- Email: [email protected]
Implementation of cyclostationary signal-based adaptive arrays
Signal Processing 80 (2000) 2249 }2254
Implementation of cyclostationary signal-based adaptive arrays Shiann-Jeng Yu, Fang-Biau Ueng* National Space Program Ozce, 8F, No. 9, Prosperity 1st Road, Science-Based Industrial Park, Hsin-Chu City, Taiwan Received 13 July 1999; received in revised form 3 January 2000
Abstract In this paper, we propose a new approach for implementing the cyclostationary signal-based adaptive array. Two methods for calculating the control vector of the adaptive array are provided. The "rst method employs the eigenvalue decomposition technique and the second method uses a simple cyclostationary correlation method to calculate the control vector from the received data. Simulation examples are illustrated to show the e!ectiveness of the proposed approach. 2000 Elsevier Science B.V. All rights reserved. Zusammenfassung In dieser Arbeit schlagen wir einen neuen Ansatz zur Realisierung von zyklostationaK ren signalbasierten adaptiven Antennenanordnungen vor. Es werden zwei Methoden zur Berechnung des Steuervektors der adaptiven Antennenanordnung vorgestellt. Die erste Methode verwendet die Zerlegung nach Eigenwerten und die zweite Methode benutzt eine einfache zyklostationaK re Korrelationsmethode, um den Steuervektor aus den Empfangsdaten zu berechnen. Simulationsbeispiele zeigen die E$zienz des vorgeschlagenen Verfahrens. 2000 Elsevier Science B.V. All rights reserved. Re2 sume2 Dans cet article, nous proposons une approche pour mettre en wuvre un reH seau adaptatif de cyclostationnariteH baseH sur le signal. Nous fournissions deux meH thodes pour calculer le vecteur de contro( le du reH seau adaptatif. La premie`re meH thode utilise une technique de deH composition en valeurs propres et la seconde meH thode utilise une meH thode simple de correH lation de la cyclostationnariteH pour calculer le vecteur de contro( le des donneH es rec7 ues. Des exemples de simulations sont preH senteH es pour montrer l'e$caciteH de l'approche proposeH e. 2000 Elsevier Science B.V. All rights reserved. Keywords: Adaptive array; Adaptive beamforming; Cyclostationary signal; Eigenvalue decomposition
1. Introduction It is known that the performance of the steered adaptive array beamforming is very sensitive to the mismatch between the steering vector and the * Corresponding author. E-mail addresses: [email protected] [email protected] (F.-B. Ueng).
(S.-J.
Yu),
true direction vector of the desired signal [5]. Many adaptive beamforming techniques without the knowledge of the true direction vector were proposed [2}4,6}10]. A technique called cyclostationary signal-based adaptive beamforming [1,3,4,8}10] employs the signal carrier or baud frequencies of the desired signal to calculate the optimal adaptive weights. The spectral self-coherence restoral (SCORE) algorithms were developed
0165-1684/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 6 8 4 ( 0 0 ) 0 0 1 0 9 - 2
2250
S.-J. Yu, F.-B. Ueng / Signal Processing 80 (2000) 2249 }2254
[1]. In the SCORE algorithms, the LS-SCORE algorithm [5] is the simplest one that estimates the steering vector by using a control vector. However, performance of the LS-SCORE strongly depends on the control vector [1]. A poor control vector results in a very slow convergence speed. The design of the control vector is therefore very critical. In this paper, we reformulate the design problem in terms of the control vector and shows that the optimal control vector can be found by performing the eigenvalue decomposition (EVD). The optimal control vector is the eigenvector with the largest eigenvalue. We also show that the optimal control vector is shown to be proportional to the true direction vector of the desired signal. Because of the heavily computational load of EVD, this paper also presents a simpler method for computing the control vector. The method utilizes cyclostationary property of the desired signal to calculate the control vector. On the consideration of implementation, we present a two-stage architecture for realizing the proposed approach. A recursive leastsquares (RLS)-based adaptive algorithm is also developed. Section 2 of this paper brie#y describes the adaptive array beamforming. In Section 3, we reformulate the design problem and propose two methods for computing the control vector. Section 4 provides simulation examples for the proposed approach. We conclude this paper in Section 5.
