Fig 3.  Block diagram of  interchannel  processing in smart antenna  base station

 It consists  L elements array,  Direction of Arrival (DOA) matrixes, tracking unit (R^-1) and weighting matrix (W).  We present the matrix of DOA as N row of the steering vectors

                                                             ,                                        (6)

where , , is i-th reference angle.

are element space and wave length respectively, T is symbol of transpose.

When in input of array are affected  signals from M users then the signal at the input of L antenna elements  is given by

                                                                                               ,                                                    (7)

where is signal from i-th  user which satisfy the "narrowband assumption",  ,  is angle of i-th user arrival signal, n(t) is zero-mean thermal noise present in the receiver, P_i  is the power of the i-th user signal.

The output signal after DOA’s matrix  when  is

                                                  ,                                                   (8)

where  ,  ,  ,

~  is conjugate and transpose symbol.

We will consider the effectiveness of the adaptive system using criteria of signal interference ration maximization (MSIR) by suppressing the signal from interference sources. In case of SIR maximization is to be used, the optimum weight vector W  is given by

                                                                 ,                                                               (9)

where  is N´N correlation matrix of the total received signal, .

From eq.(9) follows that weigh vector of i-th user which maximize SIR relation  in i-th  channel of adaptive version in comparison with no adaptive

                                                                                                            (10)

can be obtained as

                                                 .                                                    (11)

From (11) follows that

                                                                   ,                                                            (12)

where P only scale factor for each channel and can be omitted .

The output vector from N channels

                                                               ,                                                    (13)

where ,  and  x is input vector after DOA matrix

                                                                                                                       (14)

The output signal at  the i-th user channel is given by

                                                                     .                                                            (15)

4. TRACKING ADAPTIVE MODEL

When one use k sample size  for estimation of weight vector then formula (12) can be rewritten as

                                                                                                                                 (16)

where  is maximum likelihood estimation of matrix R.

In many of real scenario the input data are nonstationary. We can present this data as a sliding windows with k vectors, each for N elements and denote these data as a matrix X

where       , i = 1,2,...,k is column vector of i-th sample ("snapshot").

Presentation of data as shown in eq.(17) permits apply recursive adaptive algorithm for weight vector modification.  Consider RLS algorithm given by one rank matrix modification formula. For this purpose the matrix R_k  in (16) we  write as follows

                                                                                                 (18)

where  is forgetting factor,  ,

We can rewrite (18) as

                                                                             (19)

Introducing  of matrix of  coefficients C

                                                                      (20)

we can rewrite  the  matrix X as a weighted matrix by matrix C

.     (21)

It is well known [6],[7] that  is computed without matrix division using the matrix inversion lemma.                                      

                                                                       (22)

Even with the powerful signal processor available today it is a very challenging task to perform  eq.(22) in real time because the computational complexity of this operation O(N²). The base operation of this algorithm is matrix for vector multiplication, that sufficiently large for real time processing.

5. PROPOSED ALGORITHM

Evidently that adaptive algorithm must be implemented with number of  contradictory demands. This is, first of all, high convergence to optimum solution, since in low convergence will be seen the high level of interference at the beginning part of connection with subscriber. In the second place, the adaptive algorithm preferably must to have low hardware implementation, since adaptive processor  have to put in satellite. The first demand dictates applying of the direct matrix inverse algorithms of adaptation, and second recursive implementation of such algorithms.

The convergence of recursive direct matrix inverse algorithms of adaptation in detail is investigated in [4], [6].

We present recursive least square (RLS) vector operations efficient algorithms for real time weights optimizing and signal tracking that apply eq.(22) with computational complexity  O(N).   Applying of this algorithm for tracking allows decrease its complexity of hardware implementation.

For i = 1,2,...,k the output vector can be obtain as following

                                                 (23)

where ,  i = 1,2,...,k  is output of channels in i-th moment of time .

Making substitution                                   

                                                                                                                            (24)

eq.(23) can be rewrite  as

                                                                      (25)

where  is scalar.

Inverse of matrix then we present as

                                                                                                        (26)

Then taking into account eq.(26) we rewrite  eq.(24)

                                                                                                     (27)

6. SOFTWARE IMPLEMENTATION

The algorithm of output signal obtaining Y_i in the i-th moment of time we present in the next stage

1.      Initialization

        ,    .

2.    for i = 1,2,...,k-1              (i is currently sample number )

;  (training part)  

                  (processing part)

end.

3.    for  i = k ,k+1, k+2,...

;  (training part)

              (processing part)

          for j = 2,...,k

                     (shift in memory that memorize vectors z )

          end

In the stage 1 is being put constant  and nulling the memory that stores the vectors z.

Then in stage 2 is being filled out  the memory  by no zero elements. The stage 3 is processing  of the data in real time. At that, the new vector  obtained after processing part have to shift in the place of , the vectorin the place of and so on.

7. HARDWARE IMPLEMENTATION OF THE PROCESSOR

The architecture of processor has shown en figure 4. As follows from this figure the processor consist  k identical blocks that presented in figure 5 (k-1 blocks  for training procedure and 1 block for sample processing).  As follows from figures 4 and 5 it needs approximately 2Nk multipliers and adders to hardware implementation of processor.

8. SIMULATION RESULTS

The sample scenario was created. The antenna array consists L=20 linearly arranged elements with spacing. Number of channels N=8. Number of users M=8. Each sources was simulated as Gaussian distributed with SNR and azimuth location that have shown in table 1. Algorithm was applied with forgetting factor . Convergence of algorithm was evaluated as number of samples when parameter G_i no worse then 3 db in comparison with optimum solution.  200 independent simulations have been averaged.

Table 1.

The table shown that convergence of proposed algorithm is no worse then 5 samples in comparison with conventional  algorithm, but how follows from part above proposed algorithm have advantage en computational complexity and hardware implementation.

9. SUMMERY

Two approaches for SDMA satellite communication systems are presented: switch beam and adaptive technique. Tracking adaptive algorithm and its hardware implementation for computing the optimum weight vector in accordance with MSIR criterion for adaptive antenna station developed. The proposed algorithm has a linear complexity because one uses only vector operations and can be easy implemented in software and hardware.

REFERENCES

1.      Sklar B. Digital Communications: Fundamentals and Applications.- Prentice Hall, New Jersey, 1988.

2.   Tomasi W. Electronic Communications Systems Fundamentals Through Advanced.- Prentice Hall, New Jersey, 1996.

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

4.   MonzingoR.A., Miller W. Introduction to Adaptive Arrays. - John Wiley & Sons Inc, New York, 1980.

5.   Dudgeon D.E., Mensereau R.M. Multidimensional Digital signal processing.- Prentice Hall, Englewood Cliffs, 1984.

6.   Haykin S. "Adaptive Filter Theory", Prentice Hall, NJ, 1996.

7.   Golub G.H., Van Loan C. F. "Matrix Computations". 2nd edition, The Johns Hopkins University Press, North Oxford   Academic Publishing Co., Baltimore and London, 1989.

Authors:
V. Zaharov, email: f_zvv@rambler.ru, F. Casco, M. Gutierrez
Departamento de Ingenieria Electrica, Universidad Autonoma  Metropolitana - Iztapalapa, Mexico D.F., MEXICO

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discussion