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recursively updated Cholesky decomposition-based detector performs as well as [1] with a much reduced computational complexity. The novel algorithm can easily be extended to uplink CDMA system, where the channel paths between the base station and the different users are different.
Cognitive Amplify-and-Forward Relay Networks Over Nakagami-m Fading Trung Q. Duong, Student Member, IEEE, Daniel Benevides da Costa, Member, IEEE, Maged Elkashlan, Member, IEEE, and Vo Nguyen Quoc Bao, Member, IEEE
ACKNOWLEDGMENT The author would like to thank E. Manuel of the College of Engineering, Trivandrum, for help with part of computer simulations. The author would also like to thank the Associate Editor and the anonymous reviewers for their valuable suggestions and comments. R EFERENCES [1] T. Sartenaer and L. Vandendorpe, “Linear and DF joint detectors for DS-CDMA communication using periodic long codes,” IEEE Trans. Signal Process., vol. 52, no. 7, pp. 2080–2091, Jul. 2004. [2] A. Klein, G. K. Kaleh, and P. W. Baier, “Zero forcing and minimum mean square error equalization for multiuser detection in code division multiple access channels,” IEEE Trans. Veh. Technol., vol. 45, no. 2, pp. 276–287, May 1996. [3] S. Buzzi and M. Massaro, “Parameter estimation and multiuser detection for bandlimited long-code CDMA systems,” IEEE Trans. Wireless Commun., vol. 7, no. 6, pp. 2307–2317, Jun. 2008. [4] S. Buzzi and H. V. Poor, “On parameter estimation in long-code DS/CDMA systems: Cramer Rao bounds and least squares algorithms,” IEEE Trans. Signal Process., vol. 51, no. 2, pp. 545–549, Feb. 2003. [5] D. Guo, L. R. Rasmussen, and T. J. Lim, “Linear parallel interference cancellation in long code CDMA multiuser detection,” IEEE J. Sel. Areas Commun., vol. 17, no. 12, pp. 2074–2081, Dec. 1999. [6] S. Bhashyam and B. Aazhang, “Multiuser channel estimation and tracking for long-code CDMA systems,” IEEE Trans. Commun., vol. 50, no. 7, pp. 1081–1090, Jul. 2002. [7] Y-H. Lee and S-J. Kim, “Sequence acquisition of DS-CDMA systems employing Gold sequences,” IEEE Trans. Veh. Technol., vol. 49, no. 6, pp. 2397–2404, Nov. 2000. [8] A. Mirbagheri and Y. C. Yoon, “A linear MMSE receiver for multipath asynchronous random-CDMA with chip-pulse shaping,” IEEE Trans. Veh. Technol., vol. 51, no. 5, pp. 1072–1086, Sep. 2002. [9] A. Weiss and B. Friendlander, “Channel estimation for DS-CDMA downlink with aperiodic spreading codes,” IEEE Trans. Commun., vol. 47, no. 10, pp. 1561–1569, Oct. 1999. [10] T. Li, W. Liang, Z. Ding, and J. Tugnait, “Blind multiuser detection for long-code CDMA systems with transmission-induced cyclostationarity,” EURASIP J. Wireless Commun. Netw., vol. 2005, no. 2, pp. 206–215, Apr. 2005. [11] T. Fusco, L. Izzo, A. Napolitano, and M. Tanda, “On the second-order cyclostationarity properties of long-code DS-SS signals,” IEEE Trans. Commun., vol. 54, no. 10, pp. 1741–1746, Oct. 2006. [12] Z. Xu, P. Liu, and M. D. Zoltowski, “Diversity-assisted channel estimation and multiuser detection for downlink CDMA with long spreading codes,” IEEE Trans. Signal Process., vol. 52, no. 1, pp. 190–201, Jan. 2004. [13] Z. Yang and X. Wang, “Blind turbo multiuser detection for long-code multipath CDMA,” IEEE Trans. Commun., vol. 50, no. 1, pp. 112–125, Jan. 2002. [14] L. Tong, A.-J. Veen, P. Dewilde, and Y. Sung, “Blind de-correlating RAKE receivers for long-code WCDMA,” IEEE Trans. Signal Process., vol. 51, no. 6, pp. 1642–1655, Jun. 2003. [15] P. De and E. Manuel, “A novel fast algorithm for multiuser detection in long code CDMA system,” in Proc. IEEE Radio Wireless Symp., Long Beach, CA, Jan. 2007, pp. 399–402. [16] S. Haykin, Adaptive Filter Theory, 2nd ed. Englewood Cliffs, NJ: Prentice–Hall, 1991. [17] C. Rialan and L. L. Scharf, “Fast algorithms for computing QR and Cholesky factors of Toeplitz operators,” IEEE Trans. Acoust., Speech, Signal Process., vol. 36, no. 11, pp. 1740–1748, Nov. 1988. [18] P. De, “A delta least squares lattice algorithm for fast sampling,” IEEE Trans. Signal Process., vol. 47, no. 9, pp. 2396–2407, Sep. 1999. [19] N. Al-Dhahir and A. H. Sayed, “The finite-length multi-input multi-output MMSE-DFE,” IEEE Trans. Signal Process., vol. 48, no. 10, pp. 2921– 2936, Oct. 2000.