2. Adaptive beamforming 2.1. Adaptive array with steering vector Consider an array with N sensor elements. Let the received data vector of the array be given by X(t)"[x (t) x (t) 2 x (t)]2"s(t)S #N(t), (1) , where s(t) is the desired signal with a direction vector S "[s s 2 S ]2, N(t) is an interfer , ence-plus-noise data vector, and & T ' denotes the transpose. The adaptive beamforming feeds each array element with an adaptive weight. The adaptive weights are chosen such that the array output power is minimized while preserving a constant gain in the speci"ed direction. Let the weight vector
be expressed as ="[w w 2 w ]2. The array , output is y(t)"=&X(t), where &H' denotes the complex conjugate and transpose. The adaptive weights are designed to minimize the array output power subject to a direction constraint, i.e. 1"y(t)"2 "=&R = VV (2) subject to =&S "1, where R "1X(t)X&(t)2 and S is a steering vecVV tor. 1 2 represents the in"nite time average. The optimal solution is given by [5]
Minimize
1 = " R\S . (3) S&R\S VV VV In general, the steering vector S shall be chosen as the true direction vector S of the desired signal to achieve maximal signal-to-interference plus noise ratio (SINR) [5]. 2.2. Cyclostationary signal-based adaptive array The cyclostationary signal-based adaptive array estimates the steering vector from the received data. The technique can work based on the basic assumption that the desired signal s(t) is cyclostationary at the cycle frequency a and time delay q, but the interference and noise are not. By de"nition, s(t) is a cyclostationary signal if its cyclic correlation function R? (q)"1s(t#q)sH(t)e\ p?R2 or its QQ cyclic-conjugate correlation function R? (q)" QQH 1s(t#q)s(t)e\ p?R2 is not zero. * denotes the complex conjugate. In the following, we use the cyclic-conjugate correlation method for explanation. Consider a cyclostationary correlation matrix de"ned by R "1X(t#q);&(t)2 "R? (q) VS VVH "1X(t#q)X2(t)e\ p?R2 , (4) where ;(t)"XH(t)e p?R. Based on the cyclostationary assumption above on the desired signal, interference and noise, R can be reduced to VS R "R? (q)S S2O0. (5) VS QQH Eq. (5) implies that the true direction vector S can be found from R . The LS-SCORE algorithm seVS lects a control vector < with S2< O0 and the *1 *1
S.-J. Yu, F.-B. Ueng / Signal Processing 80 (2000) 2249 }2254
2251
steering vector S is thus given by [5] S "R < "R? (q)S S2 < JS . (6) VS *1 QQH *1 Substituting (6) into (3), the weight vector of the LS-SCORE is given by = "kR\R < , *1 VV VS *1 where k is a scalar.
(7)
3. The proposed approach 3.1. The design problem Without loss of generality, the steering vector is expressed as S "R <. Substituting S into (3) VS yields 1 R\R < (8) =" <&R R\R < VV VS SV VV VS and =&R ="(<&Q&R Q<)\, where R " VV VV SV R& and VS Q"[Q Q 2 Q ]"R\R (9) , VV VS with Q "R\R (:, i). R (:, i) is the ith column of G VV VS VS R . Therefore, the optimization problem can be VS given by Minimize =&R ="Maximize <&R <, (10) VV WW where R "Q&R Q"1>(t)>&(t)2 with >(t)" WW VV [y (t) y (t) 2 y (t)]2. It is easy to show that , the optimal control vector < is the eigenvector N with the maximum eigenvalue of R . From (5) and WW (9), we "nd that R "Q&R Q"R R\R "cSHS2 (11) WW VV SV VV VS is a rank-one matrix, where c is a constant. Therefore, the optimal control vector of the proposed EVD-based method, denoted < , is proportional to SH. Instead of performing the EVD for the control vector, a simpler method is proposed in this section. The method utilizes the cyclostationary property on >(t). The cyclostationary method calculates the control vector termed < by < "1>(t#q)fH(t)2 , (12)
Fig. 1. The proposed cyclostationary signal-based adaptive beamforming.
