Design of Distributed Space-Time Block Code for Two ...

1

Design of Distributed Space-Time Block Code for Two-hop Cooperative Wireless Relay Networks over Frequency Selective Fading Channels * Quoc-Tuan Vien, Le-Nam Tran, Ju-Hyup Kim, Een-Kee Hong, and **Yong-Seo Park *KyungHee Univ., **KyungWon Univ.

Abstract This letter proposes a new distributed space time block code (DSTBC) for two-hop cooperative wireless networks over frequency selective fading channels with amplify-and-forward (AF) relaying. The proposed DSTBC is designed to achieve maximal data rate, spatial diversity gain, and decoupling detection of data blocks. These goals are accomplished by allowing each relay to linearly process its received signals to form a column of the code matrix in the conventional space time block codes that were developed for co-located antenna systems. Moreover, the proposed DSTBC also enables the use of frequency domain equalization to reduce receiver complexity.

I. INTRODUCTION Space time block codes (STBCs) introduced in [1], [2] can greatly improve the reliability of wireless fading channels by efficiently exploiting the spatial diversity gain. The idea is to encode data symbols across a number of physically spaced antennas and time. STBCs are widely used because they can provide diversity gain with low maximum likelihood (ML) decoding algorithm. The conventional STBCs were designed for the co-located antennas. Therefore, they are easily deployed at the base station to improve the performance of the downlink transmission. However, the realization of STBCs is impractical in the uplink transmission due to the constraints on size and hardware complexity in mobile handsets. Fortunately, mobile users can cooperate to form a virtual multiple-antenna system, which is now known as “cooperative diversity” or “user cooperation” [3]-[4]. The distributed space time block codes (DSTBCs) can be thought as the distributed implementation of conventional STBCs for cooperative communications. Initially, the DSTBCs were proposed for flat fading channels [5], 0, [7]. The problem of DSTBC in frequency selective fading channels was investigated in [9] with decode-and-forward (DF) relaying, and in [10] with AF relaying. However, these DSTBCs were devised for relay networks where there exist one active relay node and a direct communication link between the interesting source and the final destination. In this paper, we design a new DSTBC for two-hop relay networks 0, [7] over frequency selective channels with AF protocol, where there are two active relay nodes. The proposed DSTBC operates as follow. In the first time slot the source sends two information symbol blocks to two relays. What is remarkable in our

proposed DSTBC is that one of two relays additionally permutates and conjugates its received signals that will be sent to the destination in the next time slot. It means each relay actively creates signals that convey a distinct column of the block Alamouti scheme (see, e.g. [1], [11], and 0). This idea allows the proposed DSTBC to obtain maximal data rate, maximal diversity gain, and decoupling detection of data blocks for low complexity of receiver structure, as we show in section II. When the DSTBC based on repetition coding in [10] is applied to the considered two-relay scenario, the source needs two time slots to send two columns of the block Alamouti scheme to the relays. This is because the relay of the method in [10] just amplifies and forwards what it receives. Thus, the proposed DSTBC has higher rate than the repetition DSTBC in [10] . Notation: Bold lower and upper case letters represent vectors and matrices, respectively;

 T

,

 

and

 

denote transpose, complex

conjugate, and Hermitian transpose operations, respectively; FM is the M -point normalized DFT K is the M  M permutation matrix carrying matrix; PM the reverse operation followed by a right cyclic shift of over K positions of a given vector of length M .

II. SYSTEM MODEL AND THE PROPOSED DSTBC We consider a wireless relay network shown in Fig. 1 where the source terminal ( S ) cannot communicate directly with the interesting destination ( D ). The data transmission from S to D is completed via two-hop protocol 0, [7] with the assistance of two relays R1 and

2 R2 . The frequency selective channel from A to B is characterized by

hAB   hAB (0),..., hAB ( LAB  1)

T

where

K PM . We choose K  B  LSR2 to ensure that, after K 2 permutation, at least last LR2 D samples of  PM r2 and

is the

LAB

K 2 PM r1 are all zeros. This is to make channel matrix

channel memory order. A time slot in this work is defined as the interval required to transmit two data blocks x1 , and x 2 shown in Fig. 1, which are created from padding a zero sequence of length L to two information data blocks si , i  1, 2 of length B . The length of the zero sequence must satisfy L  max( LSR1  LR1D , LSR2  LR2 D ) . This condition is to

from R2 to the destination become circulant. The received signal at the destination is written by

x1 s1(1) … s1(B)

LSR2 zeros

B

h R1D

x2 ZP

L

s1(1) … s1(B)

B

ZP

S

D

L

h SR2 Received from S in 1st time slot

r12

r22 K  B  LSR2

K  B  LSR2

Transmitted to D in 2nd time slot

h R2 D

R2

zeros

PMK (r22 )

K  y1  H R1D H SR1 x1  H R2 D PM H SR2 x*2  η1'

(4)

K  y 2  H R1D H SR1 x 2 + H R2 D PM H SR2 x1*  η'2 ,

(5)

where η1' and η'2 include the Gaussian noise of relays and destination. It is common to normalize the noise variance in (4) and (5) to be N 0 / 2 for fair comparison with other systems [10], which results in K  y1  1H R1D H SR1 x1   2 H R2 D PM H SR2 x*2  η1

zeros

PMK (r12 )

K  y 2  1Η R1D H SR1 x 2 +  2H R2 D PM H SR2 x1*

Fig.1. System model and data structures sent by source and relays.

