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Email: {koike, vahid}@seas.harvard.edu, [email protected] Abstract—We propose a ... modulation and adaptive network coding with precoding tech- niques.

Adaptive Modulation and Network Coding with Optimized Precoding in Two–Way Relaying ∗

Toshiaki Koike-Akino∗ , Petar Popovski† , and Vahid Tarokh∗ School of Engineering and Applied Sciences, Harvard University, 33 Oxford Street, Cambridge, MA 02138, U.S.A. † Department of Electronic Systems, Aalborg University, Niels Jernes Vej 12, DK-9220 Aalborg, Denmark Email: {koike, vahid}@seas.harvard.edu, [email protected]

Abstract—We propose a precoding strategy which controls amplitude and phase of receiving signals to improve throughput for two–stage bidirectional relaying. We consider the case when the nodes know channel state information (CSI) and can adopt adaptive modulation techniques. We introduce a novel scheme termed adaptive modulation and network coding (AMNC), which jointly optimizes modulations and network coding based on the CSI. For dynamic bit loading and power allocation, we propose a practical time–sharing method called the segmented precoding, in which a packet is split into several sub–packets, and a set of modulation and network coding is optimized in conjunction with amplitude and phase controls for each sub–packet. It is demonstrated that our proposed scheme can offer a significant improvement of achievable throughput for two–way relaying.

I. I NTRODUCTION Recently, multi–way relaying which exploits network coding [1] at the physical layer has received a significant amount of attention [2–7]. We consider a bidirectional relaying system, where there are three nodes A, B and R: The terminal A has traffic to send to the terminal B and vice versa, while the relaying node R assists the two–way communications. There are various schemes in this scenario: 4–, 3–, and 2–stage protocols with amplify–and–forward (AF), decode–and–forward (DF), joint decode–and–forward, compress–and–forward (CF), and denoise–and–forward (DNF) schemes. In this paper, we focus on the 2–stage DNF, which is introduced in [3], and deal with the problem of applying adaptive modulation. The first stage in the DNF scheme is termed multiple– access (MA) stage, during which the terminals A and B simultaneously transmit to the relay R. Due to the half–duplex constraint, neither terminal can receive anything from the other terminal. The relay R maps the received signals into symbols from a discrete constellation, and broadcasts those towards both the terminals during the second stage, termed broadcast (BC) stage. Considering the fact that A (B) knows its own information a priori, it is possible to decode the desired data from B (A) by observing the broadcasted signal from R. The information–theoretic potential of DNF is analyzed in [6, 7]. The use of structured codes to enable efficient denoising has been considered in [8, 9]. Focusing on the communication– theoretic aspects of the finite–length packets and practical modulation schemes, we have optimized signalling constellations for DNF in [10, 11] and extended it for convolutionally– coded systems in [12]. We have found that physical–layer network coding used at the relay R should be adaptively

changed according to the channel state information (CSI), and that unconventional 5–ary non–linear network coding can significantly improve throughput. In [10–12], we have only considered a relaying system, in which any CSI is not available at the transmitters for practical applications. Obviously, a significant throughput gain can be achieved if the transmitters know the CSI. The CSI enables usage of adaptive modulation scheme and an adaptive precoding technique to boost throughput. In this paper, we study a precoding strategy that efficiently uses the CSI in two–way relaying scenarios. We propose two novel approaches: 1) adaptive modulation and network coding (AMNC), which jointly optimizes modulation schemes and network coding to maximize throughput according to the CSI, and 2) segmented precoding, which splits a packet into several sub–packets (or, segments) and optimizes amplitude and phase controls as well as sub–packet lengths. In the AMNC, different network codes rather than the exclusive– or (XOR) operation are required for different modulation schemes to suppress multiple–access interference (MAI). The MAI effect can also be reduced by appropriate amplitude and phase controls. The segmented precoding makes effective use of a minimum distance property of receiving constellations: there are some cases in which the decreased transmission power can lead to the increased Euclidean distance over MA channels. Through performance evaluations, we confirm that our proposed scheme can significantly improve throughput. To the authors’ best knowledge, this paper describes an initial attempt which considers a joint design of adaptive modulation and adaptive network coding with precoding techniques. The proposed segmented precoding can be applied to any other communications system, and it enables single– carrier transmissions to adopt dynamic bit loading. II. T WO –S TAGE B IDIRECTIONAL R ELAYING WITH DNF A. Multiple Access (MA) Stage Fig. 1 depicts a system model under consideration. We consider uncoded systems. We let MA (·) and MB (·) denote signal constellation mappers for the terminals A and B, respectively. We denote by SA ∈ ZQA and SB ∈ ZQB the data symbols transmitted by A and B, where QA and QB are constellation alphabet sizes and ZQ = {0, 1, . . . , Q − 1} is an integer set. According to the CSI, the transmitters can choose proper modulation schemes out of BPSK, QPSK, 8QAM, and 16QAM.

