Smart antenna base station beamformer for mobile communications

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SMART ANTENNA BASE STATION BEAMFORMER FOR MOBILE COMMUNICATIONS

V. Zaharov, F. Casco, O. Amin

Departamento de Ingenierнa Elйctrica,

Universidad Autуnoma Metropolitana - Iztapalapa

Mйxico D.F., Mйxico

Received November 13, 2000



A novel RLS beamforming algorithm with low complexity and high accuracy for applying in mobile communications smart antenna array processor is presented. The proposed pipelined structure has a higher performing that conventional RLS algorithm because require only vector operation and can be easy implemented en VLSI. As result proposed pipelined architecture can operate en N/k times faster than conventional RLS structure, where N and k are numbers of antenna elements and training samples respectively, and more, this algorithm have a minimum variance problem in finite precision implementation because one use the Householder transform procedure of weight vector updating.

communication smart antenna speed



Introduction

The technology of smart antennas for mobile communications systems has received widely interest in the last couple of years [1],[2],[3]. The principle reason for applying smart antennas is the possibility for a large increase in capacity due to control and reduce interferences. This is accomplish through the use of narrow beams at the base site and as result the users separations at the space.

A smart antenna combines antenna array with digital signal processing unit that optimize reception and radiation patterns dynamically in response to the signal environment, e.i. mobile moving about the coverage area. It is obvious that a smart antenna base station system is much more complex that traditional omnidirectional one because must include very powerful numeric processors and beamforming control systems.

In acordens with [4] digital processor is structured in four main sections

1. Direction of arrival (DOA) estimation. From the received input data in uplink the number of incoming wavefronts and their DOAs is estimated.

2. DOA classification. In a next step are identified those wavefronts that are originated from the user: First, the spatially resolved wavefronts, each incident from an estimated DOA extract from the input data, with a spatial pre-filter. Then, a user identification decides whether a wavefront (DOA) belongs to a user or to an interfere.

3. Tracking. The user DOAs are tracked to increase the reliability of the DOA estimates.

4. Signal reconstruction - beamforming. Finally a beamforming algorithm forms an antenna pattern with a main beam steered into the direction of the user, while minimizing the influence of the interfering wavefronts.

The weights of beamforming algorithm can be optimized from two criteria: maximization of received signal from the desired user or maximization of the SIR (MSIR) by suppressing the signal from interference sources. In case of SIR maximization is to be used, the optimum weight vector Wopt is given by 

, (1)

where R is N*N correlation matrix of the total received signal, P is cross - correlation vector between input vector and desirable signal.

Even with the powerful signal processor available today it is a very challenging task to perform eq.(1) in real time because the computational complexity of this operation sufficiently large and depends of number of antenna elements and applied algorithm category. There will be a growing need for development of efficient algorithms for real time weights optimizing and signal tracking.

In this paper we propose economically vector operations RLS algorithm for beamforming and tracking which allows decrease complexity of smart antenna base station. We present the architecture of proposed algorithm and comparison of computational complexity with conventional RLS algorithm.



1. The general form of adaptive beamforming algorithms

We will consider the block diagram of the beamformer as shown at figure1.

Figure 1. Block diagram of the beamformer

The signal at the input of L antenna elements is given

, (2)

where is signal from i-th user which satisfy the "narrowband assumption", , is angle of i-th user arrival signal, are element space and wave length respectively, n(t) is zero-mean thermal noise present in the receiver, M is number of users.

Matrix of beamforming presents as N row of the steering vectors

, (3)



where , , is i-th reference angle.

The output signal of the Batler matrix is

(4)

And the output signal of the i-th user beamformer is given by

, (5)

where is weigh vector of i-th beamformer.

The recursive adaptive beamforming algorithm takes the general form

(6)

where is vector which minimize the mean square error (MSE) between a reference signal ak and its linear estimation based on an observed input vector of k-th sample time , correlated with ak , , T,H are signs of transpose and conjugate transpose accordingly.

