Aperiodic geometry design for DOA estimation
using compressive sensing
Sayed Zeeshan Asghar
Boon Poh Ng
School of Electrical and
Electronic Engineering
Nanyang Technological University
Singapore 639798
Email: SAYED1@e.ntu.edu.sg
School of Electrical and
Electronic Engineering
Nanyang Technological University
Singapore 639798
Email: EBPNG@ntu.edu.sg
Abstract—Antenna arrays used in compressive sensing based
DOA estimation algorithms are generated randomly to minimize
mutual coherence. This scheme suffers from practical limitations.
For an antenna array that is sufficiently random, some elements
of the array would almost always fall very close to each other,
which is not practicable. Rectangular arrays, although very uniform and practicable, suffer from poor performance when used
in compressive sensing algorithms that assume spatial sparsity.
Aperiodic arrays seem to offer a compromise solution. This paper
demonstrates that it is possible to design aperiodic antenna array
by using a simple disturbance optimization scheme, that can be
applied to multiple aperiodic array geometries. The optimization
scheme uses a few parameters to generate an aperiodic geometry.
We will also see that the optimized aperiodic array has better
performance than several other geometries studied in this paper
and performs very close to random array configuration.
I. I NTRODUCTION
The array geometry that is commonly used for Direction
of Arrival (DOA) estimation of signal sources is Uniform
Linear Array (ULA) or its 2-D counterpart i.e. rectangular grid
array. The neighbouring elements in these array geometries, on
each axis, are separated by a fixed distance; usually λ/2. This
configuration works well if we are looking for compression in
frequency domain. But if the design objective is to compress
the number of array elements, then this configuration is not
suitable. A random sampling of aperture is much more suitable
[7].
Herman et. al. [8] and Ender [9] studied application of
Compressive Sensing (CS) to the radar problem and gave some
initial results regarding the potential of using CS to address
DOA problem. Both of these papers however use a geometry
that is formed through random sampling of aperture. Usually,
in a randomly sampled aperture some of the elements would
lie too close to each other, which isn’t practically realisable.
The use of aperiodic geometry can reduce this problem
significantly and make the layout practically realisable. Fractal
geometries [11] have effectively been used in the design of
aperiodic arrays. These studies, however, utilize classical array
processing framework and much of the results may not be
applicable to compressive sensing scenario. In [5] Spence et.
al. introduces a design technique to generate antenna array
layouts from aperiodic tiling.
In this paper, we device an optimization technique, based
on CS, that generates an antenna array layout through a
disturbance of aperiodic tiling. This technique is well suited to
any array design methodology that can be generated through
geometrical transformations of a few initial points.
II. T HE MEASUREMENT M ATRIX
Suppose, there are m narrow-band signals with a known
centre frequency, fc , incident on an array of L sensors. We
assume that the signals originate from any of the N directions,
in which our DOA space is quantized. Then our received
signal, y, is given by
y = Ax.
(1)
y ∈ CL×1 is the measurement vector. x ∈ CN ×1 contains
source signals at the time instant t. This vector is assumed to
be sparse having just a few non-zero elements:
kxk0 = s,
(2)