where f(t) is a cyclostationary signal at cycle frequency a and time delay q. There are methods for selecting f(t). In this paper, we select f(t)"yH(t)e p?R"Q2 ;(t). In theory, < can be written as < "1Q&X(t#q);&(t)QH2 "Q&R QH. VS From (5) and (14), we have
(13)
(14)
(15) < "gSHS2QH, where g is a scalar. Using (5), we "nd that Q&S JC&S&R\S is not zero, where VV C"[1 0 2 0]2. Therefore, < of (15) is propor tional to the optimal control vector SH. This method is simple and does not require to perform the EVD. 3.2. Implementation Fig. 1 presents the architecture for implementing the proposed approach. In the upper stage, N parallel adaptive weight vectors Q are performed. G From [5], the RLS algorithm for updating the Q is G given by Q (t)"Q (t!1)#P(t)X(t)eH(t), G G G 1 P(t!1)X(t)X&(t)P(t!1) P(t)" P(t!1)! , d d#X&(t)P(t!1)X(t)
e (t)"u (t)!Q&(t!1)X(t), (16) G G G where d is a forgetting factor with 0(d(1. Next, we consider the implementation of the control vector of the lower stage of Fig. 1. An adaptive
2252
S.-J. Yu, F.-B. Ueng / Signal Processing 80 (2000) 2249 }2254
algorithm by [11] for "nding the eigenvector is used to derive the algorithm for < . The RLS based algorithm is given by < (t)"< (t!1)#(>(t) !e(t)< (t!1))eH(t)/R (t), C e(t)"<& (t!1)>(t), (17) R (t)"dR (t!1)#"e(t)". C C For the cyclostationary method, the control vector can be calculated as follows: < (t)"d< (t!1)#>(t)fH(t).
(18)
3.3. Discussion
nal cyclostationary property of the desired signal in >(t) to estimate the true direction vector. It is observed that the proposed cyclostationary method is similar to the LS-SCORE. Both are trying to estimate S . The main di!erence is that the proposed cyclostationary method estimates S from >(t), but the LS-SCORE estimates S from X(t). Since the SINR is improved in >(t), an estimate of S using the proposed cyclostationary method can achieve a better quality. The LSSCORE su!ers from strong interference in X(t) and results in slow convergence rate. Our experience also shows that the convergence rate of the proposed cyclostationary method is not sensitive to the selection of f(t).
From (9), we "nd that Q "R\R (:, i)"R\R < , (19) G VV VS VV VS /G where < is a vector with the ith entry being one /G and the other entries being zero. R (:, i) can be VS expressed as R (:, i)"1X(t#q)uH(t)2 . VS G From (19) and (17), we "nd that Q can be seen as G one of the LS-SCORE. The control vector < or < in the lower stage of the proposed architecture is equivalent to linearly combined N outputs y (t). G The adaptive array theory [5] and the LS-SCORE show that the interference can be either eliminated or reduced its strength by Q . The adaptive array G theory also show that the vector Q is adopted so G that y (t) is highly correlated to the signal G u (t)"xH(t)e p?R. Because only the desired signal G G s(t) in x (t) is cyclostationary, y (t) will be highly G G correlated to sH(t)SH , where S is the ith entry G G of S . Due to the fact that the desired signal in >(t) is much stronger than the residual interference and noise, the optimal control vector < can be selected to compensate the phase di!erence of the desired signal in >(t) in order to achieve maximal SINR. That is why the optimal control vector is proportional to SH. Due to the fact that the desired signal is dominant in >(t), it is possible to estimate SH from >(t) with more accuracy. The proposed EVD-based method estimates SH from R because WW the desired signal is dominant in >(t) and SH should be the principle eigenvector of R . For simplicity, WW the proposed cyclostationary method uses the sig-
4. Simulation examples In this section, we provide simulation examples to show the e!ectiveness of the proposed approach. A uniform linear array with 4 array elements is used for simulation. The inter-element spacing is half of wavelength. The forgetting factor is d"0.99 and the normalized sampling frequency f is equal to 1 for all of the simulations. The noise is additive white Gaussian noise with variance"0.01. The following simulation results are the average results of 30 independent runs. We "rst demonstrate the performance by using the carrier frequency recovery. Consider a BPSK as the desired signal with the carrier frequency"0, baud rate", signal-to-noise ratio (SNR)"10 dB impinges from !253 o! the broadside. Two BPSK interference signals are present in the 353 and !603 with equal interference-to-noise ratio (INR)"15 dB and equal baud rate". The car rier frequencies of the two interferers are 0.05 and 0.1, respectively. Here we employ the cyclic-conjugate correlation method with q"0 and a"0. Fig. 2 shows the output SINR versus the number of bauds. The control vector of the LS-SCORE is selected as < "[1 020]2. Fig. 2 shows that the *1 proposed approach using both the EVD-based and cyclostationary methods can e$ciently achieve the optimal performance, while the LS-CROSS algorithm converges slowly. The proposed approach using cyclostationary method has a faster
S.-J. Yu, F.-B. Ueng / Signal Processing 80 (2000) 2249 }2254
2253
Fig. 2. The output SINR versus the number of bauds. Solid curve: the proposed EVD-based method. Dash curve: the proposed cyclostationary method. Dash-dot curve: the LS-SCORE algorithm.
