Robust Adaptive Beamforming for Multiple Signals of Interest with Cycle Frequency Error
© Ju-Hong Lee and Chia-Cheng Huang. 2010
Received: 3 September 2010
Accepted: 10 December 2010
Published: 15 December 2010
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.
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) . 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 [3–5]. 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  to deal with the problem of blind adaptive signal extraction. As the number of data snapshots approaches infinity, it has been shown in  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 [6–8]. 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  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 , 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  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  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 , 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.
2.1. Signal Cyclostationarity
2.2. The MLS-SCORE Algorithm
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 (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  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 , .
4. Convergence Analysis
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.
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.
This work was supported by the National Science Council of Taiwan under Grant NSC97-2221-E002-174-MY3.
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