ITW 2010
Single Real-Symbol Decodable, High-Rate, Distributed Space-Time Block Codes K. Pavan Srinath
B. Sundar Rajan
Dept of ECE, Indian Institute of science Bangalore 560012, India Email:
[email protected]
Dept of ECE, Indian Institute of science Bangalore 560012, India Email:
[email protected]
Abstract— A scheme to apply the rate-1 real orthogonal designs (RODs) in relay networks with single real-symbol decodability of the symbols at the destination for any arbitrary number of relays is proposed. In the case where the relays do not have any information about the channel gains from the source to themselves, the best known distributed space time block codes (DSTBCs) for k relays with single real-symbol decodability offer 2 complex symbols per channel use. The an overall rate of 2+k scheme proposed in this paper offers an overall rate of 1/4 complex symbol per channel use, which is independent of the number of relays. Furthermore, in the scenario where the relays have partial channel information in the form of channel phase knowledge, the best known DSTBCs with single real-symbol decodability offer an overall rate of 1/3 complex symbols per channel use. In this paper, making use of RODs, a scheme which achieves the same overall rate of 1/3 complex symbols per channel use but with a decoding delay that is 50 percent of that of the best known DSTBCs, is presented. Simulation results of the symbol error rate performance for 10 relays, which show the superiority of the proposed scheme over the best known DSTBC for 10 relays with single real-symbol decodability, are provided.
I. I NTRODUCTION
AND
BACKGROUND
The usage of multiple transmit antennas and multiple receive antennas to exploit diversity in the wireless fading environment has been well studied [1]. With nt transmit antennas and nr receive antennas, a maximum diversity order of nt nr can be achieved with the use of suitable full-diversity space time block codes (STBCs) [2], [3]. However, depending upon the application, for instance in a network with small devices, it may not always be feasible to employ multiple transmit and receive antennas. In such a scenario, distributed space time coding [4], [5] has been shown to exploit cooperative diversity, making use of relays between the source and the destination. Each relay is made to transmit a column of an STBC, thus acting as a transmit antenna in the multiple transmit antenna system. This scheme is relevant also when there is no direct link between the source and the destination. There are several ways of processing the received signals at the relay nodes, among which the following two are widely discussed in the literature - (1) amplify and forward. (2) decode and forward. The decode and forward scheme involves additional complexity at the relays, with the relays having to decode the received signals and then re-encoding before transmitting to the destination. The amplify and forward protocol is more practical, since it involves simpler processing at the
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relays. In this paper, we consider the amplify and forward protocol. In [5], Jing and Hassibi have proposed an amplify and forward protocol for achieving cooperative diversity in wireless networks. This protocol essentially employs fulldiversity STBCs, with each relay transmitting a column of the STBC. STBCs, when used in cooperative networks, are termed Distributed STBCs (DSTBCs). An important practical issue that arises is the maximum-likelihood (ML)-decoding complexity of DSTBCs. It has been shown that low MLdecoding complexity STBCs need not necessarily be low MLdecoding complexity DSTBCs [6]. Significant amount of research has been done on developing low ML-decoding complexity DSTBCs [6], [7], [8], [9], [10]. These DSTBCs are applicable when the relays do not have any channel state information (CSI). The DSTBCs proposed in [9], which are termed distributed orthogonal space time block codes (DOSTBCs), are single real-symbol decodable (refer Section II for a formal definition), analogous to the complex orthogonal designs (CODs) [2] in the collocated MIMO set-up. These DOSTBCs have been shown to achieve a rate of 2/k complex symbols per channel use (SPCU) for k relays, with the transmission considered from the relays to the destination. It has also been shown that the upper bound on the rate of DOSTBCs is 2/k complex SPCU. The overall 2 rate of transmission from the source to the destination is 2+k complex SPCU. The DSTBCs proposed in [10] are single complex-symbol decodable and the overall rate of these is 4 complex SPCU for k relays. Here, the upper bounded by 4+k increase in rate with respect to the rate of the DOSTBCs comes at the cost of additional encoding complexity at the source and increased decoding complexity at the destination. However, for both the schemes, the rate decreases with increase in the number of relays. Additionally, the DOSTBCs have a lot of zero entries, which indicates a higher peak to average power ratio (PAPR). If we allow an additional functionality at the relay in the form of estimating the phase of the channel from the source to the relay, then, it is shown in [11] and [12] that the rate of DOSTBCs can be made 1/2 complex SPCU which is also an upper bound, with respect to transmission from the relays to destination. So, the overall rate of transmission from the source to the destination is 1/3 complex SPCU and this is independent of the number of relays. The authors of [11] and
