Open Access

Robust Adaptive Beamforming for Multiple Signals of Interest with Cycle Frequency Error

EURASIP Journal on Advances in Signal Processing20102010:873916

https://doi.org/10.1155/2010/873916

Received: 3 September 2010

Accepted: 10 December 2010

Published: 15 December 2010

Abstract

This paper deals with the problem of robust adaptive array beamforming by exploiting the signal cyclostationarity. Recently, a novel cyclostationarity-exploiting beamforming method has been proposed by J.-H. Lee and C.-C. Huang (2009) for dealing with the situation of multiple signals of interest (SOI) based on the LS-SCORE algorithm. This method is referred to as the multiple LS-SCORE (MLS-SCORE) algorithm. However, the MLS-SCORE algorithm suffers from severe performance degradation even if there is a small mismatch in the cycle frequencies of the SOIs. In this paper, we evaluate the performance of the MLS-SCORE algorithm in the presence of cycle frequency error (CFE). The output SINR of an adaptive beamforming using the MLS-SCORE algorithm deteriorates like a function as the number of data snapshots increases. To tackle this difficulty, we present an efficient method to find an appropriate estimate for each of the cycle frequencies of the SOIs iteratively to achieve robust adaptive beamforming against the CFE. Simulation results for showing the effectiveness of the proposed method are provided.

1. Introduction

For conventional array beamforming, the a  priori information required for adapting the weights is either the direction vector or the waveform of the signal of interest (SOI) [1]. A signal with cyclostationarity has the statistical property of correlating with either a frequency-shift (referred to as spectral self-coherence) or a complex-conjugate version (referred to as spectral conjugate self-coherence) of itself. For example, spectral self-coherence is induced at multiples of the symbol rate in PCM signals and spectral conjugate self-coherence is commonly induced at twice the carrier frequency in BPSK signals [2, 3]. By restoring those properties at a known value of frequency separation, it is possible to extract the SOI and suppress the signals not of interest ( ) and noise. Adaptive beamforming utilizing signal cyclostationarity has been widely considered [35]. These cyclostationarity-exploiting techniques do not need training signals, the knowledge of array manifold, or noise characteristics. The least-square spectral self-coherent restoral (LS-SCORE) algorithm has been presented by [3] to deal with the problem of blind adaptive signal extraction. As the number of data snapshots approaches infinity, it has been shown in [3] that the performance of the LS-SCORE algorithm approaches that of the conventional beamforming methods developed by maximizing the output signal-to-interference plus noise ratio (SINR). The a  priori information required by the LS-SCORE algorithm is only the cycle frequency of the SOI. Hence, its performance is sensitive to the accuracy of the presumed cycle frequency. However, the actual cycle frequency may not be known very well in some applications due to, for example, the phenomenon of Doppler shift. Accordingly, several existing works deal with the cycle frequency error (CFE) have been presented by [68]. The robust cyclostationarity-exploiting beamforming methods presented by [6, 7] are in conjunction with the SCORE algorithms. These methods developed an iterative procedure to find an appropriate estimate of the cycle frequency of the SOI. Recently, a robust cyclostationarity-exploiting direction-finding approach [8] adopts the average of the cyclic correlation matrices in a presumed range of cycle frequencies including the actual one to alleviate the performance degradation due to the CFE. Nevertheless, the aforementioned methods consider the case of one SOI with CFE.

For many practical applications, such as satellite communications [9], an antenna array is required to possess the beamforming capability that receives more than one SOI with specified gain constraints while suppressing all . This goal can be achieved using an antenna array with a multiple-beam pattern [9, 10]. Recently, a novel cyclostationarity-exploiting beamforming method has been proposed by [11] for dealing with the situation of multiple SOIs based on the LS-SCORE algorithm. This approach is referred to as the multiple LS-SCORE (MLS-SCORE) algorithm. It has been shown in [11] that the solution of the MLS-SCORE algorithm converges to the solution of the conventional linearly constrained minimum variance (LCMV) as the number of data snapshots approaches infinity. In this paper, we first evaluate the performance of the MLS-SCORE algorithm in the presence of CFE. This results in an analytical formula that demonstrates the behavior of the performance degradation for the MLS-SCORE algorithm. It is shown that the output SINR of an adaptive array beamformer using the MLS-SCORE algorithm deteriorates like a function as the number of data snapshots increases. To overcome the CFE difficulty, we then develop an efficient method to formulate the problem as an optimization problem for reducing the CFE of the SOIs iteratively. Finally, the convergence property of the proposed method is evaluated. The effectiveness of the proposed method is demonstrated by several simulation examples. It is shown that the proposed method can effectively cope with the performance degradation for the MLS-SCORE algorithm to achieve robust capability against the CFE.

