ON SPACE-FREQUENCY BLOCK CODES FOR UNEQUAL CHANNELS Berna Özbek*, Didier Le Ruyet**, Maurice Bellanger** *İzmir Institute of Technology, Electrical and Electronics Eng. Dep., İzmir, Turkey [email protected] **CNAM-Electronique, 292 rue Saint-Martin 75141 Paris cedex 03, France {leruyet, bellang}@cnam.fr ABSTRACT After a systematic derivation of space-frequency block codes, the case of two transmitter antennas where unequal channels are applied to the two symbols that belong to two adjacent subcarriers is considered. At the receiver, the coefficients of the linear combiner are calculated and the expressions are given for the signal-to-noise ratios and different approaches. We propose a scheme that gives the same performance as the ML decoding with computationally efficient structure. The performance of proposed scheme is confirmed by simulations with respect to Hiperlan/2 characteristics. The solution improves the BER performance especially at high E b / N 0 compared to ZF approach. The channel coefficients that the scheme needs can be directly estimated by using the adaptive filter in a lattice structure. 1. INTRODUCTION In personal wireless communications, the objective of space-time and space-frequency block codes is to achieve space diversity using several transmitter antennas and a single receiver antenna with the help of linear processing. A space-time block code (STBC) is defined as a T × M matrix that describes the transmission of Q data symbols over the M transmitter antennas during the time interval T. The symbol rate is Q/T. In this paper, we will restrict ourselves to the case where Q=T=M=2 and N=1, that is, 2 data symbols are transmitted through the 2 antennas at 2 consecutive channels. This space-time block code [1] can be associated to Orthogonal Frequency Division Multiplexing (OFDM) system. In that case, the symbols are replaced by OFDM symbols composed of N c data symbols. Another possible approach called space-frequency block code (SFBC) is to use adjacent subcarriers for the coding of the OFDM symbols [2]. In both cases, STBC-OFDM and SFBC-OFDM, it is generally assumed that, for each antenna, the channel transfer functions to which the symbols are applied are equal. However, for a fast moving receiver or for different OFDM subchannels, the assumption might not be true. Thus, the purpose of the present paper is to investigate the effect of applying the symbols to different channel transfer functions for each antenna. The paper is organized as follows. First, a systematic approach for the derivation of space-frequency block codes (SFBC) is introduced and developed for M=2. Then, in section 3, the approach is used to derive the coefficients of the linear combiner in the receiver, when 4 different channel transfer functions are involved. Section 4 is dedicated to simulation results, obtained with perfect channel state information (CSI) and with an adaptive filter in a lattice structure by considering Hiperlan/2 specifications. Finally, we draw conclusions and point out directions for future work in section 5.

2. A SYSTEMATIC DERIVATION FOR SFBC The space-time block code [1] has been applied to frequency selective channels by using OFDM with cyclic prefix, which transforms a frequency selective channel into N c flat fading subchannels [2]. The use of OFDM offers the opportunity for coding in the frequency domain in the form of space-frequency block codes.

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For 2 transmitter antenna, for each OFDM symbol, adjacent subcarriers k and k + 1 (k = 1,3, 4,..., N c − 1) are used in spacefrequency code. The nth OFDM symbol vector, S(n) = s1 (n) s 2 (n) ... s Nc −1 (n) s Nc (n)  , is coded into two vectors S1 (n) = s1 (n) −s*2 (n) ... s Nc −1 (n) −s*Nc (n)  and S2 (n) = s 2 (n) s1* (n) ... s Nc (n) s*Nc −1 (n)  by space-frequency block encoder. S1 (n) is transmitted from the first antenna while S2 (n) is transmitted simultaneously from the second antenna. The symbol assignment of the 1st OFDM symbol vector is shown in Figure 1.

Tx 1

Tx 2

s1 −s*2 s3 −s*4 f1

f2

f3

f4

s2

s1* s 4

s*3

s Nc −1 −s*N

...

c

fNc-1 fNc

frequency

s Nc s*N −1 c

...

Figure 1. Symbol assignment for subcarriers in SFBC By assuming that neighboring subchannels in the OFDM spectrum have the same channel transfer functions, that is, H1 (f k ) = H1 (f k +1 ) and H 2 (f k ) = H 2 (f k +1 ) , maximum diversity is achieved. However, this assumption might be far from reality at least for some section of the OFDM spectrum. Moreover, to cope with narrow-band fading, it might be advantageous to use subchannels far apart in the spectrum. Therefore, we will consider the case of unequal channels for space-frequency block codes.

