In this letter, an approach to adaptive antenna array optimisation based on genetic algorithms is assessed in a realistic scattering environment, where interfering signals are modelled using the Student's t-distribution. Simulation results demonstrate the effectiveness of the proposed array control strategy

De Natale, Francesco G.B.

Donelli, Massimo

Lommi, Andrea

Massa, Andrea

Sacchi, Claudio

English

University of Trento : UNITN-eprints

  
 
 
 
 
 
UNIVERSITY 
OF TRENTO 
 DEPARTMENT OF INFORMATION AND COMMUNICATION TECHNOLOGY
  
38050 Povo – Trento (Italy), Via Sommarive 14 
http://www.dit.unitn.it 
 
 
 
 
 
 
 
 
 
 
 
 
 
ON THE USE OF A GENETIC-BASED APPROACH FOR ANTENNA ARRAY 
CONTROL IN A SCATTERING ENVIRONMENT 
 
 
Claudio Sacchi, Massimo Donelli, Francesco G.B. De Natale, 
Andrea Massa, and Andrea Lommi 
 
 
August 2004 
 
Technical Report DIT-04-077 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
. 
  1
On the Use of a Genetic-Based Approach for Antenna Array 
Control in a Scattering Environment 
 
 
 
 
Claudio Sacchi, Massimo Donelli, Francesco G.B. De Natale, Andrea Massa, and 
Andrea Lommi 
 
 
 
Department of Information and Telecommunication Technology (DIT) 
University of Trento 
Via Sommarive 14  
I-38100, Trento, Italy 
E-mail: andrea.massa@ing.unitn.it 
  2
On the Use of a Genetic-Based Approach for Antenna Array 
Control in a Scattering Environment 
 
 
 
 
Claudio Sacchi, Massimo Donelli, Francesco G.B. De Natale, Andrea Massa, and 
Andrea Lommi 
 
 
 
 
Indexing terms:  Adaptive Antenna Arrays, Genetic Algorithms 
 
 
 
 
 
 
Abstract - In this letter, an approach to adaptive antenna 
array optimisation based on genetic algorithms is assessed 
in a realistic scattering environment, where interfering 
signals are modelled using the Student’s t-distribution. 
Simulation results demonstrate the effectiveness of the 
proposed array control strategy. 
  3
I. INTRODUCTION 
 
Since the publication of Applebaum’s landmark paper [1], a strong research interest 
has been focused on the study and development of efficient and computationally 
practicable algorithms for adaptive antenna array control. As a matter of facts, the 
optimal solution proposed in [1] requires the full inversion of the signal covariance 
matrix. Also solutions based on dynamic programming like Least-Mean-Squares 
(LMS) and Recursive-Least-Squares (RLS), which use the steepest descent theory [2], 
do not allow practical low-cost implementations [3]. In this framework, procedures 
based on the Genetic Algorithms (GA) can be regarded as an attractive solution 
[3][4]. Up to now, the effectiveness of GA-based array optimisation strategies has 
been assessed by considering simplified channel models [3]. In order to further assess 
the GA procedure, a scenario characterized by a Poisson-modelled distribution of 
interference arrivals with stochastic lifetimes is assumed in [4]. In this letter, a more 
accurate model of the signal transmission is considered, based on the statistical 
representation of interference arrival angles proposed in [5]. In particular, a GA-based 
array control strategy, namely the Learned Real-Time Genetic Algorithm (LRTGA) 
[4], is assessed by considering a multipath channel with Student’s t-distributed angles 
of arrival. The proposed model is described in Section 2, while Section 3 provides a 
set of selected simulation results and Section 4 draws the conclusions. 
 
