Modified SMI Techniques for Frequency Selective ... - CiteSeerX

Modified SMI Techniques for Frequency Selective Channels in OFDM Diego Bartolom´e, Ana I. P´erez-Neira Polytechnic University of Catalonia (UPC), Campus Nord D5-118, C/Jordi Girona 1-3, 08034 Barcelona, Spain - tel:+34 93 401 16 27 diego,anuska
@gps.tsc.upc.es Abstract— The Sample Matrix Inversion (SMI) algorithm has been widely applied in the filed of adaptive antenna arrays. However, in Time Division Multiple Access (TDMA) systems with known data sequences, a low cost solution would imply the computation of the beamformer based exclusively on this data. In some cases, this might not be possible since the known data could not be sufficient to get a stable estimate of both the sample covariance matrix and the steering vector needed for the SMI spatial filter. In this paper, windowing techniques are applied in order to calculate a beamformer per subcarrier for Orthogonal Frequency Division Multiplexing (OFDM) systems with very limited known training. This approach is compared to the subcarrier grouping approach, and the presented results show that the former method provides a further performance gain, even in channels with high delay spread.

I. INTRODUCTION For the broadband Wireless Local Area Networks (WLANs) based on the Orthogonal Frequency Division Multiplexing (OFDM), the use of an adequate cyclic prefix (CP) may mitigate the pernicious effects of multipath propagation in their typical deployment scenarios. The use of adaptive antennas at the receiver side in a standard-compliant Single Input Multiple Output (SIMO) configuration is motivated in order to cope with high interference levels or to increase the ultimate quality of the communications link in terms of Bit Error Rate (BER). In the literature, antenna array algorithms for multicarrier transmissions can be applied both in the time domain [1, 2] and in the frequency domain [3]. A comparison of techniques can be found in [4]. There it is shown that high delay spreads can be mitigated by applying antenna array algorithms in the time domain: the SMI performs an implicit temporal channel shortening, already recovering the transmitted symbols before the FFT at the receiver. A rather simple but efficient algorithm when no Inter Carrier Interference (ICI) is present is the SMI solution in the frequency domain. The optimum performance would imply computing a beamformer for each frequency in operation. However, WLAN systems provide limited known training data, usually as a preamble [5], which is not sufficient to get stable estimations of both the covariance matrix and the steering vector, leading to a poor link quality in terms of BER. In this paper, in order to attain numerically better conditioned estimations, traditional spectral estimation windowing techniques are applied to the received symbols in the frequency domain so as to compute a beamformer for each subcarrier. This method achieves good results with low complexity in freThis work was partially supported by the Spanish government FIT-0700002000-649 (Medea+ A105 UniLan), TIC99-0849, TIC2000-1025 and by the Catalan government CIRIT 2000SGR-00083.

ri(n)

x 1(k)

S/P

Q antennas

FFT

r1

P/S

x1

y(k)

w

Equalization + demapping

S/P

â(k)

y

S/P

FFT

rQ (n)

P/S xQ (k)

rQ

xQ

Fig. 1. Post-FFT combining in HL/2

quency selective channels. The performance is compared to the subcarrier grouping technique [6], where the entire set of subcarriers is partitioned into disjoint sets, applying a single beamformer for the whole group. This paper is organized as follows. In Section II, the signal model is firstly presented. Then, the beamforming methods at the receiver are described in Section III, i.e. the SMI with disjoint grouping and the windowed techniques. The channel estimation and equalization is addressed in Section IV while Section V discusses the results from simulations before the final conclusions. II. SIGNAL MODEL In the following, boldface capital letters refer to matrices and boldface lowercase letters refer to vectors. The operator   denotes conjugation,    transposition, and         . The  unitary Fourier matrix is denoted by  and 

refers to the IFFT  operation. It is well-known that if the channel is time-invariant within the symbol  and the length of the cyclic prefix is greater than the channel order, the frequency selective channel can be modeled as a frequency flat fading channel for each of the  useful subcarriers, i.e. the received signal at subcarrier  within OFDM symbol  at the  antennas can be expressed as 

