Near-Optimum Detection for Distributed Space ... - Semantic Scholar
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 11, NOVEMBER 2008
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Near-Optimum Detection for Distributed Space-Time Block Coding under Imperfect Synchronization F.-C. Zheng, A. G. Burr, and S. Olafsson
Abstract—Significant performance gain can potentially be achieved by employing distributed space-time block coding (DSTBC) in ad hoc or mesh networks. So far, however, most research on D-STBC has assumed that cooperative relay nodes are perfectly synchronized. Considering the difficulty in meeting such an assumption in many practical systems, this paper proposes a simple and near-optimum detection scheme for the case of two relay nodes, which proves to be able to handle far greater timing misalignment than the conventional STBC detector. Index Terms—Distributed space-time block coding, distributed transmit diversity, cooperative diversity, cooperative relay.
I. I NTRODUCTION
S
PACE-TIME block coding (STBC) has proved to be a very effective technique to achieve transmit diversity for wireless systems with co-located antennas at the transmitter, mainly due to its high diversity order and low decoding complexity [1]. In wireless systems such as ad hoc and sensor networks, however, it may only be possible to have a single antenna (rather than multiple co-located antennas) at the transmit/receive nodes due to cost and size constraints. As a result, there has over the past few years been a considerable research effort on creating and harnessing space diversity by applying STBC in a distributed fashion (termed distributed STBC or D-STBC) to single antenna systems [2]-[11]. So far, however, most research on D-STBC has assumed that cooperative relay nodes are perfectly synchronized so that the corresponding symbols from all the relay nodes arrive at the destination node at the same time. Such an assumption, unfortunately, is difficult or even impossible to achieve in many practical systems (e.g. ad hoc networks). With imperfect synchronization among the relay nodes (i.e. inter-relay-node synchronization), the channels may become dispersive even under flat fading conditions. There has been only limited work reported in the literature addressing D-STBC under imperfect synchronization Paper approved by N. Al-Dhahir, the Editor for Space-Time, OFDM and Equalization of the IEEE Communications Society. Manuscript received January 18, 2007; revised May 2, 2007 and August 24, 2007. This work was presented in part at the 8th IEEE International Workshop on Signal Processing Advances for Wireless Communication (SPAWC 2007), Helsinki, Finland, 17-20 June 2007. This work was supported by the UK EPSRC under Grant EP/E007139. F.-C. Zheng was with the Dept. of Electronics, the University of York, York, UK, and the School of Electrical Engineering, Victoria University, Melbourne, Australia. He is now with the School of Systems Engineering, the University of Reading, Reading, UK (e-mail: [email protected]). A. G. Burr is with the Dept. of Electronics, the University of York, York, UK. S. Olafsson is with the Mobility Research Centre, BT Group CTO, BT, Ipswich, UK, and Reykjavik University, 103 Reykjavik, Iceland. Digital Object Identifier 10.1109/TCOMM.2008.070006
among the relay nodes, chiefly using block based equalization techniques at the destination node to mitigate the impact of asynchronous signals (e.g. [8][9]). Alternative techniques to D-STBC in the presence of asynchronism also exist and these include distributed space-time trellis coding [10] and delay diversity (which relies on equalization or sequence estimation) [11]. Compared with the original STBC schemes, however, these existing methods potentially incur a much higher computational complexity at the receiver. This paper proposes a simple and near-optimum detection scheme for the case of two relay nodes under imperfect or rough synchronization. By cancelling the interference components in the received signal (caused by timing misalignment) in a decision feedback manner, a maximum likelihood (ML) detection scheme is realized on a symbol by symbol basis (assuming no feedback error), thus retaining the low computational complexity of the original STBC principle (i.e. achieving near-Alamouti simplicity). For the rest of this paper, [·]T , [·]∗ , and [·]H represent “transpose”, “conjugate”, and “transpose and conjugate”, respectively, while CN(0, σ 2 ) denotes the set of Gaussian distributed complex numbers with the standard variance of σ 2 (i.e. 0.5σ 2 per dimension). II. D-STBC U NDER I MPERFECT S YNCHRONIZATION This paper assumes the 4-node model depicted in Fig. 1. As in almost all cooperative relay systems [3][4], there are two phases involved: Phase 1 for broadcasting and Phase 2 for relaying. [Phase 1] The source node (S) transmits while relay nodes (R1 and R2 ) and destination node (D) receive. To prepare for the STBC operation in Phase 2, the data symbols at S are grouped into pairs. Denoting the ith pair of symbols in a data packet or frame transmitted by S as s(i) =[s(1, i), s(2, i)]T , the corresponding signal received at D after this direct transmission (DT) is rsd (i) = hsd s(i) + nsd (i) ,
(1)
where hsd is the channel gain between S and D, rsd (i) = [ rsd (1, i), rsd (2, i)]T , nsd (i) = [nsd (1, i), nsd (2, i)]T , and nsd (j, i) ∈CN(0, σn2 ) is the additive noise. [Phase 2] The data packet received at R1 and R2 are encoded using the Alamouti structure and transmitted to D. For clarity, this paper assumes that the channel gains hsr1 and hsr2 are such that R1 and R2 can always detect correctly [8][11]. Also, hm is the channel gain from Rm to D under perfect inter-relay-node synchronization. Denoting the encoded symbol pair corresponding to s(i) as x(i) = [x(1, i), x(2, i)]T at R1 and y(i) = [y(1, i), y(2, i)]T
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 11, NOVEMBER 2008
Phase 1
+h2 (−2)y(1, i − 1) + nrd (1, i)
R1
hsr1
and
hsd
r(2, i) = h1 x(2, i) + h2 (0) y(2, i) + h2 (−1)y(1, i)
D
S
+h2 (−2)y(2, i − 1) + nrd (2, i) ,
hsr2
R2
Phase 2
x(1,i), x(2,i)
R1
D
h2 R2
y(1,i), y(2,i)
βm = | h2 (−m)|2 /|h2 (0)|2
The 2-phase relay model with 2 relay nodes.
