bit error rate (BER) in fading channels [9]. Recently, the effectiveness of adaptive modulation in cooperative wireless
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Joint Adaptive Modulation and Distributed Switch-and-Stay for Partial Relay Selection Networks Vo Nguyen Quoc BAO†a) and Hyung Yun KONG†† , Members
SUMMARY In this letter, we propose a distributed switch-and-stay combining network with partial relay selection and show that the system spectral efficiency can be improved via adaptive modulation. Analytical expressions for the achievable spectral efficiency and average bit error rate of the proposed system over Rayleigh fading channels are derived for an arbitrary switching threshold. Numerical results are gathered to substantiate the analytical derivation showing that in terms of spectral efficiency, the system with single relay outperforms that with more than one relay at high signal-to-noise ratios (SNRs) and the optimal switching threshold can significantly improve the system performance at medium SNRs. key words: distributed switch-and-stay combining, partial relay selection, adaptive modulation
1.
Introduction
Among the many types of distributed cooperative networks, distributed switch-and-stay combining (SSC) networks [1] are considered as less complicated and more spectrally efficient networks than the conventional ones [2] since they have no actual diversity combiner installed at the destination. Operating under a rule similar to that of conventional SSC [3], the destination in distributed SSC networks will switch to and stay with the other link if the currently connected link — either the direct or the relaying link — is below the given switching threshold. In such a case, it potentially can improve the system spectral efficiency (SE) since it reduces the need of two time slots requested all the time for communication over the relaying link [4]. In cases where more than one relay is available, by incorporating the relay selection procedure [4], the distributed SSC was shown to attain diversity gain on the order of the number of relays [5]. However, as mentioned in [6], the best relay scheme is not always realized in some resource-constrained wireless systems, e.g. ad hoc or wireless sensor networks, because the need for monitoring the connectivity among all links can limit the network lifetime. To address this problem, Krikidis et al. [6] proposed the partial relay selection (PRS) in which the best relay is chosen solely based on the partial channel state information. Many recent research works such as [7], [8] are based on PRS. Adaptive transmission has long been considered as an efficient approach of achieving both high SE and acceptable Manuscript received June 17, 2011. Manuscript revised October 19, 2011. † The author is with Posts and Telecommunications Institute of Technology, Ho Chi Minh City, Vietnam. †† The author is with University of Ulsan, Korea. a) E-mail:
[email protected] (corresponding author) DOI: 10.1587/transcom.E95.B.668
bit error rate (BER) in fading channels [9]. Recently, the effectiveness of adaptive modulation in cooperative wireless communication systems in which power and/or rate at the source can be allocated to take advantage of favorable channel conditions has been demonstrated [10], [11]. In this letter, we propose the first example of distributed SSC with PRS in which variable-rate transmission is adopted. We first focus on the analytical expressions for achievable SE, and BER of the proposed system over Rayleigh fading channels. In this letter, we treat a very general model in which the communication link between nodes in the network may not be identically distributed. Therefore, this model includes the case of independent and identically distributed (i.i.d.) channels as a special case. Furthermore, this paper seeks to address the following questions: i) What is the effect of PRS on the overall system performance? ii) How do we choose the optimal switching thresholds for a given operating SNR? and iii) How much gain will we obtain as compared with the fixed ones? 2.
