1140
IEEE COMMUNICATIONS LETTERS, VOL. 14, NO. 12, DECEMBER 2010
Novel Distributed Space-Time Trellis Codes for Relay Systems over Cascaded Rayleigh Fading Hacı ˙Ilhan, Student Member, IEEE, ˙Ibrahim Altunbas¸, and Murat Uysal, Senior Member, IEEE Abstract—In this letter, we consider the deployment of distributed space-time trellis codes (STTCs) for user cooperation. We derive a pairwise error probability (PEP) expression for distributed STTCs over cascaded Rayleigh fading channels. Using the derived PEP expression, we determine a code design criterion and propose novel 4-/8-/16-state 4-PSK and 8-/16-state 8-PSK distributed STTCs through a systematic code search. We confirm the superiority of the proposed codes through extensive MonteCarlo simulations.
STTCs as a result of exhaustive computer search and compare their performance with existing ones in the literature. Notation: 𝑇 , 𝑇 𝑟(.) and 𝐻 indicate transpose, trace and Hermitian operators, respectively. 𝑄(.) stands for the Gaussian-𝑄 function. ∥.∥ denotes the Frobenius norm. 𝐺𝑝,𝑡 𝑢,𝑣 (. ∣. ) , 𝑝, 𝑡, 𝑢, 𝑣 ∈ ℕ, 𝑝 ≤ 𝑣, 𝑡 ≤ 𝑢 is the Meijer’s 𝐺-function [7]. Γ (., .) is the incomplete Gamma function [7].
Index Terms—Error rate performance, distributed space-time coding, amplify-and-forward relaying, cascaded Rayleigh fading channels.
II. S YSTEM M ODEL
C
I. I NTRODUCTION
OOPERATIVE diversity [1] exploits the broadcast nature of wireless transmission to realize spatial diversity advantages through user cooperation. The application of spacetime codes to cooperative systems has drawn much attention, see e.g. [1]-[4] and the references therein. In [2], Nabar et al. have analyzed the performance of distributed STTCs in amplify-and-forward (AF) mode through the derivation of PEP. They prove that the original design criteria for conventional STTCs (i.e., rank and determinant criteria) still apply for the design of distributed STTC schemes under the assumption of appropriate power control at relay terminals. In [3], Canpolat et al. have examined the PEP analysis for a large number of AF relays and demonstrated that the performance of distributed STTC schemes is dominated by a metric similar to the Euclidean distance. An improved design criterion for the same system has been further developed by Hong et al. in [4]. A key assumption in majority of these works is the underlying fading channel model which is characterized by Rayleigh distribution. This statistical model has been developed for urban cellular radio systems, however is not very realistic for mobile-to-mobile links such as in intervehicular communications. Through experimental and theoretical investigations, cascaded Rayleigh fading has been proposed [5], [6] for a more precise characterization of mobile-to-mobile channel. In this paper, we investigate the error rate performance of a relayassisted AF cooperative system over cascaded Rayleigh fading channels. First, we derive a Chernoff bound on the PEP. Then, based on the derived PEP, we determine a code design criterion for distributed STTCs. We present several new distributed
Manuscript received June 29, 2010. The associate editor coordinating the review of this letter and approving it for publication was R. Nabar. H. ˙Ilhan and ˙I. Altunbas¸ were supported by the Scientific and Technological Research Council of Turkey (TUBITAK) under Project 107E022. H. ˙Ilhan and ˙I. Altunbas¸ are with the Department of Electronics and Communication Engineering, Istanbul Technical University, 34469 Istanbul, Turkey (e-mail: {ilhanh, ibraltunbas}@itu.edu.tr). ¨ M. Uysal is with the Faculty of Engineering, Ozye˘ gin University, 34662, Istanbul, Turkey (e-mail:
[email protected]). Digital Object Identifier 10.1109/LCOMM.2010.102610.101133
We consider a single-relay scenario where source, relay, and destination nodes operate in half-duplex mode. Source and relay nodes are equipped with a single pair of transmit and receive antennas while the destination node contains 𝐾 receive antennas. We adopt the so-called transmit diversity (TD) protocol of [8] also referred to as Protocol I in [2]. Let 𝐿 be the total number of symbols transmitted from source node during the first transmission phase. The received signals at the relay and 𝑘 𝑡ℎ receive antenna of destination nodes are given as √ 𝑖 = 𝐸𝛼𝑆𝑅 𝑥𝑖1 + 𝑛𝑖𝑅 (1) 𝑟𝑅 √ 𝑘,𝑖 𝑟𝐷 = 𝐸𝛼𝑘𝑆𝐷 𝑥𝑖1 + 𝑛𝑘,𝑖 (2) 𝐷1 , 𝑘 = 1, 2, ..., 𝐾, 1 where 𝐸 is the average energy per transmission phase and 𝑥𝑖1 , 𝑖 = 1, 2, ..., 𝐿 denotes the unit-energy 𝑀 -ary PSK (phase shift keying) signal transmitted by the source node during the first transmission phase. The relay node scales the received signal and retransmits the resulting signal during the second transmission phase. The source node simultaneously transmits 𝑥𝑖2 . After proper normalization as in [2], the received signal at the destination is given by √ √ 𝑘,𝑖 = 𝐴1 𝐸𝛼𝑆𝑅 𝛼𝑘𝑅𝐷 𝑥𝑖1 + 𝐴2 𝐸𝛼𝑘𝑆𝐷 𝑥𝑖2 + 𝑛𝑘,𝑖 (3) 𝑟𝐷 𝐷2 2 /(
)
/( 2 ) 𝜆 + 𝛼𝑘𝑅𝐷
where 𝐴1 = 1 𝜆 + 𝛼𝑘𝑅𝐷 2 and 𝐴2 = 0.5𝜆 with 𝜆 = 2 (1 + 𝐸/𝑁0 )/(𝐸/𝑁0 ).