 !#"$ %& ('*)& &+-,/.10324 567987:3

(1) where

"$ 

are the transmitted mapped symbols, whereas  CBD is the desired 2 . user signature containing the channel frequency response and the steering vector. Note that the subindex  E refers to the antenna. In turn, the vector )&  contains the contribution from the white gaussian noise and the interference sources. Gathering the observations from the  useful subcarriers in a common matrix, the signal after the FFT at the  antennas %& ;=< >

  >3?  @A@A@ >

for OFDM symbol 

9

is denoted by

vector corresponding to the th group of subcarriers minimizes the Mean Square Error (MSE) between the output samples and the transmitted symbols according to

 
  (2) ) : ) +-, : ) ;=A@CEG where


  is a diagonal matrix con(5) B D ! FF H
!I FF taining the transmitted symbols in the frequency domain      . The operator  puts where I +-, is the all-ones . ) H vector. The solution to this the vector in his argument in the main diagonal of a matrix. problem is The rows of the matrices , and  refer to the )J ) ) ) ) +-, subcarriers, whereas the columns indicate the antenna index, : LK !  9 % (6) !2M*N !
!I e.g. The receiver processes firstly the preamble, computing the    beamformers  as in the previous equation. Then, during the         data traffic of #O OFDM symbols, the output of the 9 th space.. .. .. frequency filter is expressed as   ... . . . ) : )J +-,  P)   Q#O7 (7) ) (3) where is formed by selecting the row & vectors ) . Noteof III. BEAMFORMING METHODS corresponding to the indices in the set The algorithms here presented are computed during the that the output data from the R% weight vectors P form the signal in the frequency domain, i.e. known data, e.g. a preamble. The matrices or vectors corre- received  P P P ('  . sponding to the preamble data are denoted by subindex "! . In the following, it is assumed that the preamble consists of #$! B. Windowing Techniques - 

-

-



"

- '

-&+

 3

,



0. 

:;::

*

"$5!&"$

;@A@A@ "$ :





"

-

-

5!

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-!

2



:



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? 5!

@A@A@

? 

@A@A@





@A@A@





.

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@A@A@

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032 56

+-,



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2 -

 

OFDM symbols BPSK modulated. The classical SMI solution with subcarrier grouping applies a total of  beamformers with a partition of the whole set of frequencies into disjoint sets. On the other hand, the windowed methods apply as many beamformers as subcarriers, i.e.  , but in this latter case, the data from adjacent subcarriers is also used to stabilize the estimations. Both approaches are detailed in the next subsections.

%

A. SMI with subcarrier grouping Since the space-frequency characteristics of the received signal are different in the whole set of carrier frequencies, the optimal strategy would imply computing a different beamformer to each subcarrier. However, this means having to estimate a high number of variables from a finite sample size, which will degrade the performance. The computational complexity can be reduced and the estimates more stabilized using less number of OFDM symbols if the whole set of frequencies is partitioned into several sets, thus also exploiting the coherence bandwidth of the channel [6], i.e.

&

. )

2

A@A@A@ 

& (' 

&*)





)

@A@A@

2

-+) ,


!

(4)

/%

with the cardinality of each set. Therefore,  different beamformers are applied. Each set contains different frequencies, the subcarriers are uniquely assigned to a single set.  5!  7@A@A@    The matrices 

) ) )  ) +-,43 576 and
! !) 
! !) (
!!)  8
!!) 00#1#1!2!2  -+ ,43 576 +-, gather the #1! symbols of the preamble corresponding to subcarriers belonging to the 9 th group, which will + ,

+

,

0.



 5!



7@A@A@



?





