Co-phasing Full-Duplex Relay Link with Non-Ideal Feedback Information Taneli Riihonen #1 , Risto Wichman #2 , Jyri H¨am¨al¨ainen #3 # Helsinki University of Technology P.O. Box 3000, FI-02015 TKK, Finland 1
[email protected] [email protected] 3
[email protected] 2
Abstract—We study a wireless full-duplex amplify-and-forward relay link where a destination receives superposition of relayed and direct signals. We propose a new relay protocol that performs co-phasing of the two paths. We analyze the performance of the relay link by calculating closed-form expressions for the average end-to-end signal-to-noise ratio assuming flat Nakagami-m fading channels and by simulating the outage capacity. In the analysis, the phase feedback information is considered non-ideal due to Nph bit quantization and bit error probability δ. We show that full-duplex relaying can offer performance improvement over both direct transmission and half-duplex relaying even with few feedback bits and with a feedback channel of relatively low quality.
I. I NTRODUCTION Wireless relay is a transceiver that receives, processes, and then retransmits radio signals. The usage of relays has been found beneficial for extending coverage areas or enhancing hotspot capacity in cellular networks. This paper concentrates on a linear amplify-and-forward (AF) relay that amplifies its noisy input signal using fixed gain. Furthermore, we consider a fixed infrastructure relay that is able to operate in a fullduplex (FD) mode in which the relay receives and retransmits concurrently on the same frequency and the destination receives superposition of the direct and relayed signals. We assume that full-duplex operation is facilitated by the common countermeasures against loop interference as discussed in [1]. Without capability for full-duplex operation, the relay has to resort to a half-duplex (HD) mode [2], [3], where orthogonal channels are assigned for relay reception and transmission. Half-duplex relaying suffers from a rate pre-log factor 12 due to two allocated channels, but it enables maximum ratio combining (MRC) of relayed and direct transmissions in the destination with additional gain [4]. On the contrary in FD relaying, MRC is not possible and the direct and relayed signals are combined incoherently [5] unless the source and the relay are phase synchronized. Thus, we propose a new AF co-phasing relay protocol that facilitates a coherent combining gain. Previously similar phase rotation has been proposed for HD relaying to coherently combine signals from multiple parallel relays [6], but the application to co-phase direct and relayed transmissions in FD relaying is new. As practical implementation of co-phasing requires a feedback channel for reporting a co-phasing factor from the des-
978-1-4244-2489-4/08/$20.00 © 2008 IEEE
tination to the relay, our analysis assumes the phase feedback is non-ideal due to Nph bit quantization and bit error probability δ. Additionally, the system exploits power allocation requiring only average channel state information and thus consuming only minor amount of feedback channel capacity. The first objective of this paper is to study the performance improvement due to the proposed co-phasing algorithm and the performance loss due to non-ideal feedback channel. The second objective is to compare the performance of full-duplex and half-duplex relaying. In the analysis, performance metric is the average end-to-end signal-to-noise ratio (SNR) that is determined in closed-form. Similar average SNR analysis is published in [7] for multi-antenna transmit diversity techniques. Furthermore, we present simulation results on outage capacity. Our results show that the proposed co-phasing algorithm facilitates coherent combining gain that is comparable to the MRC gain of half-duplex relaying. Consequently, full-duplex relaying can achieve better outage capacity than half-duplex even with coarsely quantized phase information and with a large feedback bit error rate, because the rate pre-log factor 12 of half-duplex operation is avoided. II. S YSTEM M ODEL In this section, we describe the two-hop single-frequency relay network (1 × 1 × 1) having one source node (S), one relay node (R) and one destination node (D). The network model is illustrated in Fig. 1. The source broadcasts the signal x, which is received by both the relay and the destination. The transmission power is normalized to E |x|2 = PS , where E [·] is the expectation
263
nR √ ESR hSR
S
x
√ ESD hSD
Fig. 1.
r
R β
feedback channel for φ
√ ERD hRD t
nD y
D
System model of the full-duplex co-phasing relay link.