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.

of QR = 5) rather than the XOR–based network coding can bring a considerable improvement of throughput.

Step 1: Multiple Access (MA) Stage Node R XA = PA MA(SA )

HA

HB

XB = PB MB(SB)

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In the BC stage, the signal XR = MR (SR ) is broadcasted to A and B with precoder PR . Those terminals then receive

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YA = HA PR XR + ZA ,

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Fig. 1. Two-stage bidirectional relaying with physical-layer network coding C(·) and precoding PA , PB , PR for adaptive modulations MA (·), MB (·).

YB = HB PR XR + ZB .

(3)

For simplicity, we assume the reciprocal channel for both stages, and the identical noise variance σ 2 for ZA and ZB . The terminal A (and B) can extract the desired data SB (SA ) by using the own information SA (SB ). In general, the channel links for the BC stage in (3) is much more reliable than the ones for the MA stage in (1) because there is no MAI. III. T HROUGHPUT A NALYSIS

The modulated symbols in the baseband signals are written as Xk = Mk (Sk ) for k ∈ {A, B}. We assume unit–energy modulation signals, i.e., E[|Xk |2 ] = 1 with E[·] being the expectation function. Using the CSI, the transmitters A and B can control the phase and amplitude before transmitting by precoders Pk ∈ C. The precoded signals, Pk Xk , are simultaneously transmitted to R from A and B at the MA stage. Under the MA channel, the relay R receives YR = HA PA XA + HB PB XB + ZR ,

(1)

where Hk ∈ C is the channel coefficient from the terminal k. We suppose a frequency–flat slow fading for HA and HB , and a Gaussian noise for ZR with a variance of σ 2 . B. Denoise–and–Forward (DNF) with Network Coding As in [10, 11], the relay R employs a denoising function based on adaptive network coding to map the received signal YR into a quantized signal XR for the BC stage. The denoising function consists of a network coding C(·) and a constellation mapper MR (·), preceded by the maximum–likelihood (ML) estimation. Let (SˆA , SˆB ) be the ML estimates. The network coding C(·) generates a compressed data SR ∈ ZQR from the ML estimates as follows:   (2) SR = C SˆA , SˆB . Here, QR is the cardinality of network coding. Note that QR must be no smaller than max(QA , QB ) for successful persymbol denoising. Since the multiplexed signals XA and XB can interfere with each other as in (1), the ML estimates are not reliable and may have errors frequently, i.e., (SˆA , SˆB ) = (SA , SB ). The network coding C(·) is designed to suppress such an MAI by clustering unreliable pairs to yield C(SˆA , SˆB ) = C(SA , SB ) even for (SˆA , SˆB ) = (SA , SB ). This design strategy can achieve a significant improvement in the end–to–end throughput, as discussed in [10, 11]. The network coding C(·) should be adaptively changed according to the channel ratio PB HB /PA HA to minimize the error probability. Interestingly, for some specific channel conditions, an irregular network coding (e.g, with a cardinality

