doi:10.1016/j.sigpro.2006.04.007 
Copyright © 2006 Elsevier B.V. All rights reserved.
Adaptive beamforming for binary phase shift keying communication systems
aSchool of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, UK
Received 27 September 2005;
revised 21 February 2006;
accepted 11 April 2006.
Available online 8 June 2006.
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Abstract
The paper revisits adaptive beamforming assisted receiver for multiple antenna aided multiuser systems that employ binary phase shift keying (BPSK) modulation. The standard minimum mean square error (MMSE) design is based on the criterion of minimising the mean square error (MSE) between the beamformer's desired output and complex-valued beamformer's output. Since the desired output for BPSK systems is real-valued, minimising the MSE between the beamformer's desired output and real-part of the beamformer's output can significantly improve the bit error rate (BER) performance, and we refer to this alternative MMSE design as the real-valued MMSE (RV-MMSE) to contrast to the standard complex-valued MMSE (CV-MMSE) design. The minimum BER (MBER) design however still outperforms the RV-MMSE solution, particularly for overloaded systems where degree of freedom of the antenna array is smaller than the number of BPSK users. Adaptive implementation of this RV-MMSE beamforming design is realised using a least mean square (LMS) type adaptive algorithm, which we refer to as the RV-LMS, in comparison to the standard CV-LMS algorithm. The RV-LMS adaptive beamformer is shown to have a similar computational complexity as the adaptive MBER beamforming implementation known as the least bit error rate (LBER), imposing only half of the computational requirements of the CV-LMS algorithm.
Keywords: Adaptive beamforming; Binary phase shift keying modulation; Minimum mean square error; Minimum bit error rate
Fig. 1. Geometric structure of the four-element linear array having λ/2 spacing used in the simulation, where λ is the wavelength.
Fig. 2. User-1 BER comparison of three beamforming designs for the four-element array system supporting three users. BERs of the RV-MMSE and MBER beamformers are indistinguishable.
Fig. 3. Conditional probability density functions p(y|+1) (surfaces), marginal conditional probability density functions p(yR|+1) (curves), signal subsets 
and 
(points) for the four-element array system supporting three users with 
: (a) CV-MMSE, (b) RV-MMSE and (c) MBER. The beamformer weight vector is normalised to a unit length.
Fig. 4. User-1 BER comparison of three beamforming designs for the four-element array system supporting eight users.
Fig. 5. Conditional probability density functions p(y|+1) (surfaces), marginal conditional probability density functions p(yR|+1) (curves), signal subsets 
and 
(points) for the four-element array system supporting eight users with 
: (a) CV-MMSE, (b) RV-MMSE and (c) MBER. The beamformer weight vector is normalised to a unit length.
Fig. 6. User-1 BER comparison of three beamforming designs for the four-element array system supporting nine users.
Fig. 7. Conditional probability density functions p(y|+1) (surfaces), marginal conditional probability density functions p(yR|+1) (curves), signal subsets 
and 
(points) for the four-element array system supporting nine users with 
: (a) CV-MMSE, (b) RV-MMSE and (c) MBER. The beamformer weight vector is normalised to a unit length.
Fig. 8. Marginal conditional probability density functions p(yR|+1) (curves) and signal subsets 
(points) for the four-element array system supporting nine users with 
: (a) RV-MMSE and (b) MBER. The beamformer weight vector is normalised to a unit length.
Fig. 9. Learning curves of the adaptive RV-LMS and LBER algorithms averaged over 100 runs for the four-element array system supporting nine users with 
: (a) training and (b) decision-directed adaptation after 40-symbol training. The step size μ=0.005 for the RV-LMS, the step size μ=0.01 and kernel variance 
for the LBER.
Fig. 10. User-1 BER comparison of three beamforming designs for the four-element array system supporting 10 users.
Fig. 11. Conditional probability density functions p(y|+1) (surfaces), marginal conditional probability density functions p(yR|+1) (curves), signal subsets 
and 
(points) for the four-element array system supporting 10 users with 
: (a) CV-MMSE, (b) RV-MMSE and (c) MBER. The beamformer weight vector is normalised to a unit length.
Fig. 12. Marginal conditional probability density functions p(yR|+1) (curves) and signal subsets 
(points) for the four-element array system supporting 10 users with 
: (a) RV-MMSE and (b) MBER. The beamformer weight vector is normalised to a unit length.
Fig. 13. User-1 BER comparison of three adaptive beamformers for the four-element array system supporting 10 users. Training length is 1000 symbols, the CV-LMS and RV-LMS algorithms have a step size μ=0.002, while the LBER algorithm has a step size μ=0.01 and kernel variance 
.
Table 1.
Comparison of computational complexity per weight update for the three adaptive BPSK beamformers, where L is the dimension of the weight vector
Table 2.
Locations of users in terms of angle of arrival for the simulation
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