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A novel adaptive beamforming technique applied on linear antenna arrays using adaptive mutated Boolean PSO.

1. INTRODUCTION

The analysis and design of antenna arrays are very important and challenging issues in communications industry. So far, many techniques have been studied and developed in order to design arrays that satisfy specific requirements [1-9]. Due to the demanding applications in modern communications, the radiation pattern of base station arrays must be dynamically shaped according to certain requirements. Specifically, the peak of the main lobe must be steered towards a desired signal called signal-of-interest (SOI). On the contrary, pattern nulls must be formed in the directions of arrival (DOA) of interference or undesired signals. Antennas operating under the above requirements are called smart antennas [10-17] and the techniques used to calculate the excitation weights that produce the above-defined radiation pattern are called adaptive beamforming (ABF) techniques [18-25].

Most of the ABF techniques proposed so far try to recover the degradation in their performance caused by mismatches between the assumed and the actual conditions. A usual kind of mismatch is the steering vector uncertainly which is taken into account by a well-known ABF technique named robust Capon beamforming (RCB) [21]. However, the performance decrease caused by uncertainty in the interference correlation matrix is a major issue that needs careful consideration. Therefore, an ABF technique insensitive to that type of uncertainty would be desirable.

The present study introduces a new optimization technique suitable for adaptive beamforming of antenna arrays. This technique is a new variant of Particle Swarm Optimization (PSO) called Adaptive Mutated Boolean PSO (AMBPSO). The conventional PSO and all its variants are based on an update mechanism, where real number expressions are used. However, the update mechanism in the AMBPSO is implemented exclusively in Boolean form using an effectively adaptive mutation process. Both the Boolean update and the adaptive mutation process make the AMBPSO a robust technique.

The AMBPSO is utilized here as an ABF technique applied to uniform linear arrays (ULAs). The technique assumes a desired signal and several interference signals, all uncorrelated with each other, received by the array at respective directions of arrival. These directions are considered to be already estimated by well-known DOA algorithms [10,11,26-31]. A certain power level of additive Gaussian noise is also taken into account. The optimal excitation weights applied on the elements of the ULA are extracted by minimizing a suitably chosen fitness function F. In order to exhibit the robustness of the technique, the optimization process does not take into account the interference correlation matrix. In that manner, we try to develop a technique which does not depend on the knowledge of the interference signals but only on the knowledge of their DOA.

2. FORMULATION

Assume an M-element ULA that receives a SOI s(k) arriving from angle [[theta].sub.0] and N interference signals [i.sub.n](k) arriving from different angles [[theta].sub.n](n = 1,...,N) (see Figure 1). Each angle is called angle of arrival (AOA) and defines a signal DOA with respect to a reference direction normal to the array axis. The parameter k denotes the k-th time sample. Each element is consider to be an isotropic source, while all the arriving signals are monochromatic with N < M. The received signal [x.sub.m](k) at the input of every m-th element (m = 1,...,M) includes additive, zero mean, Gaussian noise [n.sub.m](k) with variance [[sigma].sup.2]. Thus, the input vector is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the array steering vector of the [[theta].sub.n] AOA, [bar.A] is the M x N matrix of steering vectors [[bar.a].sub.n], [bar.i](k) is the vector of the N uncorrelated interference signals [i.sub.n](k), [bar.n](k) is the vector of the M uncorrelated noise signals [n.sub.m](k), and q is the spacing between adjacent elements of the ULA. Also, the vector [bar.d](k) = [[bar.a].sub.0]s(k) represents the desired input signals, while the vector [bar.u](k) =[bar.A][bar.i](k) + [bar.n](k) represents the undesired (interference plus noise) input signals. Finally, the superscript T denotes the transpose operation.

[FIGURE 1 OMITTED]

The array output is given by the form:

y(k) = [[bar.w].sup.H][bar.x](k) = [[bar.w].sup.H][bar.d](k) + [[bar.w].sup.H][bar.u](k) (2)

where [bar.w] = [[[w.sub.1] [w.sub.2] ... [w.sub.M]].sup.T] is the vector of excitation weights and the superscript H denotes the Hermitian transpose operation.