(6)

 η2 .

(7)

In (6) and (7), the values of normalization factors  i are defined as

In the first time slot (first hop), the source serially transmits two data blocks to two relays. In the next time slot (second hop), one relay only amplifies and retransmits its received signals, while the other permutates, conjugates and reorders the received data blocks before transmitting to the destination as illustrated in Fig. 1. The idea behind our design is that R2 is designed to send the second column of the block Alamouti’s scheme (see, e.g. [1] and 0) to the destination. This enables the decoupling detection of two data blocks at the destination, and increases coding rate. To achieve the same goal in the considered scenario, the source of repetition code in [10] must send two columns during two time slots. Thus, the rate of this scheme is reduced to 2/3 data block/time slot. Clearly, our design can achieve a rate of 1 data block/time slot. We now proceed to proving that our proposed DSTBC can decouple the detection of two data blocks. The received signal at the relay is given by ri j  H SR j xi  ηij , i  1, 2 ; j  1, 2,

(3)

noise vector at the destination with each entry having zero-mean and variance of N 0 / 2 . From (1), we can rewrite (2) and (3) as

R1

h SR1

K 2 * y 2  H R1Dr21 + H R2 D PM (r1 )  η2 ,

the channel from R j to D , and ηi is white Gaussian

r21

LSR1 zeros

(2)

where H R j D is the M  M circulant matrix, denoting

make the channel matrices from source to relays and relays to destination circulant. r11

K 2  y1  H R1Dr11  H R2 D PM (r2 )  η1

i 

1 2   2 1

where



 j  i i ESRi LR1D

| h R1D  l  |2  1 2

l 0



i  1  ESRi / N 0





LR2D

| h R2 D  l  |2

l 0

,

(8)

 i  ERi D / N 0

,

,

and

i, j  {1, 2} . Conjugating, multiplying both sides of (7) K with the permutation matrix PM , and noting that K  K PM H PM  H for any circulant matrix H , we can rewrite (7) as   K *  y2   2 H (9) R D H SR2 x1  1H R D H SR PM x 2  η2 . 2

(1)

1

1

For mathematical convenience, we group (6) and (9) in vector-matrix form as     y1   1H R1D H SR1  2 H R2 D H SR2   x1   η1    y      P K x*   η2   2   2 H 1H H SR2 H M 2  R D SR R D   2 1 1   H eq

(10) where H SR j is the M  M circulant channel matrix from S to R j , and ηij is the white Gaussian noise vector at the j-th relay with each entry having zero-mean and variance of N 0 / 2 per dimension. Throughout this paper, the superscript j denotes the relay index, while the subscript i refers to data block index. The permutation at R2 is represented by the matrix

Let Ω

us

12

2

2

| H SR1 | | H R1D |

2 H eq H eq  I 2  Ω

is

denote

 22

| H SR2 | | H R2 D |

,

a

block-diagonal

matrix.

2

2

then

Multiplying both sides of (10) with the unitary matrix

(I 2  Ω 1 )H eq , we can decouple the detection of two data blocks. That means two data blocks can be detected independently, rather than joint detection, without any loss of optimality, which is based on the following model

3 2

2

2

H R1D   22 H SR2

2 H R2 D  xi  η i , i  1, 2, 

(11) where η i is the Gaussian noise vector resulting from the space-time decoupling. On the achievable diversity gain: The achievable diversity gain is determined by the pairwise error probability (PEP). Assuming coherent maximum likelihood (ML) detection and perfect knowledge of channel gains, the probability of transmitting s and deciding erroneously in favor of sˆ is upper bounded as [2]  d 2  s, sˆ    , (12) P s  sˆ h SR1 , h SR2 , h R1D , h R2 D  exp    4 N 0   where s  {s1, s 2 } , and d  s, sˆ  is the Euclidean





distance given by 2

2

d 2  s, sˆ   12 H R1D H SR1 Tzp (s  sˆ)   22 H R2 DH SR2 Tzp (s  sˆ) .

(13) Following the same analysis proposed in [10] for one relay system, we can easily prove that the maximal achievable diversity of our proposed system is min( LSR1 , LR1D )  min( LSR2 , LR2 D ) . Generally, ML detection or standard equalization techniques based on (13) require complex receiver. In the following, we introduce the decoupling in the frequency domain that allows for the low-complexity equalizers. We notice that any circulant matrix H can be diagonalized as H  F F where  is the diagonal matrix whose diagonal elements are the DFT of the first column of H . Taking the DFT of both sides of (10) results in   y 1   1 R1D  SR1  2  R2 D  SR2   x 1   η 1   y         . (14)  2   2 R2 D  SR2 1SR1 R1D   x 2   η 2      eq Let

us 

   eq  eq 

12

denote

|  SR1 | |  R1 D | 2

2

 22

|  SR2 | |  R2 D | 2

2

. Multiplying both side of (14) with (I 2   1/ 2 ) eq , we can decouple the detection of each data block on frequency domain as z i  x i  ωi , i  1, 2. (15) Since the equivalent channel matrix  is diagonal, (15) can be decomposed into a set of M scalar equations zk   k xk  k , k  0,1,, M  1. (16) In (16) we omit the dependency of subscript i because the detection of two data blocks is based on the same model. Typical frequency domain equalizers can be applied to the outputs of decoupling process to recover the transmitted signals. In this paper, we use frequency domain linear equalization (FD-LE) whose coefficients can be found as 0 1 Wk  2 (17)  k  N 0 /  d2