𝑘,𝑖 In (1-3), 𝑛𝑖𝑅 , 𝑛𝑘,𝑖 𝐷1 and 𝑛𝐷2 are the independent samples of zero-mean complex Gaussian random variables with variance 𝑁0 /2 per dimension, which model the additive noise terms. Here, 𝛼𝑘𝑆𝐷 , 𝛼𝑆𝑅 and 𝛼𝑘𝑅𝐷 represent, respectively, source-to-destination (𝑆 → 𝐷), source-to-relay (𝑆 → 𝑅) and relay-to-destination (𝑅 → 𝐷) links’ complex quasi static fading coefficients whose magnitudes ℎ𝑘𝑆𝐷 = 𝛼𝑘𝑆𝐷 , ℎ𝑆𝑅 = ∣𝛼𝑆𝑅 ∣, ℎ𝑘𝑅𝐷 = 𝛼𝑘𝑅𝐷 follow cascaded Rayleigh distribution. These magnitudes are assumed to be the product of statistically independent, but not necessarily identically distributed Rayleigh random variables [9].∏Specifically, we ∏𝑁𝑆𝐷 𝑁𝑆𝑅 𝑘 have ℎ𝑘𝑆𝐷 = 𝑙1 =1 ℎ𝑆𝐷,𝑙1 , ℎ𝑆𝑅 = 𝑙2 =1 ℎ𝑆𝑅,𝑙2 and ∏ 𝑁𝑅𝐷 𝑘 𝑘 𝑘 ℎ𝑅𝐷 = 𝑙3 =1 ℎ𝑅𝐷,𝑙3 where ℎ𝑆𝐷,𝑙1 , 𝑙1 = 1, 2, ..., 𝑁𝑆𝐷 , ℎ𝑆𝑅,𝑙2 , 𝑙2 = 1, 2, ..., 𝑁𝑆𝑅 and ℎ𝑘𝑅𝐷,𝑙3 , 𝑙3 = 1, 2, ..., 𝑁𝑅𝐷
c 2010 IEEE 1089-7798/10$25.00 ⃝
˙ILHAN et al.: NOVEL DISTRIBUTED SPACE-TIME TRELLIS CODES FOR RELAY SYSTEMS OVER CASCADED RAYLEIGH FADING
are Rayleigh distributed variables with normalized power and 𝑁𝑆𝐷 , 𝑁𝑆𝑅 and 𝑁𝑅𝐷 denote the degree of cascading. 𝑘,𝑖 𝑘,𝑖 𝑇 , 𝑟𝐷2 ] , the noise Introducing the received vector r𝑖𝑘 = [𝑟𝐷 1 𝑘,𝑖 𝑇 , 𝑛 ] and the transmitted vector xi = vector n𝑖𝑘 = [𝑛𝑘,𝑖 𝐷1 𝐷2 𝑖 𝑖 [𝑥1 , 𝑥2 ] for 𝑖 = 1, 2, ..., 𝐿, the received signals can be written in a matrix form as R𝑘 = A𝑘 X + N𝑘 where R𝑘 , X and N𝑘 1 2 𝐿 are defined as R𝑘 = [r1𝑘 , r2𝑘 , ..., r𝐿 𝑘 ], X = [x , x , ..., x ] and 1 2 𝐿 N𝑘 = [n𝑘 , n𝑘 , ..., n𝑘 ] with size 2×𝐿. The channel matrix A𝑘 of size 2×2 is given by ⎤ ⎡ √ 𝑘 𝐸𝛼𝑆𝐷 0 ⎦. A𝑘 = ⎣ √ √ 𝑘 𝑘 𝐴1 𝐸𝛼𝑆𝑅 𝛼𝑅𝐷 𝐴2 𝐸𝛼𝑆𝐷 (4) The received vector is then fed to a maximum likelihood decoder.
