Adaptive Beamforming with Small Samples and Low Complexity for Large-Scale Arrays

In this paper, we propose a novel adaptive beamforming method based on interference plus noise covariance matrix (INCM) reconstruction with few samples and low complexity. Unlike existing techniques, we make use of the fact that large-scale antenna arrays can significantly increase the spectral analysis’s resolution by estimating the steering vector (SV) and power of the interfering signal using the discrete Fourier transform (DFT). In addition, a simple spatial rotation operation is employed to resolve the power “leakage” problem of the DFT spectrum and increase estimation precision. The proposed method achieves a high output SINR by reconstructing an accurate INCM with the advantage of simple implementation and low computation complexity. Finally, we verify the effectiveness of the proposed method by numerical simulation and result analysis.


Introduction
Adaptive beamforming is an important sensor array processing technology, which is widely used in many fields.The desired signal (DS) is received by modifying the beamformer weights of the array dynamically while suppressing interference and noise.However, non-ideal propagation environments or conditions may be encountered in practice, causing significant degradation in the performance of conventional beamforming methods.To improve the ability to suppress interference, many effective methods have been proposed, mainly including diagonal loading (DL) methods [1], feature spacebased methods [2], uncertainty set-based methods [3], and interference plus noise covariance matrix (INCM)-based reconstruction methods [4] [10].
Modern sensor arrays are growing in size to provide improved resolution and enhanced performance.While existing adaptive beamforming methods are used for large-scale arrays, a significant number of samples are often required to obtain acceptable performance.In actual applications, especially when external disturbances are spatially diverse and time-varying, it may be difficult to satisfy the need for a large sample size.In addition, the computational complexity introduced by huge arrays and sample sizes is not suitable for the implementation of online processing.Therefore, developing adaptive beamformers with few samples and low complexity is important for large-scale sensor arrays.
The INCM reconstruction approach has attracted great attention for its potential to eliminate the DS component from the sample covariance matrix (SCM) to improve the beamformer performance.In this paper, we make use of the fact that large-scale array antennas can significantly enhance the resolution of discrete Fourier transform (DFT)-based spectral analysis to present a novel low-complexity adaptive beamforming method with few samples based on INCM reconstruction.By simultaneously estimating the steering vector (SV) and power of the interfering signal based on the DFT, we reconstruct the INCM.The accuracy of SV estimation is further improved by a spatial rotation operation for the case of spectral 'leakage' in the DFT spectrum.

Signal Model
Considering a large ULA with  ≫ 1 antenna and an array element spacing of  /2,  is the wavelength.Assuming that DS and  interferences strike the array from   , … ,  ∈ 90 , 90 , the array observation vector is where   ,  ,  , ⋯ ,  ∈ ℂ denotes the array manifold,   ∈ ℂ denotes the signal, and   ∈ ℂ is the complex Gaussian noise with variance  .The SV of the -th source is The output SINR of the array is defined as where  ∈ ℂ is the weight vector,  is the power of DS.The theoretical form of the covariance matrix  is where  represents the power of the -th interference and  ∑      is INCM.The optimal weight vector is

SVs and power estimation
According to the known array geometry, the SVs of signals can be determined by estimating their direction of arrival (DOA).By defining the DFT matrix ,   /√.Therefore, the DFT spectrum of the SV  is  [5].
If  → ∞, there is always an integer  /2 sin  such that  √ and all other elements are zero.Thus,  is sparse and all power is concentrated at the  -th DFT point, generalizing to the case of  1 incident sources.

