Improving LMS/NLMS-Based Beamforming Using Shirvani-Akbari Array

ULA is the most common geometry exp lo ited in array signal processing. In the beamforming operation, employing the ULA leads to obtaining narrower beamwidth with respect to other geometries in similar element numbers. Recently, Shirvani and Akbari proposed a new array by adding two elements to the ULA in top and bottom of the array axis, named as SAA. This new array offers a considerable improvement in DOA estimation performance in detection and resolution of signal sources placed at angles close to the array endfires. In this article, the performance of the proposed SAA is investigated especially in beamforming and compared with ULA. LMS and NLMS algorithms that are popular adaptive beamforming methods are used for evaluation and comparing the performance of SAA and ULA. Considering array factor, mean square erro r and bit error rate metrics, simulation results show improved convergence speed and higher data transmission accuracy in different signal source locations, boresight angles as well as endfire ones, for SAA with respect to ULA.


Introduction
Beamforming using an antenna array is a pro mising and necessary task in the next generation wireless and mobile systems. Sensor arrays are extensively used in signal processing and their capability has made numerous applications in different fields such as radar, sonar, radio astronomy, med ical diagnosis, acoustics and mobile o r wireless communicat ions. Co mb ination of the signals collected fro m array sensors provides temporal and spatial analysis of wave-fields and yields directional sensitivity for the system. The important tasks considered in array signal processing are beamforming, Angle Of Arrival (AOA) or Direction Of Arrival (DOA) estimation and source tracking. Variant techniques and algorithms are available with different characteristics in efficiency, accuracy and computational costs and also numerous studies are attempt ing to imp rove these techniques or proposing new efficient methods [1,2]. In addition to different algorith ms, structure of array p lays a key ro le in array signal processing. Various arrangements are studied and have demonstrated significant effects on the performance of the system.
Unifo rm Linear Array (ULA) is the most common configurat ion used for array p rocessing investigations. This geometry is simple in analysis and imp lementation and provides a good accuracy via formation of narro w beams during beamforming. However, ULA can analyse the surrounding wave-fields in a particular dimension and provides one-Dimensional (1-D) coverage. Furthermore, ULA cannot give a uniform performance in all directions. DOA estimation and beamforming with ULA leads to weak results at the angles close to the array endfires. Therefore, other array geometries are investigated to obtain a uniform performance. Unifo rm Circu lar Array (UCA) is another common geo metry used for array processing. This structure has further complexity in implementation as well as analysis with respect to ULA. Instead, UCA can provide 2-D coverage and present uniform performance in all directions. Beams in UCA are wider than ULA and this affects resolution of the array [3,4]. Concentric Circu lar Array (CCA) or mu lti-ring arrays are other configurations that provide more flexib ility in size and width of the obtained beams [5]. However, the required co mputational comp lexity for analysing the array structure increases in CCAs. Planar arrays are also used for 2-D and more precise beam-pattern generation. 3-D arrays and conformal arrays may be used in particular applications to provide M-D performance and conformity with the vehicle or system they are integrated with it [3]. However, these structures are complicated in calculat ions as well as imp lementation and research works are attending to find and develop simple arrangements which present better efficiency and perform unifo rm in all directions. Co mbination of linear arrays is considered in literature and has shown improve-ments in array performance specially the accuracy of DOA estimation algorith ms. So me of these structures are parallel linear arrays and Displaced Sensor Arrays (DSAs), Lshaped and two L-shaped arrays, V-shaped, Y-shaped, Uniform Rectangular Arrays (URA), hexagonal arrays and so on [6][7][8][9][10][11]. Each configuration has particular properties and capabilit ies. These geometries are often used for DOA estimat ion and they can provide at least 2-D DOA estimat ion capability. In Ref. [11] different types of array structures (ULA, UCA and URA) for s mart antennas, DOA estimat ion and beamforming have been examined.
This paper is following the authors' earlier papers [12][13][14], in which a simp le ULA-based geometry was introduced that could provide a significant imp rovement in DOA estimat ion capability and resolution threshold in border angles of the array endfire. In this article, by focusing on antenna beamforming, the capability of Shirvani-Akbari Array (SAA) which adds two elements to the ULA in top and bottom of the array axis is investigated. Two well-known adaptive beamforming techniques, Least Mean Squares (LM S) and Normalized Least Mean Squares (NLM S) are explo ited for evaluation of SAA performance. Simu lation results are compared with the results of beamforming using conventional ULA. Nu merical results show higher accuracy and also uniform performance through this new proposed array with respect to conventional ULA.
The rest of paper is organized as follows. Section 2 describes signal model for array processing through ULA as well as SAA. A brief overview of adaptive beamfo rming techniques and description of LMS and NLM S algorith ms are stated in sections 3, 4, respectively. In section 5, simu lation results using MATLAB software are presented and the performance of both arrays is compared together. Finally, conclusion remarks are given in section 6.    The adaptation takes place in the signal processing unit to fulfil the beamforming and adjust the complex weights in order to obtain optimized performance fo r the system. The weighted received signals are combined together to form the array output. In some of the beamforming techniques an iterative procedure may be required to adjust the weights. For statement of the signal model, ULA with N and SAA with N + 2 antenna elements are assumed that receive uncorrelated narrowband signals.