Abstract—In this paper, the outage probability (OP) of dual-hop cognitive amplify-and-forward (AF) relay networks subject to independent non-identically distributed (i.n.i.d.) Nakagami-m fading is examined. We assume a spectrum-sharing environment, where two different strategies are proposed to determine the transmit powers of the secondary network. Specifically, the transmit power conditions of the proposed spectrum-sharing network are governed by either the combined power constraint of the interference on the primary network and the maximum transmission power at the secondary network or the single power constraint of the interference on the primary network. Closed-form lower bounds and asymptotic expressions for the OP are derived. Regardless of the transmit power constraint, we reveal that the diversity order is strictly defined by the minimum fading severity between the two hops of the secondary network. This aligns with the well-known result for conventional dual-hop AF relaying without spectrum sharing. Furthermore, the impact of the primary network on the diversity–multiplexing tradeoff is investigated. We confirm that the diversity–multiplexing tradeoff is independent of the primary network. Index Terms—Amplify-and-forward (AF), dual-hop cognitive relay network (CRN), Nakagami-m fading, spectrum sharing.
I. I NTRODUCTION Over the past few years, cooperative diversity [1], [2] has rekindled enormous interest from the wireless communications community due to many perceived benefits acquired without the need for multiple antennas implemented at the terminals. In particular, by allowing several mobile terminal relay signals for each other, an antenna array is emulated, and therefore, spatial diversity can be explored. Most recently, in spectrum-sharing environments [3], the concept of cooperative diversity has been applied in cognitive networks with decodeand-forward (DF) relays [4]–[9] and amplify-and-forward (AF) relays [10]–[12]. The basics of these are outlined next. For DF relaying, Guo et al. [4] investigated the outage performance of cognitive relay networks (CRNs) within the constraint imposed on the interference suffered by the primary user (PU) receiver.1 In [5],
Manuscript received November 23, 2011; revised February 8, 2012; accepted March 19, 2012. Date of publication March 28, 2012; date of current version June 12, 2012. This work was supported by the FUNCAP under Grant BPI-0031-00090.01.00/10. This work was supported by the Vietnam National Foundation for Science and Technology Development under Grant 102.992010.10. The review of this paper was coordinated by Dr. E. K. S Au. T. Q. Duong is with Blekinge Institute of Technology, 371 79 Karlskrona, Sweden (e-mail:
[email protected]). D. Benevides da Costa is with the Federal University of Ceará, 62042-280 Sobral, Brazil (e-mail:
[email protected]). M. Elkashlan is with Queen Mary, University of London, E1 4NS London, U.K. (e-mail:
[email protected]). V. N. Q. Bao is with Posts and Telecommunications Institute of Technology, Ho Chi Minh, Vietnam (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2012.2192509 1 Hereafter,
this constraint is referred to interference power constraint.