The array output power for the desired signal is given by:

[[sigma].sup.2.sub.d] = E [[absolute value of [[[bar.w].sup.H][bar.d](k)].sup.2]] = E[[[absolute value of [[bar.w].sup.H][[bar.a].sub.0]s(k)].sup.2]] = S[[bar.w].sup.H][[bar.a].sub.0][[bar.a].sup.H.sub.0][bar.w] (3)

where S = E[[[absolute value of s(k)].sup.2]] is mean power of SOI. Also, the array output power for the undesired signal is given by:

[[sigma].sup.2.sub.u] = E[[[absolute value of [[bar.w].sup.H][bar.u](k)].sup.2]] = E[[[absolute value of [[bar.w].sup.H][[bar.A][bar.i](k) + [bar.n](k)]].sup.2]] = [[bar.w].sup.H][bar.A][[bar.R].sub.ii][[bar.A].sup.H] [bar.w] + [[bar.w].sup.H][[bar.R].sub.nn][bar.w] (4)

where [[bar.R].sub.ii] = E[[bar.i](k)[bar.i]H(k)] is the interference correlation matrix and [[bar.R].sub.nn] = E[[bar.n](k)[[bar.n].sup.H](k)] is the noise correlation matrix. Taking into account that [n.sub.m](k)(m = 1,...,M) are uncorrelated, zero mean, Gaussian noise signals with variance [[sigma].sup.2], we get [[bar.R].sub.nn] = [[sigma].sup.2]I. Therefore, (4) can be written as:

[[sigma].sup.2.sub.u] = [[bar.w].sup.H][bar.A][[bar.R].sub.ii][[bar.A].sup.H][bar.w] + [[sigma].sup.2][[bar.w].sup.H] [bar.w] (5)

Finally, the signal-to-interference-plus-noise ratio is given by:

SINR = [[sigma].sup.2.sub.d] / [[sigma].sup.2.sub.u] = S[[bar.w].sup.H][[bar.a].sub.0][[bar.a].sup.H.sub.0][bar.w] / [[bar.w].sup.H][bar.A][[bar.R].sub.ii][[bar.A].sup.H][bar.w] + [[sigma].sup.2][[bar.w].sup.H][bar.w] (6)

The fitness function F can be simply defined as the inverse of SINR. As F is minimized, SINR is maximized, which means that the peak of the main lobe is steered towards the SOI and pattern nulls are formed in the directions of arrival of all the interference signals. In order to make our technique work without the knowledge of [[bar.R].sub.ii] and S, we assume that [[bar.R].sub.ii] = I and S = 1. Then, F can be defined by the form:

F = [[bar.w].sup.H][bar.A][[bar.A].sup.H][bar.w] + [[sigma].sup.2][[bar.w].sup.H][bar.w] / [[bar.w].sup.H][[bar.a].sub.0][[bar.a].sup.H.sub.0][bar.w] (7)

It is obvious from (7) that the minimization of F performed by the AMBPSO does not depend on the knowledge of [R.sub.ii] but only on the knowledge of the interference DOA. The value of [[sigma].sup.2] can be calculated from the signal-to-noise ratio SNR in dB as follows:

[[sigma].sup.2] = [10.sup.-SNR/10] (8)

The proposed technique is compared to an efficient well-known ABF technique called Minimum Variance Distortionless Response (MVDR) which is a variant of RCB technique [11]. The MVDR beamformer seeks for the optimum weight vector [bar.w] that minimizes the power of the undesired output signal while the desired output signal is maintained. Therefore, w- is calculated by minimizing the quantity [[bar.w].sup.H][[bar.R].sub.uu][bar.w], while [[bar.w].sup.H][[bar.a].sub.0] = 1. The optimum [bar.w] is given by:

[[bar.w].sub.mvdr] = [[bar.R].sup.-1.sub.uu][[bar.a].sub.0] / [[bar.a].sup.H.sub.0][[bar.R].sup.- 1.sub.uu][[bar.a].sub.0] (9)

where [[bar.R].sub.uu] = E[[bar.u](k)[[bar.u].sup.H](k)] is the correlation matrix of [bar.u](k).

3. ADAPTIVE MUTATED BOOLEAN PSO

PSO can be found in many studies in the literature [14,32-36]. A brief description of PSO is given in [32]. The Boolean PSO (BPSO) is a binary version of PSO [37] based on the swarm behavior as well. The AMBPSO is an improved version of BPSO proposed by the authors.

In the AMBPSO, the position [[bar.x].sub.n] = [[x.sub.n1] ... [x.sub.nb] ... [x.sub.nB]] and the velocity [[bar.v].sub.n] = [[v.sub.n1] ... [v.sub.nb] ... [v.sub.nB]] of every n-th (n = 1,...,[N.sub.P]) particle of the swarm are represented as binary strings of B bits. Every position [[bar.x].sub.n] must be inside the search space defined by a lower and an upper boundary, respectively [[bar.l].sub.n] and [[bar.u].sub.n]. If a particle goes outside the search space, a large fitness value is assigned as a penalty to the particle. Since the AMBPSO aims at minimizing the fitness function, these particles are gradually moved inside the search space.