3. THE CASE OF UNEQUAL CHANNELS FOR SFBC According to the symbol assignment given in Figure 1, the discrete Fourier transform (DFT) outputs at the receiver for subcarriers k and k + 1 for unequal channel transfer functions can be written as R k = s k H1 (f k ) + s k +1H 2 (f k ) + N k R k +1 = −s∗k +1H1 (f k +1 ) + s∗k H 2 (f k +1 ) + N k +1

(1)

where N k denotes the DFT of additive white Gaussian noise corresponding to subcarrier k . Now, for simplification, we will consider that the signals belong to the 1st and 2nd subcarriers whose channel transfer functions are defined as follows H1 (f1 ) = H11 = α1e jθ

H1 (f 2 ) = H12 = a1α1e j( θ1 + ∆θ1 )

H 2 (f1 ) = H 21 = α 2e jθ 2

H 2 (f 2 ) = H 22 = a 2 α 2 e j( θ2 +∆θ2 )

1

(2)

where a 1e j∆θ1 and a 2 e j∆θ2 are the differences between channel transfer functions of adjacent subcarriers for each antenna. After arranging (1) in the matrix form, the outputs will be  R1   s1   N1   R *  = H  s  +  N*   2  2  2 H where the channel transfer matrix is H =  *11  H 22

(3)

H 21  . *  −H12 

The first solution to obtain the reconstructed signals is the zero forcing (ZF) approach, which is to multiply the complex conjugate of the channel transfer matrix with signals in (3) and detect the symbols separately. *  y1   H11 = y   *  2   H 21

H 22   H11  * − H12   H 22

* H 21   s1   H11 +   * *  − H12  s 2   H 21

2

H 22   N1   * − H12   N 2 

(4)

2 2  y1   H11 + H 22 y  =  *  2   H 21H11 − H12 H*22

* *  %  H11 H 21 − H 22 H12  s1   N 1  +    2 2 %  H 21 + H12  s 2   N 2

(5)

This detection method is denoted as SFBC-OFDM-ZF [3]. In this solution, maximum diversity is achieved, however the symbol carried by the adjacent subcarrier causes interference that adds up to the noise. In order to examine the effect of interference, we compute the signal-to-noise ratios (SNRs) at the receiver side for the symbols that belong to consecutive subcarriers. 2

SNR 1,ISI =

2

[ H11 + H 22 ]2 Ps 2

2

2

* * [ H11 + H 22 ]2σ n2 + H11 H 21 − H 22 H12 Ps 2

SNR 2,ISI =

(6)

2

[ H 21 + H12 ]2 Ps 2

2

2

[ H 21 + H12 ]2σ 2n + H*21H11 − H12 H*22 Ps

where Ps denotes the symbol transmit power. The second detection method is SFBC-OFDM-ML [3], which is based on the maximum likelihood (ML) criterion given in (7). However, it needs more computational time than the SFBC-OFDM-ZF approach.  sˆ1  sˆ  = arg min  2  {s1 ,s2 }∈Α

 R1   s1   R *  − H s   2  2

(7)

We propose a new solution in (8) that eliminates the interference terms in (5) while providing diversity gain with linear processing.  y1   A 0   s1  (8)  y  =  0 B s  + noise terms  2  2  The system matrix is rewritten similar to (4) by  y1   γ11  =  y 2   γ 21

γ12   H11  * γ 22   H 22

H 21   s1   γ11 + *  −H12  s 2   γ 21

γ12   N1   * γ 22   N 2 

(9)

The conditions of interference cancellation and diversity gain are presented below, Equations for Interference Cancellation * γ11H 21 − γ12 H12 =0

Equations for Diversity Gain γ11H11 + γ12 H *22 = A

γ 21H11 + γ 22 H *22 = 0

* γ 21H 21 − γ 22 H12 =B

By using interference cancellation equations, γ12 and γ 21 can be written in terms of γ11 and γ 22 as H H* γ12 = γ11 *21 γ 21 = − γ 22 22 H12 H11

(10)

(11)

Substituting γ12 and γ 21 into diversity gain equations, we obtain A and B expressions as,  H*  * − H 21 22  B = γ 22 − H12 H11  

 H  A = γ11 H11 + H*22 *21  H12  

(12)

* Now, we choose γ11 as H11 and γ 22 as −H12 , to obtain maximum diversity for the signals coming from transmitter antenna 1.

2

* A = H11 + H11 H *22

2

B = H12 + H12 H 21

j( ∆θ − ∆θ H 21 2 2 e H H = + 11 22 * a 1a 2 H12 1

2)

H *22 2 2 = H12 + H 21 a 2 a 1e j( ∆θ −∆θ H11 1

3

(13) 2)

Then, we rewrite γ ij values in terms of the channel transfer functions * γ11 = H11

γ 21 = H12

* γ12 = H11

H*22 = H*21a1a 2e j( ∆θ H11

1 − ∆θ 2

H 21 e j( ∆θ − ∆θ H = 22 * H12 a1 a 2 1

2

)

γ 22 = − H12

)

(14)

This result shows that it is possible to eliminate the interference terms by multiplying H 22 and H *21 in complex conjugated channel transfer matrix in (4) with two coefficients depending on a1a 2 and ∆θ1 − ∆θ 2 . In order to investigate the overall performance, we calculate the SNR values at the receiver side for both symbols. 2

SNR1,proposed =

2

2

H11 + H 22 e j( ∆θ1 −∆θ2 ) / a1a 2 2

2

H11 + H 22 / a a 2

SNR 2,proposed =

Ps 2σ 2n

2 2 1 2

2

H12 + H 21 e j( ∆θ1 −∆θ2 ) a1a 2 2

2

H12 + H 21 a a 2 1

2 2

2

(15)

Ps 2σn2

Notice that SNR values depend on the differences between the channel transfer functions of the adjacent subcarriers for each antenna.

