Advanced
Optimum Array Processing with Variable Linear Constraint
Optimum Array Processing with Variable Linear Constraint
Journal of information and communication convergence engineering. 2014. Sep, 12(3): 140-144
Copyright © 2014, The Korea Institute of Information and Commucation Engineering
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
  • Received : January 29, 2014
  • Accepted : April 25, 2014
  • Published : September 30, 2014
Download
PDF
e-PUB
PubReader
PPT
Export by style
Article
Author
Metrics
Cited by
TagCloud
About the Authors
Byong Kun Chang
chang@incheon.ac.kr

Abstract
A general linearly constrained adaptive array is examined in the weight vector space to illustrate the array performance with respect to the gain factor. A narrowband linear adaptive array is implemented in a coherent signal environment. It is shown that the gain factor in the general linearly constrained adaptive array has an effect on the linear constraint gain of the conventional linearly constrained adaptive array. It is observed that a variation of the gain factor of the general linearly constrained adaptive array results in a variation of the distance between the constraint plane and the origin in the translated weight vector space. Simulation results are shown to demonstrate the effect of the gain factor on the nulling performance.
Keywords
I. INTRODUCTION
If the desired signal is uncorrelated with incoming interference signals, a linearly constrained adaptive array successfully estimates the desired signal by reducing the interference signals [1] . If the desired signal is correlated partially or totally (i.e., coherent) with the interference signals, the desired signal is partially or totally cancelled in the array output depending on the extent of correlation between the desired signal and the interference signals.
Some methods, such as the spatial smoothing approach [2 , 3] , master-slave type array processing [4] , alternate mainbeam nulling method [5] , and general linearly constrained adaptive array [6] , have been proposed to prevent the signal cancellation phenomenon in a correlated signal environment. A drawback of the methods proposed in [2 - 5] is that they employ additional hardware or algorithms to reduce the effect of the coherent interferences.
In this study, the general linearly constrained adaptive array is examined in the weight vector space to find the nulling performance in [6] in terms of the gain factor, which turns out to be the reduction of the gain in the look direction in the translated weight vector space. It is shown that the variation of the gain factor results in the variation of the distance between the constraint plane and the origin in the translated weight vector space, which has a geometric effect of shifting the constraint plane with respect to the origin.
The simulation results are shown to illustrate the nulling performance with respect to the gain factor. It is shown that the general linearly constrained adaptive array performs better than the linearly constrained adaptive array with respect to the elimination of the coherent interferences.
II. OPTIMUM WEIGHT VECTOR
It is assumed that a desired signal is incident from a known direction (i.e., the look direction) while coherent interferences come from unknown directions on the narrowband general linearly constrained adaptive array with N sensor elements, as shown in Fig. 1 . The weights wn , 1 ≤ n N , are adjusted to find an optimum weight vector to estimate the desired signal with unit gain in the look direction. It is to be noted in the figure that the conventional beamformer output is multiplied by the gain factor g to yield the desired response dk .
PPT Slide
Lager Image
Narrowband general linearly constrained adaptive array. LMS: least mean squares.
To find the array weights that optimally estimate the desired signal in the look direction while eliminating the undesired interference signals as much as possible, we solve the following optimization problem in which the mean square error is minimized subject to the unit gain constraint in the look direction.
PPT Slide
Lager Image
where the error signal between the adaptive array output and the desired response is given by
PPT Slide
Lager Image
the output of the adaptive array is represented as
PPT Slide
Lager Image
the desired response is given by
PPT Slide
Lager Image
and the signal vector, the weight vector w , and the steering vector s for the look direction are respectively given by
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
where β denotes the radian frequency of the desired signal, τ o = d sin θ o / c , θ o , indicates the incident angle of the desired signal from the array normal, d refers to the interelement spacing, c represents the signal propagation velocity, k indicates the iteration index, and E , T , and H denote the expectation, transpose, and complex conjugate transpose, respectively.
It can be shown that the mean squared error E [| ek | 2 ] is given by
PPT Slide
Lager Image
where
PPT Slide
Lager Image
The method of Lagrange multipliers [1] is used for finding the optimal solution by solving the unconstrained optimization problem with the objective function
PPT Slide
Lager Image
where λ denotes a Lagrange multiplier.
