paper presents a design method for robust sparse antenna arrays based on an extension of compressive sensing. An extra constraint is applied to the process to ...
Robust Sparse Antenna Array Design Via Compressive Sensing Matthew B. Hawes and Wei Liu Communications Research Group Department of Electronic and Electrical Engineering University of Sheffield, UK {elp10mbh, w.liu}@sheffield.ac.uk
Abstract—In sparse arrays, the randomness of antenna locations avoids the introduction of grating lobes, while allowing adjacent antenna spacings to be greater than half a wavelength. This means a larger array size can be implemented using a relatively small number of antennas. However, careful consideration has to be given to antenna locations to ensure that an acceptable performance level is achieved. Model perturbations can also cause steering vector errors, which in turn cause discrepancies in the array’s response and this makes a robust array desirable. This paper presents a design method for robust sparse antenna arrays based on an extension of compressive sensing. An extra constraint is applied to the process to ensure that the change in response due to a norm-bounded steering vector error is kept below an acceptable level. Design examples are presented to validate the effectiveness of the method.
I. I NTRODUCTION For Uniform Linear Arrays (ULAs) the adjacent antenna spacing has to be less than half a wavelength in order to avoid unwanted grating lobes [1], [2]. This can be problematic when considering arrays with a large aperture size, due to the cost associated with the number of antennas required. As a result, sparse arrays become a desirable alternative [3], which allow separations greater than half a wavelength, while still avoiding grating lobes due to the randomness of antenna locations. This means an array of a given aperture size can be achieved with less antennas. Moreover, even with the same number of sensors and a similar aperture size, the nonuniform nature of a sparse array also provides more degrees of freedom to achieve a better beam response.
nonlinear optimisation methods such as Genetic Algorithms (GAs) [4], [5], [6], [7], [8], and Simulated Annealing (SA)[9], have been regularly used to achieve the required optimisation. The disadvantage of these methods can be the potentially long computation time and the possibility of convergence to a nonoptimal solution. More recently, the area of Compressive Sensing (CS) has been explored [10], and CS-based methods have been proposed in the design of sparse arrays [11], [12], [13], [14]. CS theory tells us that when certain conditions are met it is possible to recover some signals from fewer measurements than used by traditional methods [10]. This can then form the basis of sparse array design methods by trying to get an exact, or almost exact, match to a reference pattern while using as few antennas as possible. Previous work using CS to design sparse arrays has focused on the traditional beamforming scenario where the steering vector of the array is known exactly. In practice, this may not always be the case, as there are various possible model perturbations such as errors in antenna locations, mutual coupling and discrepancies in individual antenna responses. If one or more of these are present, there will be an error associated with the array’s steering vector, which in turn affects the array’s response. As a result it is desirable to have an array that is robust to the mismatch between designed and achieved steering vectors. Various methods have been used to design robust adaptive
However, the tradeoff in using sparse arrays is the un-
beamformers, such as diagonal loading, worst case optimisation and robust Capon beamformers [15], [16], [17], [18], [19],
predictable sidelobe behaviour and it is often necessary to
[20], [21], [22], [23], where it is usually assumed that there is
optimise antenna locations in order to achieve a desired performance, (e.g. minimising the peak sidelobe level). Some
a norm-bounded steering vector error. In this paper this idea is used to place an extra constraint on the CS-based design
978-1-4673-5807-1/13/$31.00 ©2013 IEEE
DSP2013
d M−1 θ
x 0 [n]
θ
where {·}H is the Hermitian transpose and w is the coefficient vector of the array
θ
x 1 [n]
w = [w0 w1 ... wM−1 ]T .
x M−1 [n]
(3)
B. CS-Based Design
w0
w M−1
w1
As previously mentioned, CS can be used to design sparse arrays by trying to match the array’s response to a desired/reference response Pr (Ω, θ). First, consider Fig. 1 as being a grid of potential active antenna locations. In this
y [n] Fig. 1.
instance, dM−1 is the aperture of the array and the values of dm , for m = 1, 2, . . . , M − 2, are selected to give a
A general narrowband array structure.
uniform grid, with M being a very large number. Sparseness is then introduced by selecting the weight coefficients to give as process. As a result, the difference between the designed and achieved array responses can be kept below an acceptable level.
few active antennas as possible, while still giving a designed response that is close to the desired one. This problem is formulated as
The remainder of this paper is structured as follows: Section II gives details of the proposed design method, including the array model being used in Section II-A, and the extended CS process for designing sparse arrays in Section II-B. Two design examples are presented in section III, with conclusions drawn in section IV.