STBC pair
From R1 ......
From R2 ......
x(2,i-1) x(1,i) τ
T
x(2,i)
.....
T
y(2,i-1) y(1,i)
y(2,i)
(3b)
where h2 (l), l = 0, −1, −2, is the channel gain between R2 and D under imperfect inter-relay-node synchronization, nrd ∈CN(0, σn2 ) is the additive noise. Here, terms h2 (−m), m = 1, 2, reflect the inter symbol interference from the previous symbols (due to imperfect synchronization). For practical pulse shaping waveforms (PSW’s) such as the raised cosine, h2 (−2) is already much less dominant. All the other terms such as h2 (−3) are even smaller or zero in value (e.g. due to digital generation of the PSW) and have therefore been truncated in the above expressions (otherwise they can be dealt with similarly using the procedure in this paper). The relative strength of h2 (−m) will be represented by ratio
h1
Fig. 1.
(3a)
.....
Fig. 2. Time delay (imperfect synchronization) at the destination node (D) between 2 relay nodes.
at R2 , the STBC matrix (Alamouti scheme) transmitted by R1 and R2 is therefore x(1, i) y(1, i) s(1, i) s(2, i) = . (2) x(2, i) y(2, i) −s∗ (2, i) s∗ (1, i)
(3c)
in this paper. The value of βm reflects the composite effect of timing delay τ and the particular pulse shaping waveform used. However, we normally have βm = 0 for τ = 0, and β1 = 1 for τ = 0.5T . All the channel gains above are assumed to remain constant over the whole data packet, but to be subject to Rayleigh fading from packet to packet, i.e. hsd ∈CN(0, σs2 ), h1 and h2 ∈CN(0, σr2 ). For a fair comparison with the non-relay schemes, R1 and R2 transmit at half power, i.e. σr2 = 0.5σs2 . The above signal model may be viewed as a special case of some more general channel models (see [12] and the references therein). In [12], for example, an optimum scheme for STBC was proposed for co-located transmit antennas using zero-padding block transmission. The detection procedure in this paper, however, is based on symbol pairs only, and hence achieves a near-Alamouti simplicity (due to the special signal model in (3a) and (3b)). Substituting (2) into (3a) and (3b), we have (4) r(i) = Hs(i) + I(i) + nrd (i) , h1 h2 (0) ∗ T , where r(i) = [r(1, i), r (2, i)] , H= h∗2 (0) −h∗1 T ∗ I(i) = [I(1, i), I(2, i)] = [h2 (−1)s (1, i − 1) + h2 (−2)s(2, i − 1), h∗2 (−1)s∗ (2, i) + h∗2 (−2)s(1, i − 1)]T , and nrd (i) = [nrd (1, i), n∗rd (2, i)]T . From (4), the conventional STBC detection can be carried out via the following standard 2-step procedure (assuming perfect channel state information at D). Step 1: Linear transform.
Due to factors such as different propagation delays, x(i) and y(i) will most likely arrive at D at different time instants. Since accurate synchronization is difficult or impossible [8][11], there is normally a timing misalignment of τ between the received versions of these signals. Considering the effort of (rough) synchronization which is always required and the fact that R1 and R2 are normally close to S (to ensure correct detection at relay nodes), we assume at this stage that τ is no greater than the symbol period T . This assumption is easy to meet in practice. Such a relative time delay, as is shown in Fig. 2, will still cause “inter symbol interference (ISI)” from neighboring symbols at the receiver, due to sampling or matched filtering (whatever kind of pulse shaping is used). Without the loss of generality, we can assume that the receiver at D is perfectly synchronized to R1 . The received signal at D over the two symbol periods can then be represented by
g(i) [g(1, i), g(2, i)]T Θr(i) = Ds(i) + ΘI(i) + v(i), (5) where Θ =HH , and v(i) = [v(1, i), v(2, i)]T = Θnrd (i). Also, “” above means “is defined as”. λ 0 Due to the Alamouti structure, D= ΘH= , where 0 λ
r(1, i) = h1 x(1, i) + h2 (0) y(1, i) + h2 (−1)y(2, i − 1)
λ = |h1 |2 + |h2 (0)|2 .