Here, we analyze the end–to–end throughput for two–stage bidirectional relaying. To simplify the analysis, we first focus on the packet success probabilities in the MA stage and in the BC stage, and then derive the end–to–end throughput. A. Success Probability in Multiple Access (MA) Stage Let us consider the relay’s hypothesis of the transmitted data, (SˆA , SˆB ), as well as the actual transmitted data (SA , SB ). If the network coded versions of these data are identical, i.e., C(SA , SB ) = C(SˆA , SˆB ), the relaying node can successfully forward the network coded data to the terminals. Otherwise, the relay forwards erroneous data. Supposed that C(SA , SB ) = C(SˆA , SˆB ), the pairwise error probability that the receiver quantizes a constellation for (SA , SB ) into an erroneous constellation for C(SˆA , SˆB ) is written as  d2MA (S A , S B ) 1 pMA (S A , S B ) = erfc , (4) 2 4σ 2 where S k  (Sk , Sˆk ) for k ∈ {A, B}. The squared distance of the pair is expressed as  2 d2MA (S A , S B ) = HA PA ΔA (S A ) + HB PB ΔB (S B ) , (5) where Δk (Sk , Sˆk )  Mk (Sk ) − Mk (Sˆk ). It is known that the symbol error rate (SER) can be well approximated by a weighted sum over all the possible pairwise error probabilities. Among the summation, the most dominant factor to determine the SER is the squared minimum distance: d2min,MA =

min

C(SA ,SB )   = C(SA ,SB )

d2MA (S A , S B ).

(6)

Using the minimum distance, the SER p¯MA is bounded as    d2min,MA d2min,MA 1 erfc . (7) ≤ p¯MA ≤ exp − 2 4σ 2 4σ 2 Assuming that the packet length is N symbols long, we can write the probability of successful forwarding as tMA = (1 − p¯MA )N .

(8)

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.

B. Success Probability in Broadcast (BC) Stage During the BC stage, the pairwise error probability that the terminal k ∈ {A, B} fails decoding the network coded data from SR to SˆR (for any SR = SˆR ) is expressed as  d2BC,k (S R ) 1 pBC,k (S R ) = erfc , (9) 2 4σ 2 where S R  (SR , SˆR ). The squared distance is written as  2 (10) d2BC,k (S R ) = Hk PR ΔR (S R ) , where ΔR (S R )  MR (SR ) − MR (SˆR ). The SER lies in    d2min,BC,k d2min,BC,k 1 erfc , (11) ≤ p¯BC,k ≤ exp − 2 4σ 2 4σ 2 where d2min,BC,k is the squared minimum distance: d2min,BC,k = min d2BC,k (S R ). ˆR SR =S

(12)

The packet success rate is given as tBC,k = (1 − p¯BC,k )N . C. End–to–End Throughput The probability that the source data SA can be successfully forwarded to B is approximately given by the multiplication of tMA and tBC,B . Likewise, log2 (QB ) bits per symbol are successfully transmitted from B to A via R with a probability of tMA tBC,A . Consequently, the end–to–end throughput is approximately expressed as 1 1 T  log2 (QA )tMA tBC,B + log2 (QB )tMA tBC,A . (13) 2 2 It should be noted that the end–to–end throughput is highly dependent on the modulation schemes Mk (·), network coding C(·) and precoding Pk , as well as the CSI Hk .

precoder design to have |HA PA Δmin,A |2 = |HB PB Δmin,B |2 is one of the best candidates. The squared distance in (5) can be rewritten as  2 γB jθ  2 d2MA (S A , S B ) = γA e ΔB (S B ) , (15) ΔA (S A ) + γA where γk = |Hk Pk | is the equivalent channel amplitude and θ = ∠(HB PB /HA PA ) is the phase difference of the equivalent channels. When we fix γA , the normalized distance dMA /γA is dependent on the channel amplitude ratio γB /γA and the phase difference θ as well as the modulation schemes. Note that increasing the transmission power |PB |2 does not always contribute to the increase of the squared distance over the MA channels due to the MAI effect. This is illustrated in Figs. 2 (a) through (f), in which we present the normalized minimum distance at the MA stage, dmin,MA /γA , as a function of γB /γA for several different pairs of modulation schemes used at the terminals A and B: (a) both the MA (·) and MB (·) are QPSK, (b) both are 8QAM, (c) both are 16QAM, (d) the terminal A uses 8QAM and B uses QPSK, (e) A uses 16QAM and B uses QPSK, and (f) 16QAM and 8QAM are used at A and B, respectively. In these figures, the network coding is optimized by the closest–neighbor clustering method, which is proposed in [10, 11]. As mentioned in (14), the normalized minimum distance dmin,MA /γA does not exceed either |Δmin,A | or (γB /γA )|Δmin,B |. From Fig. 2 (a), we can see the following: •

• •

IV. O PTIMIZED P RECODING FOR A DAPTIVE M ODULATION AND N ETWORK C ODING (AMNC) We propose a joint optimization scheme of modulation mappers Mk (·), network coding C(·) and precoder Pk to maximize the end–to–end throughput.