Fig. 3. The output SINR versus the number of bauds. Solid curve: the proposed EVD-based method. Dash curve: the proposed cyclostationary method. Dash-dot curve: the LS-SCORE algorithm.
convergence rate than that using the EVD-based method. The cyclostationary method needs about 20 bauds to converge to 3 dB lower than the optimal SINR and the EVD-based method needs about 40 bauds to achieve the same performance. From this "gure, the proposed EVD-based method significantly outperforms the LS-SCORE when the number of bauds is larger than 20. Next, we demonstrate the tracking performance of the proposed approach. The scenario of Example 1 is considered again, but the angle of arrival of the desired signal is linearly varying from !253 to !53. Fig. 3 shows the tracking performance of the proposed approach. The proposed two-stage architecture can e$ciently track the desired signal. Again, the proposed approach outperforms the LS-SCORE. In this case, the cyclostationary method has a better convergence behavior than the EVD-based method. The proposed cyclostationary and EVD-based methods achieve almost the same SINR after 90 bauds.
problem in terms of the control vector. Two methods for calculating the optimal control vector are provided. The "rst method derives the control vector from the EVD. Due to the heavily computational load of the EVD, the second method calculates the control vector using the simple cyclostationary correlation of >(t). A two-stage architecture for implementing the proposed approach is presented. An RLS-based adaptive algorithm for real-time computing the control vector is also developed.
5. Conclusion This paper extends the LS-SCORE and presents a new approach for the cyclostationary signalbased adaptive array. We reformulate the design
References [1] B.G. Agee, S.V. Schell, W.A. Gardner, Spectral selfcoherence restoral: a new approach to blind adaptive extraction using antenna arrays, Proc. IEEE 78 (April 1990) 753}766. [2] J.F. Cardoso, A. Souloumiac, Blind beamforming for nonGaussian signals, IEE Proc., Pt. F 140 (December 1993) 362}370. [3] L. Castedo, A.R. Figueiras, An adaptive beamforming technique based on cyclostationary signal properties, IEEE Trans. Signal Process. 43 (July 1995) 1637}1650. [4] L. Castedo, C.-Y. Tseng, L.J. Gri$ths, A new cost function for adaptive beamforming using cyclostationary signal properties, Proceedings of the International Conference on Acoustics, Speech and Signal Processing, Minneapolis, MN, April 1993, pp. 284}287.
2254
S.-J. Yu, F.-B. Ueng / Signal Processing 80 (2000) 2249 }2254
[5] R.T. Comption Jr., Adaptive Antennas, Concepts, and Performance, Prentice-Hall, Englewood Cli!s, NJ, 1989. [6] M.C. Dogan, J.M. Mendel, Cumulant-based blind optimum beamforming, IEEE Trans. Aerospace Electron. Systems 30 (July 1994) 722}740. [7] R.P. Gooch, J.D. Lundell, The CM array: an adaptive beamformer for constant modulus signals, Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, Tokyo, Japan, April 1986, pp. 2523}2526. [8] R. Ho, Q. Wu, K.M. Wong, Implementation of cyclic beamforming technique on mobile communication sys-
tems, Proceedings of the second Workshop on Cyclostationary Signals, Monterey, CA, July 1994, pp. 16.1}16.9. [9] Q. Wu, K.M. Wong, Adaptive beamforming of cyclic signal and fast implementation, Proceedings of the International Conference on Acoustics, Speech and Signal Processing, Adelaide, South Australia, April 1994, pp. 157}160. [10] Q. Wu, K.M. Wong, Blind adaptive beamforming for cyclostationary signals, IEEE Trans. Signal Process. 44 (November 1996) 2757}2767. [11] B. Yang, Projection approximation subspace tracking, IEEE Trans. Signal Process. 43 (January 1995) 95}107.