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ZHENG et al.: NEAR-OPTIMUM DETECTION FOR DISTRIBUTED SPACE-TIME BLOCK CODING UNDER IMPERFECT SYNCHRONISATION
Also, note that v(i)vH (i) = σn2
λ 0 0 λ
IV. C OMBINING WITH D IRECT T RANSMISSION
.
(6)
Step 2: Least square (LS) detection. s(j, i) = arg{ min |g(j, i) − λsm |2 },
(7)
sm ∈S
where j = 1, 2, sm is an arbitrary symbol in the corresponding symbol alphabet S, and M is the number of elements or symbols within S (similarly hereinafter). Due to the component of ΘI(i) in (5), however, the above procedure can suffer from significant detection errors, unless h2 (−m) = 0 (i.e. τ = 0, the case of perfect synchronization). III. M AXIMUM L IKELIHOOD D ETECTION Upon examining (3) and (4), we notice that s(m, i−1), m = 1, 2, are in fact already known if the detection process has been initialized properly (e.g. through the use of pilot symbol(s) at the start of the packet). As such, I(1, i) = h2 (−1)s∗ (1, i − 1) + h2 (−2)s(2, i − 1) and component h∗2 (−2)s(1, i − 1) can be removed (“decision feedback”) before applying the linear transform of Θ =HH in (5). This leads to
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∗
Even if the direct transmission (DT) fails, the received signal at D during Phase 1 (i.e. Eq.(1)) still contains valuable information and should therefore be retained and combined appropriately with its counterpart during Phase 2 wherever possible. A. The conventional STBC detector Applying maximum ratio combining (MRC) to (1) and (4), corresponding to (5) we now have g(i) = [g(1, i), g(2, i)]T = Θr(i) + h∗sd rsd (i) = Ds(i) + ΘI(i) + v(i) , λ 0 where D= with 0 λ
λ = |hsd |2 + |h1 |2 + |h2 (0)|2 , and
(13)
(14)
v(i) = Θnrd (i) + h∗sd nsd (i).
Here, Θ =HH as before. Note that we still have H 2 λ 0 v(i)v (i) = σn , 0 λ
g (i) = [g (1, i), g (2, i)] = Θr (i) = Ds(i)+zs (2, i)+v(i) , (8) where (9) where λ is now from (14). Step 2 is the same as (7) except z = [z1 , z2 ]T = Θ[0, h∗2 (−1)]T , for the new λ value in (14) above. and r(1, i) − I(1, i) . (10) B. The maximum likelihood detector r (i) = ∗ r (2, i) − h∗2 (−2)s(1, i − 1) The same maximum likelihood detection can now be apEquation (8) can then be written as plied in a similar manner as follows. (11a) g (1, i) = λs(1, i) + z1 s∗ (2, i) + v(1, i) , − Calculate g (i) = [g (1, i), g (2, i)]T = Θr (i) + h∗sd rsd (i). and − Carry out (12a) and (12b) using the above g (i) and the (11b) g (2, i) = λs(2, i) + z2 s∗ (2, i) + v(2, i) . new λ value in (14). Obviously, the two key properties of the STBC principle In (11b), g (2, i) is only related to s(2, i), and therefore simplicity and optimality (or near optimality in this case) - still s(2, i) can be detected as hold. Also, as can be seen from (14), the utilisation of the DT (12a) sˆ(2, i) = arg{ min |g (2, i) − λsm − z2 s∗m |2 }. signal has increased the diversity order of the system from 2 sm ∈S Then, s(i, 1) can be detected by substituting s(2, i) back to to 3. This increased diversity order, together with the fact that DT is at “full power”, will deliver a significant performance (11a): gain, as will be confirmed next. sˆ(1, i) = arg{ min |g (1, i) − λsm − z1 sˆ∗ (2, i)|2 }. (12b) T
sm ∈S
The above procedure totally eliminates the ISI caused by imperfect synchronization, but still retains the following two key properties of the original STBC principle. − Optimality: If there is no decision feedback error for s(m, i − 1), m = 1, 2, then the above procedure is optimum in terms of maximum likelihood (ML) . Since (6) holds, (12a) is an ML detector, and so is (13b). Our work shows that the impact of decision feedback error is normally very small (see Section V), hence the term “near-optimum detection” for the above procedure. − Simplicity: The above ML search involves one symbol only and thus has a linear computational complexity. Compared with the conventional Alamouti detector, it only involves M/2 extra multiplications per symbol, hence achieving a nearAlamouti simplicity.