𝐅𝐱 𝑡
,   ,   , ⋯ ,       ( 7 ) which should have a clear peak at the position  ,  0,1, . and Therefore, we can estimate the DOAs and power of the signal according to the peaks and positions of .In fact, we can directly calculate  ˆ  ,  , … ,  according to the DFT spectrum     of   to further reduce the computational complexity Then, we distinguish DS and interferences by incorporating the prior angular sector information of DS.Suppose that  is the angular sector where DS is located, and  ‾ is the complement sector of  in the entire spatial domain [6].By letting    ,   and substituting into Equation ( 8) to get     , ⌈  ⌉ , ⌊. ⌋ and ⌈. ⌉ denote rounding operation.As a result, the locations of the peaks on the set   and its complement can be used to calculate the respective DOAs of the DS and interferences.Furthermore, the interference's power can be evaluated simultaneously from the peak value

DOAs and Power Estimate Correction
While the DOAs of the incident signals do not exactly satisfy Equation ( 6) for a given integer  , the power "leakage" will occur, leading to a DOA estimate error.By further concentrating the signal power at the DFT locations  ,  0,

INCM reconstruction
Based on the known array geometry and the estimated DOAs of the signals, SV estimations of DS and interference  ˆ ,  ˆ , … ,  ˆ can be calculated.Combined with the corrected interference power estimates  ,  , . . .,  , the INCM can be reconstructed as [6]  ˆ ∑   ˆ  ˆ   (18) By using  ˆ and  ˆ to replace the  and  in Equation ( 5), the proposed beamformer is The computational complexity of the proposed method mainly includes    for calculating the DFT spectrum,   for calculating  ˆ,   for the spatial rotation operation, and the remaining operations are relatively negligible.Generally, higher output SINR can be achieved for smaller  and  , and we set  10 in subsequent simulations.Therefore, the total computational complexity is about      .

Example 1: Ideal case
We analyze the beam pattern of the proposed method for the ideal case and plot it in Figure 1.It can be seen that even in the case of a single snapshot, the proposed method can still form a deeper zero trap in the interference direction.The obtained beam pattern is close to the optimal weights vector (Equation ( 5)), resulting in high output SINR.

Example 2: mismatch due to signal look direction error
We simulate and analyze the effect of the signal look direction error on the beamformer performance.
The random DOA error of the DS is uniformly distributed in 3 ∘ , 3 ∘ .Figure 2 shows the output SINR of all methods versus the input SNR.It can be observed that the proposed method achieves better estimation results by estimating the SVs of the interfering signals and power reconstruction INCM to achieve higher output SINR even for a single snapshot.The INCM-Integral method performs poorly for a small number of snapshots, but the output SINR improves with the increasing number of snapshots.The Tri-DL, WCOB, and LSMI methods use nominal DOA of DS to calculate SV, and a slight look direction error leads to a severe decrease in output SINR in the case of large-scale array elements.Figure 3 depicts the output SINR curves for all beamformers with the number of snapshots.The Tri-DL, WCOB and LSMI methods cause poor performance due to their sensitivity to the line-ofsight error.As the input SNR increases, the proposed method can estimate the SV of the DS to achieve a high output SINR and outperforms the rest of the methods.

Example 3: Running time comparison
We compare the average running times of all methods at different array sizes, and the results are shown in Figure 4, where SNR = 10 dB and  100.It can be observed that the running times of the proposed method at different array sizes are higher than those of the LSMI method, but lower than those of the other compared methods.This is because the LSMI method requires only a scale unit matrix to be added to the SCM without additional operations and thus has very low computational complexity.However, the LSMI method calculates the weight vector based on the SV of the nominal DS, and as shown in Example 2, a slight look direction error can cause a serious loss of performance.In contrast, the proposed method achieves better performance while having very low computational complexity.

Conclusion
In this paper, we propose a novel beamforming method using the feature that large-scale arrays can greatly improve the DFT resolution.The method uses the DFT to estimate the SVs of the interfering signals and the power to reconstruct the INCM, which can be effectively applied to large-scale arrays with a small number of samples.

Figure 1 .
Figure 1.The beam pattern of the proposed method, where SNR = 10 dB and  1.
[5]can reduce the computational complexity by determining the optimal rotational phase  at each peak.Specifically, by dividing / , / into  grids, the corresponding  is extracted by searching such that[5] 1, … ,  via the spatial rotation operation   , we can reduce leakage and improve the accuracy of DOAs and power estimate.By letting   diag 1,  , … ,  ,  0,1, … ,  , according to Equation (6) and Equation (9),  should satisfy ,:     ,  ,: denotes the  -th row of .