IEEE ISWCS 2008
operator. Consequently, the relay receives the signal r = ESR hSR x + nR ,
III. P ERFORMANCE OF F ULL -D UPLEX R ELAYING (1)
where nR denotes the additive receiver √ noise component (of 2 power σR ) at the relay input and ESR hSR is the channel coefficient between the source and the relay. The relay amplifies and forwards the received signal r, i.e., its output signal is t = βr, where β is an AF gain factor. Since single-frequency operation is assumed, the relay transmits using the same frequency band as the source node. In this paper, we concentrate on a fixed gain relay protocol [2] that is improved The average relay input power by phase rotation. 2 σ , and to guarantee average relay is E |r|2 = ESRPS + R transmit power E |t|2 = PR , the fixed gain amplification factor becomes PR jφ (2) β=e 2 , ESR PS + σR where centralized power allocation is used to determine PR and PS , and the co-phasing factor φ is obtained from the destination via a feedback channel. Finally, the destination receives a superposition y of direct link and relayed signals: y = ERD hRD β ESR hSR + ESD hSD x + ERD hRD βnR + nD , (3) where nD denotes the additive receiver noise component √ 2 ) at the destination, ERD hRD represents the (of power σD channel coefficient between the relay and the destination, and √ ESD hSD is a direct link channel coefficient. The system also incorporates a feedback channel. The destination estimates the co-phasing factor and quantizes it with uniform 2Nph levels. The Nph feedback bits are transmitted to the relay via independent uncoded delay-free binary symmetric channels with equal error probabilities δ ∈ [0, 12 ]. Next we summarize assumptions that hold for all channel coefficients hxy and Exy and link SNRs γxy , where xy ∈ {SR, RD, SD}. The constant average channel energy factors Exy are defined by the network geometry, i.e., by a path-loss model. The complex channel coefficients hxy represent the fast fading of the channels. Each channel hxy is assumed to be independently Nakagami-mxy distributed (mxy ≥ 12 ) with uniform phase distribution between 0 and 2π. Their mean powers are normalized to unity, i.e., E |hxy |2 = 1, where E [·] is the expectation over fast-fading channel distributions. P E Thus, the instantaneous link SNRs γxy = xσ2xy |hxy |2 are y Gamma-distributed random variables with the averages of P E γ¯xy = xσ2xy and their probability density functions are y
m
pγxy (z) =
z mxyxy z mxy −1 − mγ¯xy e xy , mxy γ¯xy Γ(mxy )
(4)
In this section, we present closed-form expressions for the average end-to-end SNR in the full-duplex (FD) co-phasing relay link exploiting non-ideal feedback information. The instantaneous end-to-end SNR is the received signal power divided by the noise power and conditioned on channel states: √ √ √ ERD hRD β ESR hSR + ESD hSD 2 PS FD = (5) γ √ ERD hRD β 2 σ 2 + σ 2 R
D
γSR + 1)γSD γSR γRD + (¯ = γ¯SR + γRD + 1
=γ incoh
√ √ √ √ 2 γ¯SR + 1 γSR γRD γSD cos(ψ) + , γ¯SR + γRD + 1
(6)
=γ coh
where the phase difference is ψ = (hSR hRD ) − (hSD ) + φ. Based on the instantaneous SNR, Shannon capacity theorem gives the expression RFD = log2 (1 + γ FD ) [bps/Hz]
(7)
for the instantaneous rate. The average end-to-end SNR in full-duplex relaying is γ¯ FD = E γ FD = γ¯ incoh + γ¯ coh , (8) where E [·] denotes the average over the three link SNR and the phase difference distributions. The expectation of the first rational term in (6) is given by [5, Eq. 10] γ¯ incoh = γ¯SD + (¯ γSR − γ¯SD )mRD eA EmRD +1 (A) ,
(9)
γ ¯SR +1 where A = γ¯RD /mRD . This term corresponds to the average SNR when the relay does not exploit any transmit channel state information (Nph = 0), and the direct and relayed signals are thus incoherently combined in the destination. The expectation of the second rational term in (6) corresponds to the average SNR improvement due to co-phasing in the relay resulting in coherent combining at the destination. By conducting the derivation explained in Appendix A we get BSR BRD BSD √ A γ¯ coh = 2 Ae EmRD + 12 (A) cNph ,δ , (10) γ¯RD /mRD
Γ(mxy + 12 ) γ ¯ where Bxy = mxy for xy ∈ {SR, RD, SD}, and Γ(mxy ) xy En (·) is the exponential integral defined in (25). Next we determine cNph ,δ = E [cos(ψ)] that depends on the co-phasing algorithm, the number of feedback bits Nph and the bit error rate δ in the feedback channel.