Implementation of cyclostationary signal-based adaptive arrays Shiann-Jeng Yu, Fang-Biau Ueng* National Space Program Ozce, 8F, No. 9, Prosperity 1st Road, Science-Based Industrial Park, Hsin-Chu City, Taiwan Received 13 July 1999; received in revised form 3 January 2000
Abstract In this paper, we propose a new approach for implementing the cyclostationary signal-based adaptive array. Two methods for calculating the control vector of the adaptive array are provided. The "rst method employs the eigenvalue decomposition technique and the second method uses a simple cyclostationary correlation method to calculate the control vector from the received data. Simulation examples are illustrated to show the e!ectiveness of the proposed approach. 2000 Elsevier Science B.V. All rights reserved. Zusammenfassung In dieser Arbeit schlagen wir einen neuen Ansatz zur Realisierung von zyklostationaK ren signalbasierten adaptiven Antennenanordnungen vor. Es werden zwei Methoden zur Berechnung des Steuervektors der adaptiven Antennenanordnung vorgestellt. Die erste Methode verwendet die Zerlegung nach Eigenwerten und die zweite Methode benutzt eine einfache zyklostationaK re Korrelationsmethode, um den Steuervektor aus den Empfangsdaten zu berechnen. Simulationsbeispiele zeigen die E$zienz des vorgeschlagenen Verfahrens. 2000 Elsevier Science B.V. All rights reserved. Re2 sume2 Dans cet article, nous proposons une approche pour mettre en wuvre un reH seau adaptatif de cyclostationnariteH baseH sur le signal. Nous fournissions deux meH thodes pour calculer le vecteur de contro( le du reH seau adaptatif. La premie`re meH thode utilise une technique de deH composition en valeurs propres et la seconde meH thode utilise une meH thode simple de correH lation de la cyclostationnariteH pour calculer le vecteur de contro( le des donneH es rec7 ues. Des exemples de simulations sont preH senteH es pour montrer l'e$caciteH de l'approche proposeH e. 2000 Elsevier Science B.V. All rights reserved. Keywords: Adaptive array; Adaptive beamforming; Cyclostationary signal; Eigenvalue decomposition
1. Introduction It is known that the performance of the steered adaptive array beamforming is very sensitive to the mismatch between the steering vector and the * Corresponding author. E-mail addresses: [email protected] [email protected] (F.-B. Ueng).
(S.-J.
Yu),
true direction vector of the desired signal [5]. Many adaptive beamforming techniques without the knowledge of the true direction vector were proposed [2}4,6}10]. A technique called cyclostationary signal-based adaptive beamforming [1,3,4,8}10] employs the signal carrier or baud frequencies of the desired signal to calculate the optimal adaptive weights. The spectral self-coherence restoral (SCORE) algorithms were developed
0165-1684/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 6 8 4 ( 0 0 ) 0 0 1 0 9 - 2
2250
S.-J. Yu, F.-B. Ueng / Signal Processing 80 (2000) 2249 }2254
[1]. In the SCORE algorithms, the LS-SCORE algorithm [5] is the simplest one that estimates the steering vector by using a control vector. However, performance of the LS-SCORE strongly depends on the control vector [1]. A poor control vector results in a very slow convergence speed. The design of the control vector is therefore very critical. In this paper, we reformulate the design problem in terms of the control vector and shows that the optimal control vector can be found by performing the eigenvalue decomposition (EVD). The optimal control vector is the eigenvector with the largest eigenvalue. We also show that the optimal control vector is shown to be proportional to the true direction vector of the desired signal. Because of the heavily computational load of EVD, this paper also presents a simpler method for computing the control vector. The method utilizes cyclostationary property of the desired signal to calculate the control vector. On the consideration of implementation, we present a two-stage architecture for realizing the proposed approach. A recursive leastsquares (RLS)-based adaptive algorithm is also developed. Section 2 of this paper brie#y describes the adaptive array beamforming. In Section 3, we reformulate the design problem and propose two methods for computing the control vector. Section 4 provides simulation examples for the proposed approach. We conclude this paper in Section 5.