The update of [[bar.v].sub.n] and [[bar.x].sub.n] is made by using "and", "or" and "xor" operators:

[v.sub.nb] = [c.sub.1] * [v.sub.nb] + [c.sub.2] * ([P.sub.nb] [direct sum] [x.sub.nb]) + [c.sub.3] * ([g.sub.b] [direct sum] [x.sub.nb]) (10)

[x.sub.nb] = [x.sub.nb] [[direct sum] [v.sub.nb] (11)

where [p.sub.nb] is the 6-th bit of the best position [[bar.p].sub.n] achieved so far by the n-th particle and [g.sub.b] is the b-th bit of the best position [bar.g] achieved so far by the swarm. In addition, [c.sub.1], [c.sub.2], and [c.sub.3] are random bits with probabilities of being '1' respectively equal to [C.sub.1], [C.sub.2], and [C.sub.3]. The exclusively Boolean update of [[bar.v].sub.n] and [[bar.x].sub.n] makes the AMBPSO more efficient than the popular binary PSO version of [38], where the velocity update is made by using a real number expression.

In order to control the convergence speed of the process, the AMBPSO utilizes a parameter [v.sub.max] called maximum allowed velocity and defined as the maximum number of '1's allowed in [[bar.v].sub.n]. The actual number of '1's in [[bar.v].sub.n] is the "velocity length" l([[bar.v].sub.n]) and is controlled by the "negative selection" (NS), which is a basic mechanism of Artificial Immune Systems (AISs) [37]. AISs are inspired by the biological immune systems. The NS is responsible for eliminating T-cells that recognize self antigens in the thymus. According to the NS, [[bar.v].sub.n] is considered as self antigen when l([[bar.v].sub.n]) > [v.sub.max] and then randomly chosen '1's in [[bar.v].sub.n] change into '0's until l([[bar.v].sub.n]) = [v.sub.max]. If l([[bar.v].sub.n]) [less than or equal to] [v.sub.max], [[bar.v].sub.n] is considered as non-self antigen and is not changed.

In order to increase the exploration ability of the particles, after the completion of the NS, an adaptive mutation process is applied by changing the '0's of every [[bar.v].sub.n] to '1's with "mutation probability" m. The mutation process starts from relatively small values of m to avoid pure random search. In every iteration, m undergoes a linear reduction until it reaches zero at the end of the optimization process. The reduction in the values of m provides the AMBPSO with the adaptation feature.

The AMBPSO is a technique of high computational complexity like all the other evolutionary techniques and thus needs much more CPU time than the MVDR technique to find an optimal solution. In the cases studied here, an Intel Core 2 Duo computer was used and the CPU time per execution was measured around 2 seconds. However, this problem can be overcome by using Graphics Processing Units (GPUs), which provide cheap access to high-performance parallel computing resources and make the algorithm execution 10-100 times faster [25].

A brief description of the AMBPSO algorithm is given below:

1. Choose the values of [N.sub.P], B, [C.sub.1], [C.sub.2], [C.sub.3], [v.sub.max], m, [[bar.l].sub.n] and [[bar.u].sub.n] (n = 1,...,[N.sub.P]), and the maximum number of iterations [T.sub.max] of the optimization process.

2. Initialize random values for [[bar.v].sub.n] (n = 1,...,[N.sub.P]) and apply the NS to correct them. Also, initialize random values for [[bar.x].sub.n] (n = 1,...,[N.sub.P]) inside the search space and calculate their fitness values F([[bar.x].sub.n]).