 
II. MATHEMATICAL FORMULATION 
 
Let xk(t) be the modulated signals radiated by a mobile transmitter belonging to a K-
user family given by: 
( ){ }tfjkk cettx πγ 2Re)( =  Kk ..1=     (1) 
where ( )tkγ  is the complex envelope and cf  is the working frequency. For a mobile 
channel, characterised by scattering phenomena, the received complex signal by the i-
th antenna element of a equally-spaced V-element array  (being d the inter-element 
distance) is given by [6]: 
tfj
i
R
r
ditfj
krrkr
i
k
c
kr
c
rc
etne
c
ditty π
θλ
πφπθτγα 2
1
sin22
)(sin)( +

 −−=∑
=



 −−
  (2) 
  4
where R is the number of signal paths; rα  and rφ  are the r-th attenuation and phase 
delay, respectively; rτ  is the propagation time delay; rk ,θ  is the angle of arrival 
related to the k-th signal and to the r-th path; )(tni  is the additive white Gaussian 
background noise; c is the speed-light and λc is the free-space wavelength. Under 
narrowband assumption [6], starting from (2) it is possible to define the following 
low-pass signal: 
( )∑
=
−− −=
R
r
idj
rk
j
r
i
k
kr
cr etet
1
sin2
)(
θλ
π
φ τγαξ    (3) 
According to [4], the total amount of co-channel interference impinging at the i-th 
array element is given by:  
∑
=
=
)(
1
)()(
tM
k
i
ki ttI ξ     (4) 
where M(t) is a stochastic process characterised by a Poisson distribution. Without 
loss of generality, let us assume a two Rayleigh fading paths  (R=2) with known 
deterministic attenuations and phase-delays [5]. Moreover, the stochastic 
characterisation of the arrival angles,{ }krθ , be modelled with a Student’s t-distribution 
[5] and the interfering signal lifetime limited to GTχ , χ  being an integer random 
value uniformly distributed in the range [ ]ζ,0 .  
The array control problem consists in the optimal choice of the array weights vector, 
( )Vieaww ijii ,..,1 ; === ϕ , in order to maximize the signal-to-noise-plus-interference-
ratio (SINR) at the array output [1]:  
( )
ww
w
SINR
u
T
T
Λ
Ω= *
2
12
θγ     (5) 
where ( )1θΩ  is a phase-related term referred to the desired signal defined in [4] being 
( )R,11,11 ,...,ˆ θθθ = ; uΛ  is the undesired signal covariance matrix [1] [3] related both to 
the Gaussian background noise, )(tni , and to the co-channel interference, )(tI i ; 2γ  is 
the squared mean value of the slowly time-varying signal envelope. This problem is 
equivalent to maximizing the following cost function [3]: 
( ) ( )
ww
w
wf
y
T
T
Λ
Ω= *
2
1θ     (6) 
  5
where yΛ  is the covariance matrix  related to the observation vector y  [3] [4] (being 
( ) ( ) ( ) ( )tntItty iiii ++= 1ξ ).  
As far as the beamforming system is concerned, the array control is achieved by 
means of digital phase shifters with fixed Dolph-Chebyschev amplitude coefficients, 
{ai}. Consequently, the LRTGA-based optimization procedure maximizes (5) by 
searching for an optimal configuration of the discrete phase coefficients only. To this 
end, the GA-chromosome, { }Nϕϕ=φ ,...,1 , encodes the phase coefficients, iϕ  being the 
coded-representation of i-th phase shift trial solution. The adaptive strategy is carried 
out by a real-time tuning of GA probabilities cP  (crossover probability) and mP  
(mutation probability) according to a “learning strategy” defined in [4]. 
 