where k.k0 denotes the number of non-zero elements or in
other words the sparsity level of the vector. A contains vectors
from N directions as its columns:
A = a0 . . ai . . aN −1 , i ∈ {0, ..., N − 1},
(3)
and the steering vector, ai , is given by
iT
T
T
1 h T
ai = ejkθi P = √ ejkθi p1 , ..., ejkθi pL ,
(4)
L
where pj is the three dimensional position vector for j th
sensor and kθi is the wave-number vector for source i. It is
given by kθi = 2π
λ vθi . vθi is a unit vector in the direction of
ith source.
The estimate of source vector, x̂, can be found by:
minimize
x
kxk0
(5)
subject to y = Ax,
But (5) is a combinatorial problem and is NP hard. Candes
and Romberg have proposed to use l1 −norm instead of using
l0 − norm. The theorem presented in [7] states that instead
of solving the problem given by (5), if we solve the problem
(this problem is called Basis Pursuit):
kxk1
minimize
x
(6)
subject to y = Ax,
(b) Type II
we are guaranteed to recover x with a very high probability.
This problem now becomes a linear programming optimization
problem which is relatively easy to solve.
If there is noise in the system the new model becomes y =
Ax + n. n ∈L×1 is a vector realization of random Gaussian
process. Candes and Romberg [7] suggested a solution to this
problem called Basis Pursuit deNoising (BPDN):
minimize
kxk1
subject to
ky − Axk2 < .
x
2
2
(1 − δk ) kxk ≤ kAxk ≤ (1 + δk ) kxk ,
(8)
for all x ∈ Rn such that kxk0 ≤ k.
One of the few matrices that observe this property is the
Fourier matrix [1]. Another significant parameter is the mutual
coherence, µ, which indicates the level of sparsity that can be
recovered with very high probability.
For the system y = ΦΨx, µ is defined as follows:
√
µ(Φ, Ψ) = n. max |hϕk , ψj i| .
(9)
1≤k,j≤n
If the number of measurements, m, for s-sparse vector obey
the following inequality
m ≥ Cµ(Φ, Ψ)2 s log(n),
(c) Type III
(7)
The measurement matrix, A, has a central importance to CS
theory. Its properties are crucial to the exact recovery of sparse
vectors from a very few observations. One important property
that this matrix has to observe is the restricted isometry
property [1] given by:
2
(a) Type I
(10)
for some positive constant, C, then the solution to this problem
is exact with “overwhelming probability” [1].
One such Φ, Ψ pair with low µ is the Fourier matrix and
the identity matrix pair. In our case, i.e y = Ax, where Φ =
A, Ψ = I.
Suppose there is a matrix B = AH A, the diagonal entries
of B must be unity as the vectors are normalized. The offdiagonal entries should be as close to zero as possible, to get
a lower value of µ. This depends on the column vectors of
matrix A. For a regularly sampled aperture the values of µ
are higher than that of irregular sampled aperture.
What we explore in the next section is a kind of aperiodic
geometry obtained by tiling of very simple base shapes. This
aperiodic structure was discovered by Roger Penrose in 1970
and has since been found in naturally occurring crystalline
materials as well [10]. This structure, being aperiodic, is
naturally sparse (average sensor spacing is greater than λ/2)
as it doesn’t contain any redundancies that are signature of
periodic arrays like rectangular arrays or its 1-D counterpart
i.e. uniform linear arrays.
Figure 1: Basic Danzer tiles
III. A PERIODIC T ILING
A 2-D aperiodic tiling consists of a collection of tiles that
divide the 2-D plane in such a way that it lacks horizontal
or vertical translational symmetry. A periodic tiling, however,
contains both horizontal and vertical translational symmetry. A