In this case for MSE and MSIR criterions the optimal vector Wopt is given by the same form [5]

(7)

where 

When one used finite sample size then formula (7) can be rewritten as

(8)

where , .

The increment depends on the algorithm. The popular least mean square (LMS) algorithm has a form

(9)

This algorithm has good performers but when covariance matrix has large eigenvalue spread, LMS algorithm has a rather slow convergence[5]. The situation are changed if one should use in the increment a inverse sampling matrix which is close to . In this case is obtained recursive least square algorithm (RLS) and least square solution

(10)

The convergence rate of this algorithm more fast and not depends of eigenvalue spreading, but the computational complexity of this algorithm is large. Consider RLS algorithm given by one rank matrix modification formula

(11)

where µ is forgetting factor, . It is well known that is computed without matrix division using the matrix inversion lemma.

(12)

This updating algorithm (12) requires complex multiplication per one iteration because the base operation is multiplication of matrix on vector with dimension N and high sensitivity to computational errors.

2. Synthesis of the algorithm

We consider beamforming algorithm which has a computational complexity and high accuracy implementation, where k is number of samples in matrix Rk .For this purpose the matrix Rk in (11) we write as follows

(13)

where X is matrix N*k consisted from k input vectors, and matrix T is a k*k square matrix .

In (11) make a substitution and given

(14)

According to matrix inversion lemma



(15)

If then

(16)

We may rewrite eq.(15) as

(17)

where (18)

Matrix A may be present as L - factorization

(19)

where L and are (k*?k) lower and upper triangular matrixes accordingly. Omitting scale factor in eq.(17) we get

(20)

where is matrix N*k which can be obtain solving system

(21)

Using eq.(8) and eq.(20) the optimal vector can be obtained in form



, (22)

3. Recursive performens of weight vector

If input data are a sliding windows with N elements then matrix X for the m-th window has a structure

(23)

where .

Matrix for the (m+1)-th window can be obtain as

(24)

where - matrix without 1-th column, .

In this case matrix A from eq.(18) has the following structure (here shown only lower triangular part and diagonal for k=3 because matrix A is eremite and upper part will be conjugated



As follows from eq. (25)

(26)

From the matrix can be obtained recursively the matrix (see part 5).

Its easy that matrix has a following structure

(28)

where is k-th row of matrix . is matriz without 1th column. Solving the system of linear eq. (21) we obtain the vector , j=1,2,...,k

(29)

Using eq. (8),(22) and eq.(29) we have following recursion for updating the value of adaptive vectors Wj, j = 1,2,…,k .

, (30)

4. Recursive updating of matrix

The matrixes and can be obtain using Cholesky factorization [6].Then the elements of matrix is defined by



(31)

How follows from eq.(28) for matrix no need calculate all elements but only for i=k, becouse other elements was calculated at privies matrix . The elements of k-th row could be calculate as

(32)

However in case of obtaining matrix using Choletsky factorization improvement of numerical stability is insignificant and the matrix R should be formed in an obvious form (i.e. it requires additional memory). Then must be made it factorization that requires also additional computational complexity.

The proposed algorithm allows to make recursive updating of vector W no form a matrix R in an obvious form but elements of a matrix L to receive directly from a matrix of input data X using Householder transform [7]. In this case thealgorithm will have a minimum variance problem in finite precision implementation because the Householder transform has a big immunity against the computational errors [8].

Let us consider recursive procedure of updating of elements of a matrix with Householder transform application. We assume, that the matrix X is presented as 

(33)

so that . Then

, (34)

where Q - (N+k)*=(N+k) orthogonal Householder matrix. From eq.(34) follows, that the elements of a matrix can be received using orthogonal transformation above a matrix X, not forming in an obvious form of a matrix A.