[12] both make use of the rate-1/2 COD [2] to achieve this. The decoding delay (refer Section II for a formal definition), however, of both the schemes, is large. The contributions of this paper are: • For the case where there is no CSI at the relays, we propose a scheme that makes use of the rate-1 Real Orthogonal Designs (RODs) to obtain DSTBCs that offer an overall rate of 1/4 complex SPCU with single realsymbol decodability at the destination for any number of relays. So, when the number of relays is greater than 6, the proposed scheme achieves a higher bandwidth efficiency than the DOSTBCs proposed in [9]. • For the case where there is channel phase information (CPI) at the relays, we propose another scheme that again makes use of the rate-1 RODs and provides an overall rate of 1/3 complex SPCU with single real-symbol decodability, the same as that of the DOSTBCs with CPI [11], [12], but with a decoding delay that is reduced by 50 percent. • For the case where there is no CSI at the relays, the simulation results for the scheme proposed in this paper show a large coding gain over the comparable DOSTBCs. The paper is organized as follows. In Section II, we give the system model and the relevant definitions. Section III explains the coding technique for applying RODs in relay networks without CSI at the relays and Section IV details the scheme to apply RODs in relay networks with CPI at the relays. Section V provides the simulation results for the 10-relay network without CSI at relays and the 8-relay network with CPI at relays. Discussion and concluding remarks constitute Section VI. Notations: Throughout, bold, lowercase letters are used to denote vectors and bold, uppercase letters are used to denote matrices. Let X be a complex matrix. Then, XH , XT and X∗ denote the Hermitian, √the transpose and the conjugate of X, resp. and j represents −1. The set of all real and complex numbers are denoted by R and C, respectively. The real and the imaginary part of a complex number x are denoted by xI and xQ , respectively. kXk denotes the Frobenius norm of X, |x| denotes the absolute value of a complex number x, and IT and OT denote the T × T identity matrix and the null matrix, respectively. bxc denotes the largest integer not greater than x and dxe denotes the smallest integer not lesser than x. The Kronecker product is denoted by ⊗. For a complex random variable X, E[X] denotes the mean of X. II. S YSTEM M ODEL We consider a source node, a destination node and k relays which aid in communicating information from the source to the destination, as shown in Fig. 1. All the nodes and the relays are assumed to be equipped with one antenna and communication is half-duplex, i.e. no node can send and receive simultaneously in the same frequency band. Transmission of data to the destination occurs in two phases. In the first phase, the source transmits to the relays and in the second phase, the relays transmit to the destination, while the source remains
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Fig. 1.
Network model
switched off. The wireless channels between the terminals are assumed to be quasi-static and flat fading. The channel gain from the source to the ith relay is denoted by gi and the channel gain from the ith relay to the destination is denoted by fi . The channel gains are all assumed to be independent and identically distributed (i.i.d.) complex Gaussian random variables with zero mean and unit variance. The destination is assumed to have perfect knowledge of all the channel gains. Let P be the total average transmit power used by the source and the relays. Let x ∈ CT1 ×1 denote the information vector that the source sends to the relays in T1 time slots, with the entries of x being information symbols taking values from a suitable signal constellation. Further, let x contain m independent information symbols, m ≤ T1 . In literature, generally, m is assumed to be equal to T1 , but we need and consider this generalized scenario. We further assume that E[xH x] = T1 . Then, p ri = π1 P gi x + ni ∈ CT1 ×1 , (1)
where, ri is the received vector at the ith relay, π1 P is the power used by the source for transmission to the relays (the average power used by the source will be lesser than π1 P , since the source remains switched off when the relays transmit to the destination), ni is the noise vector, the entries of which are i.i.d. Gaussian random variables with zero mean and unit variance, i.e. E[ni ni H ] = IT1 . At each relay, processing of the received signal vector is done and the relay transmits the processed signal vector to the destination in T2 time slots. To do the processing, the ith relay uses two complex matrices Ai and Bi , each of size T2 × T1 and with the constraint that kAi k2 + kBi k2 = T2 , and sends the signal vector Ai ri + Bi ri ∗ to the destination. Let π2 P be the power that each relay uses to transmit to the destination. The total average transmit power used by the source and the relays together is thus given by P =