3. Set [[bar.p].sub.n] = [[bar.x].sub.n] and F([[bar.x].sub.n]) = F([[bar.x].sub.n]) (n = 1,...,[N.sub.P]).

4. Find [F.sub.min] = F([bar.g]) among F([bar.p]n)(n = 1,...,[N.sub.P]).

5. Update [[bar.v].sub.n] (n = 1,...,[N.sub.P]) using (10) and apply NS to correct them.

6. Mutate the '0's of [[bar.v].sub.n] (n = 1,...,[N.sub.P]) according to the value of m.

7. Update [[bar.x].sub.n] (n = 1,...,[N.sub.P]) using (11).

8. Calculate the fitness values F([[bar.x].sub.n]) (n = 1,...,[N.sub.P]).

9. Assign a large fitness value for [[bar.x].sub.n] lying outside the search space.

10. For n = 1,...,[N.sub.P], if F([[bar.X].sub.n]) < F([[bar.p].sub.n]) then [[bar.p].sub.n] = [[bar.x].sub.n].

11. For n = 1,...,[N.sub.P], if F([[bar.p].sub.n]) < F([bar.g]) then [bar.g] = [[bar.p].sub.n].

12. Reduce the value of m according to a linear decrease expression.

13. If [T.sub.max] is not reached, repeat the algorithm from step (5), or else report results and terminate.

4. NUMERICAL RESULTS

The AMBPSO algorithm was applied on a 10-element ULA. The parameters used by the algorithm were: [N.sub.P] = 20, [C.sub.1] = 0.1, [C.sub.2] = [C.sub.3] = 0.5, [v.sub.max] = 4, m = 0.10, and [T.sub.max] = 10000. The ULA receives a SOI arriving from angle [[theta].sub.0] = 30[degrees] and 8 interference signals arriving from respective angles [[theta].sub.n] [member of] {-70[degrees],-40[degrees],-30[degrees],-10[degrees],0[degrees], 10[degrees],50[degrees],70[degrees]}. All the above signals are uncorrelated with each other. Four cases are studied with different spacing q between adjacent elements and different SNR. In the first case, SNR = 30 dB and q = 0.5[lambda] which is the usual spacing for most of the ABF techniques. In the second case, our technique is tested for q [not equal to] 0.5[lambda]. Therefore, q is set to 0.6[lambda], while SNR = 30 dB. In order to explore the efficiency of our technique for smaller and larger values of SNR, two more cases are studied. In the third case, SNR = 15 dB and q = 0.5[lambda], and in the fourth case SNR = 50dB and q = 0.5[lambda].

Initially, the AMBPSO was compared to the conventional BPSO in terms of convergence. Both algorithms use the same fitness function F given in (7). For each case, the AMBPSO and BPSO algorithms were executed 100 times in order to derive comparative graphs that depict the average convergence of F (see Figure 2). Although the AMBPSO converges a little slower than the BPSO, it finally gives better solutions.

[FIGURE 2 OMITTED]

Then, the AMBPSO was compared with the MVDR technique. The optimal excitation weights of the four cases are given respectively in Tables 1-4, while the radiation patterns are shown respectively in Figures 3-6. All the cases show the superiority of the AMBPSO algorithm over a robust ABF technique such as the MVDR. Both techniques succeed to steer the peak of the main lobe towards the SOI and form pattern nulls in the DOA of every interference signal. However, the AMBPSO provides deeper nulls and that's why all the radiation patterns produced by the AMBPSO have lower side lobe level (SLL) than the patterns produced by the MVDR technique. In order to achieve specific values of SLL for certain angular regions, a properly defined term must be added to the fitness function F. Of course, the additional term increases the CPU time required by the AMBPSO to find an optimal solution.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

Finally, the AMBPSO is compared in terms of SINR with the MVDR technique for various SNR values considering a 10-element ULA with q = 0.5[lambda]. For each value of SNR, the AMBPSO algorithm is executed 100 times and statistical results concerning the SINR are extracted (see Table 5). The results show low standard deviation and mean values of SINR close to the respective best values. Therefore, the AMBPSO algorithm seems to have stable and good performance regardless of the SNR values. In addition, the mean SINR achieved by the AMBPSO is always greater than the SINR achieved by the MVDR technique, and their difference increases with increasing SNR.

5. CONCLUSION

The cases studied in the present work show that the AMBPSO converges a little slower than the conventional BPSO, but it finally leads to better solutions. Also, the AMBPSO can be used as an efficient ABF technique capable of producing radiation patterns better than patterns produced by a robust ABF technique such as the MVDR. The AMBPSO succeeds not only to steer the main lobe towards the SOI and form nulls in the DOA of all the interference signals but also to reduce the SLL more than the MVDR technique does. As an ABF technique, the AMBPSO does not need the knowledge of the interference correlation matrix but only the knowledge of the interference DOA. In addition, the AMBPSO algorithm exhibits stable and good behavior for every value of SNR, providing better SINR values than those obtained by the MVDR technique. By using GPUs, the computational complexity can be overcome and then the AMBPSO algorithm can be used by adaptive beamforming networks in real-time applications. Therefore, the AMBPSO seems to be quite promising in the smart antenna technology. As a future work, the AMBPSO will be applied on more complex fitness functions in order not only to control the pattern nulls but also to achieve specific values of SLL.