V. S IMULATIONS With an 8-PSK system and assuming that the relays R1 and R2 can always detect correctly, D-STBC is employed regardless of DT success or failure. Also, the signal to noise ratio (SNR) is defined as SNR= σs2 /σn2 (dB). Also, the detection process is initialized only once, i.e., at the beginning of each symbol packet/stream. [The impact of h2 (−1) ] As h2 (−m) reflects the composite effect of both time delay τ and the pulse shaping waveform (PSW) and h2 (−1) is the most dominant term, we set β2 = 0 and β1 = −10, 0, 5, 10 (dB) in this example. The bit error rates (BER’s) of the proposed ML scheme are displayed in Fig. 3 (without DT combining (DTC)) and Fig. 4 (with DTC). For comparison, the corresponding results of the conventional detector and those of “DT alone” and “STBC with perfect
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-1
With EP & no DTC with no EP & no DTC With EP & DTC With no EP & DTC
-1
10
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DT alone ML: β 1 = 10dB Con: β1 = 10dB
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ML: β 1 = 5dB Con: β1 = 5dB ML: β 1 = 0dB
-4
10
-4
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Con: β1 = 0dB ML: β 1 = -10dB Con: β1 = -10dB
-5
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perfect synch
5
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30
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SNR
Fig. 3. The BER’s of the ML and conventional detectors under different values (without DT combining).
DT alone ML: β1 = 10dB
-1
10
Fig. 5. The impact of error propagation (EP) on the BER’s of the ML detector with and without DT combining (DTC): =5dB.
DT alone ML: τ = 0.8T Con: τ = 0.8T ML: τ = 0.6T Con: τ = 0.6T ML: τ = 0.4T Con: τ = 0.4T ML: τ = 0.2T Con: τ = 0.2T perfect synch
-1
10
Con: β 1 = 10dB ML: β1 = 5dB Con: β 1 = 5dB
-2
10
-2
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ML: β1 = 0dB Con: β 1 = 0dB -3
ML: β1 = -10dB
-3
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Con: β 1 = -10dB perfect synch -4
-4
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SNR
Fig. 4. The BER’s of the ML and conventional detectors under different values (with DT combining).
Fig. 6. The BER’s of the ML and conventional detectors under different values (with DT combining).
synchronization” (i.e. τ = 0) are also shown. Fig. 3 indicates that the ML detector alone greatly outperforms the conventional STBC detector and DT with β1 up to 5 dB, while Fig. 4 demonstrates that DTC has resulted in a significant improvement for the ML detector: it can now deal with much larger β1 (or τ ) values. [Propagation of decision feedback errors] To examine this critical issue, the ML detection is carried out (i) with error propagation (EP, i.e. as it is - with the natural propagation of any feedback errors), and (ii) with no EP (i.e. using the true symbols of the previous pairs for ISI removal). The BER results for β1 = 5dB (β2 = 0) are shown in Fig. 5. Clearly, the impact of error propagation is very minor indeed, showing the near-optimum nature of the detector in this paper. [Practical pulse shaping waveforms] To gain some insight into the specific impact of time mismatch τ alone, a raisedcosine pulse with the 3GPP roll-off factor of 0.22 is employed, and the results are presented in Fig. 6 (with DTC) for τ =
0.2T, 0.4T, 0.6T, and 0.8T , showing the effectiveness of the ML detector again. VI. C ONCLUDING R EMARKS It has been shown that the near-optimum detection scheme proposed in this paper is indeed very effective for 2-relaynode systems even under some relatively large βm or τ values while the conventional STBC detector fails even under small βm or τ values. DT combining can greatly improve the system performance due to the higher diversity order. Due to the reduced effective diversity order under imperfect synchronization, the proposed procedure does show gradual performance degradation with increasing βm or τ values. However, τ ≤ 0.5T (or β1 = 0 dB) will guarantee an excellent performance. In practice, this (i) represents a much relaxed requirement for inter–relay-node synchronization, and (ii) can always be realized by simply instructing R1 to delay its transmission by T if τ ∈ [0.5T, T ].
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ZHENG et al.: NEAR-OPTIMUM DETECTION FOR DISTRIBUTED SPACE-TIME BLOCK CODING UNDER IMPERFECT SYNCHRONISATION
R EFERENCES [1] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inform. Theory, vol. 45, no. 5, pp. 1456-67, July 1999. [2] M. Dohler, M. Hussain, A. Desai, and H. Aghvami, “Performance of distributed space-time block codes,” in Proc. IEEE VTC 2004-S, vol. 2, pp. 742-746, May 2004. [3] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: efficient protocols and outage behaviour,” IEEE Trans. Inform. Theory, vol. 50, no. 12, pp. 3062-3080, Dec. 2004. [4] J. N. Laneman and G. W. Wornell, “Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2415-2425, Oct. 2003. [5] P. Mitra, H. Ochiai, and V. Tarokh, “Space-time diversity enhancements using collaborative communications,” IEEE Trans. Inform. Theory, vol. 51, no. 6, pp. 2041-57, June 2005. [6] P. A. Angel and M. Kaveh, “On the performance of distributed spacetime coding systems with or two non-regenerarive relays,” IEEE Trans. Wireless Commun., vol. 5, no. 3, pp. 682-692, Mar. 2006.