Here, the optimum weight vector is given by
PPT Slide
Lager Image
It is observed in (11) that the optimum weight vector lies between the uniform weight of the conventional beamformer and the optimum weight vector for the unit gain constraint depending on the value of the gain factor.
III. OPTIMUM WEIGHT VECTOR IN THE TRANSLATED WEIGHT VECTOR SPACE
We designate the weight vector w translated by g
PPT Slide
Lager Image
as v . Then, the optimization problem in (1) is formulated in terms of v as
PPT Slide
Lager Image
Solving (12) by using the Lagrange multiplier method, we obtain the optimum weight vector as
PPT Slide
Lager Image
The optimum weight vector in (13) is for the unit gain constraint scaled by (1 – g ).
IV. GENERAL LINEARLY CONSTRAINED LMS ALGORITHM
The steepest descent method [7] is employed to find the iterative solution for the optimum weight vector, where the next weight vector is given by the current one added by the negative gradient with respect to v and scaled by a convergence parameter as follows:
PPT Slide
Lager Image
where μ denotes the convergence parameter.
The objective function is given by
PPT Slide
Lager Image
Applying the constraints in (12) to (14) to find the value of λ and manipulating the resulting equation, we find the following iterative solution:
PPT Slide
Lager Image
where
PPT Slide
Lager Image
I denotes the N × N identity matrix. If R is estimated using an instantaneous approximation, i.e., R = xk
PPT Slide
Lager Image
, a stochastic adaptive algorithm is derived; it can be represented as
PPT Slide
Lager Image
where ∗ denotes a complex conjugate.
Eq. (18) is called the general linearly constrained least mean square (LMS) algorithm. In (19), it is observed that the updated weight vector vk is projected onto the constrained subspace, which is an orthogonal complement of the subspace spanned by the steering vector of the desired signal. Then, the projected weight vector is added by the scaled version of the steering vector for the look direction. Notice that the steering vector is orthogonal to the constrained plane.
Thus, a variation of the gain factor results in a variation of the distance between the constraint plane and the origin in the translated weight vector space (i.e., an increase in the gain factor results in a decrease in the distance). This phenomenon has an effect on the nulling performance of the general linearly constrained adaptive array in terms of the gain factor.
V. SIMULATION RESULTS
To illustrate the nulling performance of the general linearly constrained adaptive array in terms of the gain factor, the simulation results in [6] are redisplayed for the cases of one and two coherent interferences.
A narrowband linear array with 7 equispaced sensor elements is employed to examine the performance of the general linearly constrained adaptive array. The incoming signals are assumed to be plane waves. The desired signal is assumed to be a sinusoid incident on the linear array at the array normal. The cases for one and two coherent interferences are simulated. The nulling performances are compared with respect to the gain factor g and with the linearly constrained adaptive array. The convergence parameter μ is assumed to be 0.001.
- A. Case of One Coherent Interference Case
It is assumed that a coherent interference is incident at 30° from the array normal. The variation of the error power between the array output and the desired signal is displayed in terms of the gain factor g in Fig. 2 . The optimum value of g , which yields the minimum error power, is shown to be 0.331. The comparison of the array performance for g = 0.331, 0.01, and the linearly constrained adaptive array is shown in Figs. 3 and 4 with respect to the array output and the desired signal for k = 1 − 1000 and 29001 − 30000 smaples, respectively. It is demonstrated that the case for g = 0.331 performs best while the case for g = 0.01 performs better than the general linearly constrained adaptive array. It is observed that the desired signal of the linearly constrained adaptive array disappears (i.e., is cancelled out) in the array output for 29002 ≤ k ≤ 30000.
PPT Slide
Lager Image
Variation of the error power in terms of the gain factor for the case of one coherent interference.
PPT Slide
Lager Image
Comparison of the array output and the desired signal for the case of one coherent interference for 1 ≤ k ≤ 1000.
PPT Slide
Lager Image
Comparison of the array output and the desired signal for the case of one coherent interference for 29001 ≤ k ≤ 3000.
The beam patterns are shown in Fig. 5 , in which the case for g = 0.331 forms a deep null (-51 dB) in the direction of the interference (30°).