TABLE I C ODE SEARCH RESULTS FOR 4-PSK MODULATION . 2𝑣
[
( ) ˆ X, X∣A ≤ exp 𝑘
𝐾 −1 ∑ 𝑇𝑟 4𝑁0 𝑘=1
Applying method, the term ( ) ( the )diagonalization 𝐻 ˆ ˆ X−X X−X in (5) can be written as V𝐻 ΔV 𝐻 where V V = I (I: identity matrix) and Δ is a real ∑𝐿 𝑖 ˆ𝑖1 2 and diagonal matrix with entries 𝜒1 = 𝑖=1 𝑥1 − 𝑥 ∑𝐿 𝑖 2 ˆ𝑖2 . Therefore, (5) can be rewritten as 𝜒2 = 𝑖=1 𝑥2 − 𝑥 ( )2 ( ( ) 𝐾 0.5𝜆 ℎ𝑘 ∏ 𝑆𝐷 ˆ 𝑘 , ℎ𝑆𝑅 , ℎ𝑘 𝑃 X, X∣ℎ exp − ≤ 𝑆𝐷 𝑅𝐷 𝜆+(ℎ𝑘 )2 𝑘=1 ( ( ) 𝑅𝐷) ( )2 𝑘 ( 𝑘 )2 𝐸 ℎ2 𝑆𝑅 ℎ𝑅𝐷 𝐸 ℎ𝑆𝐷 4𝑁 + × exp − 𝜒1 . 2 4𝑁 0 0 𝜆+(ℎ𝑘 𝑅𝐷 )
−
𝑦∣ 1,...,1 with 𝑦¯ = E[𝑦] = 1 follow 𝑓 (𝑦) = (1/𝑦) 𝐺𝑁,0 0,𝑁 𝑦/¯ 𝑘 𝑘 [9]. Averaging (6) with respect to 𝑦𝑆𝐷 , 𝑦𝑆𝑅 and 𝑦𝑅𝐷 by using [7, eq. 7.811.4] we obtain ⎛
1
𝑁
𝑆𝐷 𝐺1,𝑁
,1
⎜ ⎜ ⎝
1
(
2 0
10
8
]
] 1 2
8
8
Chen et al. [11]
[ 0 2
2 3
1 2
] 2 0
10
3.1
NewCode1
[ 2 2
3 3
2 0
] 1 2
10
8
12
10.65
10
10.65
12
10.65
12
10.75
12
10.66
[ Canpolat et al. [3]
2 2
3 3
2 1
] 1 0
3 2
[ 1 2
2 0
2 1
0 2
1 2
Chen et al. [11]
[ 2 2
2 0
2 1
1 2
0 2
NewCode2
[ 1 2
2 1
1 3
0 2
1 1
8
[ Canpolat et al. [3] Chen et al. [11] NewCode3
16
3 2
2 3
2 0
3 2
] ] ]
2 1
] 0 2
[ 1 2
2 0
1 3
2 2
3 2
] 2 0
16
18.6
[ 2 2
1 3
3 3
2 3
1 2
] 2 0
16
22
)
𝐸 𝜒 4𝑁0 2
(6) )2 ( 𝑘 𝑘 𝑘 Let 𝑦𝑆𝐷 , 𝑦𝑆𝑅 and 𝑦𝑅𝐷 denote 𝑦𝑆𝐷 = ℎ𝑘𝑆𝐷 , 𝑦𝑆𝑅 = ℎ2𝑆𝑅 )2 ( 𝑘 and 𝑦𝑅𝐷 = ℎ𝑘𝑅𝐷 . Their ( probability ) density functions
( ) 𝐾 ∞ ∫ ∏ ˆ ≤ 𝑃 X, X
3 2 2 0
4
) ( )( )𝐻 ) ˆ ˆ A𝐻 . X−X 𝑘 A𝑘 X − X
(5)
2 2
𝑑2
2 2
The PEP represents the probability of choosing the coded ˆ when indeed X was transmitted. Under the sequence X assumption of perfect channel state information, a Chernoff bound on PEP is given by [3], 𝑃
3 3
𝑑2𝑐𝑙
[ 1 1
Hong et al. [4]
Hong et al. [4]
(
Generator Matrix (G𝑇 )
Canpolat et al. [3]
III. PEP D ERIVATION AND D ISTRIBUTED STTC D ESIGN C RITERION
(
1141
)
1
⎞ ⎟ ⎟ ⎠
using [10, eq. 07.34.03.0723.01]. Under sufficiently high 𝐸/𝑁0 values, (8) simplifies to ( ) ( / ) ˆ ≤ 16 𝑑2 𝐾 (𝐸/𝑁0 )−2𝐾 𝑃 X, X (9) / where 𝑑2 = 𝜒21 3 + 𝜒1 𝜒2 /9. In order to minimize the derived upper bound in (9), the minimum value of 𝑑2 should be maximized for all path pairs of the codes. Although derivation has been made under static 𝑅 → 𝐷 link, it can be still used as a sub-optimal design criterion [2], [3]. IV. C ODE S EARCH
In this section, we perform a code search to find out optimal distributed STTCs in the sense of minimizing PEP. We adopt 1 ( ) a similar approach with [11] which uses a systematic code − 𝑁𝑆𝑅 ,1 ⎝ 𝑁 ,0 1 𝑘 𝑘 . ⎠ 𝐺 𝑅𝐷 ×𝐺1,𝑁 𝑑𝑦𝑅𝐷 𝐸 𝑦𝑘 𝜒 1 0,𝑁𝑅𝐷 𝑦𝑅𝐷 1,...,1 𝑆𝑅 4𝑁0 𝑅𝐷 1 search for conventional STTC design. Let G be a generator 𝜆+𝑦𝑘 𝑅𝐷 1,...,1 (7) matrix of size (𝑣 + log2 𝑀 ) × 2 where 𝑣 refers to the number A closed form solution for (7) is unfortunately not available of memory elements in the code and 𝑀 is the signal constelfor the general case. However, assuming 𝑆 → 𝐷, 𝑆 → 𝑅 lation size. Our code search algorithm first conducts a search links experience double Rayleigh distribution (𝑁 = 2) and to find out the set of codes with largest minimum squared static 𝑅 → 𝐷 link (i.e., no fading, only AWGN), (7) can be Euclidean distance (i.e. 