System Description
Let us consider a distributed SSC combining network having one source, N amplify-and-forward relays and one desN and D, respectively. As a tination, denoted as S, {Ri }i=1 distributed version of dual-branch SSC, the data transmission between the source and the destination is performed either by the direct link (S → D) or by the relaying link (S → Ri → D). Stated another way, only one link is active in each transmission slot. Similar to the operational rule of SSC [3], the destination will continue receiving data signals on the currently connected link if the SNR is above the given threshold, T . Otherwise, the destination switches to, and stays with the alternate link regardless of the SNR on that link. The selection of the appropriate link is governed based on the limited feedback signal from the destination. The switching is effective on the following transmission slot under the assumption that the channel is approximately timeinvariant over at least two consecutive transmission slots. Since the relaying link is in active, the transmission slot is divided into two-sub timeslots in which the communication takes place. During the first sub slot, the source broadcasts its signal, s, with the average transmit power P1 to all relays. In the second subslot, the relaying transmission is performed by the best relay selected by PRS procedure† [6]. In particular, the relay having highest SNR of the links from the source, indexed by i∗ , will become
c 2012 The Institute of Electronics, Information and Communication Engineers Copyright
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the forwarder in the second sub-slot, i.e., it will amplify and then forward the received signal towards the destination with the average transmit power P2 . Assuming that the additive noise at each receiver is represented by a zero mean, complex Gaussian random variable with variance N0 , the instantaneous SNRs for S → Ri and Ri → D links are denoted as γ1,i = |hSRi |2 P1 /N0 and γ2,i = |hRi D |2 P2 /N0 , respectively where hSRi and hRi D are the corresponding channel coefficients. Under Rayleigh channels, they are therefore independent exponentially distributed random variN = γ¯ 1 = λ1 P1 /N0 and ables with expected values {¯γ1,i }i=1 N {¯γ2,i }i=1 = γ¯ 2 = λ2 P2 /N0 where λ1 = E{|hSRi |2 } and λ2 = E{|hRi D |2 }. According to [2], the equivalent instantaneous SNR for the relaying link, γR , is given by γR = Δ γSRi∗ γRi∗ D /(γSRi∗ + γRi∗ D + 1) where γSRi∗ = maxi=1,...,N {γ1,i } and γRi∗ D is the instantaneous SNR from the best relay to the destination. It can be observed that in medium-to-high SNR regimes, the dual-hop amplify-and-forward relaying link is dominated by the weakest link; therefore, γR can be wellapproximated by γR ≈ min(γSRi∗ ; γRi∗ D ) [2]. If all the links from the source are independently faded, thanks to [12], we have the probability density function (PDF) and cumulative distribution function (CDF) of γR respectively being of the forms N N 1 − μγ fγR (γ) = (−1)i−1 e i, (1) i=1 i μi N N −γ FγR (γ) = (−1)i−1 1 − e μi , (2) i=1 i where μi = (i/¯γ1 + 1/¯γ2 )−1 . In another case, i.e., the direct link is active, the corresponding PDF and CDF of the direct link, γD , in Rayleigh fading channel are fγD (γ) =
γ
1 − γ¯ 0 γ¯ 0 e
, FγD (γ) = 1 − e
− γ¯γ
0
,
(3)
where γ¯ 0 = λ0 P1 /N0 with λ0 = E{|hSD |2 }. Assume that there are K mode adaptive M-QAM available for the system in which the modulation constellation used in each transmission slot is determined by the end-toend SNR and the target BER, BERT , with an aim to maximize the SE by using the largest possible constellation size. To that effect, the entire effective received SNR range is partitioned into K non-overlapping intervals, determined by a set of boundary values, 0 = γT0 < γT1 < · · · < γTk < · · · < γTK = +∞. Note that each of fading regions, [γTk−1 γTk ), is associated with the particular constellation size, Mk -QAM. Denoting γΣ as the instantaneous SNR of the currently active link, we have γΣ ∈ {γD , γR }. Since γΣ is in between γTk−1 and γTk , the destination will choose the modulation scheme Mk -QAM among the K possible modulation modes. However, to avoid deep fades, no data is sent when γΣ is lower than γT1 [9]. Through a feedback channel, the destination feedbacks this information to the source. To facilitate the analysis, we assume that the feedback channel is error free with no delay. To determine the boundary values, we start with the
approximate BER of M-QAM with Gray coding over additive white Gaussian noise (AWGN) channels [3, Eq. (9.31)], namely (k, γΣ ) ≈ αk Q βk γΣ , (4) PQAM b
2 1, mk = 1, 2 , mk = 1, 2 Δ , and mk = , βk = mk3 where αk = 4 , m ≥ 3 k mk 2mk −1 , mk ≥ 3 log2 (Mk ). With (4), it is possible to calculate γTk to satisfy the target BER, BERT , by setting to the SNR required as γTk = [Q−1 (BERT /αk )]2 /βk . 3.