This paper is organized as follows: in Section 2, we briefly describe the property of signal cyclostationarity, the MLS-SCORE algorithm [11], and the performance analysis of the MLS-SCORE algorithms under CFE. The proposed method is presented in Section 3. The convergence analysis of the proposed method is presented in Section 4. Simulation examples for confirming the effectiveness of the proposed method are provided in Section 5. Finally, we conclude the paper in Section 6.

2. Preliminaries

2.1. Signal Cyclostationarity

Many man-made communication signals exhibit cyclostationarity with cycle frequency equal to the twice of the carrier frequency or multiples of the baud rate or combinations of these [2]. According to [2], a signal is said to possess the second-order periodicity with cycle frequency α if and only if the cyclic or the cyclic conjugate autocorrelation function given by
(1)
or
(2)
does not equal zero at cycle frequency α for some time delay , where the superscript " " denotes the complex conjugate and represents the finite-time average operation. Moreover, is addressed as possessing the self-coherent or conjugate self-coherent properties for or does not equal zero at cycle frequency α. In matrix form, the cyclic conjugate autocorrelation matrix associated with the data vector can be expressed as
(3)

where the superscript " " denotes the transpose operation.

2.2. The MLS-SCORE Algorithm

Consider that there are total ( ) far-field narrowband signals including SOIs and impinging on an -element antenna array. Assume that the background noise is spatially white. The received data vector is given by [1]
(4)

where and are the waveforms of the th SOI and the th , and represent the direction vectors of the th SOI and the th , and includes the and noise, respectively.

Without loss of generality, we suppose that the SOIs are cyclostationary and have the cycle frequencies , where denotes the set of cycle frequencies of the SOIs. However, is not cyclostationary at and is temporally uncorrelated with the SOIs. Based on the MLS-SCORE algorithm [11], the optimal weight vector is given by
(5)
where the array output , and the reference signal is given by
(6)
where denotes the control signal and c a fixed control vector. Moreover, the superscript " " denotes the conjugate transpose. Let the sampling interval be . The optimal weight vector of (5) with data snapshots (i.e., ) used is given by
(7)
where and are the sample autocorrelation matrix of and the sample cross-correlation vector of and , respectively. It has been shown in [11] that converges to the solution of the conventional optimum array beamforming based on the linearly constrained minimum variance (LCMV) criterion when approaches infinity. We note from (4) and (6) that , where the sample cross-correlation matrix is defined as
(8)

where is the sample cyclic conjugate autocorrelation matrix of . We observe from (7) that the a priori information required for computing the optimal vector is only the cycle frequencies of the SOIs. Hence, the performance of the MLS-SCORE algorithm is sensitive to the accuracy of the presumed cycle frequency for each of the SOIs. However, the actual cycle frequency may not be known very well in some applications due to, for example, the phenomenon of Doppler shift. Next, we evaluate the performance of the MLS-SCORE algorithm in the presence of CFE.

2.3. Performance Analysis under CFE

From (8), the sample cyclic conjugate correlation matrix at the presumed frequencies is approximately equal to
(9)
as the number of data snapshots is large enough, where is the sample cyclic conjugate autocorrelation function of the th SOI, and denotes the sample cyclic conjugate autocorrelation matrix of . In fact, also includes the sample cyclic cross-correlations between the SOIs, the SOIs and the , the SOIs and noise, and the and noise. However, they are negligible when is large enough. Due to the fact that the cyclic spectrum of a cyclostationary signal is discrete in the cycle frequency, the cyclic conjugate correlation function of the th SOI is given by
(10)
where 's are the cycle frequencies of th SOI, denotes the strength of the th SOI at cycle frequency , and represents the Kronecker delta. According to (10), the sample cyclic conjugate autocorrelation function of the th SOI is given by
(11)
where is a function. Substituting (11) into (9) yields
(12)
Let the presumed cycle frequency of the th SOI be denoted by (i.e., ), where and represent the actual cycle frequency and the amount of CFE of the th SOI, respectively. Then, (12) becomes
(13)
with . Equation (13) reveals that
(14)

From (14), we note that the effect of the and noise is negligible when . Accordingly, is approximately equal to , where is a constant. It has been shown by [11] that of (7) converges to the LCMV beamformer as long as for all . On the contrary, when , we note that there exists the effects of cycle leakage on through a window due to the fact that when equals to an integer or approaches infinity. Consequently, the output SINR of an adaptive array beamformer based on the MLS-SCORE algorithm exists periodic nulls as the number of data snapshots increases. This leads to performance degradation for the MLS-SCORE algorithm.

3. The Proposed Method

From (14), we observe that both of the and approach zero as increases when CFE exists. Hence, the performance degradation for the MLS-SCORE algorithm becomes severer as increases. To overcome the difficulty, we present an efficient method in conjunction with the MLS-SCORE algorithm to find an appropriate estimate for , .