Received 19 April 2011, Accepted 25 May 2011, Scheduled 5 June 2011

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Z. D. Zaharis (1), * and T. V. Yioultsis (2)

(1) Telecommunications Center, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece

(2) Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece

* Corresponding author: Zaharias D. Zaharis (zaharis@auth.gr).
Table 1. Optimal weight values for
q = 0.5[lambda] and SNR = 30 dB.

m     [w.sub.mvdr]      [w.sub.ambpso]

1    0.352 - j0.227    0.191 + j 0.197
2   0.017 + j 0.405    -0.195 + j 0.490
3   -0.750 + j 0.152   -0.692 - j0.202
4   -0.220 - j0.900     0.165 - j0.983
5     1.000 + j 0        1.000 + j 0
6   0.342 + j 0.940    0.070 + j 1.033
7   -0.921 + j 0.101   -0.905 + j 0.228
8   -0.114 - j0.757    -0.214 - j0.622
9    0.386 - j0.122     0.435 - j0.239
10  -0.092 + j 0.409   0.144 + j 0.096

Table 2. Optimal weight values for
q = 0.6[lambda] and SNR = 30 dB.

m      [w.sub.mvdr]      [w.sub.ambpso]

1     0.387 - j0.377     0.316 - j0.104
2     0.483 - j0.070    0.252 + j 0.256
3    -0.711 + j 0.223   -0.722 + j 0.095
4    -0.668 - j0.518    -0.369 - j0.782
5      1.000 + j 0        1.000 + j 0
6    0.392 + j 0.920     0.181+ j 1.010
7    -0.738 - j0.412    -0.767 - j0.202
8    -0.073 - j0.741    -0.025 - j0.717
9    0.124 + j 0.472    0.223 + j 0.159
10   -0.195 + j 0.503   -0.089 + j 0.285

Table 3. Optimal weight values for
q = 0.5[lambda] and SNR = 15 dB.

m    [w.sub.mvdr]      [w.sub.ambpso]

1   0.354 - j0.226    0.303 + j 0.030
2   0.018 + j0.407    -0.093 + j 0.476
3   -0.751 + j0.152   -0.752 - j0.053
4   -0.219 - j0.901   -0.011 - j0.986
5     1.000 + j0         1.000 + j0
6   0.340 + j0.940     0.195 + j1.020
7   -0.922 + j0.100   -0.901 + j0.171
8   -0.113 - j0.758   -0.153 - j0.671
9   0.389 - j0.122     0.405 - j0.192
10  -0.092 + j0.410    0.010 + j0.210

Table 4. Optimal weight values
for q = 0.5[lambda] and SNR = 50dB.

 m     [w.sub.mvdr]      [w.sub.ambpso]

 1    0.352 - j0.227    0.213 + j 0.034
 2    0.017 + j0.405    -0.127 + j 0.442
 3    -0.750 + j0.152   -0.683 - j0.072
 4    -0.220 - j0.900    0.036 - j0.921
 5      1.000 + j0         1.000 + j0
 6    0.342 + j0.940     0.159 + j0.978
 7    -0.921 + j0.101   -0.921 + j0.185
 8    -0.114 - j0.757   -0.190 - j0.678
 9    0.386 - j0.122     0.427 - j0.194
10    -0.092 + j0.409    0.086 + j0.226

Table 5. SINR derived from MVDR and
AMBPSO for various values
of SNR, considering a 10-element
ULA with q = 0.5[lambda].

SNR    SINR (dB) derived         SINR (dB) derived from AMBPSO
(dB)       from MVDR
                             Best      Worst       Mean      STD

-20        -10.0540        -10.0522   -10.0548   -10.0523   0.0004
-15         -5.1601        -5.1395    -5.1512    -5.1399    0.0020
-10         -0.3701        -0.2975    -0.3692    -0.2998    0.0098
 -5         4.3345          4.5321     4.3422     4.5269    0.0218
 0          8.8967          9.4241     8.5481     9.3749    0.1643
 5          13.4522        14.3768    12.0647    14.2676    0.3628
 10         18.2011        19.3598    15.1371    19.2810    0.4463
 15         22.6889        24.3542    16.5370    24.1008    1.0643
 20         27.3035        29.3509    17.2416    29.0332    1.2290
 25         31.7012        34.3515    22.4314    33.6680    1.4163
 30         36.4811        39.3341    30.2715    38.7648    0.8722
 35         40.4633        44.3440    32.6393    43.1564    1.5287
 40         45.2813        49.3461    36.6898    48.1781    1.2409
 45         49.3217        54.3358    43.2386    52.5134    1.5788
 50         54.8269        59.3317    47.6499    58.3221    1.7439
 55         59.1356        64.3458    52.7630    63.0660    1.6119
 60         63.4345        69.3468    56.3379    67.5858    1.8369
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Author:Zaharis, Z.D.; Yioultsis, T.V.
Publication:Progress In Electromagnetics Research
Article Type:Report
Geographic Code:4EUGR
Date:Jul 1, 2011
Words:5103
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