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[7] P. A. Angel and M. Kaveh, “On the diversity of cooperative systems,” in Proc. IEEE ICASSP 2004, pp. IV-577-580. [8] X. Li, “Space-time coded multi-transmission among distributed transmitters without perfect synchronisation,” IEEE Signal Processing Lett., vol. 11, no. 12, pp. 948-951, Dec. 2004. [9] X. Li, F. Ng, J.-T. Hwu, and M. Chen, “Channel equalization for STBCencoded cooperative transmission with asynchronous transmitters,” in Proc. 39th Asilomar Conf. on Signals, Systems and Computers, pp. 457461, Oct. 2005. [10] Y. Li and X.-G. Xia, “A family of distributed space-time trellis codes with asynchronous cooperative diversity,” preprint, www.ee.udel.edu∼xxiaLiXia.pdf. [11] S. Wei, D. L. Goeckel, and M. Valenti, “Asynchronous cooperative diversity,” IEEE Trans. Wireless Commun., vol. 5, no. 6, pp. 1547-1557, June 2006. [12] 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, 2001.
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Near-Optimum Detection for Distributed Space-Time Block Coding under Imperfect Synchronization F.-C. Zheng, A. G. Burr, and S. Olafsson
Abstract—Significant performance gain can potentially be achieved by employing distributed space-time block coding (DSTBC) in ad hoc or mesh networks. So far, however, most research on D-STBC has assumed that cooperative relay nodes are perfectly synchronized. Considering the difficulty in meeting such an assumption in many practical systems, this paper proposes a simple and near-optimum detection scheme for the case of two relay nodes, which proves to be able to handle far greater timing misalignment than the conventional STBC detector. Index Terms—Distributed space-time block coding, distributed transmit diversity, cooperative diversity, cooperative relay.
I. I NTRODUCTION
S
PACE-TIME block coding (STBC) has proved to be a very effective technique to achieve transmit diversity for wireless systems with co-located antennas at the transmitter, mainly due to its high diversity order and low decoding complexity [1]. In wireless systems such as ad hoc and sensor networks, however, it may only be possible to have a single antenna (rather than multiple co-located antennas) at the transmit/receive nodes due to cost and size constraints. As a result, there has over the past few years been a considerable research effort on creating and harnessing space diversity by applying STBC in a distributed fashion (termed distributed STBC or D-STBC) to single antenna systems [2]-[11]. So far, however, most research on D-STBC has assumed that cooperative relay nodes are perfectly synchronized so that the corresponding symbols from all the relay nodes arrive at the destination node at the same time. Such an assumption, unfortunately, is difficult or even impossible to achieve in many practical systems (e.g. ad hoc networks). With imperfect synchronization among the relay nodes (i.e. inter-relay-node synchronization), the channels may become dispersive even under flat fading conditions. There has been only limited work reported in the literature addressing D-STBC under imperfect synchronization Paper approved by N. Al-Dhahir, the Editor for Space-Time, OFDM and Equalization of the IEEE Communications Society. Manuscript received January 18, 2007; revised May 2, 2007 and August 24, 2007. This work was presented in part at the 8th IEEE International Workshop on Signal Processing Advances for Wireless Communication (SPAWC 2007), Helsinki, Finland, 17-20 June 2007. This work was supported by the UK EPSRC under Grant EP/E007139. F.-C. Zheng was with the Dept. of Electronics, the University of York, York, UK, and the School of Electrical Engineering, Victoria University, Melbourne, Australia. He is now with the School of Systems Engineering, the University of Reading, Reading, UK (e-mail: [email protected]). A. G. Burr is with the Dept. of Electronics, the University of York, York, UK. S. Olafsson is with the Mobility Research Centre, BT Group CTO, BT, Ipswich, UK, and Reykjavik University, 103 Reykjavik, Iceland. Digital Object Identifier 10.1109/TCOMM.2008.070006