First, we derive a squared error based on the solution of the MLS-SCORE algorithm shown by (7) and replace with to obtain
(15)
where . We note that of (15) is greater than or equal to zero. Hence, minimizing is equivalent to maximizing the second term of (15) which is redefined as follows:
(16)
We note that the objective function achieves its maximum when and decays to zero as the number of snapshots increases when . Therefore, (16) can be used as an objective function for formulating an optimization problem. Then, we resort to solving the following optimization problem:
(17)
Using the steepest descent method for solving (17), we take the derivative of with respect to as follows:
(18)
where denotes the real part of and . In vector form, we can rewrite of (18) as follows:
(19)
with , , is given by
(20)
Accordingly, we update at the time instant as follows:
(21)
where is a diagonal matrix. The diagonal entries , , are positive real-valued parameters referred to as the step-size parameters. To ensure the convergence of the steepest-descent algorithm used by (21), we set the th step-size parameter equal to
(22)
where denotes the maximum singular value of the matrix , and is the appropriate positive real value determined by experiment. Substituting of (21) into (7), the corresponding optimal weight vector at the time instant is given by
(23)
For practical implementation, we compute the required sample correlation matrix and the cross-correlation vector by utilizing (24), where and is a preset positive integer. Since becomes more appropriate as increases, we use data snapshots to update the corresponding correlations for increasing the effect on the estimates of the considered correlations shown by  (24)
(24)

4. Convergence Analysis

Here, the convergence property of the proposed method is evaluated. For simplicity, we set . The objective function of (16) can be rewritten as
(25)
where the term in (25) can be expressed as
(26)
We note from (26) that the cross-terms disappear due to the assumed uncorrelation between and . As to the term , we have
(27)
The approximation is obtained due to when as the number of data snapshots is large enough. According to (26) and (27), we have
(28)
Substituting (28) into (25), we obtain an approximation for as follows:
(29)
We note from (29) that the last term vanishes asymptotically due to (note that ) becomes more appropriate as increases. Accordingly, the th entry in (19) can be further expressed as
(30)
For binary phase-shift-keying (BPSK) signals, we assume that
(31)
where and denote the amplitude and the random phases equal to for the th SOI, respectively. Consequently, we have
(32)
Consider the case of . Then, (32) leads to
(33)
Therefore, (30) becomes
(34)
Next, substituting (31) into (20) and performing some algebraic manipulations, we have
(35)
At the time instant , the time interval and the estimated cycle frequencies are given by . In order to make the influence due to negligible, it is appropriate to make sure that the relationship given by
(36)
is kept, where . Accordingly, the maximum singular value of is approximately equal to
(37)
where . As a result, the th step size parameter of (22) is equal to
(38)
It follows from (21) that
(39)
since . Substituting (34) and (38) into (39) yield
(40)
Here, we prove that if to ensure the convergence of the proposed method. Substituting the extreme value for into (40) and performing some necessary algebraic manipulations yields
(41)
where . Hence, under the condition of , we have
(42)
It follows from (42) that
(43)
From (43), we note that if ,   if , and if . Accordingly, there exists some that can be appropriately chosen between 1 and 2 to make that . Similarly, substituting the other extreme value for into (40), we have
(44)
when is large enough. It is easy to show that
(45)

for . Hence, there exists some between 1 and 2 such that if . As a result, by exploiting the spectral conjugate self-coherence property of the BPSK signals, the convergence of the proposed method can be guaranteed. However, for other types of cyclostationary signals such as QPSK signals or QAM signals, the convergence property may be different from that of BPSK signals. As a result, we have to find from the complex waveforms of QPSK signals or QAM signals. Then, we follow the similar procedure as described by (33) to (44) to show the convergence property for other types of cyclostationary signals such as QPSK signals or QAM signals.

5. Simulation Examples

Here, we present two simulation examples to show the effectiveness of the proposed method. For all simulations, we use a uniform linear array (ULA) with array elements and interelement spacing equal to , where is the wavelength of the SOIs. Assume that the SOI and are BPSK signals with rectangular pulse shape. The SOI have the signal-to-noise ratio (SNR) and baud rate equal to 5 dB and 5/11, respectively. Two SNOIs with cycle frequencies equal to 4.6 and 7.8 impinge on the array from −20° and 40° off broadside, respectively. Moreover, the SNOI have the interference-to-noise ratio (INR) and baud rate equal to 10 dB and 5/11, respectively. The sampling interval is set to 0.1, , the control vector is given by , and , for simplicity.

Example 1.

