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Robust Adaptive Beamforming for Multiple Signals of Interest with Cycle Frequency Error
EURASIP Journal on Advances in Signal Processing volume 2010, Article number: 873916 (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 [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 [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 [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 [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
or
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
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]
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
where the array output 
, and the reference signal 
is given by
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
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
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
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
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
where 
is a 
function. Substituting (11) into (9) yields
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
with 
. Equation (13) reveals that
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
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:
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:
Using the steepest descent method for solving (17), we take the derivative of 
with respect to 
as follows:
where 
denotes the real part of 
and 
. In vector form, we can rewrite 
of (18) as follows:
with 
, 
, is given by
Accordingly, we update 
at the time instant 
as follows:
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
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
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)
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
where the term 
in (25) can be expressed as
We note from (26) that the cross-terms disappear due to the assumed uncorrelation between 
and 
. As to the term 
, we have
The approximation is obtained due to 
when 
as the number of data snapshots 
is large enough. According to (26) and (27), we have
Substituting (28) into (25), we obtain an approximation for 
as follows:
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
For binary phase-shift-keying (BPSK) signals, we assume that
where 
and 
denote the amplitude and the random phases equal to 
for the 
th SOI, respectively. Consequently, we have
Consider the case of 
. Then, (32) leads to
Therefore, (30) becomes
Next, substituting (31) into (20) and performing some algebraic manipulations, we have
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
is kept, where 
. Accordingly, the maximum singular value of 
is approximately equal to
where 
. As a result, the 
th step size parameter 
of (22) is equal to
It follows from (21) that
since 
. Substituting (34) and (38) into (39) yield
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
where 
. Hence, under the condition of 
, we have
It follows from (42) that
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
when 
is large enough. It is easy to show that
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.
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.
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.
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Acknowledgment
This work was supported by the National Science Council of Taiwan under Grant NSC97-2221-E002-174-MY3.
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Lee, JH., Huang, CC. Robust Adaptive Beamforming for Multiple Signals of Interest with Cycle Frequency Error. EURASIP J. Adv. Signal Process. 2010, 873916 (2010). https://doi.org/10.1155/2010/873916
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DOI: https://doi.org/10.1155/2010/873916