0

serve for the computation of the beamformers. The SMI weight

-



-$@A@A@

-





The windowing techniques take a different approach to the problem. Instead of computing a single beamformer to each set of subcarriers, the aim is to obtain a weight vector for each individual subcarrier. However, due to the limited preamble, it is not possible to use exclusively the data corresponding to the selected frequency. Therefore, the information from adjacent subcarriers helps stabilizing the estimations for a given frequency. This is possible due to the fact that the channel is not uncorrelated in frequency, there is a certain coherence bandwidth. The procedure is presented below. Without loss of generality, the symmetric window    5!  ;@A@A@   is assumed to have odd length and its center located at index ? 2 . The samples from this window during the symbols of the preamble are gathered in the diagonal matrix

T

T 

S

T UWV1  ZX Y N #[!

  ]S ^  ^  ]S \ ..  ... . ^ ^ ba Let ! be the #1! UWV "

 9

"



 9

^   ^ .. .. . . _  ]S

576`3 XZY 75 63 ZX Y

@A@A@

@A@A@

@A@A@

"

+

,

0

/@

 9

(8)

3   matrix of received symbols containing the data from frequency index  (the center frequency of the group, denoted by the superindex) together with the frequency symbols corresponding to the ? 2 previous and next subcarriers. Then, given the matrix , the MSE that shall be minimized for each individual subcarrier  is:

c

\ XZY N

   

FF \d !a : a  \
!8a I ZX Y FF

? 

(9)


.

.

.

.

.

.


.

.




.

.

.

.

θ k-1

.

.

.

θk

.

.

& $

.

Fig. 2. The overlapping groups of the windowing technique. Note that the size of the window is  , and the center frequency has always the maximum window value. The subcarriers corresponding to the group  are denoted by filled boxes, the center of which is slightly less dark.

UWV 

\ !a !a \ d
!a

where is the all-ones  vector. Denoting the windowed received samples during the preamble by  and the weighted transmitted symbols as  , the beamformer solution to the minimization of (9) is

!a

: Va LK !a !a M*N !a 8!a I ZX Y 2





56





67:

@

(10)

This approach is in fact a kind of subcarrier grouping, but instead of designing disjoint groups of subcarriers, overlapping sets are used. This can be expressed analogous to (4) as

&

2

& 

A@A@A@ 

&*)





)



@A@A@

) )ZX Y

2


!@

(11)

In this case, though, the subcarriers   belong to several groups, see example in Fig. 2. Note also that the subcarriers close to the lower and upper bounds cannot always use all the adjacent subcarriers. Indeed, the solution in (10) is equivalent to

UWV

: Va

    

N

2   

  

  56





67:





(12)

where the covariance matrix      and steering vector      at subcarrier  are estimated according to a weighted averaging of the adjacent subcarriers, i.e.

     

  3

  3

57 6

N , T ] !,  a 57 6   T N , ]   a  !, 2

2



!





? 



? 

!

 

)



 





)

! 

)

  (13)

 

=!

" 3



)

@

(14)

In fact, assuming a symmetrical window, the previous equations can be expressed as a frequency convolution with the window, i.e.

     

   

   

ZX  Y N ] %$ &  #" XZ Y N ]  & $ 2

 

2

 #"

 

 

 

'

'

N ! 57 6  N !

  3

 & $

Nu

I XZY

57 6

θ k+1

Lw



T ]

]

where the multiplying factor   is   ? , and the instan" taneous estimations of the covariance matrix and the steering vector of the received signal are respectively:

UWV '  [  UWV '  [





(15)

2



2

  3





! 

  

  

(17)

=!

  " 3 @

(18)

In this case, : beamformers are applied. Analogous to (7) in the previous subsection, the received signal in the frequency domain is filtered as

(



  

: Va

  

56



967:

56



#OZ

6



(19)

where  &  is the row vector corresponding to the  th row of matrix - . The output from the  weight vectors form the post-FFT received signal, i.e. -  ( 5! (   @A@A@ ( :   , which shall be equalized.





P

 

IV. CHANNEL EQUALIZATION In the previously exposed methods, equalization is needed due to the subcarrier grouping. Then, channel estimation is performed by a classical Least Squares approach with a constrain on the channel temporal duration, fixed by the CP of the preamble ( *) ). The estimated frequency response according to the previous criterion is

U(!

 



K 2



2

!

 2

M=N


! !  (20) 3 7 5 6  stacks the received sym-

2 



2

+

,



04. 