 
III. NUMERICAL ASSESSMENT 
 
In the numerical validation, the transmission carrier-to-single interference ratio (C/I) 
has been set to –30dB, whereas the signal-to-background noise ratio is equal to 30dB. 
The parameter ζ  has been set equal to 10. As an example, Figure 1 shows a sample 
realization of Student’s t-distributed angle of arrivals with two degrees of freedom 
[5]. 
As far as the digital phase shifter is concerned, the number of bits for phase 
discretization has been set equal to 7. In this scenario, the proposed GA-based 
approach has been compared with a standard GA (SGA), with the Applebaum’s 
method (using unfeasible continuous phases [1]), and with a realistic implementation 
of the Applebaum’s criterion (with discrete phases). Figure 3 shows the resulting 
running averages of the SINR. It can be observed that LRTGA can significantly 
improve the adaptation capabilities of the SGA-based control algorithm, achieving an 
average SINR equal to 12.48dB (average SINR = 2.22dB for the SGA method). 
Moreover, despite the optimality of the method in [1], the SINR attained by the 
LRTGA is almost equivalent on average to the one achieved by Applebaum’s method 
with discrete phases (average SINR = 11.65dB), even if the latter considers 
continuous modules. Table 1 shows the corresponding average SINR values and 
confirms the improvement in robustness of LRTGA with respect to SGA. For the sake 
of completeness, it is to be mentioned that SINR values provided by sub-optimal 
  6
Applebaum’s strategy get closer to optimal ones for an increasing number of phase 
quantization levels, whereas LRTGA does not improve significantly its performance 
(see Tab. 1). Such phenomenon becomes evident in phase shifters using 8 bits or 
more. Nevertheless, a large number of phase quantization levels inevitably involve 
unacceptable hardware costs, making them unsuitable for marketable 
implementations. 
 
 
IV. CONCLUSIONS 
 
In this letter, a suitable GA-based strategy for adaptive antenna array control was 
assessed in a realistic mobile scattering environment. The performance attained by 
LRTGA was compared with the optimal solution in terms of SINR values. Simulation 
results showed the robustness of the proposed array control strategy with respect to 
conventional approaches, particularly when low-cost hardware implementations are 
required. 
 
 
  7
REFERENCES 
 
 [1] S. P. Applebaum, “Adaptive arrays,” IEEE Trans. Antennas Propagat., Vol. 
A.P-24, No. 5, pp. 585-598, September 1976. 
[2]  B. Widrow, P. E. Mantey, L. J. Griffiths, B. B. Goode, “Adaptive antenna 
systems,” IEEE Proc., 1967, Vol. 55, pp. 2143-2159. 
[3] D. S. Weile, E. Michielssen, “The control of adaptive antenna arrays with 
genetic algorithms using dominance and diploidy,” IEEE Trans. Antennas 
Propagat., vol. 49, No. 10, pp. 1424-1433, Oct. 2001. 
[4] M. Donelli, A. Lommi, A. Massa, C. Sacchi, “Assessment of the GA-Based 
Adaptive Array Control Strategy: The Case of Stochastic Life-Time Co-
Channel Interferences”, Microwave and Optical Technology Letters, Vol. 37, 
No. 3, May 5, 2003, pp. 198-201. 
[5] J. Bach-Andersen, K. Ingemann-Pedersen, ”Angle-of-Arrival Statistics for 
Low Resolution Antennas,” IEEE Trans. Antennas Propagat., vol. 50, No. 3, 
pp. 391-395, March 2002. 
[6] D.R. Van Rheeden, S.C. Gupta, “A Temporal Smoothing Approach to 
Direction of Arrival Estimation of Coherent Signals in Fading Channels”, 
Proc. of IEEE WNCN 1999 Conf., Vol.1, pp. 286-290.  
  8
FIGURE CAPTIONS 
 
• Figure 1.  
Scattered interferers direction of arrivals vs. GA iteration number. 
• Figure 2. 
Running average SINR vs. GA iteration number (7-bit digital phase shifter).  
 
 
 
TABLE CAPTIONS 
 
• Table I. 
Average SINR values. 
 
  9
 
 
 
 
 
 
 
 
-90
-60
-30
0
30
60
90
0 100 200 300 400 500 600 700 800 900
θ
Iteration Number (k)  
 
 
 
 
 
 
 
 
 
 
 
Fig. 1 – C. Sacchi et al., “On the use of a Genetic-based approach...” 
 