2. Adaptive beamforming 2.1. Adaptive array with steering vector Consider an array with N sensor elements. Let the received data vector of the array be given by X(t)"[x (t) x (t) 2 x (t)]2"s(t)S #N(t), (1) , where s(t) is the desired signal with a direction vector S "[s s 2 S ]2, N(t) is an interfer , ence-plus-noise data vector, and & T ' denotes the transpose. The adaptive beamforming feeds each array element with an adaptive weight. The adaptive weights are chosen such that the array output power is minimized while preserving a constant gain in the speci"ed direction. Let the weight vector
be expressed as ="[w w 2 w ]2. The array , output is y(t)"=&X(t), where &H' denotes the complex conjugate and transpose. The adaptive weights are designed to minimize the array output power subject to a direction constraint, i.e. 1"y(t)"2 "=&R = VV (2) subject to =&S "1, where R "1X(t)X&(t)2 and S is a steering vecVV tor. 1 2 represents the in"nite time average. The optimal solution is given by [5]
Minimize
1 = " R\S . (3) S&R\S VV VV In general, the steering vector S shall be chosen as the true direction vector S of the desired signal to achieve maximal signal-to-interference plus noise ratio (SINR) [5]. 2.2. Cyclostationary signal-based adaptive array The cyclostationary signal-based adaptive array estimates the steering vector from the received data. The technique can work based on the basic assumption that the desired signal s(t) is cyclostationary at the cycle frequency a and time delay q, but the interference and noise are not. By de"nition, s(t) is a cyclostationary signal if its cyclic correlation function R? (q)"1s(t#q)sH(t)e\ p?R2 or its QQ cyclic-conjugate correlation function R? (q)" QQH 1s(t#q)s(t)e\ p?R2 is not zero. * denotes the complex conjugate. In the following, we use the cyclic-conjugate correlation method for explanation. Consider a cyclostationary correlation matrix de"ned by R "1X(t#q);&(t)2 "R? (q) VS VVH "1X(t#q)X2(t)e\ p?R2 , (4) where ;(t)"XH(t)e p?R. Based on the cyclostationary assumption above on the desired signal, interference and noise, R can be reduced to VS R "R? (q)S S2O0. (5) VS QQH Eq. (5) implies that the true direction vector S can be found from R . The LS-SCORE algorithm seVS lects a control vector < with S2< O0 and the *1 *1
S.-J. Yu, F.-B. Ueng / Signal Processing 80 (2000) 2249 }2254
2251
steering vector S is thus given by [5] S "R < "R? (q)S S2 < JS . (6) VS *1 QQH *1 Substituting (6) into (3), the weight vector of the LS-SCORE is given by = "kR\R < , *1 VV VS *1 where k is a scalar.
(7)
3. The proposed approach 3.1. The design problem Without loss of generality, the steering vector is expressed as S "R <. Substituting S into (3) VS yields 1 R\R < (8) =" <&R R\R < VV VS SV VV VS and =&R ="(<&Q&R Q<)\, where R " VV VV SV R& and VS Q"[Q Q 2 Q ]"R\R (9) , VV VS with Q "R\R (:, i). R (:, i) is the ith column of G VV VS VS R . Therefore, the optimization problem can be VS given by Minimize =&R ="Maximize <&R <, (10) VV WW where R "Q&R Q"1>(t)>&(t)2 with >(t)" WW VV [y (t) y (t) 2 y (t)]2. It is easy to show that , the optimal control vector < is the eigenvector N with the maximum eigenvalue of R . From (5) and WW (9), we "nd that R "Q&R Q"R R\R "cSHS2 (11) WW VV SV VV VS is a rank-one matrix, where c is a constant. Therefore, the optimal control vector of the proposed EVD-based method, denoted < , is proportional to SH. Instead of performing the EVD for the control vector, a simpler method is proposed in this section. The method utilizes the cyclostationary property on >(t). The cyclostationary method calculates the control vector termed < by < "1>(t#q)fH(t)2 , (12)
Fig. 1. The proposed cyclostationary signal-based adaptive beamforming.