+ , where the matrix 04. bols during the preamble in the frequency domain. The transmitted symbols in the preamble are contained in the matrix 3 5!  ;@A@A@ 3   whereas the Fourier 9 Matrix  is obtained by selecting the rows of  corresponding 2 to the useful subcarriers and its first *) columns. The mapped symbols "$  can be recovered by equalization of the received signal in the frequency domain, i.e. + (  -, . +    , where . +    denotes the estimated "$  total response for subcarrier  . For the  useful subcarriers,


! 7
! 


!



+


! 0#1!   UA!

P

- 

1+

-0/

+

,

 

032 

(21)

where / denotes the element by element division. In turn, the + . +  5! . +  $@A@A@ . +  :   contains the estimated vector 1  channel coefficients, which will be specified for each approach.



 

A. SMI with subcarrier grouping

9

In this case, the frequency response vector for subcarriers in the th group for equalization in (21) is: (16)

1+

)

) )  :  9 %

2



6

67

!

(22)

TABLE I C HANNEL C HARACTERISTICS



Maximum delay [ns] 390 1050 1760

1

rms delay spread[ns] 50 150 250

0.8

value

Channel A C E

Windows

)

&=)



where  is formed by selecting the rows of  corresponding to the indices in . The total vector is formed by concatenating the responses from the  groups.

R%

The total channel response is now computed on a subcarrier basis as:    

%    

: Va

567967:





(23)

where %     is obtained selecting the  th row of the estimated channel matrix  .



V. SIMULATIONS The simulated system corresponds to HL/2 at a carrier fre
 4@  GHz, i.e. it consists of a   point quency of IFFT/FFT, with a CP of   samples, thus  5 . Translating those values to temporal duration, the OFDM symbol time ' is  , whereas the CP lasts 5 5  .   are useful carriers, while the rest are padding zeros. Simulations have been conducted in several environments from BRAN [7] (see Table I), comprising from a typical office (channel ’A’) to large open spaces with large delay spread both indoor (’C’) and outdoor (’E’). ' 
) A uniform linear array is used at the receiver (   with inter-element spacing of 0.5  . Two randomly distributed interferences disturb permanently the desired user, causing a C/I range from -10 dB to 20 dB. The SNR is 20 dB. Although the signal model is presented for time-invariant signals, the terminals from simulations are moving at the maximum speed allowed in HL/2 scenarios, i.e. 3 m/s, while the angular spread is  . The selected signal mapping is QPSK for the ; 5 transmitted OFDM symbols, and no channel coding is imple'  OFDM symbols for the mented. Note that there are ' preamble, with a cyclic prefix of *)  samples. The used windows are shown in Fig. 3, [8]. Simulations are conducted in order to evaluate the optimum performance of several windows and the subcarrier grouping. The results are summarized in Table II, where the optimum number of subcarriers for the proposed methods is displayed. Note that for the partition into disjoint sets, each one may not use the same number of frequencies. For example, with 5 frequencies per set, 8 groups contain 5 subcarriers and 3 of them consist of 4 subcarriers. It is observed that when the number of antennas is increased, there is a need of utilizing more subcarriers to reach the optimum accuracy. Additionally, as the delay spread of the channel increases, yielding to ICI in models ’C’ and ’E’ and a more

U 



0.4

0.2

0 −30

B. Windowing techniques

.+

0.6

=9

#[!

#O

UA!