among the relay nodes, chiefly using block based equalization techniques at the destination node to mitigate the impact of asynchronous signals (e.g. [8][9]). Alternative techniques to D-STBC in the presence of asynchronism also exist and these include distributed space-time trellis coding [10] and delay diversity (which relies on equalization or sequence estimation) [11]. Compared with the original STBC schemes, however, these existing methods potentially incur a much higher computational complexity at the receiver. This paper proposes a simple and near-optimum detection scheme for the case of two relay nodes under imperfect or rough synchronization. By cancelling the interference components in the received signal (caused by timing misalignment) in a decision feedback manner, a maximum likelihood (ML) detection scheme is realized on a symbol by symbol basis (assuming no feedback error), thus retaining the low computational complexity of the original STBC principle (i.e. achieving near-Alamouti simplicity). For the rest of this paper, [·]T , [·]∗ , and [·]H represent “transpose”, “conjugate”, and “transpose and conjugate”, respectively, while CN(0, σ 2 ) denotes the set of Gaussian distributed complex numbers with the standard variance of σ 2 (i.e. 0.5σ 2 per dimension). II. D-STBC U NDER I MPERFECT S YNCHRONIZATION This paper assumes the 4-node model depicted in Fig. 1. As in almost all cooperative relay systems [3][4], there are two phases involved: Phase 1 for broadcasting and Phase 2 for relaying. [Phase 1] The source node (S) transmits while relay nodes (R1 and R2 ) and destination node (D) receive. To prepare for the STBC operation in Phase 2, the data symbols at S are grouped into pairs. Denoting the ith pair of symbols in a data packet or frame transmitted by S as s(i) =[s(1, i), s(2, i)]T , the corresponding signal received at D after this direct transmission (DT) is rsd (i) = hsd s(i) + nsd (i) ,
(1)
where hsd is the channel gain between S and D, rsd (i) = [ rsd (1, i), rsd (2, i)]T , nsd (i) = [nsd (1, i), nsd (2, i)]T , and nsd (j, i) ∈CN(0, σn2 ) is the additive noise. [Phase 2] The data packet received at R1 and R2 are encoded using the Alamouti structure and transmitted to D. For clarity, this paper assumes that the channel gains hsr1 and hsr2 are such that R1 and R2 can always detect correctly [8][11]. Also, hm is the channel gain from Rm to D under perfect inter-relay-node synchronization. Denoting the encoded symbol pair corresponding to s(i) as x(i) = [x(1, i), x(2, i)]T at R1 and y(i) = [y(1, i), y(2, i)]T
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Phase 1
+h2 (−2)y(1, i − 1) + nrd (1, i)
R1
hsr1
and
hsd
r(2, i) = h1 x(2, i) + h2 (0) y(2, i) + h2 (−1)y(1, i)
D
S
+h2 (−2)y(2, i − 1) + nrd (2, i) ,
hsr2
R2
Phase 2
x(1,i), x(2,i)
R1
D
h2 R2
y(1,i), y(2,i)
βm = | h2 (−m)|2 /|h2 (0)|2
The 2-phase relay model with 2 relay nodes.
STBC pair
From R1 ......
From R2 ......
x(2,i-1) x(1,i) τ
T
x(2,i)
.....
T
y(2,i-1) y(1,i)
y(2,i)
(3b)
where h2 (l), l = 0, −1, −2, is the channel gain between R2 and D under imperfect inter-relay-node synchronization, nrd ∈CN(0, σn2 ) is the additive noise. Here, terms h2 (−m), m = 1, 2, reflect the inter symbol interference from the previous symbols (due to imperfect synchronization). For practical pulse shaping waveforms (PSW’s) such as the raised cosine, h2 (−2) is already much less dominant. All the other terms such as h2 (−3) are even smaller or zero in value (e.g. due to digital generation of the PSW) and have therefore been truncated in the above expressions (otherwise they can be dealt with similarly using the procedure in this paper). The relative strength of h2 (−m) will be represented by ratio
h1
Fig. 1.
(3a)
.....
Fig. 2. Time delay (imperfect synchronization) at the destination node (D) between 2 relay nodes.
at R2 , the STBC matrix (Alamouti scheme) transmitted by R1 and R2 is therefore x(1, i) y(1, i) s(1, i) s(2, i) = . (2) x(2, i) y(2, i) −s∗ (2, i) s∗ (1, i)
(3c)
in this paper. The value of βm reflects the composite effect of timing delay τ and the particular pulse shaping waveform used. However, we normally have βm = 0 for τ = 0, and β1 = 1 for τ = 0.5T . All the channel gains above are assumed to remain constant over the whole data packet, but to be subject to Rayleigh fading from packet to packet, i.e. hsd ∈CN(0, σs2 ), h1 and h2 ∈CN(0, σr2 ). For a fair comparison with the non-relay schemes, R1 and R2 transmit at half power, i.e. σr2 = 0.5σs2 . The above signal model may be viewed as a special case of some more general channel models (see [12] and the references therein). In [12], for example, an optimum scheme for STBC was proposed for co-located transmit antennas using zero-padding block transmission. The detection procedure in this paper, however, is based on symbol pairs only, and hence achieves a near-Alamouti simplicity (due to the special signal model in (3a) and (3b)). Substituting (2) into (3a) and (3b), we have (4) r(i) = Hs(i) + I(i) + nrd (i) , h1 h2 (0) ∗ T , where r(i) = [r(1, i), r (2, i)] , H= h∗2 (0) −h∗1 T ∗ I(i) = [I(1, i), I(2, i)] = [h2 (−1)s (1, i − 1) + h2 (−2)s(2, i − 1), h∗2 (−1)s∗ (2, i) + h∗2 (−2)s(1, i − 1)]T , and nrd (i) = [nrd (1, i), n∗rd (2, i)]T . From (4), the conventional STBC detection can be carried out via the following standard 2-step procedure (assuming perfect channel state information at D). Step 1: Linear transform.