where f(t) is a cyclostationary signal at cycle frequency a and time delay q. There are methods for selecting f(t). In this paper, we select f(t)"yH(t)e p?R"Q2 ;(t). In theory, < can be written as < "1Q&X(t#q);&(t)QH2 "Q&R QH. VS From (5) and (14), we have
(13)
(14)
(15) < "gSHS2QH, where g is a scalar. Using (5), we "nd that Q&S JC&S&R\S is not zero, where VV C"[1 0 2 0]2. Therefore, < of (15) is propor tional to the optimal control vector SH. This method is simple and does not require to perform the EVD. 3.2. Implementation Fig. 1 presents the architecture for implementing the proposed approach. In the upper stage, N parallel adaptive weight vectors Q are performed. G From [5], the RLS algorithm for updating the Q is G given by Q (t)"Q (t!1)#P(t)X(t)eH(t), G G G 1 P(t!1)X(t)X&(t)P(t!1) P(t)" P(t!1)! , d d#X&(t)P(t!1)X(t)
e (t)"u (t)!Q&(t!1)X(t), (16) G G G where d is a forgetting factor with 0(d(1. Next, we consider the implementation of the control vector of the lower stage of Fig. 1. An adaptive
2252
S.-J. Yu, F.-B. Ueng / Signal Processing 80 (2000) 2249 }2254
algorithm by [11] for "nding the eigenvector is used to derive the algorithm for < . The RLS based algorithm is given by < (t)"< (t!1)#(>(t) !e(t)< (t!1))eH(t)/R (t), C e(t)"<& (t!1)>(t), (17) R (t)"dR (t!1)#"e(t)". C C For the cyclostationary method, the control vector can be calculated as follows: < (t)"d< (t!1)#>(t)fH(t).
(18)
3.3. Discussion
nal cyclostationary property of the desired signal in >(t) to estimate the true direction vector. It is observed that the proposed cyclostationary method is similar to the LS-SCORE. Both are trying to estimate S . The main di!erence is that the proposed cyclostationary method estimates S from >(t), but the LS-SCORE estimates S from X(t). Since the SINR is improved in >(t), an estimate of S using the proposed cyclostationary method can achieve a better quality. The LSSCORE su!ers from strong interference in X(t) and results in slow convergence rate. Our experience also shows that the convergence rate of the proposed cyclostationary method is not sensitive to the selection of f(t).
From (9), we "nd that Q "R\R (:, i)"R\R < , (19) G VV VS VV VS /G where < is a vector with the ith entry being one /G and the other entries being zero. R (:, i) can be VS expressed as R (:, i)"1X(t#q)uH(t)2 . VS G From (19) and (17), we "nd that Q can be seen as G one of the LS-SCORE. The control vector < or < in the lower stage of the proposed architecture is equivalent to linearly combined N outputs y (t). G The adaptive array theory [5] and the LS-SCORE show that the interference can be either eliminated or reduced its strength by Q . The adaptive array G theory also show that the vector Q is adopted so G that y (t) is highly correlated to the signal G u (t)"xH(t)e p?R. Because only the desired signal G G s(t) in x (t) is cyclostationary, y (t) will be highly G G correlated to sH(t)SH , where S is the ith entry G G of S . Due to the fact that the desired signal in >(t) is much stronger than the residual interference and noise, the optimal control vector < can be selected to compensate the phase di!erence of the desired signal in >(t) in order to achieve maximal SINR. That is why the optimal control vector is proportional to SH. Due to the fact that the desired signal is dominant in >(t), it is possible to estimate SH from >(t) with more accuracy. The proposed EVD-based method estimates SH from R because WW the desired signal is dominant in >(t) and SH should be the principle eigenvector of R . For simplicity, WW the proposed cyclostationary method uses the sig-
4. Simulation examples In this section, we provide simulation examples to show the e!ectiveness of the proposed approach. A uniform linear array with 4 array elements is used for simulation. The inter-element spacing is half of wavelength. The forgetting factor is d"0.99 and the normalized sampling frequency f is equal to 1 for all of the simulations. The noise is additive white Gaussian noise with variance"0.01. The following simulation results are the average results of 30 independent runs. We "rst demonstrate the performance by using the carrier frequency recovery. Consider a BPSK as the desired signal with the carrier frequency"0, baud rate", signal-to-noise ratio (SNR)"10 dB impinges from !253 o! the broadside. Two BPSK interference signals are present in the 353 and !603 with equal interference-to-noise ratio (INR)"15 dB and equal baud rate". The car rier frequencies of the two interferers are 0.05 and 0.1, respectively. Here we employ the cyclic-conjugate correlation method with q"0 and a"0. Fig. 2 shows the output SINR versus the number of bauds. The control vector of the LS-SCORE is selected as < "[1 020]2. Fig. 2 shows that the *1 proposed approach using both the EVD-based and cyclostationary methods can e$ciently achieve the optimal performance, while the LS-CROSS algorithm converges slowly. The proposed approach using cyclostationary method has a faster
S.-J. Yu, F.-B. Ueng / Signal Processing 80 (2000) 2249 }2254
2253
Fig. 2. The output SINR versus the number of bauds. Solid curve: the proposed EVD-based method. Dash curve: the proposed cyclostationary method. Dash-dot curve: the LS-SCORE algorithm.