A. Co-phasing Algorithm With Nph feedback bits available for transmit phase adjustment in the relay, the destination selects the uniformly quantized phase shift using the condition φ∗ = arg max γ FD = arg max | (hSR hRD ) − (hSD ) + φ| , φ∈Φ
where Γ(·) is the Gamma function.
264
φ∈Φ
(11)
Nph where Φ = { 22πn − 1}. If the feedback Nph | n = 0, 1, . . . , 2 channel is error-free (δ = 0), the phase difference ψ in the destination becomes uniformly distributed in [− 2Nπph , 2Nπph ] and π 2Nph sin( Nph ). cNph ,0 = E [cos(ψ)] = (12) π 2 Taking into account feedback errors, the performance depends on the feedback labeling. Let us denote that wij is the number of erroneous bits, when feedback codeword i is transmitted but codeword j is received. Thus, by averaging over all feedback error transitions we get
cNph ,δ = cNph ,0
Nph Nph 2 2
i=1 j=1 1
δ wij (1 − δ)Nph −wij CNph ,i−j , (13)
2πk 2Nph
=
c2,δ
=
c3,δ
=
as shown in [4]. The corresponding instantaneous rate is RHD =
. In this paper, we assume where CNph ,k = 2Nph cos that the feedback labeling is the standard binary reflected Gray code. By determining wij , substituting them into (13) and simplifying the formulas, we can settle the following closedform expressions for cNph ,δ when Nph = 1, 2, 3: c1,δ
power in direct link transmission is fixed to PS = 1 throughout this paper. In half-duplex (HD) relaying, two time slots are allocated for transmission. The source transmits only during the first time slot. The relay receives during the first slot and retransmits during the second slot. Half-duplex relaying allows the destination to perform maximum ratio combining of the direct link signal received during the first slot and the relayed signal received during the second slot. Thus the instantaneous endto-end SNR is expressed by γSR γRD (18) γ HD = γSD + γ¯SR + γRD + 1
2 (1 − 2δ) , (14) π√ 2 2 (1 − 2δ) , and (15) π √ √ 2 4 2− 2 1 − (1 − )δ (1 − 2δ) . (16) π 2
Closed-form expressions for Nph > 3 become quite complicated. Fortunately (12) shows that co-phasing with 3 bit feedback already offers almost the same SNR improvement as optimal unquantized co-phasing. B. Power Allocation Algorithm In addition to the phase feedback information in the relay, it is assumed that the transmitters can exploit centralized power control information based on average channel strengths. Let us denote PR = pPS , which along with normalizing the total transmit power in the network to unity (PS + PR = 1) results p 1 and PR = p+1 . We employ power allocation p∗ in PS = p+1 that maximizes the average end-to-end SNR γ¯ : p∗ = arg max γ¯ . p>0
(17)
In this paper, p∗ is determined by numerically finding the maximum of the closed-form average SNR expression, but a nearly optimal closed-form solution is available [5] for incoherent full-duplex relaying (Nph = 0). IV. P ERFORMANCE IN R EFERENCE S YSTEMS We compare the performance of full-duplex relaying to two reference systems being direct link transmission without a relay and half-duplex relaying. In direct link (DL) transmission the instantaneous and average end-to-end SNRs are γ DL = γSD and γ¯ DL = γ¯SD , respectively. The instantaneous rate becomes RDL = log2 (1 + γ DL ) [bps/Hz]. The transmit
1 log2 (1 + γ HD ) [bps/Hz], 2
(19)
where the pre-log factor 12 is due to two allocated time slots. Furthermore, the average end-to-end SNR becomes γ¯ HD = γ¯SD + γ¯SR mRD eA EmRD +1 (A) ,
(20)
which is previously published in [4, Eq. 12] for Rayleigh fading channels, i.e., for mRD = 1. The total energy consumption of the system over time is normalized to unity, and as both nodes transmit only half of the time, they can use double power, i.e., PS + PR = 2 in half-duplex relaying. Power allocation is done in the same way as with full-duplex relaying, see (17). A nearly optimal power allocation can be determined in closed-form as shown in [4], but we use numerically determined optimal values. V. E VALUATION In this section, we evaluate the performance of full-duplex relaying by comparing it to the reference systems. Analytical evaluation is conducted by using the closed-form expressions of the average end-to-end SNR derived in the previous sections. Furthermore, we present simulation results that illustrate outage capacity improvement. For the performance evaluation, we consider a network geometry defined as follows: The distance between the source and the destination is normalized to 1. The relay is located on a line segment whose end points are the source and the destination. The normalized distance between the source and the relay is denoted by d. We assume Rayleigh channels and constant ESD = 20 dB, and normalize noise powers to unity, 2 2 = σD = 1. An exponential model with exponent 3 is i.e., σR assumed for path loss. Thus, the average link SNRs become R ESD γ¯SD = PS ESD , γ¯SR = PSdE3SD , and γ¯RD = P(1−d) 3 . A. Analytical Comparison We use the average end-to-end SNR for measuring the performance gain of the proposed co-phasing and power allocation algorithms and the performance loss due to the nonideal feedback channel. First the average end-to-end SNR is calculated for uniform power allocation (p = 1) and results are shown in Fig. 2. When the relay is close to the source, uniform power allocation