set of shapes, called prototiles, are used to populate the whole
2-D plane. Roger Penrose discovered a two-prototile set (kite
and dart) that could fill up a 2-D plane aperiodically, if the
tiles are placed together with certain predefined rules.
A robust iterative method called “inflation” is used for
tiling the plane aperiodically. A tile is first enlarged and then
subdivided into its constituent tiles. This process is repeated
several times iteratively until the whole plane is covered with
tiles. Figure 1 shows three prototiles know as Danzer tiles.
These three prototiles, through the process of inflation can
fill any 2-D plane aperiodically [3]. To form an antenna
array based on this geometry, we can place individual antenna
sensors at the indices of each triangle as shown in Figure 2.
IV. O PTIMIZATION SCHEME
The measurement matrix A contains an exponential at row
k, and column l
T
A(k, l) = ejk0 pk rl , k = {1, ..., L}, l = {1, ..., N },
(11)
where rl is the normalized direction vector for direction index
0
l and pk is the position vector for sensor k. k0 = 2πf
c .
Matrix B can be written as
B(u, v) =
L
X
T
T
ejk0 (pξ rv −pξ ru ) u, v ∈ {1, ..., N }.
(12)
ξ=1
There are N 2 entries in the matrix B. P is a 3 × L matrix
containing position 3-vectors of L sensors as its columns. We
choose three points p0 , p1 , p2 inside Danzer triangles of
Type-I, II and III, respectively. These initial three points pass
through a transformation matrix involving translation, rotation
and scaling sub matrices to form the next level Danzer triangle.
Now P = T p, where p = [pT0 pT1 pT2 ]T . We pose an
When u 6= v, we have N (N − 1) equations:
L
X
T
T
ejk0 ([T p]ξ rv −[T p]ξ ru ) = 0,
u 6= v.
(16)
ξ=1
For a particular u and v, u 6= v, the equation in expanded
form is
T
T
ejk0 ([T p]1 (rv −ru )) + ... + ejk0 ([T p]N (rv −ru )) = 0.
(17)
We can choose the arguments of exponentials (on a unit circle)
in such a way that their sum becomes equal to zero
ej1 + ej2 + ... + ejN = 0.
(18)
Then,
a)
[T p]T1 (rv − ru )
=
[T p]T2 (rv − ru )
=
.
1
,
k0
2
,
k0
.
.
[T p]TL (rv
b)
Figure 2: a) Level-3 Danzer aperiodic tiling b) Sensors array
antenna formed by placing sensors at the indices of each
triangle.
optimization problem:
minimize
kB − δIk2
subject to
Mu pu + Cu ≤ 0.
p
(13)
Here, Mu and Cu contains slopes and y-intercepts, respectively, of the base triangle boundaries. In its current form,
this problem does not seem to be solvable. What we do next
would pave the way for finding a solution to this problem.
Our objective is to bring B as close as possible to δI, subject
to certain conditions. There are N 2 equations to satisfy, apart
from the inequality conditions on p. Each equation looks like:
(P
T
L
jk0 ([T p]T
ξ rv −[T p]ξ ru ) = δ
if u = v
ξ=1 e
(14)
PL
T
T
jk0 ([T p]ξ rv −[T p]ξ ru )
= 0 otherwise.
ξ=1 e
When u = v, we have diagonal entries and the equations are:
L
X
ejk0 0 = δ.
(15)
ξ=1
This condition can be satisfied if we multiply each column of
A by Lδ .
− ru )
=
L
.
k0
There are L equations for each u, v pair. As [T p]Ti (rv −ru )
is a scalar, we can transpose it to get (rv − ru )T [T p]i or
equivalently (rv −ru )T [T ]i p, as the initial points denoted by p
remain constant, and only the transformation matrix changes.
As we have L such equations, we can write them in a matrix
form
T̂u,v p = ,
(19)
here T̂u,v = (rv − ru )T T and = [ k10 ... kN0 ]T , we have
L2 − L equations of the form T̂u,v p = for each u, v =
{1, 2, 3, ..., N }, u 6= v. All the combined equations could be
written into one huge matrix equation
Υp = E.
(20)