π1 P T1 + kπ2 P T2 T1 + T2
(2)
Generally, in literature, it is assumed that π1 = 1 and π2 = 1/k. The relay has to scale Ai ri + Bi ri ∗ by a factor of π1πP2 P+1
before transmission. Hence, at the destination, the received signal is as follows. r k π2 P X fi .(Ai ri + Bi ri ∗ ) + w ∈ CT2 ×1 , (3) y= π1 P + 1 i=1
where, w is the noise vector, the entries of which are i.i.d Gaussian random variables with zero mean and unit variance and E[wwH ] = IT2 . Hence, s π1 π2 P 2 y= Xf + n, (4) π1 P + 1 where, f , [f1 f2 · · · fk ]T , X is the T2 ×k sized DSTBC whose column-vector representation matrices [13] are {Ai , Bi }, i = 1, 2, · · · , k, and r k π2 P X fi .(Ai ni + Bi ni ∗ ) + w. (5) n, π1 P + 1 i=1
The following definitions are relevant to wireless relay network. Definition 1: Rate of transmission: If the source transmits m independent complex symbols to the relays in T1 time slots and the relays transmit these symbols to the destination in T2 time slots, the overall rate of transmission is defined to be m T1 +T2 complex symbols per channel use. Definition 2: ML-decoding complexity : The MLdecoding complexity is a measure of the number of computations required in minimizing the ML-metric, which in our case is H
−1
− 12
2
This is true if −1 −1 CH Cj + CH Ci = Ok , 1 ≤ i 6= j ≤ 2m. i R j R
(10)
It must be noted that even if (10) is true, the DSTBC is single real-symbol decodable only if the constellation employed is such that the real and the imaginary parts of each symbol can be independently decoded, like in the case of rectangular QAM. In the case where QAM is employed, single realsymbol decodable codes have an ML-decoding complexity that is independent of the constellation size [15], [16]. The DOSTBCs [9] are single real-symbol decodable codes. Definition 4: Single complex-symbol decodable DSTBCs: A DSTBC is said to be single complex-symbol decodable [10] if the ML-metric can be expressed as follows. M (X) =
m X
1
1
kR− 2 (y − ρ(xiI C2i−1 + xiQ C2i )fk − (m − 1)kR− 2 yk.
i=1
This is true if −1 −1 CH Cj + CH Ci = O k , i R j R
∀i 6= j, j + 1, ∀i 6= j, j − 1,
if j is odd if j is even.
(11)
Single complex-symbol decodable DSTBCs have an MLdecoding complexity of the order of M , where M is the size of the signal constellation. The codes presented in [10] are single complex-symbol decodable. Definition 5: Decoding Delay : This is the time that the destination has to wait, starting from the time of transmission of symbols from the source, before it can start decoding the DSTBC matrix that it receives. With respect to our system model, the decoding delay is equal to T1 + T2 time slots. III. R ATE -1/4, SINGLE REAL - SYMBOL DECODABLE DSTBC S WITHOUT CSI AT RELAYS
M (X) = (y − ρXf) R (y − ρXf) = kR (y − ρXf)k , (6) q In this section, we propose a scheme to construct single π1 π2 P 2 where, ρ = π1 P +1 and R is the covariance matrix of n, real-symbol DSTBCs with rate 1/4 for any number of relays, i.e., E[nnH ]. Precisely, with no CSI at the relays. We make use of the rate-1 RODs, which exist for any number of transmit antennas [2]. Consider k π2 P X T ×k denote the rate-1 ROD R= (7) k relays in the network. Let X ∈ R |fi |2 (Ai Ai H + Bi Bi H ) + IT2 . π1 P + 1 i=1 for k transmit antennas, with T being the number of channel Since the DSTBC X has m independent information symbols uses. Since T is always a power of 2 [2], the source transmits which take values from a signal constellation of size M , the T /2 complex symbols to the relays in T time slots as follows. maximum number of searches required in minimizing the ML- Let x1 , x2 , ...xT /2 be the symbols to be transmitted, which take values from a suitable rectangular QAM constellation. metric is M m . Let m With the notations as in Section II, the source transmits a T X xiI C2i−1 + xiQ C2i , (8) length vector x = [x1 x∗1 x2 x∗2 ..... xT /2 x∗T /2 ]T . Let U , X= i=1 1 1 √1 and M = IT /2 ⊗ U. Clearly, 2 −j j where, Ci , i = 1, 2, ...2m are the dispersion matrices [14] or √ the weight matrices of the DSTBC. Mx = 2[x1I x1Q .... x(T /2)I x(T /2)Q ]T . Definition 3: Single real-symbol decodable DSTBCs: A DSTBC is said to be single real-symbol decodable [10] if the Consider the rate-1 ROD X for k transmit antennas, with entries x1I , x1Q ,... ,x(T /2)I and x(T /2)Q . Each column of X, ML-metric can be expressed as follows. denoted by xi , i = 1, 2, · · · , k, is a T length real vector. Let 1 x m i = √2 Di (Mx), with Di , i = 1, 2, · · · , k being T × T unitary X −1 M (X) = kR 2 (y − ρxiI C2i−1 f)k matrices with only one non-zero entry in each row, the noni=1 zero entry being ±1 [13]. Each relay with index i is assigned m X 1 −1 −2 only one relay matrix, which is Di M. The relay uses this relay (y − ρxiQ C2i f)k − (2m − 1)kR 2 yk. (9) + kR i=1 matrix to premultiply the received signal vector ri with. The
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received signal vector ri at each relay with index i is p ri = π1 P gi x + ni , [x1 x∗1
x2 x∗2 .....