min
subject to ||pr − wH S||2 ≤ α
where ||w||0 is the number of nonzero weight coefficients in w, pr is the vector holding the desired beam response at the sampled angular points θl for the frequency of interest corresponding steering vectors and α places a limit on the allowed difference between the desired and the designed
A. Array Model A narrowband array structure consisting of M antennas is shown in Fig. 1. The antennas are assumed to be omnidi-
responses. In this constraint || · ||2 denotes the l2 norm. In detail, pr and S are respectively given by
rectional and have identical responses. Also shown are the received array signals, xm [n], for m = 0, 1, . . . , M − 1, with a desired signal arriving at an angle of θ. A plane wave signal model is assumed, i.e. the signal impinges upon the array from the far field. The distance from the zeroth antenna to subsequent antennas is denoted as dm for m = 0, 1, . . . , M −1, with d0 = 0, i.e. the distance from the zeroth antenna to itself.
(1)
where Ω = ωTs is the normalized frequency with Ts being dm cTs
pr
= [Pr (Ω, θ0 ), Pr (Ω, θ1 ), · · · , Pr (Ω, θL−1 )],
and S = [ˆs(Ω, θ0 ), ˆs(Ω, θ1 ), · · · , ˆs(Ω, θL−1 )]. Here the desired response Pr (Ω, θ) can be an ideal one, i.e., one at the mainlobe area and zero for the sidelobe area, or obtained from that of a traditional ULA design method.
The steering vector of the array is given by
the sampling period, μm =
(4)
Ω, l = 0, 1, · · · , L − 1, S is the matrix composed of the
II. P ROPOSED D ESIGN M ETHOD
ˆs(Ω, θ) = [1, e−jμ1 Ω cos θ , . . . , e−jμM −1 Ω cos θ ]T ,
||w||0
for m = 1, . . . , M − 1, and
{·} denotes the transpose operation. T
However, (4) is computationally expensive and the problem can be more efficiently expressed as a minimisation of the l1 norm of the weight coefficients [10], i.e. min subject to
||w||1 ||pr − wH S||2 ≤ α .
(5)
The response of the array is then given by P (Ω, θ) = wHˆs(Ω, θ),
(2)
Suppose that the actual steering vector is related to the assumed steering vector by s = ˆs + ˜s, where ˆs is the designed
steering vector (given by (1)) and ˜s is the error introduced by model perturbations. If this error is norm-bounded, i.e. ||˜s||2 ≤ ε,
(6)
where ε is the upper bound, then the possible difference between designed and achieved array responses is given by |wH s − wHˆs| = |wH˜s| ≤ ε||w||2 .
(7)
TABLE I N ARROWBAND ANTENNA LOCATION DESIGN RESULT. n 1 2 3 4 5 6 7
dn /λ 0 0.75 1.56 2.36 3.13 3.90 4.63
wn 0.0060 0.0177 0.0351 0.0557 0.0754 0.0894 0.0961
n 8 9 10 11 12 13 14
dn /λ 5.37 6.10 6.87 7.64 8.44 9.25 10
wn 0.0961 0.0894 0.0754 0.0557 0.0351 0.0177 0.0060
It is possible to add this as an extra constraint to (5), leading to min
that meet the condition given in (6). For the nth error vector the achieved response at angle θk , Rn (θk ), is found and the
||w||1
subject to ||pr − wH S||2 ≤ α ε||w||2 ≤ β,
average achieved response is given by (8)
where the second constraint ensures that the difference in array response, caused by the norm-bounded error, remains below a predetermined acceptable level specified by β. This problem can be solved using cvx, a package for specifying and solving convex programs [24], [25]. It is also possible to extend this method to solve multiband design problems. To do this, pr and S can be specified in the following way pr
= [Pr (Ω0 , θ0 ), · · · , Pr (Ω0 , θL−1 ), · · · , Pr (ΩF , θ0 ), · · · , Pr (ΩF , θL−1 )],
and S =
[ˆs(Ω0 , θ0 ), · · · , ˆs(Ω0 , θL−1 ), · · · , ˆs(ΩF , θ0 ), · · · , ˆs(ΩF , θL−1 )],
N −1 1 P¯ (θk ) = Pn (θk ), N n=0
(9)
which is then used to find the normalised variance of the achieved array response, var(θk ) =
N −1 1 |Pn (θk ) − P¯ (θk )|2 , N n=0 |P¯ (θML )|
(10)
where N is the number of different error vectors used. A close match between mean achieved and designed responses, along with low variance levels, would indicate that robustness has been achieved. A. Narrowband Design Example First, a narrowband (Ω = π) design example will be given. For this example a grid of 300 potential active antenna locations was uniformly spread over an aperture of 10λ, where λ is the wavelength of the signal of interest. The values
where F + 1 frequencies of interest are being considered.