rectangular triangular hamming

0 samples

30

Fig. 3. Shape of the selected windows

rapidly channel variation in frequency, the optimum number of subcarriers decreases for all methods except for the rectangular window, whose optimum value is always  ' . The BER curves for   antennas are shown in Figs. 4,5 and 6. For the sake of conciseness, figures for 2 antennas are not shown. One can state, though, that the performances between the proposed algorithms do not differ excessively in the case of 2 antennas at the receiver. The performance of the rectangular windowing is very close to that of the optimum SMI with disjoint sets. In fact, it is a simple averaging both forwards and backwards. In all methods, the best performance is obtained in channel ’A’, where the maximum delay of the channel does not exceed the CP. When the delay spread increases, the performance is degraded, especially if the maximum delay of the channel doubles the CP duration (channel ’E’). The optimum performance is achieved with the triangular and Hamming windows, which attain approximately equal BER. In fact, this is logical as the shape is very similar (Fig. 3). These windows outperform the optimum disjoint grouping in more than 3 dB, even in channels with high delay spread (Fig. 6) where the frequency domain methods seem to have significant losses with respect to the time domain methods [4] due to the ICI. Table II shows that less subcarriers are needed for the triangular window, which offers the best trade-off between performance and complexity among the proposed methods.

Q

VI. CONCLUSIONS In this paper, modified versions of the SMI algorithm have been proposed and analyzed via simulations. The first approach is the partition of the subcarriers into disjoint sets. In order to improve the performance, windowing techniques are applied to compute a spatial filter for each individual subcarrier, helped by the adjacent subcarriers data. The triangular window offers the best performance, although the rectangular window offers a fixed number of subcarriers independently of the environment. These windows improve the performance of antenna array algorithms even in ICI channels.

TABLE II O PTIMUM GROUPING ( NUMBER OF SUBCARRIERS ) Q

Disjoint sets 4 2 2 5 5 4

channel A C E A C E

2 4

Rectangular 3 3 3 5 5 5

Windows Triangular 5 3 3 7 5 5

Hamming 7 5 5 9 7 7

Performance comparison, channel C,SNR=20 dB,AS=15º,Q=4

0

10

SMI Rectangular Triangular Hamming

−1

uncoded BER

10

Performance comparison, channel A,SNR=20 dB,AS=15º,Q=4

−1

10

SMI Rectangular Triangular Hamming

−2

10

−2

uncoded BER

10

−3

10 −10

−5

0

5 C/I [dB]

10

15

20

Fig. 5. BER vs. C/I in channel C

−3

10

−4

10 −10

−5

0

5 C/I [dB]

10

15

20

Fig. 4. BER vs. C/I in channel A

R EFERENCES

Performance comparison, channel E,SNR=20 dB,AS=15º,Q=4

0

10

SMI Rectangular Triangular Hamming

[1] D. Bartolom´e, A.I. P´erez-Neira, and A. Pascual-Iserte, “Blind and Semiblind Spatio-Temporal Diversity for OFDM Systems,” Accepted for ICASSP’02, Orlando, USA, May 2002.

[3] F.W. Vook and K.L. Baum, “Adaptive Antennas for OFDM,” in Proceedings of 48th VTC, 1998, vol. 1, pp. 606–610. [4] D. Bartolom´e and A.I. P´erez-Neira, “Pre- and Post-FFT SIMO Array Techniques in Hiperlan/2 Environments,” Accepted for VTC’02 Spring, Birmingham, USA, May 2002.

−1

10 uncoded BER

[2] M. Okada and S.Komaki, “Pre-DFT Combining Space Diversity Assisted COFDM,” IEEE Transactions on Vehicular Technology, vol. 50, no. 2, pp. 487–496, March 2001.

−2

10

[5] ETSI TS 101 475 V.1.2.1, “Broadband Radio Access Networks (BRAN); HIPERLAN Type 2; Physical (PHY) layer,” Tech. Rep., European Telecommunications Standards Institute (ETSI), April 2000. −3

[6] D. Bartolom´e, X. Mestre, and A.I. P´erez-Neira, “Single Input Multiple Output techniques for Hiperlan/2,” in Proc. IST Mobile Communications Summit, Sitges, Spain,, Sep. 2001. [7] J. Medbo and P. Schramm, “Channel Models for Hiperlan/2 in Different Indoor Scenarios,” BRAN 3ERI085B, March 1998. [8] A.V. Oppenheim. R.W. Schafer, Discrete-Time Signal Processing, Prentice Hall, 1989.

10 −10

−5

0

5 C/I [dB]

10

Fig. 6. BER vs. C/I in channel E

15

20