0018-9545/$31.00 © 2012 IEEE
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the outage probability (OP) of CRNs with a suitable relay selection was examined. Adopting both interference and maximum allowable transmit power constraints, it was shown that full diversity order can be achieved. In [6], the capacity of reactive DF schemes in CRNs subject to an interference power constraint was evaluated. In [7], assuming the presence of a direct link in the secondary network, a tight lower bound expression for the OP of CRNs was derived. Recently, Si et al. [8] proposed to employ the PU’s outage constraints (instead of interference power constraints) to reduce the need for channel state information of interference links from the PU receiver. Common to the works in [4]–[8] is that Rayleigh fading was considered. More recently [9], an outage analysis of CRNs over Nakagami-m fading was performed, assuming both interference and maximum allowable transmit power constraints. Some other works in the literature have considered CRNs with AF relaying. In [10], three low-complexity relay selection strategies were proposed, in which closed-form asymptotic expressions for the OP were derived, assuming that the transmit power of the secondary network is controlled by both the primary network and the secondary transmitters. In [11], an exact closed-form expression for the OP was derived within an interference power constraint. This was later extended in [12] by considering a direct link in the secondary network. In that case, a selection-combining receiver at the secondary destination was employed. Once again, common to the works in [10]–[12] is that they considered Rayleigh fading channels. While all of the aforementioned works substantially provide a good understanding of CRNs, most of them assumed Rayleigh fading channels. This may not be useful in a wide range of fading scenarios that are typical in realistic wireless relay applications. Owing to this fact, the aim of this paper is to investigate the outage performance of dual-hop CRNs with AF relays and subject to independent nonidentically distributed (i.n.i.d.) Nakagami-m fading channels. From a realistic viewpoint, the choice of Nakagami-m fading is to characterize more versatile fading scenarios that are more or less severe than Rayleigh fading via the m fading parameter, which includes the Rayleigh fading (m = 1) as a special case. The Nakagami-m fading also approximates the Hoyt fading, for m < 1, and the Rice fading, for m > 1. Furthermore, PUs and secondary users (SUs) are often far from each other; as such, i.n.i.d. fading is assumed with distinct fading parameters in the respective links. To determine the transmit powers of the secondary network, two strategies are employed. More specifically, the transmit power conditions of the proposed spectrumsharing network are governed by either the combined power constraint of the interference on the primary network and maximum transmission power at the secondary network or the single power constraint of the interference on the primary network. Closed-form lower bounds and easy-to-evaluate asymptotic solutions for the OP are derived. Our solutions reveal important design insights and the impact of some key network parameters on the network behavior, such as fading severity, power constraints, and PU position. Our outcomes are outlined as follows: First, we show that the diversity order is strictly defined by the minimum fading severity between two hops of the secondary network. This result is in line with those obtained for dual-hop AF relaying without spectrum sharing. Second, the secondary network achieves full diversity order of min(mg1 , mg2 ), regardless of the transmit power constraint, where mg1 and mg2 are the fading severity parameters of the first and second hops, respectively. Third, by relaxing the transmit power constraint at the SUs, we show that the outage performance is much improved by locating the PU away from the secondary network. Fourth, we confirm that the diversity–multiplexing tradeoff is independent of the primary network. This result is independent of the transmit power constraint.