α = 0.25, β = 1.5 and ε = 1 were also used.
Pr (Ω0 , θ0 ) and S(Ω0 , θ0 ) are the desired response and steering vector for the first frequency at the angular point θ0 . Similarly
The resulting array has 14 active antenna locations and makes use of the full potential array length, as detailed in
Pr (ΩF , θ0 ) and S(ΩF , θ0 ) are the desired response and steering vector for the F th frequency at the same angular point.
Table I. The designed and mean achieved responses are shown in Fig. 2, while the variance of responses is shown in Fig. 3.
III. D ESIGN E XAMPLES
From Fig. 2 we can see the designed mainbeam at the expected location with sufficient attenuation in the sidelobe
In this section, two design examples, one narrowband and one multiband, are presented to verify the effectiveness of
regions, which indicates that a sparse array has successfully been designed. There is also an acceptable match between the
the proposed design method. For both of the examples the desired response is the ideal one, with the mainbeam at the
designed and mean achieved responses, which, coupled with the low variance levels illustrated in Fig. 3, shows that some
direction θML = 90◦ and the sidelobe region at θSL = [0◦ , 80◦ ] [100◦ , 180◦], sampled every 1◦ . The average achieved response and variance of responses are calculated when assessing the robustness of an array. When doing so 1000 different error vectors are generated
degree of robustness has been achieved. B. Multiband Design Example For the multiband design example, two frequencies of interest are considered, Ω1 = 0.8π and Ω2 = π. In what
TABLE II M ULTIBAND ANTENNA LOCATION DESIGN RESULT.
0 Designed Response Mean Achieved Response −10
n 1 2 3 4 5 6 7
Beam pattern (dB)
−20
−30
−40
−50
−60
dn /λ 0 0.79 1.62 2.46 3.29 4.16 5.00
wn 0.0114 0.0283 0.0493 0.0727 0.0942 0.1093 0.1147
n 8 9 10 11 12 13
dn /λ 5.84 6.71 7.54 8.38 9.21 10
wn 0.1093 0.0942 0.0727 0.0493 0.0283 0.0114
−70 0
1
21
41
61
81 101 θ (degrees)
121
141
161
−20
Fig. 2. The desired, designed and mean achieved responses for the narrowband design example.
0.04
−30 −40 −50 −60 −70
0.035 Normalised variance of beam pattern
Designed Response Mean Achieved Response
−10
181
Beam pattern (dB)
−80
−80 0.03
−90 0.025
−100 0.02
0.015
1
21
41
61
81 101 θ (degrees)
121
141
161
181
Fig. 4. The desired, designed and mean achieved responses for Ω1 = 0.8π.
0.01
0.005
0
0
Fig. 3.
20
40
60
80 100 θ (degrees)
120
140
160
180
The variance of responses for the narrowband design example.
a reasonable match between designed and mean achieved responses, together with the low variance levels suggesting that some degree of robustness has again been achieved, although the level of robustness is not as high as in the narrowband example, i.e. higher variance levels and not as well matched
follows, λ is the wavelength associated with the signal that has the normalised frequency Ω2 = π. A grid of 300 potential active antenna locations was uniformly spread over a distance of 10λ, and the values α = 0.25, β = 1.5 and ε = 1 were used. Again the resultant sparse array has 13 active antenna locations with the aperture being the maximum 10λ, as detailed in Table II. Figs. 4 and 5 show the designed and mean achieved responses for the two different frequencies, while Fig. 6 gives the variance of responses at both frequencies. It can be seen that the mainlobe for both the designed and mean achieved responses are in the correct location. However, the sidelobe levels of the designed responses for
response shape in the sidelobe regions. IV. C ONCLUSIONS In this paper a novel design method for a robust sparse array based on an extension of compressive sensing has been proposed. The number of active antennas is kept to a minimum while ensuring a reasonable match between desired and designed responses. An extra constraint is added in order to ensure that the change in response due to a norm-bounded steering vector error does not exceed a predefined level. Two design examples, one narrowband and one multiband, have been presented to illustrate the validity of the proposed method.
both frequencies are higher compared to the narrowband
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0 Designed Response Mean Achieved Response
−10 −20
Beam pattern (dB)
−30 −40 −50 −60 −70 −80 −90 −100
1
Fig. 5.
21
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61
81 101 θ (degrees)
121
141
161
181
The desired, designed and mean achieved responses for Ω2 = π.
0.05 Ω=0.8π Ω=π
Normalised Variance of beam pattern
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0
Fig. 6.
20
40
60
80 100 θ (degrees)
120
140
160
180
The variance responses for the multiband design example.
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