We present the output SINR versus the number of data snapshots for comparison. In this example, we consider the case of two SOIs ( ) with cycle frequencies impinging on the array from {10°, 50°} off broadside. Moreover, we assume that the CFE of the SOIs is , that is, . We observe from Figure 1 that there are periodic nulls for the original MLS-SCORE algorithm with CFE as the number data snapshots increases. In contrast, the proposed method can effectively cope with the performance degradation due to the CFE and provides the performance very close to that of the original MLS-SCORE algorithm without CFE.
Figure 1

Output SINR versus number of data snapshots for Example 1 .

Example 2.

Consider that there are three SOIs ( ) two of the SOIs are the same as those used by Example 1, and the other SOI has different cycle frequency equal to 9 and is impinging on the array from −20° off broadside. Here, the CFE is set to , that is, . As expected, we observe from Figure 2 that the performance degradation of the original MLS-SCORE algorithm becomes severer as the number data snapshots increases when the CFE exists. The proposed method works satisfactorily in the same circumstances.
Figure 2

Output SINR versus number of data snapshots for Example 2 .

6. Conclusion

This paper has evaluated the performance degradation of the original MLS-SCORE algorithm in the presence of CFE. An efficient method in conjunction with the MLS-SCORE algorithm has been proposed to overcome the CFE difficulty for achieving robust adaptive beamforming. Based on the proposed method, an appropriate estimate for each of the cycle frequencies of the signals of interest is found iteratively by utilizing the steepest-descent method. The convergence of the proposed method has been shown for the case of using BPSK signals. Simulation results demonstrate that an adaptive beamforming using the proposed method can effectively cure the performance deterioration due to CFE and provides the performance very close to that of using the original MLS-SCORE algorithm without CFE.

Declarations

Acknowledgment

This work was supported by the National Science Council of Taiwan under Grant NSC97-2221-E002-174-MY3.

Authors’ Affiliations

(1)
Department of Electrical Engineering, Graduate Institute of Communication Engineering, and Graduate Institute of Biomedical Electronics and Bioinformatics, National Taiwan University
(2)
Graduate Institute of Communication Engineering, National Taiwan University

References

  1. van Trees HL: Optimum Array Processing, Part IV of Detection, Estimation, and Modulation Theory. John Wiley & Sons, New York, NY, USA; 2002.Google Scholar
  2. Gardner WA, Napolitano A, Paura L: Cyclostationarity: half a century of research. Signal Processing 2006, 86(4):639-697. 10.1016/j.sigpro.2005.06.016View ArticleMATHGoogle Scholar
  3. Agee BG, Schell SV, Gardner WA: Spectral self-coherence restoral: a new approach to blind adaptive signal extraction using antenna arrays. Proceedings of the IEEE 1990, 78(4):753-767. 10.1109/5.54812View ArticleGoogle Scholar
  4. Wu Q, Wong KM: Blind adaptive beamforming for cyclostationary signals. IEEE Transactions on Signal Processing 1996, 44(11):2757-2767. 10.1109/78.542182View ArticleGoogle Scholar
  5. Du KL, Swamy MNS: A class of adaptive cyclostationary beamforming algorithms. Circuits, Systems, and Signal Processing 2008, 27(1):35-63. 10.1007/s00034-007-9009-4View ArticleMATHGoogle Scholar
  6. Lee J-H, Lee YT: Robust adaptive array beamforming for cyclostationary signals under cycle frequency error. IEEE Transactions on Antennas and Propagation 1999, 47(2):233-241. 10.1109/8.761062View ArticleGoogle Scholar
  7. Lee J-H, Lee YT, Shih WH: Efficient robust adaptive beamforming for cyclostationary signals. IEEE Transactions on Signal Processing 2000, 48(7):1893-1901. 10.1109/78.847776View ArticleGoogle Scholar
  8. Zhang J, Liao G, Wang J: Robust direction finding for cyclostationary signals with cycle frequency error. Signal Processing 2005, 85(12):2386-2393. 10.1016/j.sigpro.2005.01.015View ArticleMATHGoogle Scholar
  9. Mayhan JT: Area coverage adaptive nulling from geosynchronous satellites: phased arrays versus multiple-beam antennas. IEEE Transactions on Antennas and Propagation 1986, 34(3):410-419. 10.1109/TAP.1986.1143824View ArticleGoogle Scholar
  10. Yu KB: Adaptive beamforming for satellite communication with selective earth coverage and jammer nulling capability. IEEE Transactions on Signal Processing 1996, 44(12):3162-3166. 10.1109/78.553494View ArticleGoogle Scholar
  11. Lee J-H, Huang C-C: Blind adaptive beamforming for cyclostationary signals: a subspace projection approach. IEEE Antennas and Wireless Propagation Letters 2009, 8: 1406-1409.View ArticleGoogle Scholar

Copyright

© Ju-Hong Lee and Chia-Cheng Huang. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.