A. Minimum Distance Performance The success probability at the MA stage, tMA , depends on the CSI Hk and the precoder Pk , as in (5). Hence, the terminals A and B should perform the adaptive modulation by using the instantaneous CSI, in conjunction with precoder optimization. The squared minimum distance, d2min,MA in (6), does not surpass the smaller distance of the individual receiving constellations from the terminals A and B, i.e.,

d2min,MA ≤ min |HA PA Δmin,A |2 , |HB PB Δmin,B |2 , (14) where |Δmin,k |2 = minS k |Δk (S k )|2 is the squared minimum distance of the modulation schemes. We should control Pk and C(·) to fulfil the equality of the above relation, whenever possible. If we have |HA PA Δmin,A |2 = |HB PB Δmin,B |2 , there is an energy waste for either terminal. It suggests that the

Without network coding and precoding, the minimum √ distance can fall into zero at γB /γA = 1/2, 1, 2 even when γA = 0, γB = 0. One of such distance shortening at γB /γA = 1 can be avoided by using the XOR–based network coding. With phase–synchronizing precoder to have θ = 0 (but without network √ coding), the distance shortening at γB /γA = 1/2, 2 can be avoided, whereas it cannot avoid for γB /γA = 1. Adaptive network coding with 5–ary cardinality proposed in [10, 11] can exclude all the undesired zero–distance shortening even without precoding. Phase–synchronizing precoder together with XOR–based network coding can offer the maximum available distance of min(γA |Δmin,A |, γB |Δmin,B |).

From the observations above, the phase–synchronizing precoder with the XOR–based network coding seems promising to realize highly reliable two–way relaying. However, as shown in Fig. 2 (c), the phase synchronizing precoder does not always offer the best performance; at around γB /γA = 0.7, the optimal phase is not θ = 0. Moreover, the minimum distance at γB /γA = 0.8 can be less than the one at γB /γA = 0.7 even when we use the optimal phase control. It implies that we should decrease the power |PB |2 to have γB /γA = 0.7 rather than γB /γA = 0.8 in order to increase the minimum distance. Such a characteristic (decreasing transmission power can increase the minimum distance) is a rather unique feature of the two–stage bidirectional relaying.

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.

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B. Adaptive Modulation and Network Coding (AMNC) With higher level modulations, we can send more bits per symbol of log2 (Qk ) at maximum; 1, 2, 3 and 4 bits for BPSK, QPSK, 8QAM and 16QAM. However, the packet success probability tMA can be decreased because the squared minimum distance, |Δmin,k |2 , decreases for higher level modulations; more specifically, 4, 2, 0.8 and 0.4 for BPSK, QPSK, 8QAM and 16QAM, respectively. Therefore, there exists a tradeoff between the achievable throughput and the cardinality of modulation schemes. Correspondingly, an appropriate choice of modulations for Mk (·) can maximize the end–to– end throughput T in (13). It can be done by calculating tMA and tBC,k for all the possible pairs of modulation schemes. From Fig. 2, we propose an appropriate joint set of adaptive modulations, network coding and precoding. In Fig. 2 (a) for the QPSK–by–QPSK case, the most energy–efficient precoding is to achieve γB /γA = 1, in which the optimal network coding is based on the XOR network coding and

the optimal phase is θ = 0. Similarly, for the 16QAM–by– 16QAM case in Fig. 2 (c), we use an appropriate precoder for γB /γA = 1, in which we require a modified XOR network some local coding whose cardinality is QR = 16. Although summit points (γB /γA = 1/3, 1/8, 1/5, 1/2, 1/2) can be good candidates for the precoder operation, we use only one point for each modulation pair; more specifically, we control the amplitude ratio to have γB /γA = |Δmin,A |/|Δmin,B | for the most energy–efficient possibility. Shown in Fig. 2 (d) and (f), we require optimal phase control rather than θ = 0. Table I lists all the set of appropriate modulation, network coding and precoding. There are 16 combinations of modulation schemes used at the terminals A and B to select a pair of BPSK, QPSK, 8QAM and 16QAM. Note that the optimal network coding (C1 , C2 , . . . , C16 ) is different from each other (well– known XOR is not used for high level modulations). Those are designed by the method proposed in [10, 11]. We select the best AMNC set from Table I to maximize the throughput T in (13). Using the minimum distance per-

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.