265
26
24
24
Average end-to-end SNR [dB]
Average end-to-end SNR [dB]
26
22
20
FD (Nph = ∞, δ = 0%) FD (Nph = 2, δ = 5%) FD (Nph = 1, δ = 10%) FD (Nph = 0) FD without direct link HD HD without direct link Direct link
18
16
14 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
22
20
FD (Nph = ∞, δ = 0%) FD (Nph = 2, δ = 5%) FD (Nph = 1, δ = 10%) FD (Nph = 0) FD without direct link HD HD without direct link Direct link
18
16
14 0.8
0.9
1
0
0.1
0.2
0.3
Relay location d Fig. 2. The average end-to-end SNR for varying relay location with uniform power allocation (p = 1).
15
FD (Nph = ∞, δ = 0%) FD (Nph = 2, δ = 5%) FD (Nph = 1, δ = 10%) FD (Nph = 0) FD without direct link HD HD without direct link
10
p∗ [dB]
5
0.4
0.5
0.6
0.7
0.8
0.9
1
Relay location d Fig. 4. The average end-to-end SNR for varying relay location with optimal power allocation (p = p∗ ).
relay is roughly in the half-way between the source and the destination, but a half-duplex relay should be placed closer to the destination. We see that the proposed co-phasing relay protocol improves SNR over incoherent relaying also with non-ideal phase feedback. Assuming 2 bit quantization and 5% bit error rate in the feedback channel, the average SNR loss is at maximum 0.4 dB with both uniform and optimal power allocation. The average SNR is smaller in full-duplex than in half-duplex, but this comparison is not fair, because the half-duplex mode uses higher instantaneous transmit power, but lower end-to-end rate due to two time slots and different power normalization.
0
−5
−10
B. Simulation Results −15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Relay location d Fig. 3. The optimal power allocation factor p∗ as a function of the relay location. The transmit powers are then PS , and PR = p∗ PS for the source and the relay, respectively. The optimal power allocation for the direct link without a relay is p∗ = −∞ dB (not shown here).
The average end-to-end SNR considered in the previous section does not reveal the effect of the rate pre-log factor 12 that half-duplex relaying suffers from, but full-duplex does not. Therefore, for conducting a fair performance comparison between half-duplex and full-duplex, we simulate the 1%outage capacity: C1% = arg max P (R ≤ ρ) ≤ 0.01, ρ
performs well and the ideal coherent combining gain of fullduplex relaying is less than 1 dB below the MRC gain of half-duplex relaying. When the relay is close to the destination, full-duplex relaying becomes worse than even mere direct link transmission and power balancing would be crucial. Thus, we determine the optimal power allocation illustrated in Fig. 3 by numerically solving (17) for the relaying protocols. The average end-to-end SNR using the optimal power allocation is shown in Fig. 4. We see that power allocation benefits are the largest, if the relay is close to the destination. Furthermore, power balancing guarantees that all relaying schemes are better than mere direct link transmission for all relay locations. With the optimal power allocation, the best location for a full-duplex
(21)
where R is the instantaneous rate. In other words, we study the rate that can be supported with 99% probability. Fig. 5 and Fig. 6 illustrate the simulated 1%-outage capacities with uniform and SNR maximizing power allocations, respectively. We see that already single bit co-phasing improves the performance tremendously (up to 2 bps/Hz) even if the feedback bit error probability is high (10%), because then the direct and relayed signals are rarely combined destructively. The outage capacity deteriorates, if a full-duplex relay is close to the destination and transmit powers are not balanced. Thus, the SNR maximing power allocation is beneficial for making full-duplex perform well in all relay locations. However, the SNR maximizing power allocation slightly decreases
266
1%-outage capacity [bps/Hz]
6
(Nph (Nph (Nph (Nph
= = = =
∞, δ = 0%) 2, δ = 5%) 1, δ = 10%) 0)
6
FD without direct link HD HD without direct link Direct link
5 4
3
2
1 0
FD FD FD FD
7
1%-outage capacity [bps/Hz]
FD FD FD FD
7
(Nph (Nph (Nph (Nph
= = = =
∞, δ = 0%) 2, δ = 5%) 1, δ = 10%) 0)
FD without direct link HD HD without direct link Direct link
5 4
3
2
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1
Relay location d
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Relay location d
Fig. 5. The 1%-outage capacity for varying relay location with uniform power allocation (p = 1).