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Fig. 1. Network model for cognitive spectrum sharing with AF relaying.
II. N ETWORK AND C HANNEL M ODELS Consider a dual-hop cooperative spectrum-sharing network where PUs and SUs share the same spectrum band in a given propagation environment. Our network is composed of one SU source S, one AF SU relay R, one SU destination D, and one PU receiver P, as shown in Fig. 1. All nodes are equipped with a single antenna and operate in half-duplex mode. A time-division multiple-access scheme is employed for the SU communication, which is performed into two time slots. In the first time slot, S sends its signal to R with transmit power PS . In the second time slot, R amplifies the received signal from S and forwards the resulting signal to D with transmit power PR . The transmit powers at S and R are constrained, so that the interference impinged on the PU receiver remains below the maximum tolerable interference power Q. In addition, if the SUs are powerlimited terminals, S and R may transmit up to the maximum allowable power P. Therefore, the transmit powers at S and R can be mathematically written as PS = min(Q/|h1 |2 , P) and PR = min(Q/|h2 |2 , P), where h1 and h2 are the channel coefficients of the interference links S → P and R → P, respectively. Taking these two transmit power constraints into account, the endto-end instantaneous signal-to-noise ratio (SNR) at D can be written as γdA =
γ1 γ2 PS |g1 |2 PR |g2 |2 = N0 (PS |g1 |2 + PR |g2 |2 ) γ1 + γ2
where γ1 and γ2 are given by
γ1 = min
γ2 = min
γQ ,γ |h1 |2 P γQ ,γ |h2 |2 P
(1)
|g1 |2
|g2 |2
(2)
with γ¯Q = Q/N0 , γ¯P = P/N0 , and N0 representing the noise variance. By its turn, when S and R are not power-limited terminals, i.e., when they have full freedom to exploit their respective powers, the transmit power constraint only depends on the interference level arriving at the PU. In this case, PS = Q/|h1 |2 , and PR = Q/|h2 |2 , so that the end-to-end instantaneous SNR at D can now be written as γdB =
γ3 γ4 γ3 + γ4
(3)
where γ3 =
γ Q |g1 |2 |h1 |2
γ4 =
γ Q |g2 |2 . |h2 |2
(4)
Herein, we assume that all channel coefficients undergo i.n.i.d. Nakagami-m fading. As a result, |g1 |2 , |g2 |2 , |h1 |2 , and |h2 |2 are Gamma distributed with fading severity parameters mg1 , mg2 , mh1 , and mh2 and channel powers Ωg1 , Ωg2 , Ωh1 , and Ωh2 , respectively. Thus, the probability density function and cumulative distribution
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function (cdf) of X, for X ∈ {|g1 |2 , |g2 |2 , |h1 |2 , |h2 |2 }, can be formulated in compact form as fX (x) =
αm m−1 x exp(−αx) Γ(m)
Applying the power series expansion of the lower incomplete Gamma function, (9) can be rewritten as I2 =
h ∞ αh1 1
×
y mg1 +mh1 +i−1 exp − αh1 +
γQ
1 Υ m g1 ,
Fγ1 (γ) = Pr min
γQ ,γ |h1 |2 P
γ γ = Pr |g1 |2 ≤ , Q ≥ γP γ P |h1 |2
I1
+ Pr
I2
γ I1 = Pr |h1 |2 ≤ Q γP
= F|h1 |2
γQ γP
γ Pr |g1 |2 ≤ γP
F|g1 |2
γ γP
.
(7)
∞ I2 =
f|h1 |2 (y)
γQ
Γ(mh1 )Γ(mg1 + i + 1)
αg1 γ γQ
αh1 +
γQ γP
Fγ1 (γ) =
1 Υ m g1 ,
αg1 γ γP
Γ(mg1 )
Pout ≥ 1− ⎣1−
mg1 +mh1 +i
(11)
αgi αhmh1 γ i 1 1
Γ(mh1 )i!γ iQ
i=0
Γ mh1 + i,
1 Υ m g1 ,
αg1 γ γP
γQ γP
× ⎣1 −
αh1 +
αg1 γ γQ
αg1 γ γQ
mh1 +i
(12)
Γ mh1 + i,
γQ γP
αh1 +
2 Υ m g2 ,
αg2 γ γP