TABLE I A DAPTIVE M ODULATION AND N ETWORK C ODING (AMNC) S ET WITH P RECODING Index i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

MA (·) BPSK QPSK BPSK QPSK 8QAM BPSK 8QAM QPSK 8QAM 16QAM BPSK 16QAM QPSK 16QAM 8QAM 16QAM

MB (·) BPSK BPSK QPSK QPSK BPSK 8QAM QPSK 8QAM 8QAM BPSK 16QAM QPSK 16QAM 8QAM 16QAM 16QAM

Modulation QA QB |Δmin,A |2 2 2 4 4 2 2 2 4 4 4 4 2 8 2 0.8 2 8 4 8 4 0.8 4 8 2 8 8 0.8 16 2 0.4 2 16 4 16 4 0.4 4 16 2 16 8 0.4 8 16 0.8 16 16 0.4

|Δmin,B |2 4 4 2 2 4 0.8 2 0.8 0.8 4 0.4 2 0.4 0.8 0.4 0.4

Physical–Layer C(·) QR ˆA , S ˆB ) C1 (S 2 ˆA , S ˆB ) C2 (S 4 ˆB , S ˆA ) C2 (S 4 ˆA , S ˆB ) C4 (S 4 ˆA , S ˆB ) C5 (S 8 ˆB , S ˆA ) C5 (S 8 ˆA , S ˆB ) C7 (S 8 ˆB , S ˆA ) C7 (S 8 ˆA , S ˆB ) C9 (S 8 ˆA , S ˆB ) 16 C10 (S ˆB , S ˆA ) 16 C10 (S ˆA , S ˆB ) 16 C12 (S ˆB , S ˆA ) 16 C12 (S ˆA , S ˆB ) 16 C14 (S ˆB , S ˆA ) 16 C14 (S ˆA , S ˆB ) 16 C16 (S

Network Coding MR (·) |Δmin,R |2 BPSK 4 QPSK 2 QPSK 2 QPSK 2 8QAM 0.8 8QAM 0.8 8QAM 0.8 8QAM 0.8 8QAM 0.8 16QAM 0.4 16QAM 0.4 16QAM 0.4 16QAM 0.4 16QAM 0.4 16QAM 0.4 16QAM 0.4

Precoding 2 2 γB /γA θ 1 0 1/2 0 2 0 1 0 5 π/4 1/5 −π/4 2/5 π/4 5/2 −π/4 1 0 1/10 0 10 0 1/5 0 5 0 2 π/4 1/2 −π/4 1 0

formance in Fig. 2, we can calculate the success probabilities tMA , tBC,A and tBC,B . The precoding Pk controls the amplitude ratio γB /γA to have the best point with the power constraint of |PA |2 + |PB |2 + |PR |2 = 3Es where Es is the transmission symbol energy. However, even if we use such an optimal power allocation, the end–to–end throughput curves can become a step–like function because we only use one modulations pair for one whole packet. To improve throughput, we propose a novel approach termed segmented precoding which is described in the following.

where QA (i) and QB (i) are the cardinality of the modulation schemes used at the terminals A and B, respectively, for the i–th segment. Focusing on the upper bound of SER, we obtain the segment success probabilities tMA (i), tBC,A (i) and tBC,B (i) as 