III. NUMERICAL RESULTS The uncoded BER performance of the proposed DSTBC is evaluated in this section. Each data block consists of 64 symbols including the zero sequence and information-carrying data which is modulated by QPSK with Gray mapping. To demonstrate the achievable diversity gain, the maximum likelihood detection should be involved. However, computational complexity of such ML algorithms is always prohibitively high. Alternatively, we consider FD-LE in (17) in our simulations for our proposed DSTBC with perfect synchronization and channel estimation. Fig. 2 shows the BER performances of the proposed DSTBC for various combinations of channel lengths. We assume that the value of ESR1 / N 0 is fixed at

25dB , and ER1D  ER2 D  10dB (power balance), and plot the BER curves as a function of ESR2 / N 0 . For the same value of LSR1 and LSR2 , the better performance occurs as the number of paths from the relays to destination increases, and the best performance is obtained when such channels become non-fading, i.e. AWGN channel. Although, there are differences on BER performances, the slopes of BER curves in high ESR2 / N 0 are identical, provided that LSR1  LSR2 . This is because the achievable diversity gain of our proposed DSTBC is min( LSR1 , LR1D )  min( LSR2 , LR2 D ) . The similar observation can also be found in [10] 10-1

10-2

10-3

Average BER

 z i   12 H SR1 

ESR /No = 25dB 1

ER D = ER D = 10dB

10-4

1

2

LSR = LSR = LR D = LR D = 1 1 2 1 2 LSR = LSR = 1, LR D = LR D = 2

10-5

1

2

1

2

LSR = 1, LSR = LR D = LR D = 2 1 2 1 2 LSR = LSR = 2, LR D = LR D = 4 1

10-6

2

1

2

LSR = LSR = 4, LR D = LR D = 6 1 2 1 2 LSR = LSR = LR D = 4, LR D = Nonfading 1

2

1

2

LSR = LSR = 4, LR D = Nonfading, LR D = Nonfading 1 2 1 2

10-7 0

2

4

6

8

10

12

14

16

18

20

ESR /N0 (dB) 2

Fig.2. BER performance of DSTBC with FD-LE for various combinations of channel memory order.

IV. CONCLUSION A new DSTBC scheme for two-hop cooperative systems over frequency selective fading channels has been proposed. The proposed DSTBC can achieve high rate, maximum spatial diversity gain, and low-complexity receiver. The main idea in our design is that each relay conveys a distinct column of the conventional STBC developed for co-located antennas. Namely, one relay will permutate and

4 conjugate its received signals. Although we consider the case of two relay nodes, the extension of the proposed DSTBC to cope with the systems where there are more than two relay nodes is straightforward. This is accomplished by applying the idea of linear dispersion codes where each column of the code matrix is a linear summation of transmit data and their conjugations [8]. We also have demonstrated the possibility of implementing FDE for our proposed DSTBC to reduce the complexity of the equalizer.

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[6] Y. Jing and B. Hassibi, “Distributed space–time coding in wireless relay networks,” IEEE Trans. Wireless Commun., vol. 5, no. 12, pp. 3524–3536, Dec. 2006. [7] Zhihang Yi and Il-Min Kim, “Single-Symbol ML Decodable Distributed STBCs for Cooperative Networks,” IEEE Trans. Inform. Theory, vol. 53, no. 8, pp. 2977-2985, Aug. 2007. [8] B. Hassibi and B. Hochwald, “High-rate codes that are linear in space and time,” IEEE Trans. Inform. Theory, vol. 48, pp. 1804–1824, July 2002. [9] G. Scutari, and S. Barbarossa, “Distributed space-time coding for regenerative relay networks,” IEEE Trans. Wireless Commun. , vol. 4, no. 5, pp. 2387 – 2399, Sep. 2005. [10] H. Mheidat, M.Uysal, and N. Al-Dhahir, “Equalization Techniques for Distributed Space-Time Block Codes With Amplify-and-Forward Relaying,” IEEE Trans. Signal Process., vol.55, no.5, pp.1839-1852, May 2007. [11] S. Zhou and G. B. Giannakis, “Space-time coding with maximum diversity gains over frequency-selective fading channels,” IEEE Signal Processing Lett., vol. 8, no. 10, pp. 269-272, Oct. 2001. [12] N. Al-Dhahir “Single carrier frequency domain equalization for space time block coded transmissions over frequency selective fading channels,” IEEE Commun. Lett., vol. 5, pp. 304, Jul. 2001.