γQ γP
αh2 +
αgi αhmh1 γ i 1 1 i=0
αh1 +
αg1 γ γQ
αg1 γ γQ
⎤ ⎥ ⎦
mg2 −1
− ¯2 +
αgi αhmh2 γ i 2 2 i=0
αh2 +
αg2 γ γQ
Γ(mh1 )i!γ iQ
mh1 +i
Γ(mg2 )
Γ mh2 + i,
mg1 −1
− ¯1 +
Γ(mg1 )
⎡
×
f|g1 |2 (x) dx dy
αg1 γ γQ
mg1 −1
+ ¯1 −
⎡
×
y
αh1 +
γP ))/Γ(mh1 ). The cdf of γ2 can where ¯1 = Γ(mh1 , (αh1 γ¯Q /¯ be directly derived from the cdf of γ1 after substituting the respective parameters by their counterparts (i.e., mh1 → mh2 , mg1 → mg2 , Ωg1 → Ωg2 , Ωh1 → Ωh2 ). Finally, knowing that Pout ≥ 1 − (1 − Fγ1 (γth ))(1 − Fγ2 (γth )), the OP can be lower bounded by
(8)
γP ))/Γ(mh1 ). Assuming integer valwhere 1 = Υ(mh1 , (αh1 γ¯Q /¯ ues for the fading parameters m, with the help of [13, Eq. (8.352.2)] and following the same steps previously described, (11) can be rewritten as
where F|h1 |2 (·) and F|g1 |2 (·) are obtained from (6) by making the appropriate substitutions. By its turn, to evaluate I2 in (7), we invoke the concepts of probability theory [15] so that γ γQ
+
αh1 +
Due to the independence between g1 and h1 , the first term of (7), i.e., I1 , can be rewritten as
×
×
y dy. (10)
mh ∞ αh1 1 (αg1 γ/γ Q )mg1 +i
Γ mg1 + mh1 + i,
γ |g1 |2 γ ≤ , Q ≤ γP |h1 |2 γ Q |h1 |2
i=0
|g1 |2 ≤ γ
αg1 γ γP
Γ(mg1 )
A. Combined Power Constraints: Interference at the PU and Maximum Transmit Power at the SUS
From [13, Eq. (3.351.2)], the integral in (10) can be easily solved. Using this result and substituting (8) into (7), the cdf of γ1 can be derived as2
1) Tight Lower Bound Expression for OP: The OP is defined as the probability that the instantaneous SNR at D is below a certain threshold γth , i.e., Pout = Pr(γdA ≤ γth ). In [14], it has been proven that γdA in (1) can be tightly upper bounded by γup = min(γ1 , γ2 ). Therefore, it follows that the OP can be lower bounded by Pout ≥ Fγup (γth ) = 1 − (1 − Fγ1 (γth ))(1 − Fγ2 (γth )). In addition, it is easy to see that the cdfs of γ1 and γ2 are similar, so that one cdf can be easily attained from the other. Then, focusing on the derivation of the cdf of γ1 , we have
αg1 γ γQ
γP
Fγ1 (γ) =
III. P ERFORMANCE A NALYSIS AND D ISCUSSIONS
mg1 +i
∞
(6)
respectively, where Γ(·) and Γ(·, ·) denote the Gamma function [13, Eq. (8.310.1)] and the upper incomplete Gamma function [13, Eq. (8.350.2)], respectively; Υ(·, ·) indicates the lower incomplete Gamma function [13, Eq. (8.350.1)]; α ∈ {αg1 = mg1 /Ωg1 , αg2 = mg2 /Ωg2 , αh1 = mh1 /Ωh1 , αh2 = mh2 /Ωh2 }; and m ∈ {mg1 , mg2 , mh1 , mh2 }.
αg1 γ γQ
Γ(mh1 )Γ(mg1 + i + 1)
i=0
Γ(m, αx) 1 Υ(m, αx) = 1 − FX (x) = Γ(m) Γ(m)
m
(5)
αg2 γ γQ
mh2 +i
Γ(mh2 )i!γ iQ
⎤
⎥ ⎦.
(13)
0
γP
∞ f|h1 |2 (y)F|g1 |2
= γQ γP
γy γQ
dy.
(9)
2 Although (11) is expressed in terms of an infinite sum, it converges very fast to the exact result. For example, only five terms are required in the summation to achieve a high accuracy of 10−15 for a representative scenario with the following parameters: γ ¯ = 30 dB, γth = 3 dB, {mg1 = mg2 = 3.5, mh1 = 2, mh2 = 3}, {Ωg1 = Ωg2 = 1, and Ωh1 = Ωh2 = 7.3}.
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2) Asymptotic Analysis for OP: To provide insights about how the fading parameters affect the network performance, we now derive the OP in the high-SNR regime, from which the diversity and coding gains γP , are obtained.3 Without loss of generality, we assume that γ¯Q = μ¯ where μ is a positive constant, and define the average SNR as γ¯ = γ¯P . Utilizing the series representation of the incomplete Gamma function [13, Eq. (8.354.1)], we have that Υ(m, αx) = (αx)m
∞ (−1)n (αx)n n=0
(αx)m . = m
x→0
n!(m + n)
Fγ1 (γ) =
(15)
αg1 μαh1
mg1 mg1 γ γ¯
. (16)
Similarly, an asymptotic expression for the cdf of Fγ2 (γ) can be attained as γ ¯ →∞
Fγ2 (γ) =
F|g1 |2
γ y γQ
f|h1 |2 (y) dy.