We present a matrix Q in factorized form as . Then eq.(34) can be rewrite as

, (35)

where , ,

i=1,2..., k, thus column , j=1,2...,k of matrix have passed the orthogonalization by matrixes Q1, Q2..., Qi-1 (i-1) times. The matrixes Qi according to [7] are determined as follows

, i = 1,2..., k, (36)

where is (N+k)*(N+k) identity matrix. In eq.(36) scalar Ci is determined as follows

, i = 1,2..., k , (37)

where is i-th column of a matrix . The column vector Ui in eq.(36) has dimension N+k and is determined by expression

, (38)

where , is the real part of an element , is column-vector which dimension same as a vector ,where unit is in the i-th place. The value ui in (36) is i-th element of a vector Ui. Then i-th step of column ortogonalization of a matrix will be

. (39)

We will present expression eq.(39) with the account eq.(36) as operations above vectors

. (40)

With the account eq.(38) we present eq.(40) as

, (41)

where , .

The expression eq.(41) determines vector algorithm of reception of elements lij and orthogonalization of vectors using Householder transform. Proceeding from eq.(41) we obtain expression for elements lij of a matrix as

, (42)

where i = 1,2..., k, j = i,i+1..., k.

Expression for vectors orthogonalization yielding

. (43)

Thus recursive computing process of reception of elements lij of a matrix with use of Householder transform on i-th iteration (i = 1,2..., k) consists of the following stages:

a) Forming of a scalar Ci according to expression (37);

b) Forming of a vector Ui agrees eq.(38);

c) The calculation lij agrees eq.(42);

d) Ortogonalization of vectors according to eq. (43).

5. The computational complexity and architecture of algorithm

The computational steps of adaptive weighting modifications algorithm could be presented as follows1.

Inicialization W0 =1,. R0 = IN, Z0 = X1

for i=1,2,3,…

for j=1,2,…k

2. Compute of triangular decomposition

elements

2.1. Choletsky factorization

1. a) Obtain of i-th row of matrix A (eq.26)

b) Obtain of i-th row of matrix L (eq.32)

2.2. Householder transform

a)Obtain of i-th row of matrix L (eq.42)

3. Calculate of vector Z elements

4. Update of adaptive vectors elements Wj,

Figure 2. The architecture of the tracking algorithm



Figure 2 illustrates the architecture of the proposed algorithm and its computational complexity in terms of multiplications and adders. As it can be observed from the figure 3 the number of operations on the each stage no more than N and total computational complexity equal 3N+k multiplicationes and adders per one iteration. It means that pipelined architecture will operate N/k times faster then conventional structure.



Summary

Presented beamforming adaptive algorithm for computing the optimum weight vector in accordance with MSIR criterion for mobile communications smart antenna base station beamformer. The proposed algorithm can be implemented with a linear complexity using only vector operations and can be easy implemented en VLSI. Pipelined architecture can operate en N/k times faster than conventional RLS structure. The novel algorithm has a minimum variance problem in finite precision implementation because one use in the procedure of weight vector updating well known stable Householder transform which has a big immunity against the computational errors.



References

1. Liberti J.C., Rappaport T.S., Smart Antennas for Wireless Communications: IS-95 and Third-Generation CDMA Applications. Prentice Hall, NJ, 1999.

2. Barrett M., Arnott R. "Adaptive Antennas for Mobile Communications". Electronics and Communications Engineering Journal, Aug, 1994, pages 203214.

3. Winters J.H. “Smart antennas for wireless systems". IEEE Personal Com. Magazine, pages 23 -27, Feb, 1998.

4. Kuchar A. at all “Real-time smart antenna processing for GSM1800 base station” in IEEE Vehicular Technology Conference '99, Houston, Texas.

5. Haykin S. ”Adaptive Filter Theory”, Prentice Hall, NJ, 1996.

6. Golub G.H., Van Loan C. F. ”Matrix Computations”. 2nd edition, The Johns Hop-kins University Press, North Oxford Academic Publishing Co., Baltimore and London, 1989.

7. Householder A.S. “A class of method for inverting matrices”.J. Sos. Indust. Appl. Math, 1958, V.6, # 2, pages 189-195.

8. Wilkinson J..H. The algebraic eigenvalue problem: monographs in Numerical Analysis. Clarendon Press, Oxford, 1988.

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