Signal Model for Array Processing
Received signals, can be exp ressed as a linear co mb ination of incident signals and zero mean Gaussian noise with variance σ n 2 . The incident signals are assumed to be direct Line Of Sight (LOS) and uncorrelated with the noise. The input signal vector denoted by (t) can be written as (1).
is M × 1 vector concerning to the m-th source located at direction θ m fro m the array boresight. ( θ m ) is the N × 1 steering vector or response vector of the array for direction of θ m . For ULA, the steering vector is as: (2) whered is the inter-element spacing and k = 2π/λ.
For SAA, the steering vector has (N + 2) element as (3). The first rows of ( θ m ) for SAA are equivalent with steering vector of ULA. By addition of two remained ro ws with respect to ULA, SAA is exp ressed as: For ULA, is a N × M matrix of steering vectors, written as: (4) For SAA, SAA with (N + 2) × M dimensions is defined as: The array output signal is given by: where is the 1 × N, 1×(N+2) weight vector for ULA and SAA. H denotes the Hermit ian, conjugate transpose, operator and is the received signal vector defined in (1).

Adaptive Array Beamforming
Beamforming that is also referred as spatial filtering is the process of adjusting the antenna radiation pattern so that the antenna focuses energy towards Signal Of Interest (SOI) and nulls directed to Signals Not Of Interest (SNOIs). Th is task can be fu lfilled through different approaches. Although some earlier techniques such as classical beamforming or fixed-beam methods carry out this process with lower co mplexity, adaptive beamforming techniques are mo re capable. These methods adjust array weights dynamically with respect to signal environment and perform interference mit igation as well as desired signal tracking. Therefore, adaptive beamforming methods have been more considerable in recent years.
In an overview, adaptive algorith ms are classified in two categories: non-blind and blind algorithms. Adaptive beamforming techniques generally need some in itial informat ion about the signal characteristics to obtain an appropriate performance. Th is init ial in formation is provided via a reference signal, DOA estimation methods or some particu lar properties of the desired signal. The beamforming methods that utilize a reference signal, known as train ing sequence, are called training-based or non-blind algorith ms. LM S and its variants, Recursive Least Squares (RLS) and Samp le Matrix Inversion (SMI) are train ing-based algorithms. Blind algorith ms use the desired signal properties and their DOA instead of training sequences to determine the weight vector. Constant Modulus Algorithm (CMA), Decision Directed (DD), Conjugate Gradient (CG), Least Squares (LS) and Spectral self-Coherence REstoral (SCORE) are so me examp les of blind algorith ms [2]. In this research, two popular and reliab le non blind methods, LMS and NLMS are discussed and simulated in both ULA and SAA.