(20)
Fγ3 (γ) =
k=0
mh
∞
αh1 1 Γ(mg1 +mh1 +k)
αg1 γ γQ
αg1 γ γQ
Γ(mh1 )Γ(mg1 +k+1) αh1 +
mg1+k
mg1+mh1 +k . (21)
mg
Υ(mh1 , μαh1 )αg1 1 Γ(mg1 + 1)Γ(mh1 )
Γ(mg1 + mh1 , μαh1 ) Γ(mg1 + 1)Γ(mh1 )
+
∞ Fγ3 (γ) = 0
Then, by plugging (15) into (8) and (9), an asymptotic expression for Fγ1 (γ) can be derived as γ ¯ →∞
1) Tight Lower Bound and Exact Closed-Form Expression for OP: From (3), it can be seen that the cdfs of γ3 and γ4 are required for the OP analysis. Once again, these cdfs are very similar, and herein, we focus on the derivation of the cdf of γ3 = γ¯Q |g1 |2 /|h1 |2 . Then, we have
γQ )y) in the power series, (20) can be By expanding F|g1 |2 ((γ/¯ solved as
(αx)m . Γ(m + 1)
x→0
B. Single Power Constraint: Interference at the PU
(14)
By substituting (14) into (6), we get FX (x) =
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Again, as claimed in footnote 2, (21) converges very fast, and therefore, few terms are required to achieve high accuracy. However, assuming integer values for the fading parameters m, by utilizing [13, Eq. (8.352.4)], we can express (21) as
Fγ3 (γ) = 1 −
k=0
mg
Υ(mh2 , μαh2 )αg2 2 Γ(mg2 + 1)Γ(mh2 )
Γ(mg2 + mh2 , μαh2 ) + Γ(mg2 + 1)Γ(mh2 )
αg2 μαh2
mg2 mg2 γ γ¯
. (17)
Then, an asymptotic expression for the OP can be achieved using the fact that Pout = Fγ1 (γth ) + Fγ2 (γth ) − Fγ1 (γth )Fγ2 (γth ) and γ ¯ →∞
knowing that Fγ1 (γth )Fγ2 (γth ) Fγ1 (γth ) + Fγ2 (γth ), where this latter can be easily attested from (16) and (17). Thus, after some rearrangements and omitting the small terms, it follows that γ ¯ →∞
Pout = βA where βA is given by βA =
γth γ¯
β1 , β1 + β2 , β2 ,
min(mg1 ,mg2 )
⎡
(19)
and β1 and β2 are expressed as mg
Υ (mh1 , μαh1 ) αg1 1 β1 = Γ(mg1 + 1)Γ(mh1 ) mg1 Γ(mg1 + mh1 , μαh1 ) αg1 + Γ(mg1 + 1)Γ(mh1 ) μαh1 mg
Υ(mh2 , μαh2 )αg2 2 β2 = Γ(mg2 + 1)Γ(mh2 ) mg2 Γ(mg2 + mh2 , μαh2 ) αg2 + Γ(mg2 + 1)Γ(mh2 ) μαh2 respectively. 3 It is important to note that, in this section, the asymptotic analysis for OP relaxes the assumption of integer m. In other words, herein, the diversity and coding gains hold for any value of the fading parameters.
k!Γ(mh1 ) αh1 +
mh
mg1 −1
⎢ Pout ≥ 1 − ⎣
αh1 1 Γ(mh1 + k)
k!Γ(mh1 ) αh1 +
k=0
⎡
αg1 γ γQ
αg1 γ γQ
αg1 γth γQ
αg1 γth γQ
mh
⎢ ×⎣ k=0
if mg1 < mg2 if mg1 = mg2 if mg1 > mg2
αh1 1 Γ(mh1 + k)
k
mh1 +k .