Due to factors such as different propagation delays, x(i) and y(i) will most likely arrive at D at different time instants. Since accurate synchronization is difficult or impossible [8][11], there is normally a timing misalignment of τ between the received versions of these signals. Considering the effort of (rough) synchronization which is always required and the fact that R1 and R2 are normally close to S (to ensure correct detection at relay nodes), we assume at this stage that τ is no greater than the symbol period T . This assumption is easy to meet in practice. Such a relative time delay, as is shown in Fig. 2, will still cause “inter symbol interference (ISI)” from neighboring symbols at the receiver, due to sampling or matched filtering (whatever kind of pulse shaping is used). Without the loss of generality, we can assume that the receiver at D is perfectly synchronized to R1 . The received signal at D over the two symbol periods can then be represented by
g(i) [g(1, i), g(2, i)]T Θr(i) = Ds(i) + ΘI(i) + v(i), (5) where Θ =HH , and v(i) = [v(1, i), v(2, i)]T = Θnrd (i). Also, “” above means “is defined as”. λ 0 Due to the Alamouti structure, D= ΘH= , where 0 λ
r(1, i) = h1 x(1, i) + h2 (0) y(1, i) + h2 (−1)y(2, i − 1)
λ = |h1 |2 + |h2 (0)|2 .
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ZHENG et al.: NEAR-OPTIMUM DETECTION FOR DISTRIBUTED SPACE-TIME BLOCK CODING UNDER IMPERFECT SYNCHRONISATION
Also, note that v(i)vH (i) = σn2
λ 0 0 λ
IV. C OMBINING WITH D IRECT T RANSMISSION
.
(6)
Step 2: Least square (LS) detection. s(j, i) = arg{ min |g(j, i) − λsm |2 },
(7)
sm ∈S
where j = 1, 2, sm is an arbitrary symbol in the corresponding symbol alphabet S, and M is the number of elements or symbols within S (similarly hereinafter). Due to the component of ΘI(i) in (5), however, the above procedure can suffer from significant detection errors, unless h2 (−m) = 0 (i.e. τ = 0, the case of perfect synchronization). III. M AXIMUM L IKELIHOOD D ETECTION Upon examining (3) and (4), we notice that s(m, i−1), m = 1, 2, are in fact already known if the detection process has been initialized properly (e.g. through the use of pilot symbol(s) at the start of the packet). As such, I(1, i) = h2 (−1)s∗ (1, i − 1) + h2 (−2)s(2, i − 1) and component h∗2 (−2)s(1, i − 1) can be removed (“decision feedback”) before applying the linear transform of Θ =HH in (5). This leads to
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∗
Even if the direct transmission (DT) fails, the received signal at D during Phase 1 (i.e. Eq.(1)) still contains valuable information and should therefore be retained and combined appropriately with its counterpart during Phase 2 wherever possible. A. The conventional STBC detector Applying maximum ratio combining (MRC) to (1) and (4), corresponding to (5) we now have g(i) = [g(1, i), g(2, i)]T = Θr(i) + h∗sd rsd (i) = Ds(i) + ΘI(i) + v(i) , λ 0 where D= with 0 λ
λ = |hsd |2 + |h1 |2 + |h2 (0)|2 , and
(13)
(14)
v(i) = Θnrd (i) + h∗sd nsd (i).
Here, Θ =HH as before. Note that we still have H 2 λ 0 v(i)v (i) = σn , 0 λ
g (i) = [g (1, i), g (2, i)] = Θr (i) = Ds(i)+zs (2, i)+v(i) , (8) where (9) where λ is now from (14). Step 2 is the same as (7) except z = [z1 , z2 ]T = Θ[0, h∗2 (−1)]T , for the new λ value in (14) above. and r(1, i) − I(1, i) . (10) B. The maximum likelihood detector r (i) = ∗ r (2, i) − h∗2 (−2)s(1, i − 1) The same maximum likelihood detection can now be apEquation (8) can then be written as plied in a similar manner as follows. (11a) g (1, i) = λs(1, i) + z1 s∗ (2, i) + v(1, i) , − Calculate g (i) = [g (1, i), g (2, i)]T = Θr (i) + h∗sd rsd (i). and − Carry out (12a) and (12b) using the above g (i) and the (11b) g (2, i) = λs(2, i) + z2 s∗ (2, i) + v(2, i) . new λ value in (14). Obviously, the two key properties of the STBC principle In (11b), g (2, i) is only related to s(2, i), and therefore simplicity and optimality (or near optimality in this case) - still s(2, i) can be detected as hold. Also, as can be seen from (14), the utilisation of the DT (12a) sˆ(2, i) = arg{ min |g (2, i) − λsm − z2 s∗m |2 }. signal has increased the diversity order of the system from 2 sm ∈S Then, s(i, 1) can be detected by substituting s(2, i) back to to 3. This increased diversity order, together with the fact that DT is at “full power”, will deliver a significant performance (11a): gain, as will be confirmed next. sˆ(1, i) = arg{ min |g (1, i) − λsm − z1 sˆ∗ (2, i)|2 }. (12b) T
sm ∈S
The above procedure totally eliminates the ISI caused by imperfect synchronization, but still retains the following two key properties of the original STBC principle. − Optimality: If there is no decision feedback error for s(m, i − 1), m = 1, 2, then the above procedure is optimum in terms of maximum likelihood (ML) . Since (6) holds, (12a) is an ML detector, and so is (13b). Our work shows that the impact of decision feedback error is normally very small (see Section V), hence the term “near-optimum detection” for the above procedure. − Simplicity: The above ML search involves one symbol only and thus has a linear computational complexity. Compared with the conventional Alamouti detector, it only involves M/2 extra multiplications per symbol, hence achieving a nearAlamouti simplicity.