LMS and NLMS Description
LMS is an adaptation technique based on the steepestdescent method. The algorith m updates the weights recursively by estimating grad ient of the erro r surface and changing the weights in the direct ion opposite to the gradient to minimize the Mean Square Error (M SE) [2,15]. Error signal is the difference between desired signal, ( n ) , and array output, (n), computed by the expression: ( ) ( ) n n (n) = − e d y (7) According to the LM S algorithm, the final recursive equation for updating the weight vector is: n μ (n) (n) + = + w w x e (8) μ is the step size parameter wh ich controls the convergence speed of the algorith m. The s mall step size causes slow convergence but high stability around the optimu m value. On the other hand, large step size results in rapid convergence and lower stability. Hence, step size is a majo r parameter that makes a trade-off between the convergence speed and the LMS stability. The convergence rate of LMS also depends on the Eigen-value variance of the input correlation matrix. Variable step size algorith ms are used to overcome the slow rate of convergence [16,17]. LMS has a simp le iterat ive equation which does not require correlat ion matrix calculat ion nor does require matrix inversions. This makes LMS a co mmon widely-used adaptation algorithm. According to equation (8), the correction term μ. (n) * (n) applied to the weight vector during LMS algorithm, is proportional to the input vector ( n ) . Therefore, a gradient noise amplification problem occurs in the standard LMS algorithm. Th is problem can be solved by normalized LMS, in which a data dependent step size is used for adaptation. NLMS normalizes the weight vector correction term with respect to the squared Euclidean norm of the input vector ( n ) at time n. So its equation can be written as: ( n + 1 ) = ( n ) + μ ‖ ( n )‖2 ( n ) * ( n ) NLMS can present better behaviour in convergence and stability than LMS [15,18]. To prevent the convergence problem due to division by a small nu mber, a positive constant ε may be added to the Euclidean norm of the input vector in (9) and the weight vector is computed through: (n + 1) = (n) + μ ε +‖ (n)‖ 2 (n) * (n) (10) The algorith m is known as ε-NLMS and results in mo re reliable implementation [18]. In addition to NLMS, other versions of LMS such as sign-LMS, Constrained LMS (CLM S), Block-LMS, Partial Update LMS (PULM S) and Variable Step Size LMS (VSSLM S) have been proposed with different properties in convergence speed and efficiency. In this research, standard LMS and ε-NLMS algorith ms are emp loyed but new proposed SAA configuration can be applied for all above mentioned versions of LMS.

Numerical Results
For simulation of LMS and NLMS in ULA and SAA, two arrays with 7 and 9 elements and half wavelength interelement spacing are assumed. The performance of beamforming algorithms is evaluated in both arrays and in different signal source locations using MATLAB. Beamfo rming performance may be evaluated with d ifferent criteria. In this investigation, absolute value of Array Factor (AF), MSE, and Bit Erro r Rate (BER) are considered as performance metrics. Fig. 4 depicts the radiation pattern when the desired signal is located at 15° and two interference signals are assumed to be located at −45°, 75°and the Signal to Noise Ratio (SNR) and Signal to Interference Ratio (SIR) are assumed to be 10 dB and 3 d B, respectively. Both arrays have tracked desired signal successfully and placed appropriate nulls in the interferers' directions. Beamwidth in SAA is a litt le wider than ULA in middle angles. Mean square error diagram during training process for both arrays is demonstrated in Fig. 5. Beamformers converge more rap idly in SAA than ULA. Fig. 6 shows the simu lation results in the sense of BER fo r middle DOAs. According to that, exploitation of SAA in beamfo rming purposes leads to better results in convergence rate of beamformer and accuracy of system. To investigate beamforming performance at the angles close to the array endfire the desired signal source is assumed at 80° and two interference signals are assumed at −45°, 45°. Radiation pattern of both ULA and SAA arrays is shown in Fig. 7. Main lobes that are formed at border angles have better tracking accuracy at desired DOA and their widths are less than main beams formed in these directions by ULA.    Fig. 9. The level of BER is lo wer in SAA than ULA. As shown in above mentioned numerical results, the level of AF and MSE during data transmission in SAA and ULA is equivalent. Besides, exp loitation of SAA in beamfo rming purposes leads to better results in convergence rate of beamformer and accuracy of system in the sense of BER. In addition, the performance of beamformer at boresight as well as endfire directions is better than ULA, especially in low SNRs. For h igher SNRs, greater than 5d B, performance of both beamforming algorith ms, LM S and NLM S, are the same for ULA as well as SAA.

Conclusions
In this paper, the performance of a new proposed array, SAA, for adaptive beamforming task was investigated. The new array is based on ULA configuration with two extra elements at the top and bottom o f the array axis. Two popular training-based beamforming algorithms, LMS and NLMS were considered for this purpose.
Nevertheless, the performance o f SAA and ULA is the same fo r the antenna beamforming in the case of source and interferers located at midd le angles, SAA offers a well performance in detecting and resolving sources located at array endfires. Simulat ion results show improved convergence speed and accuracy in data transmission considering AF, MSE and BER performance metrics for the new array, especially for low SNRs. This research focused on LMS and NLMS algorith ms for both ULA and SAA, but the similar procedure can be applied for other beamforming algorith ms.