PPT Slide
Lager Image
Comparison of the beam patterns for the case of one coherent interference.
- B. Case of Two Coherent Interferences
It is assumed that two coherent interferences are incident at -54.3° and 57.5° from the array normal. The variation of the error power between the array output and the desired signal is displayed in Fig. 6 . The optimum value of g is shown to be 0.632. The comparison of the array performances for g = 0.632, 0.01 and the linearly constrained adaptive array is shown in Figs. 7 and 8 with respect to the array output and the desired signal for k = 1 − 1000 and 29001 − 3000 samples, respectively. It is shown that the case for g = 0.632 performs best, while the case for g = 0.01 performs similar to the linearly constrained adaptive array. The beam patterns are shown in Fig. 9 , in which the case for g = 0.632 forms two deep nulls (-36.6 dB and -30.4 dB) at the incident angles (-54.3° and 57.5°) of the two coherent interferences, while the gains for the linearly constrained adaptive array are -21.0 dB and - 21.8 dB and the gains for g = 0.01 are -22.7 dB and -25.1 dB, respectively.
PPT Slide
Lager Image
Variation of the error power in terms of the gain factor for the case of two coherent interferences.
PPT Slide
Lager Image
Comparison of the array output and the desired signal for the case of two coherent interferences for 1 ≤ k ≤ 1000.
PPT Slide
Lager Image
Comparison of the array output and the desired signal for the case of two coherent interferences for 29001 ≤ k ≤ 30000.
PPT Slide
Lager Image
Comparison of the beam patterns for the case of two coherent interferences.
VI. CONCLUSIONS
A narrowband general linearly constrained adaptive array is examined in the weight vector space to calculate the array performance with respect to the gain factor in a coherent signal environment. It is observed that a variation of the gain factor results in a variation of the distance between the constraint plane and the origin in the translated weight vector space. This phenomenon has an effect on the nulling performance of the general linearly constrained adaptive array. Further, it is shown that the general linearly constrained adaptive array performs better than the linearly constrained adaptive array.
Acknowledgements
This work was supported by the University of Incheon Research Grant, 2011.
BIO
Byong Kun Chang
Dr. Chang received his B.E. degree from Dept. of Electronics Engineering, Yonsei University, in 1975, his MS degree from Dept. of Electrical and Computer Engineering, University of Iowa, in 1985, and his Ph.D. degree from Dept. of Electrical and Computer Engineering, University of New Mexico, in 1991. Currently, he is Professor in the Dept. of Electrical Engineering at University of Incheon. His research interests include adaptive signal processing, array signal processing, and microcomputer applications.
References
Frost O. L. 1972 “An algorithm for linearly constrained adaptive array processing,” Proceedings of the IEEE 60 (8) 926 - 935    DOI : 10.1109/PROC.1972.8817
Shan T. J. , Kailath T. 1985 “Adaptive beamforming for coherent signals and interference,” IEEE Transactions on Acoustics, Speech and Signal Processing 33 (3) 527 - 536    DOI : 10.1109/TASSP.1985.1164583
Reddy V. U. , Paulraj A. , Kailath T. 1987 “Performance analysis of the optimum beamformer in the presence of correlated sources and its behavior under spatial smoothing,” IEEE Transactions on Acoustics, Speech and Signal Processing 35 (7) 927 - 936    DOI : 10.1109/TASSP.1987.1165239
Chang B. K. , Ahmed N. , Youn D. H. 1988 “Fast convergence adaptive beamformers with reduced signal cancellation,” in Proceedings of the 22nd Asilomar Conference on Signals, Systems and Computers Pacific Grove, CA 823 - 827
Chang B. K. , Jeon C. H. , Song D. H. 2009 “Performance improvement in alternate mainbeam nulling by adaptive estimation of convergence parameters in linearly constrained adaptive arrays,” International Journal of KIMICS 7 (3) 392 - 398
Chang B. K. , Kim T. Y. , Lee Y. K. 2012 “A novel approach to general linearly constrained adaptive arrays,” Journal of Information & Communication Convergence Engineering 10 (2) 108 - 116    DOI : 10.6109/jicce.2012.10.2.108
Widrow B. , Stearns S. D. 1985 Adaptive Signal Processing Prentice-Hall Englewood Cliffs, NJ