𝑑2𝑐𝑙 = 𝜒1 +𝜒2 ). Then, among this set a / written as second search is conducted to maximize 𝑑2 = 𝜒21 3+𝜒1 𝜒2 /9. [ ( )]𝐾 Otherwise the search yields codes with higher 𝑑2 , but with ( ) −2 16(𝐸/𝑁)( 4(𝐸/𝑁0 )−1 0) ˆ ( ) 𝑃 X, X ≤ Γ 0, lower 𝑑2𝑐𝑙 compared to conventional ones and ultimately results 1 𝜒 𝜒1 + 0.5𝜆 𝜒 𝜒1 + 0.5𝜆 𝜒 𝜆+1 2 𝜆+1 2 𝜆+1 1 [ ( ) ( )]𝐾 in worse error rate performance. 4(𝐸/𝑁0 )−1 4(𝐸/𝑁0 )−1 4(𝐸/𝑁0 )−1 × exp + Γ 0, 1 𝜒 1 𝜒 Table I presents search results for 4-PSK. The code search 𝜒1 + 0.5𝜆 𝜒 𝜆+1 1 𝜆+1 1 𝜆+1 2 (8) yields new 4-, 8- and 16-state distributed STTCs with values 𝑘=1 0
⎛
𝑘 𝑦𝑅𝐷
𝑆𝐷
𝜒1 +
⎞
0.5𝜆 𝜆+𝑦𝑘 𝑅𝐷
𝜒2
𝐸 4𝑁0
1,...,1
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IEEE COMMUNICATIONS LETTERS, VOL. 14, NO. 12, DECEMBER 2010
TABLE II C ODE SEARCH RESULTS FOR 8-PSK MODULATION . 2𝑣
Generator Matrix (G𝑇 ) [
Chen et al. [11] [ NewCode4
8 [
Chen et al. [11] [ NewCode5
16
𝑑2𝑐𝑙
𝑑2
2 1
4 6
0 4
3 4
2 0
] 4 0
7.17
2.67
0 4
4 5
0 2
4 0
1 2
] 2 4
7.17
4.2
2 4
4 0
7 2
3 7
6 6
0 7
] 4 4
8
2.47
6 0
4 1
5 4
2 4
4 0
7 6
] 0 0
8
6.17
Figure 1 illustrates the FER performance of the proposed 8-state 4-PSK code (NewCode2), 8-state 8-PSK (NewCode4) and 16-state 8-PSK code (NewCode5). As benchmarks for NewCode2, the performance of 8-state conventional STTC (Chen et al. [11]) and distributed STTCs (Canpolat et al. [3] and Hong et al. [4]) are provided. As observed from the figure, NewCode2 outperforms the competing codes of [11], [3], [4] and brings SNR gains of 0.8 dB, 0.3 dB and 0.3 dB, respectively. Similarly, proposed NewCode4 and NewCode5 outperform the benchmark codes taken from [11]. Specifically, we obtain SNR gains of 0.6 dB and 0.8 dB, respectively for 8 and 16-state codes. VI. C ONCLUSIONS In this letter, we have investigated the performance of distributed STTCs over cascaded Rayleigh fading channels through the derivation of PEP. Although such PEP expressions are readily available in the literature for Rayleigh fading, the derived PEP expression provides useful insight into inner performance mechanisms of cascaded Rayleigh fading channels which are particularly important for inter-vehicular communication applications. Using the derived PEP expression, we have determined a code design criterion and proposed novel distributed STTCs through a systematic code search. Through a Monte-Carlo simulation study, it has been demonstrated that the proposed codes outperform the existing codes in the literature. R EFERENCES
Fig. 1. Performance of distributed STTCs for 8-state 4-PSK and 8-/16-state 8-PSK modulation.