(12)
xT /2 x∗T /2 ]T
with x = and the other notations as in Section II. The received signal at the destination is r k π2 P X fi Di Mri + w = ρXh + n, (13) y= π1 P + 1 i=1 q 2π1 π2 P 2 where, ρ = π1 P +1 , X is the rate-1 ROD for k T
transmit antennas, h , [g1q f1 g2 f2 .... gk fk ] and n , Pk ρ1 i=1 fi Di Mni + w, ρ1 = π1πP2 P+1 . The covariance matrix R is as follows. R = E[nnH ] = ρ21
k X i=1
|fi |2 IT + IT = (ρ21
k X i=1
|fi |2 + 1)IT .
Hence, R is a scaled identity matrix. This makes the conditions H (10) reduce to CH i Cj + Cj Ci = Ok , 1 ≤ i 6= j ≤ 2m, which are satisfied since X as an STBC which is single real-symbol decodable in the co-located MIMO scenario. Further, since the scheme transmits T /2 independent complex symbols in 2T time slots (T time slots from the source to the relays and T time slots from the relays to the destination), the overall rate of transmission is 1/4 complex SPCU, and this is independent of the number of relays. The overall rate of the 2 scheme using DOSTBCs proposed in [9] is 2+k . Hence, for k > 6, our scheme has a higher rate of transmission. Further, the DOSTBCs have a lot of zero entries, making the PAPR very high for transmission from the relays. Since the rate-1 RODs have no zero entries, the PAPR is much lower than with the case of the DOSTBCs. However, the only drawback of our scheme is the higher decoding delay than that of the DOSTBCs. This is due to the fact that the rate-1 RODs have a transmission time T that increases exponentially with the number of transmit antennas k and is as follows. ( k 2( 2 −1) if k is divisible by 8, T = k k 2(4b 8 c+dlog2 (k−8b 8 c)e) if k is not divisible by 8. On the contrary, the DOSTBCs have a transmission time that increases linearly with the number of relays. IV. R ATE -1/3, SINGLE REAL - SYMBOL DECODABLE DSTBC WITH CPI AT RELAYS
with x = [x1 x2 ... xT /2 ]T . Each relay with index i compensates for the phase of gi . The ith relay multiplies the received signal vector ri by gi∗ /|gi |. Let 1 1 1 1 , U2 , √ U1 , √ 2 −j 2 j and M1 = IT /2 ⊗ U1 , M2 = IT /2 ⊗ U2 . Clearly, √ M1 x + M2 x∗ = 2[x1I x1Q .... x(T /2)I x(T /2)Q ]T Considering the rate-1 ROD X for k transmit antennas, with entries x1I , x1Q ,... ,x(T /2)I and x(T /2)Q , each column of X, denoted by xi , i = 1, 2, · · · , k, can be expressed as xi = √1 Di (M1 x + M2 x∗ ), with Di , i = 1, 2, · · · , k as explained in 2 the previous section. So, each relay with index i is assigned the unitary relay matrices Di M1 and Di M2 and sends the vector q (gi∗ M1 ri +gi M2 r∗ π2 P i) to the destination, in T time slots. π1 P +1 Di |gi | Hence, the destination receives the vector r k π2 P X fi Di (gi∗ M1 ri +gi M2 r∗i )+w = ρXh+n, y= π1 P + 1 i=1 |gi | q 2π1 π2 P 2 where, ρ = π1 P +1 , X is the rate-1 ROD for k transmit antennas, h , [|g1 |f1 |g2 |f2 .... |gk |fkq ]T and n , Pk fi ∗ ∗ ρ1 i=1 |gi | Di (gi M1 ni + gi M2 ni ) + w, ρ1 = π1πP2 P+1 . The covariance matrix R is as follows. R = E[nnH ] = ρ21
k X i=1
|fi |2 IT + IT = (ρ21
k X i=1
|fi |2 + 1)IT .