(5)
Performance Analysis
Before proceeding with the evaluation of the system performance in terms of average BER and SE, we first specify the occurrence probability of each mode. For a given arbitrary value of T , the occurrence probability of mode k with k ∈ {1, 2, · · · , K} can be generally expressed by taking into account all possible cases of T relative to (γTk−1 , γTk ]†† as [13] ⎧ (1) ⎪ πk , γTk < T ⎪ ⎪ ⎪ ⎪ ⎨ (2) (6) πk = ⎪ πk , γTk−1 < T ≤ γTk , ⎪ ⎪ ⎪ ⎪ ⎩ π(3) , T < γk−1 T k where
k−1 π(1) < γR ≤ γTk ) k = pD Pr(γD ≤ T ) Pr(γT +pR Pr(γR ≤ T ) Pr(γTk−1 < γD ≤ γTk ) , k−1 k k π(2) k = pD Pr(γD ≤ T )Pr(γT < γR ≤ γT )+Pr(T < γD ≤ γT ) +pR Pr(γR ≤ T )Pr(γTk−1 < γD ≤ γTk )+Pr(T < γR ≤ γTk ) k k = π(1) k + pD Pr(T < γD ≤ γT )+ pR Pr(T < γR ≤ γT ), k−1 k k−1 k π(3) k = pD Pr(γD ≤ T )Pr(γT < γR ≤ γT )+Pr(γT < γD ≤ γT ) +pR Pr(γR ≤ T )Pr(γTk−1< γD ≤ γTk )+Pr(γTk−1< γR ≤ γTk ) k−1 k k−1 k = π(1) k + pD Pr(γT < γD ≤ γT )+ pR Pr(γT < γR ≤ γT )
with Pr(x < γZ ≤ γTk ) = FγZ (γTk ) − FγZ (x), Z ∈ {D, R} and x ∈ {T, γTk−1 }. Furthermore, pZ is the steady state selection probability for each link defined as the fraction of time that link will serve as an active link, namely (7) pR = Pr(γD ≤ T ) Pr(γD ≤ T ) + Pr(γR ≤ T ) , pD = Pr(γR ≤ T ) Pr(γD ≤ T ) + Pr(γR ≤ T ) , (8) where Pr(γZ ≤ T ) = FγZ (T ). 3.1 Spectral Efficiency The achievable SE is simply the sum of the data rate in † Suppose that the source is able to periodically monitor the channel state information of its connections to relays through its local feedback. †† It is noted that the case of T < γTk−1 , i.e., π(3) 1 , does not exist for k = 1.
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each partition regions weighted by the corresponding occurrence probability. Making use the fact that the SE for γΣ ∈ [γTk−1 , γTk ) with k ≥ 2 is log2 (Mk ) and log2 (Mk ) 2 bps/Hz for the direct link and the relaying link respectively, we have ⎧ (1) k+1 ⎪ ηk , γT < T ⎪ ⎪ ⎪ K ⎪ ⎨ (2) k ASE = ηk with ηk = ⎪ ηk , γT < T ≤ γTk+1 , (9) ⎪ k=2 ⎪ ⎪ ⎪ ⎩ η(3) , T < γk T k where
k−1 k mk η(1) k = pD Pr(γD ≤ T )Pr(γT < γR ≤ γT ) 2 + pR Pr(γR ≤ T ) Pr(γTk−1 < γD ≤ γTk )mk ,
x
(1) k η(2) k = ηk + pD Pr(T < γD ≤ γT )mk
3.3 Switching Optimization for Maximizing ASE
+ pR Pr(T < γR ≤ γTk ) m2k ,
Subject to a constraint on the average BER, the system ASE may achieve its maximum by fixing the average system BER to be no greater than BERT instead of the instantaneous BER. In particular, it is the optimization problem, which is mathematically stated as
(1) k−1 π(3) < γD ≤ γTk )mk k = πk + pD Pr(γT
+ pR Pr(γTk−1 < γR ≤ γTk ) m2k . 3.2 Bit Error Rate We are now in a position to derive the average system BER, which is considered as the most important metric in wireless systems. Denoting BERk as the average BER in a specific region of [γTk−1 γTk ) and with reference to the total probability theory, the average BER for the system can be calculated as [9] K k=1 mk BERk BER = . (10) K k=1 mk πk Because the source stops data transmission in mode 1 to avoid deep fade, i.e. m1 = 0, (10) can be reduced as K k=2 mk BERk BER = , (11) K k=2 mk πk where ⎧ (1) ⎪ ⎪ BERk , γTk < T ⎪ ⎪ ⎪ ⎪ ⎨ (2) BERk=⎪ ⎪ BERk , γTk−1 < T ≤ γTk . ⎪ ⎪ ⎪ ⎪ ⎩ BER(3) , T < γk−1 k T
(12)
()
In the above, BERk with ∈ {1, 2, 3} is of the form (1)
BERk = pD Pr(γD ≤ T )IkR (γTk−1 , γTk )+ pR IkD (γTk−1 , γTk ), (2)
(1)
(3)
(1)
BERk =BERk + pD IkD (T, γTk ) + pR IkR (T, γTk ), BERk =BERk + pD IkD (γTk−1 , γTk ) + pR IkR (γTk−1 , γTk ). Furthermore, IkZ (x) is defined as IkZ (x) =
γTk x
PQAM (k, γ) fZ (γ)dγ. b
Substituting (1) and (3) into (13), after some manipulation, we obtain IkD (x) = J(αk , βk , γ¯ 0 , x, γTk ) and N k i−1 N IR (x) = (−1) (14) J(αk , βk , μi , x, γTk ), i=1 i √ γk γ where J(αk , βk , γ¯ , x, γTk+1 ) = x T αk Q βk γ γ1¯ e− γ¯ dγ is de∞ rived in the closed-form with Γ(a, x) = x ta−1 e−t dt [14, Eq. (8.350-2)] as √ γ J(αk , βk , γ¯ , x, γTk ) = αk Q βk γ 1 − e− γ¯ β β 1 β γTk (15) 1 1 1 βk γ 1 −2 1 1 k k k Γ 2 , 2 + γ¯ γ . − 2 πΓ 2 , 2 − 2π 2 + γ¯