  10
 
 
 
 
 
 
 
-20
-10
0
10
20
30
40
50
60
50 150 250 350 450 550 650 750 850
S
I
N
R
 
[
d
B
]
Iteration Number (k)
LRTGA
SGA
Optimal Applebaum
Sub. Optimal Applebaum
 
 
 
 
 
 
 
 
 
 
Fig. 2 – C. Sacchi et al., “On the use of a Genetic-based approach...” 
 
  11
 
 
 
 
 
 
 
  
AVERAGE SINR (dB) 
 
Bit Bo. 
 
SGA 
 
LRTGA 
 
APPLEBAUM 
(DISCRETE PHASES) 
 
APPLEBAUM [1] 
 
5 
 
- 0.39 
 
6.03 
 
1.41 
 
41.5 
 
6 
 
 
- 3.21 
 
10.50 
 
5.56 
 
41.5 
 
7 
 
 
2.22 
 
12.48 
 
11.65 
 
41.5 
 
8 
 
 
8.04 
 
12.71 
 
17.36 
 
41.5 
 
 
 
 
 
 
Tab. 1 – C. Sacchi et al., “On the use of a Genetic-based approach...” 
 


On the Use of a Genetic-Based Approach for Antenna Array Control in a Scattering Environment

Unitn-eprints Research

       UNIVERSITY OF TRENTO  DEPARTMENT OF INFORMATION AND COMMUNICATION TECHNOLOGY  38050 Povo – Trento (Italy), Via Sommarive 14 http://www.dit.unitn.it              ON THE USE OF A GENETIC-BASED APPROACH FOR ANTENNA ARRAY CONTROL IN A SCATTERING ENVIRONMENT   Claudio Sacchi, Massimo Donelli, Francesco G.B. De Natale, Andrea Massa, and Andrea Lommi   August 2004  Technical Report DIT-04-077                                                         .   1On the Use of a Genetic-Based Approach for Antenna Array Control in a Scattering Environment     Claudio Sacchi, Massimo Donelli, Francesco G.B. De Natale, Andrea Massa, and Andrea Lommi    Department of Information and Telecommunication Technology (DIT) University of Trento Via Sommarive 14  I-38100, Trento, Italy E-mail: andrea.massa@ing.unitn.it   2On the Use of a Genetic-Based Approach for Antenna Array Control in a Scattering Environment     Claudio Sacchi, Massimo Donelli, Francesco G.B. De Natale, Andrea Massa, and Andrea Lommi     Indexing terms:  Adaptive Antenna Arrays, Genetic Algorithms       Abstract - In this letter, an approach to adaptive antenna array optimisation based on genetic algorithms is assessed in a realistic scattering environment, where interfering signals are modelled using the Student’s t-distribution. Simulation results demonstrate the effectiveness of the proposed array control strategy.   3I. INTRODUCTION  Since the publication of Applebaum’s landmark paper [1], a strong research interest has been focused on the study and development of efficient and computationally practicable algorithms for adaptive antenna array control. As a matter of facts, the optimal solution proposed in [1] requires the full inversion of the signal covariance matrix. Also solutions based on dynamic programming like Least-Mean-Squares (LMS) and Recursive-Least-Squares (RLS), which use the steepest descent theory [2], do not allow practical low-cost implementations [3]. In this framework, procedures based on the Genetic Algorithms (GA) can be regarded as an attractive solution [3][4]. Up to now, the effectiveness of GA-based array optimisation strategies has been assessed by considering simplified channel models [3]. In order to further assess the GA procedure, a scenario characterized by a Poisson-modelled distribution of interference arrivals with stochastic lifetimes is assumed in [4]. In this letter, a more accurate model of the signal transmission is considered, based on the statistical representation of interference arrival angles proposed in [5]. In particular, a GA-based array control strategy, namely the Learned Real-Time Genetic Algorithm (LRTGA) [4], is assessed by considering a multipath channel with Student’s t-distributed angles of arrival. The proposed model is described in Section 2, while Section 3 provides a set of selected simulation results and Section 4 draws the conclusions.   