𝑑2 = 8, 10.75 and 22, respectively. For comparison, we include conventional STTCs proposed by Chen et al. [11] along with distributed STTCs proposed by Canpolat et al. [3] and Hong et al. [4]. The new codes provide identical or larger 𝑑2𝑐𝑙 and 𝑑2 values than the corresponding codes those in [3], [4] and [11]. Table II presents new distributed 8- and 16-state 8-PSK STTCs along with reference codes from [11]. Similar to 4-PSK codes, the new 8-PSK codes provide identical 𝑑2𝑐𝑙 and larger 𝑑2 values with respect to the reference codes. V. S IMULATION R ESULTS In this section, we present frame error rate (FER) results for the proposed codes. In the simulations, we assume quasistatic double Rayleigh fading channels for all underlying links and 𝐾 = 4 receive antennas deployment at the destination terminal. We further assume that 𝐿 = 130 modulation symbols are transmitted from source in each transmission phase. SNR is defined as 𝐸/𝑁0 and all performance comparisons are made at a FER of 10−3 . Due to space limitations, performance results are provided only for selected codes.
[1] J. N. Laneman and G. W. Wornell, “Distributed space-time coded protocols for exploiting cooperative diversity in wireless networks,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2415-2525, Oct. 2003. [2] R. U. Nabar, H. B¨olcskei, and F. W. Kneubhler, “Fading relay channels: performance limits and space-time signal design,” IEEE J. Sel. Areas Commun., vol. 22, no. 6, pp. 1099-1109, Aug. 2004. [3] O. Canpolat, M. Uysal, and M. M. Fareed, “Analysis and design of distributed space-time trellis codes with amplify-and-forward relaying,” IEEE Trans. Veh. Technol., vol. 56, pp. 1649-1660, July 2007. [4] S. K. Hong and J. M. Chung, “Improved design criterion for distributed space-time trellis codes with AF relaying,” Electron. Lett., vol. 44, pp. 1212-1213, Sep. 2008. [5] I. Z. Kovacs, “Radio channel characterisation for private mobile radio systems: mobile-to-mobile radio link investigation,” Ph.D. thesis, Aalborg University, Sep. 2002. [6] J. Salo, H. El-Sallabi, and P. Vainikainen, “Statistical analysis of the multiple scattering radio channel,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 3114-3124, Nov. 2006. [7] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th edition. Academic Press, 2007. [8] H. Ochiai, P. Mitran, and V. Tarokh, “Variable-rate two-phase collaborative communication protocols for wireless networks,” IEEE Trans. Inf. Theory, vol. 52, no. 9, pp. 4299-4313, Sep. 2006. [9] G. K. Karagiannidis, N. C. Sagias, and P. T. Mathiopoulos, “N*Nakagami: a novel stochastic model for cascaded fading channels,” IEEE Trans. Commun., vol. 55, no. 8, pp. 1453-1458, Aug. 2007. [10] http://functions.wolfram.com. [11] Z. Chen, J. Yuan, and B. Vucetic, “Improved space-time trellis coded modulation scheme on slow Rayleigh fading channels,” Electron. Lett., vol. 37, pp. 440-441, Mar. 2001.