V. S IMULATIONS With an 8-PSK system and assuming that the relays R1 and R2 can always detect correctly, D-STBC is employed regardless of DT success or failure. Also, the signal to noise ratio (SNR) is defined as SNR= σs2 /σn2 (dB). Also, the detection process is initialized only once, i.e., at the beginning of each symbol packet/stream. [The impact of h2 (−1) ] As h2 (−m) reflects the composite effect of both time delay τ and the pulse shaping waveform (PSW) and h2 (−1) is the most dominant term, we set β2 = 0 and β1 = −10, 0, 5, 10 (dB) in this example. The bit error rates (BER’s) of the proposed ML scheme are displayed in Fig. 3 (without DT combining (DTC)) and Fig. 4 (with DTC). For comparison, the corresponding results of the conventional detector and those of “DT alone” and “STBC with perfect
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 11, NOVEMBER 2008
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Fig. 3. The BER’s of the ML and conventional detectors under different values (without DT combining).
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Fig. 5. The impact of error propagation (EP) on the BER’s of the ML detector with and without DT combining (DTC): =5dB.
DT alone ML: τ = 0.8T Con: τ = 0.8T ML: τ = 0.6T Con: τ = 0.6T ML: τ = 0.4T Con: τ = 0.4T ML: τ = 0.2T Con: τ = 0.2T perfect synch
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Fig. 4. The BER’s of the ML and conventional detectors under different values (with DT combining).
Fig. 6. The BER’s of the ML and conventional detectors under different values (with DT combining).
synchronization” (i.e. τ = 0) are also shown. Fig. 3 indicates that the ML detector alone greatly outperforms the conventional STBC detector and DT with β1 up to 5 dB, while Fig. 4 demonstrates that DTC has resulted in a significant improvement for the ML detector: it can now deal with much larger β1 (or τ ) values. [Propagation of decision feedback errors] To examine this critical issue, the ML detection is carried out (i) with error propagation (EP, i.e. as it is - with the natural propagation of any feedback errors), and (ii) with no EP (i.e. using the true symbols of the previous pairs for ISI removal). The BER results for β1 = 5dB (β2 = 0) are shown in Fig. 5. Clearly, the impact of error propagation is very minor indeed, showing the near-optimum nature of the detector in this paper. [Practical pulse shaping waveforms] To gain some insight into the specific impact of time mismatch τ alone, a raisedcosine pulse with the 3GPP roll-off factor of 0.22 is employed, and the results are presented in Fig. 6 (with DTC) for τ =
0.2T, 0.4T, 0.6T, and 0.8T , showing the effectiveness of the ML detector again. VI. C ONCLUDING R EMARKS It has been shown that the near-optimum detection scheme proposed in this paper is indeed very effective for 2-relaynode systems even under some relatively large βm or τ values while the conventional STBC detector fails even under small βm or τ values. DT combining can greatly improve the system performance due to the higher diversity order. Due to the reduced effective diversity order under imperfect synchronization, the proposed procedure does show gradual performance degradation with increasing βm or τ values. However, τ ≤ 0.5T (or β1 = 0 dB) will guarantee an excellent performance. In practice, this (i) represents a much relaxed requirement for inter–relay-node synchronization, and (ii) can always be realized by simply instructing R1 to delay its transmission by T if τ ∈ [0.5T, T ].
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ZHENG et al.: NEAR-OPTIMUM DETECTION FOR DISTRIBUTED SPACE-TIME BLOCK CODING UNDER IMPERFECT SYNCHRONISATION
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[7] P. A. Angel and M. Kaveh, “On the diversity of cooperative systems,” in Proc. IEEE ICASSP 2004, pp. IV-577-580. [8] X. Li, “Space-time coded multi-transmission among distributed transmitters without perfect synchronisation,” IEEE Signal Processing Lett., vol. 11, no. 12, pp. 948-951, Dec. 2004. [9] X. Li, F. Ng, J.-T. Hwu, and M. Chen, “Channel equalization for STBCencoded cooperative transmission with asynchronous transmitters,” in Proc. 39th Asilomar Conf. on Signals, Systems and Computers, pp. 457461, Oct. 2005. [10] Y. Li and X.-G. Xia, “A family of distributed space-time trellis codes with asynchronous cooperative diversity,” preprint, www.ee.udel.edu∼xxiaLiXia.pdf. [11] S. Wei, D. L. Goeckel, and M. Valenti, “Asynchronous cooperative diversity,” IEEE Trans. Wireless Commun., vol. 5, no. 6, pp. 1547-1557, June 2006. [12] 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, 2001.
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