II. MATHEMATICAL FORMULATION  Let xk(t) be the modulated signals radiated by a mobile transmitter belonging to a K-user family given by: ( ){ }tfjkk cettx πγ 2Re)( =  Kk ..1=     (1) where ( )tkγ  is the complex envelope and cf  is the working frequency. For a mobile channel, characterised by scattering phenomena, the received complex signal by the i-th antenna element of a equally-spaced V-element array  (being d the inter-element distance) is given by [6]: tfjiRrditfjkrrkrikckrcrcetnecditty πθλπφπθτγα 21sin22)(sin)( + −−=∑= −−  (2)   4where R is the number of signal paths; rα  and rφ  are the r-th attenuation and phase delay, respectively; rτ  is the propagation time delay; rk ,θ  is the angle of arrival related to the k-th signal and to the r-th path; )(tni  is the additive white Gaussian background noise; c is the speed-light and λc is the free-space wavelength. Under narrowband assumption [6], starting from (2) it is possible to define the following low-pass signal: ( )∑=−− −=Rridjrkjrikkrcr etet1sin2)(θλπφ τγαξ    (3) According to [4], the total amount of co-channel interference impinging at the i-th array element is given by:  ∑==)(1)()(tMkiki ttI ξ     (4) where M(t) is a stochastic process characterised by a Poisson distribution. Without loss of generality, let us assume a two Rayleigh fading paths  (R=2) with known deterministic attenuations and phase-delays [5]. Moreover, the stochastic characterisation of the arrival angles,{ }krθ , be modelled with a Student’s t-distribution [5] and the interfering signal lifetime limited to GTχ , χ  being an integer random value uniformly distributed in the range [ ]ζ,0 .  The array control problem consists in the optimal choice of the array weights vector, ( )Vieaww ijii ,..,1 ; === ϕ , in order to maximize the signal-to-noise-plus-interference-ratio (SINR) at the array output [1]:  ( )wwwSINRuTTΛΩ= *212θγ     (5) where ( )1θΩ  is a phase-related term referred to the desired signal defined in [4] being ( )R,11,11 ,...,ˆ θθθ = ; uΛ  is the undesired signal covariance matrix [1] [3] related both to the Gaussian background noise, )(tni , and to the co-channel interference, )(tI i ; 2γ  is the squared mean value of the slowly time-varying signal envelope. This problem is equivalent to maximizing the following cost function [3]: ( ) ( )wwwwfyTTΛΩ= *21θ     (6)   5where yΛ  is the covariance matrix  related to the observation vector y  [3] [4] (being ( ) ( ) ( ) ( )tntItty iiii ++= 1ξ ).  As far as the beamforming system is concerned, the array control is achieved by means of digital phase shifters with fixed Dolph-Chebyschev amplitude coefficients, {ai}. Consequently, the LRTGA-based optimization procedure maximizes (5) by searching for an optimal configuration of the discrete phase coefficients only. To this end, the GA-chromosome, { }Nϕϕ=φ ,...,1 , encodes the phase coefficients, iϕ  being the coded-representation of i-th phase shift trial solution. The adaptive strategy is carried out by a real-time tuning of GA probabilities cP  (crossover probability) and mP  (mutation probability) according to a “learning strategy” defined in [4].   III. NUMERICAL ASSESSMENT  In the numerical validation, the transmission carrier-to-single interference ratio (C/I) has been set to –30dB, whereas the signal-to-background noise ratio is equal to 30dB. The parameter ζ  has been set equal to 10. As an example, Figure 1 shows a sample realization of Student’s t-distributed angle of arrivals with two degrees of freedom [5]. As far as the digital phase shifter is concerned, the number of bits for phase discretization has been set equal to 7. In this scenario, the proposed GA-based approach has been compared with a standard GA (SGA), with the Applebaum’s method (using unfeasible continuous phases [1]), and with a realistic implementation of the Applebaum’s criterion (with discrete phases). Figure 3 shows the resulting running