Fig. 3. The output SINR versus the number of bauds. Solid curve: the proposed EVD-based method. Dash curve: the proposed cyclostationary method. Dash-dot curve: the LS-SCORE algorithm.
convergence rate than that using the EVD-based method. The cyclostationary method needs about 20 bauds to converge to 3 dB lower than the optimal SINR and the EVD-based method needs about 40 bauds to achieve the same performance. From this "gure, the proposed EVD-based method significantly outperforms the LS-SCORE when the number of bauds is larger than 20. Next, we demonstrate the tracking performance of the proposed approach. The scenario of Example 1 is considered again, but the angle of arrival of the desired signal is linearly varying from !253 to !53. Fig. 3 shows the tracking performance of the proposed approach. The proposed two-stage architecture can e$ciently track the desired signal. Again, the proposed approach outperforms the LS-SCORE. In this case, the cyclostationary method has a better convergence behavior than the EVD-based method. The proposed cyclostationary and EVD-based methods achieve almost the same SINR after 90 bauds.
problem in terms of the control vector. Two methods for calculating the optimal control vector are provided. The "rst method derives the control vector from the EVD. Due to the heavily computational load of the EVD, the second method calculates the control vector using the simple cyclostationary correlation of >(t). A two-stage architecture for implementing the proposed approach is presented. An RLS-based adaptive algorithm for real-time computing the control vector is also developed.
5. Conclusion This paper extends the LS-SCORE and presents a new approach for the cyclostationary signalbased adaptive array. We reformulate the design
References [1] B.G. Agee, S.V. Schell, W.A. Gardner, Spectral selfcoherence restoral: a new approach to blind adaptive extraction using antenna arrays, Proc. IEEE 78 (April 1990) 753}766. [2] J.F. Cardoso, A. Souloumiac, Blind beamforming for nonGaussian signals, IEE Proc., Pt. F 140 (December 1993) 362}370. [3] L. Castedo, A.R. Figueiras, An adaptive beamforming technique based on cyclostationary signal properties, IEEE Trans. Signal Process. 43 (July 1995) 1637}1650. [4] L. Castedo, C.-Y. Tseng, L.J. Gri$ths, A new cost function for adaptive beamforming using cyclostationary signal properties, Proceedings of the International Conference on Acoustics, Speech and Signal Processing, Minneapolis, MN, April 1993, pp. 284}287.
2254
S.-J. Yu, F.-B. Ueng / Signal Processing 80 (2000) 2249 }2254
[5] R.T. Comption Jr., Adaptive Antennas, Concepts, and Performance, Prentice-Hall, Englewood Cli!s, NJ, 1989. [6] M.C. Dogan, J.M. Mendel, Cumulant-based blind optimum beamforming, IEEE Trans. Aerospace Electron. Systems 30 (July 1994) 722}740. [7] R.P. Gooch, J.D. Lundell, The CM array: an adaptive beamformer for constant modulus signals, Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, Tokyo, Japan, April 1986, pp. 2523}2526. [8] R. Ho, Q. Wu, K.M. Wong, Implementation of cyclic beamforming technique on mobile communication sys-
tems, Proceedings of the second Workshop on Cyclostationary Signals, Monterey, CA, July 1994, pp. 16.1}16.9. [9] Q. Wu, K.M. Wong, Adaptive beamforming of cyclic signal and fast implementation, Proceedings of the International Conference on Acoustics, Speech and Signal Processing, Adelaide, South Australia, April 1994, pp. 157}160. [10] Q. Wu, K.M. Wong, Blind adaptive beamforming for cyclostationary signals, IEEE Trans. Signal Process. 44 (November 1996) 2757}2767. [11] B. Yang, Projection approximation subspace tracking, IEEE Trans. Signal Process. 43 (January 1995) 95}107.