 3D Phased Array Imaging - Design Methods," IEEE Trans.
Ultrason., Ferroelec. Freq. Contr., Aug. 2002, pp. 1073-1086.
[8] A. Austeng and S. Holm, "Sparse 2D Arrays for 3D Phased Array Imaging - Experimental Validation," IEEE
Trans. Ultrason., Ferroelec. Freq. Contr., Aug. 2002, pp. 1087-1093.
[9] A. Austeng, S. Holm, "The impact of 'non-half-wavelength' element spacing on sparse array optimization,"
Proc. IEEE Nordic Signal Processing. Conf., NORSIG-02, Norway, Oct. 2002.
[10] A. Austeng, J. E. Kirkebø and S. Holm, "A Flexible Algorithm for Layout-Optimized Sparse cMUT Arrays",
Proc. IEEE Ultrasonics Symp., (Montreal, Canada), vol. 2, pp. 1266-1269, Aug. 2004.
[11] J. E. Kirkebø, A. Austeng and S. Holm,"Layout-optimized Cylindrical Sonar Arrays", Proc. IEEE OCEANS,
(Kobe, Japan), vol. 2, pp. 598-602, Nov. 2004.
[12] J. E. Kirkebø and M. L. R. de Campos, "Weight and Layout Optimized Arrays Using Least Angle Regression",
Proc. IEEE ISSPA, (Sharjah, UAE), Feb. 2007.
[13] J. E. Kirkebø and A. Austeng, “Improved beamforming using curved sparse 2D arrays in ultrasound,” accepted
for publication in Ultrasonics, 2007.