averages of the SINR. It can be observed that LRTGA can significantly improve the adaptation capabilities of the SGA-based control algorithm, achieving an average SINR equal to 12.48dB (average SINR = 2.22dB for the SGA method). Moreover, despite the optimality of the method in [1], the SINR attained by the LRTGA is almost equivalent on average to the one achieved by Applebaum’s method with discrete phases (average SINR = 11.65dB), even if the latter considers continuous modules. Table 1 shows the corresponding average SINR values and confirms the improvement in robustness of LRTGA with respect to SGA. For the sake of completeness, it is to be mentioned that SINR values provided by sub-optimal   6Applebaum’s strategy get closer to optimal ones for an increasing number of phase quantization levels, whereas LRTGA does not improve significantly its performance (see Tab. 1). Such phenomenon becomes evident in phase shifters using 8 bits or more. Nevertheless, a large number of phase quantization levels inevitably involve unacceptable hardware costs, making them unsuitable for marketable implementations.   IV. CONCLUSIONS  In this letter, a suitable GA-based strategy for adaptive antenna array control was assessed in a realistic mobile scattering environment. The performance attained by LRTGA was compared with the optimal solution in terms of SINR values. Simulation results showed the robustness of the proposed array control strategy with respect to conventional approaches, particularly when low-cost hardware implementations are required.     7REFERENCES   [1] S. P. Applebaum, “Adaptive arrays,” IEEE Trans. Antennas Propagat., Vol. A.P-24, No. 5, pp. 585-598, September 1976. [2]  B. Widrow, P. E. Mantey, L. J. Griffiths, B. B. Goode, “Adaptive antenna systems,” IEEE Proc., 1967, Vol. 55, pp. 2143-2159. [3] D. S. Weile, E. Michielssen, “The control of adaptive antenna arrays with genetic algorithms using dominance and diploidy,” IEEE Trans. Antennas Propagat., vol. 49, No. 10, pp. 1424-1433, Oct. 2001. [4] M. Donelli, A. Lommi, A. Massa, C. Sacchi, “Assessment of the GA-Based Adaptive Array Control Strategy: The Case of Stochastic Life-Time Co-Channel Interferences”, Microwave and Optical Technology Letters, Vol. 37, No. 3, May 5, 2003, pp. 198-201. [5] J. Bach-Andersen, K. Ingemann-Pedersen, ”Angle-of-Arrival Statistics for Low Resolution Antennas,” IEEE Trans. Antennas Propagat., vol. 50, No. 3, pp. 391-395, March 2002. [6] D.R. Van Rheeden, S.C. Gupta, “A Temporal Smoothing Approach to Direction of Arrival Estimation of Coherent Signals in Fading Channels”, Proc. of IEEE WNCN 1999 Conf., Vol.1, pp. 286-290.    8FIGURE CAPTIONS  • Figure 1.  Scattered interferers direction of arrivals vs. GA iteration number. • Figure 2. Running average SINR vs. GA iteration number (7-bit digital phase shifter).     TABLE CAPTIONS  • Table I. Average SINR values.    9        -90-60-3003060900 100 200 300 400 500 600 700 800 900θIteration Number (k)             Fig. 1 – C. Sacchi et al., “On the use of a Genetic-based approach...”    10       -20-10010203040506050 150 250 350 450 550 650 750 850SINR [dB]Iteration Number (k)LRTGASGAOptimal ApplebaumSub. Optimal Applebaum          Fig. 2 – C. Sacchi et al., “On the use of a Genetic-based approach...”    11         AVERAGE SINR (dB)  Bit Bo.  SGA  LRTGA  APPLEBAUM (DISCRETE PHASES)  APPLEBAUM [1]  5  - 0.39  6.03  1.41  41.5  6   - 3.21  10.50  5.56  41.5  7   2.22  12.48  11.65  41.5  8   8.04  12.71  17.36  41.5       Tab. 1 – C. Sacchi et al., “On the use of a Genetic-based approach...”  

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On the Use of a Genetic-Based Approach for Antenna Array Control in a Scattering Environment

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