Fig. 6. The 1%-outage capacity for varying relay location with power allocation that maximizes the average end-to-end SNR (p = p∗ ).
the outage capacity when the relay is close to the source. This effect is most severe in half-duplex relaying. To summarize, we see that outage capacity in full-duplex is much higher than in half-duplex even with only 1 or 2 available feedback bits and with 5-10% bit error rate in feedback channel.
γ ¯SR +1 where A = γ¯RD /mRD and the exponential integral is defined by [9, Eq. 5.1.4] ∞ −zt e En (z) = dt. (25) tn 1 √ γxy are NakagamiFurthermore, the random variables distributed with the averages given by √ γ¯xy Γ(mxy + 12 ) (26) γxy = E mxy Γ(mxy )
VI. C ONCLUSION We proposed a co-phasing algorithm for a full-duplex amplify-and-forward relay link, and modeled a realistic feedback channel for conveying the co-phasing information to the relay. The main result of the paper is the derivation closedform expressions for the average end-to-end SNR in the relay link with quantized phase feedback information and bit errors in the feedback channel. The full-duplex co-phasing relay link is compared to a half-duplex maximum ratio combining relay link and to direct link communication without a relay. The results show that with average SNR maximizing power allocation, full-duplex co-phasing relaying is favorable even with few (1 or 2) available feedback bits and with a relatively high (5-10%) bit error rate in feedback channel. A PPENDIX A P ROOF OF (10) By exploiting the assumption that all fast-fading channels are independent, the expectation of the second rational term in (6) can be expressed as √ √ γ¯ coh = 2 γ¯SR + 1E [ γSR ]E [ γSD ]E [cos(ψ)] I1 . (22) First, the integral I1 in (22) can be written in terms of the incomplete Gamma function using [8, Eq. 8.353.3] and further simplified using [9, Eq. 5.1.45] and [9, Eq. 6.1.15]. Thus, ∞ √ √ γRD z pγRD (z) dz (23) I1 = E = γ¯SR + γRD + 1 z + γ¯SR + 1 0 mRD Γ(mRD + 12 ) A = e EmRD + 12 (A) , (24) γ¯RD Γ(mRD )
for xy ∈ {SR, RD, SD}. Finally, (10) is obtained by substituting (24) and (26) into (22). R EFERENCES [1] H. Hamazumi, K. Imamura, N. Iai, K. Shibuya, and M. Sasaki, “A study of a loop interference canceller for the relay stations in an SFN for digital terrestrial broadcasting,” in IEEE Global Telecommunications Conference, vol. 1, November 2000, pp. 167–171. [2] R. U. Nabar, H. B¨olcskei, and F. W. Kneub¨uhler, “Fading relay channels: performance limits and space-time signal design,” IEEE J. Sel. Areas Commun., vol. 22, no. 6, pp. 1099–1109, August 2004. [3] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3062–3080, December 2004. [4] X. Deng and A. M. Haimovich, “Power allocation for cooperative relaying in wireless networks,” IEEE Commun. Lett., vol. 9, no. 11, pp. 994–996, November 2005. [5] T. Riihonen and R. Wichman, “Power allocation for a single-frequency fixed-gain relay network,” in 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, September 2007. [6] P. Larsson, “Large-scale cooperative relaying network with optimal coherent combining under aggregate relay power constraints,” in Future Telecommunications Conference, December 2003. [7] J. H¨am¨al¨ainen and R. Wichman, “Multiple antenna transmission utilizing side information for WCDMA systems,” in The 5th CDMA International Conference and Exhibition, November 2000. [8] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. Academic Press, 1994. [9] M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, 1972.
267