N (i) d2 min,MA (i) , (19) tMA (i)  1 − exp − 4σ 2 

N (i) d2 min,BC,k (i) tBC,k (i)  1 − exp − . (20) 4σ 2

C. Segmented Precoding for Seamless AMNC In the segmented precoding technique, we split a packet into 16 sub–packets (referred to as “segments” hereafter). The i–th segment consists of N (i)–symbol modulated signals, whose constellation is chosen from the i–th index of the AMNC set presented in Table I. For each segment, optimal network coding and precoding are employed according to this table. We let Pk (i) be the precoder for the node k ∈ {A, B, R} at the i–th segment. We have a total transmission power constraint:

Using the Lagrangian multipliers method, we can solve the optimization problem in terms of Pk (i) and N (i) to maximize T , subject to the power constraint in (16) and the packet length constraint in (17) as well as N (i) ≥ 0. Of course, we allow N (i) = 0 for some segments; which indicates that the modulations pair of the i–th index in Table I is of no use to maximize the throughput. For example, BPSK is not useful for high SNR regimes and 16QAM is not required for low SNR regimes. This segmented precoding is advantageous for switching the modulations seamlessly, which means that the throughput curve can become a smoothly increasing function (not step–like) with the increased SNR. Since the analytical solution is cumbersome, we use numerical optimization method to derive the optimal power allocations and segment lengths.

16   1 N (i) |PA (i)|2 + |PB (i)|2 + |PR (i)|2 = Es , (16) 3N i=1

where N is the total packet length, i.e., N=

16

N (i).

(17)

i=1

Under the power constraint, we optimize Pk (i) and N (i) such as to maximize the end–to–end throughput. Using the i– th segment success probabilities tM A (i), tBC,k (i), the whole packet success probability becomes tM A (i)tBC,k (i). Therefore, the end–to–end throughput T is rewritten as   16 16 i=1 N (i) log2 (QA (i)) tMA (i)tBC,B (i) T  2N i=1  16  16 i=1 N (i) log2 (QB (i)) + tMA (i)tBC,A (i), (18) 2N i=1

V. E ND – TO –E ND T HROUGHPUT E VALUATION We evaluate the throughput performance of the proposed approach in Figs. 3 and 4 for 10 dB and 0 dB Rician factor, respectively. We set N = 512 symbols long. We assume the path loss of the channels HA and HB is identical. In these figures, we plot the throughput curves achieved by the 2–stage protocol (with segmented precoding, with phase–controlling precoder, and without any precoding), the 3–stage protocol (with XOR network coding), and the 4–stage protocol. Using appropriate modulation sets (BPSK through 16QAM), the throughput curves can increase with the increased SNR up to 4, 3 and 2 bps/Hz for 2–, 3– and 4–stage protocols, respectively. These curves are step–like functions. Meanwhile, it is confirmed

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.

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Fig. 4. End–to–end throughput as a function of average SNR in Nakagami– Rice fading channels (0 dB Rician factor).

that our proposed approach can offer a smoother increase of the throughput curve, and it can significantly improve the end–to–end throughput. Note that 2–stage protocol achieves higher throughput than does the other protocol. However, if we do not employ adaptive precoding nor adaptive network coding, the fixed (XOR–based) network coding (only using adaptive modulation) exhibits performance degradation. In consequence, we can see the considerable advantage of joint optimization of modulation, network coding, and precoding.

ACKNOWLEDGMENT

VI. C ONCLUSION In this paper, we have investigated optimized precoding schemes for adaptive modulation with physical–layer network coding in two–way relaying systems. Based on the throughput analysis with the squared minimum distance of the receiving constellations, we have introduced an appropriate set of modulation schemes, network coding, and precoding. To further improve throughput, we have proposed a novel approach termed segmented precoding, which can offer a seamless switching of the adaptive modulation and network coding (AMNC) set by splitting a packet into several segments. This approach can be interpreted as a practical way of the so–called time–sharing method. Through performance evaluations, we confirmed that the proposed approach can significantly improve the end–to– end throughput performance in two–way relaying. In this paper, we assumed a perfect knowledge of CSI for all the links, in advance of transmissions. In practice, it should be evaluated for the case when we only have a partial knowledge of CSIs with a limited feedback. The extension towards a joint optimization of the AMNC and channel coding with precoding also remains as a future work.

This work is partially supported by SCAT, JSPS Postdoctoral Fellowships for Research Abroad, and the Danish Research Council for Technology and Production.

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978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.

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