Hence, as in the case where the relays do not have CSI, R is a scaled identity matrix. As a result, X as a DSTBC continues to be single real-symbol decodable. Further, since the scheme transmits T /2 independent complex in 3T /2 time slots (T /2 time slots from the source to the relays and T time slots from the relays to the destination), the overall rate of transmission is 1/3 complex SPCU, and this is independent of the number of relays. The overall rate of the scheme using the rate-1/2 CODs, proposed in [11], [12] is also 1/3 complex SPCU, but in these schemes, the source has to transmit T complex symbols to the relays in T time slots, and the relays transmit the columns of the rate-1/2 COD in 2T time slots. Hence, the overall decoding delay is twice that of our scheme. V. S IMULATION R ESULTS
In this section, we assume that the relays have a knowledge of the phase of the channel gains from the source to themselves and hence, can compensate for the phase. Here again, for the k-relay network, we consider the rate-1 ROD for k transmit antennas that use T channel uses, but unlike in the previous section where the source transmits T /2 independent complex symbols in T time slots to the relays, the source now transmits T /2 symbols in T /2 time slots to the relays. Hence, the received signal vector at the ith relay is p ri = π1 P gi x + ni ,
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Considering quasi-static, Rayleigh flat-fading channels between all links, we compare the symbol Error Rate (SER) performance of our proposed scheme for the case where there is no CSI at the relays with that of the DOSTBCs [9] by providing simulation results for the case of 10 relays. We consider 10 relays for two reasons. Firstly, our proposed scheme provides a higher rate than the DOSTBCs do only when the number of relays is greater than 6. Secondly, choosing suitable QAM constellations for transmission at 1 bit per channel use (BPCU) is easier for the case of 10 relays, in which case our scheme has a rate of 1/4 complex SPCU and the DOSTBC has a rate of 1/6 complex SPCU. Our DSTBC is the rate-1 ROD
0
0
10
10
Proposed scheme−8 Relays with CPI
Proposed scheme−10 Relays 10
−1
10 Symbol Error Rate
Symbol Error Rate
Il−Min Kim DSTBC−8 Relays with CPI
DOSTBC−10 Relays
−1
−2
10
−3
10
−4
10
−2
10
−3
10
−4
10
−5
10
−5
−6
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5
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20
10
25
Average transmit power in dB
Fig. 2.
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6
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24
Average transmit power in dB
SER performance at 1 BPCU for 10 relays without CSI at relays
for 10 transmit antennas [2]. Fig. 2 shows the performance of our DSTBC and the DOSTBC for 10 relays at 1 BPCU. Our scheme employs 16-QAM and the DOSTBC employs 64QAM. Our scheme beats the DOSTBC handsomely. Moreover, the PAPR at the relays using our scheme is 1, while that at the relays using the DOSTBC is 5. Fig. 3 provides a comparison of the SER performance at 1 BPCU of our proposed scheme for 8 relays with CPI at relays and the single real-symbol decodable DSTBC (which we refer to as Il-Min Kim DSTBC) proposed in [11], [12], using the rate-1/2 COD for 8 transmit antennas. Both the schemes, with rate 1/3 complex SPCU, use 8-QAM and the performance is the same. However, the decoding delay for our scheme is 12 time slots as against a decoding delay of 24 time slots for the scheme in [11], [12]. VI. D ISCUSSION In this paper, we have shown that by cleverly choosing the information vector to be transmitted by the source to the relays, the overall rate of transmission can be increased while maintaining single real-symbol decodability at the destination, when the relays do not have CSI. Our proposed scheme has a higher rate of transmission than the best existing schemes for more than 6 relays. Further, for the case where the relays know the phase of the channel gain from the source to the relays, we have shown that without losing out in the rate of transmission and single real-symbol decodability, decoding delay can be reduced by 50 percent with regards to the best existing schemes. Finding out if such similar gains are obtainable for the case of single complex-symbol decodable DSTBCs could constitute future research. ACKNOWLEDGEMENT This work was partly supported by the DRDO-IISc program on Advanced Research in Mathematical Engineering, through research grants to B. Sundar Rajan.
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Fig. 3.
SER performance at 1 BPCU for 8 Relays with CPI
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