T̂1,2


.




.


 T̂1,L−1 


 T̂2,1 
 , E = [T T ...T ]T .
Here Υ = 


.




.




.




.
T̂L,L−1
The optimization problem now becomes (constrained leastsquared optimization problem):
minimize
kΥp − Ek2
subject to
Mu pu + Cu ≤ 0.
p
(21)
V. R ESULTS
We use Danzer level-2, type-I triangle as the base geometry.
There are L = 48 elements in this array. We can see the
array geometry in Fig. 3d. After optimizing the array geometry
according to the optimization problem given by (21), we get
the array shown in Fig. 3e. We compare these geometries with
three other given in Figs. 3 a, b and c. Fig. 3a shows an array of
48 elements, their positions were chosen randomly but with
a constraint that no two elements are closer to each other
than λ/3. This constraint was chosen to avoid the practical
difficulties of designing arrays that are too close to each other.
Another geometry, which is very common, is the rectangular
grid array, but its aperture has been chosen to be triangular to
match with the one chosen for Danzer array. It is shown in
Fig. 3c.
Next, We study different parameters for different geometrical configurations. There are three parameters: the minimum
nearest neighbour distance, dmin , the maximum nearest
neighbour distance, dmax and the average nearest neighbour
distance, dav , defined as following:
dmin = min(dn ),
(a)
(b)
(22)
n = 1, 2, ..., N
dmax = max(dn ),
(23)
n = 1, 2, ..., N
davg =
N
1 X
dn .
N n=1
(24)
Table I shows values of these parameters for the different
types of geometries studied. In the table, these values are
normalized through dividing each value by λ. Rectangulargrid array has all values equal to 0.5, since the separation
between the nearest neighbour for each sensor is fixed to λ/2.
For random array without any constraints dmin is the lowest as
compared to all other configurations, while dmax is the highest
as compared to the rest of the configurations. dav , the average
nearest neighbour distance, is comparable to rectangular grid
array. This shows that the utilization of aperture in this
configuration is not uniform as some of the array elements
are too close and some are too much far apart. The values
of parameters for random array configuration, but with a
Table I: Geometrical properties of different antenna array
designs. The minimum nearest neighbour distance(22). The
average nearest neighbour distance(24). The maximum nearest
neighbour distance(23)
Array Configuration
Random
Random - constrained
Danzer
Danzer - optimized
Rectangular grid
No. of
elements
48
48
48
48
179
dmin
λ
dav
λ
dmax
λ
0.0918
0.36
0.14
0.316
0.5
0.52
0.59
0.60
0.69
0.5
1.62
1.03
1.47
1.27
0.5
(c)
(d)
(e)
Figure 3: a) 48-element random array, b) 48-element constrained random array, c) 94-element triangular array (rectangular grid sampling), d) 48-element Danzer array, e) 48element optimized aperiodic array.
geometries, whether based on inflation process or some other
geometrical transformations (translation, scaling and rotation)
of base geometry.
Usually, aperiodic arrays are generated through either turning off array elements or disturbing their positions on a large
grid of array elements. Such schemes require optimization of
a large number of parameters. Our scheme requires a very
few parameters that needs optimization. This scheme can be
utilized in arrays that consist of several sub-arrays. Due to the
aperiodic structure, such arrays are inherently sparse and well
suited for applications that are using powerful capabilities of
compressive sensing algorithms that exploits sparsity in some
domain (in our case it is source-space.)
Figure 4: Mean Squared Error plot for different geometries.
constraint of λ/3 on minimum nearest neighbour separation,
are somewhat conservative as compared to completely random
configuration. For Danzer array configuration, the value of
dmin /λ is 0.14, greater than completely random configuration
but still less than desirable. For optimal configuration dmin is
quite close to λ/3 and dmax is close to 1.27λ. The value of
davg (0.69λ) for optimal configuration is largest as compared
to the rest of the configurations.
Next we compare the performance of each of these array
configurations in regard to estimating the direction of arrival of
three closely located sources. We use these array geometries
to estimate three sources, located at 19, 20 and 21 degrees
azimuth with amplitudes 1.5, 1.0 and 1.5, respectively; all at
the same elevation, 25◦ .
Fig. 4 shows the mean squared error (MSE) plot for all four
geometries using 50 iterations of monte carlo simulation. MSE
has been calculated using the following equation:
N
1 X
(xn − xnˆ)2 .
M SE =
N n=1
(25)
x̂ is the estimate of x, which is the output of the l1 minimization program (7).
We can see in Fig. 4 that the triangular array (with rectangular grid sampling, Fig 3c) has the largest MSE, followed by
Danzer array (Fig 3d), although the number of elements (179)
in the rectangular grid array is far greater than the number of
elements (48) in the Danzer array. Constrained random array
(Fig 3b) performed better than the Danzer array. Random array
configuration (Fig 3a) has slightly better performance than
optimized aperiodic array configuration (Fig 3e).
VI. C ONCLUSION
In this paper, we have introduced a design technique that
is capable of being used for generation of antenna sensor
layouts based on aperiodic tilings, specifically Danzer tilings,
in the context of compressive sensing. This technique generates antenna sensor layouts through a slight disturbance of
the inflation process through which aperiodic geometries can
be generated. We have devised an optimization scheme that
is generic in nature and can be applied to other aperiodic
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