4. SIMULATION RESULTS The performance evaluation of the proposed solution for wideband channels was performed using channel model A provided by Hiperlan/2 standard [4,5]. The total OFDM symbol duration is T = 4µs including Tg = 0.8µs cyclic prefix. Channel model A corresponds to a typical office environment and its power delay profiles where a classical Jakes’ Doppler spectrum and Rayleigh fading statistics are assumed for all taps. So, each tap has a complex gain with Rayleigh amplitude and uniformly distributed phase. The channel transfer function changes between subcarriers in a single OFDM frame while the whole function changes very slowly from frame to frame due to the fact that a low Doppler frequency is chosen.

4.1. SNR Performance The performance of the proposed scheme and the ZF scheme which causes interference have been compared with respect to their signal-to-noise-ratios for all subcarriers. A representative example is given in Figures 2 and 3 at Es/N0=25 dB.

Figure 2. Comparison of SNR1 values

Figure 3. Comparison of SNR2 values

As can be seen from the figures, the proposed solution gives better signal-to-noise-ratios for both symbols in all subcarriers.

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4.2. BER Performance with Perfect CSI Considering the overall system performance, we simulated the SFBC-OFDM system with ZF, ML and proposed schemes assuming perfect channel state information (CSI) available at the receiver. At each OFDM frame, 52 information symbols drawn from QPSK constellation are used without channel coding.

Figure 4. The BER performance of different approaches with perfect CSI Figure 4 confirms that the proposed scheme improves the bit-error-rate performance especially at high Eb/N0 compared to ZF detection. The proposed scheme gives the same performance as the ML decoding with computationally efficient structure that provides flexibility especially for high constellation size.

4.3. BER Performance with Adaptive Channel Estimation The channel coefficients γ ij given in (14) can be directly estimated by using the adaptive filter in lattice structure shown in Figure 5. R2

(.)

(.)

*

-1

R 1 γ 11

y1

γ 12

e1

γ 21 y2

R *2

γ 22

Decision Device

Decision Device

e2

~ d1

d1 ~ d2

d2

Figure 5. The Adaptive filter scheme for channel estimation The filter coefficients are adapted by a recursive least square (RLS) algorithm which has a faster converge time than least mean square (LMS) algorithm [6]. The RLS algorithm converges toward the proposed solution that eliminates the interference terms. While evaluating BER versus Eb/N0, the length of the initial training sequence was taken as 20 OFDM symbols. Again 20 OFDM symbols were used as retraining sequence after each 100 transmitted OFDM symbols in order to avoid divergence since the channel is a slowly time varying channel. BER performance of the proposed solution with the RLS equalizer approaches to perfect CSI model with a reasonable degree as shown in Figure 6.

5

Figure 6. The BER performance of proposed scheme for adaptive channel estimation with RLS algorithm compared to perfect CSI

5. CONCLUSION We have developed a systematic approach for SFBC allowing flexibility in choosing the system parameters to derive an optimal solution for the case of unequal channels. The proposed solution was validated by simulations in terms of different performance criteria. From the results of channel estimator we can suppress the interference terms using two correction coefficients. We have shown that an adaptive RLS algorithm converges toward the proposed solution. In conclusion, the proposed method gives better performance than the ZF solution and the same performance as the ML decoding with computationally efficient structure. The extension of these results to higher order systems will be examined in future works. ACKNOWLEDGEMENT The authors would like to thank for their support İzmir Institute of Technology in Turkey and the Institut Aeronautique et Spatial (IAS) in France.

6. REFERENCES [1] [2] [3] [4] [5] [6]

S. M. Alamouti, “A Simple Transmitter Diversity Scheme for Wireless Communications,” IEEE J. Select. Areas Communication, vol. 16, no. 8, pp. 1451-1458, 1998. K. F. Lee, D. B. Williams, “A Space-Frequency Transmitter Diversity Technique for OFDM Systems”, in Proc. 2000 Global Telecommunications Conf., San Francisco, CA, pp.1473-7, Nov. 2000. R. Narasimhan, “Performance of Diversity Schemes for OFDM systems with Frequency Offset, Phase Noise and Channel Estimation Errors”, IEEE Trans. on Communications, vol.50, no.10, pp.1561-1565, Oct. 2002. J. Khun-Jush, P. Schramm, U. Wachsmann, F. Wenger, “Structure and Performance of the Hiperlan/2 Physical Layer”, in Proc. of the IEEE Vehicular Technology Conf. Amsterdam, June 1999, vol. 5, pp. 2667-2671. ETSI Normalization Committee, “Channel Models for Hiperlan/2 in Different Indoor Scenarios”, France, 1998. M. Bellanger, Adaptive Digital Filters, Marcel Dekker Inc., New York - Basel, 2002.

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