(13)
K−1 ASE subject to BER = BERT . max{γTk }k=1
(16)
For a given target BER and the destination threshold (T ), the set of optimized switching thresholds is found by solving K − 1 nonlinear equations as [10] ⎧ K N ⎪ ⎪ k=1 mk BERk − BERT k=1 mk πk = 0 ⎪ ⎪ ⎪ ⎪ υ1 (γT1 ) − υk (γT2 ) = 0 ⎪ ⎨ , (17) ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎩ υ1 (γT1 ) − υk (γTK−1 ) = 0 where υk (γTk ) = 4.
mk PQAM (k,γTk )−mk−1 PQAM (k−1,γTk ) b b . mk −mk−1
Numerical Results and Conclusion
In this section, we perform computer simulations to validate the above theoretical analysis. In all the simulations, the channel parameters are set as λ0 = 1, λ1 = 2 and λ2 = 3. We vary the number, N, of relays in the network N = 1, 3 and 5, and five-mode adaptive M-QAM [10] with BERT = 10−3 is employed. For simplicity, we assume that the transmission power of the source and the relays is the same, i.e., P1 = P2 = P. The effect of the number of relays on the SE and BER is shown in Figs. 1 and 2, respectively. From these figures, there are three observations worth noting. First, the SE of the system improves as the transmission power (P) increases, as expected. Second, at low-to-medium SNRs, the considered PRS with adaptive modulation outperforms the distributed SSC with single relay (N = 1) around 2 dB. Thirdly, at high SNRs the system with more than one relay suffers a certain loss in the SE as compared to that with one relay while the improvement on the BER is not significant as indicated in Fig. 2. This behavior can be explained by
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that, contrarily to conventional distributed cooperative networks, no actual combiner is used at the destination thus making it an attractive choice for implementation. Finally, it should be noted that the analysis results in this letter can also be used as an approximate solution for decode-and-forward distributed SSC networks with PRS under adaptive modulation since the output SNR of the dual-hop DF relaying link at high SNRs has the same form as reported in the literature. Acknowledgment
Fig. 1
Average SE, T = 10.
This research was supported by the Vietnam’s National Foundation for Science and Technology Development (NAFOSTED) (No. 102.99-2010.10). References
Fig. 2
Average BER, T = 10.
using the fact that in such conditions, the destination connects the relaying link most of the time thereby increasing the use of two timeslots requested for one data packet over relaying communications. Before closing this section, it is appropriate to comment on the fixed and optimal switching thresholds. Recalling that the fixed switching thresholds (Eq. (5)) are designed to ensure that the instantaneous BER always remains below the target BER while optimal switching thresholds are obtained by optimizing the system SE under the constraint of the target BER. As can be seen in Fig. 2, the average BER of the system with fixed SNR thresholds is always much smaller than the target BER requirement. And this smaller average BER is obtained at the cost of a loss in the average SE (around 2 dB) as we can observe in Fig. 1. Meanwhile the average BER of the system using optimal switching threshold remains equal to the target BER up to the certain value of average SNR (about 33 dB) and its corresponding SE provides 2 dB gain as compared to the case of fixed switching thresholds. In conclusion, we have proposed adaptive modulation for distributed SSC combining networks with PRS. From the numerical results, it can be seen that implementing the optimal switching threshold is advisable to achieve better SE. An important point to stress on distributed SSC system is
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