(22)
As before, the cdf of γ4 can be directly derived from the cdf of γ3 after substituting the respective parameters by their counterparts (i.e., mh1 → mh2 , mg1 → mg2 , Ωg1 → Ωg2 , and Ωh1 → Ωh2 ). Thus, making use of these results, the OP can be lower bounded by
mg1 −1
(18)
mh
mg1 −1
αh2 2 Γ(mh2 + k)
k!Γ(mh2 ) αh2 +
⎤
k
⎥
mh1 +k ⎦
αg2 γth γQ
αg2 γth γQ
k
⎤
⎥
mh2 +k ⎦ . (23)
In addition to (23), an exact closed-form expression for the OP can be derived by rewriting the SNR γdB given in (3) as γdB =
γ Q |g1 |2 |g2 |2 . |g1 |2 |h2 |2 + |g2 |2 |h1 |2
(24)
From (24) and invoking the concepts of probability theorem [15], the OP can be expressed4 as
Pout = E|h2 |2
⎧ ⎪ ⎨ ∞ ⎪ ⎩
F|g2 |2
γth |h2 |2 γQ
+ E|h1 |2 ,|h2 |2
f|g2 |2 |g2 |2 F|g1 |2
γth |h2 |2
⎛
×⎝
γth |g2 |2 +
γth |h2 |2 γQ
γ Q |g2 |2
|h1 |2
⎞ ⎠ d|g2 |2
⎫ ⎬ ⎭
(25)
4 Due to the space limitation, the detailed derivations are mostly omitted, and only the key steps are shown.
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where E{·} denotes expectation. By applying the binomial theorem [13, Eq. (1.111)] and with the help of [13, Eq. (3.471.9)], [13, Eq. (6.643.3)], and [13, Eq. (7.621.3)], an exact closed-form expression can be derived as mg1 −1
g2 u m
Pout = 1 −
u=0
where βB is given by βB =
mh
Γ(mg1 + mh1 ) Γ(mg1 + 1)Γ(mh1 )
β4 =
Γ(mg2 + mh2 ) Γ(mg2 + 1)Γ(mh2 )
mh
u!Γ(mg2)Γ(mh1)Γ(mh2)Γ(p2 + p3 )
u v
×
m g2 − 1 w
αg2 γth + αh2 γQ
αg1 γth + αh1 γQ
−p3
−p1
p 5
γth γQ
Pout =
(26)
×
+
γth γQ
mg2 +k
+ k)
αg2 γth + αh2 γQ
1
k=0 =0
Γ(q6 )Γ(mg2 )
mh
×
−q3 −k
q q1 −mh1 mg2 mh1 ∞ αg2 αh q αg1
l
αh2 2 Γ(q1 )Γ(q2 )Γ(q3 )Γ(q4 )
Γ(mh1 )Γ(mh2 )Γ(q2 + q3 )
×
αg2 γth + αh2 γQ
−q3
γth γQ
αg1 γth + αh1 γQ
−q1
γ ¯ →∞
γ ¯ →∞
αg1 αh1
mg1 mg1 γ γ¯
. (28)
The preceding equation can also be used for approximating the cdf of γ4 . For such, we just need to perform the appropriate substitutions. Then, using the same approach as adopted in the previous section, the OP can be asymptotically approximated by γ ¯ →∞
Pout = βB
γth γ¯
βA βB
min(m 1 ,m g1
g2 )
.
(31)
− log Pout (¯ γ , r) . log γ¯
(32)
Now, plugging γth = (1 + γ¯ )2r − 1 into (18) and (29), and replacing the respective resulting expressions into (32), we get (27)
Γ(mg1 + mh1 ) Γ(mg1 + 1)Γ(mh1 )
mg2
2) Diversity–Multiplexing Tradeoff: Making use of the asymptotic expressions previously derived for the OP, the diversity–multiplexing tradeoff will now be evaluated. As well known, the outage threshold γth can be written in terms of the spectral efficiency R (in bits per second per hertz) as γth = 22R − 1. In addition, from the definition in [1], the spectral efficiency R can be expressed in the form of the normalized spectral efficiency r with respect to the channel capacity as R = r log2 (1 + γ¯ ). Consequently, we have that γth = (1 + γ¯ )2r − 1 so that the diversity–multiplexing tradeoff can be formulated as [1] Δ
where q = mg1 +mg2 +k−1, q1 = mg1 +mg2 +mh1 +k−, q2 = mg1 + mh1 + k, q3 = mg2 + mh2 , q4 = mh2 + , q5 = mg1 + 2mg2 + k − , and q6 = mg1 + k + 1. 2) Asymptotic OP Analysis: For high values of γ¯ , Fγ3 · can be approximated by Fγ3 (γ) =
d(r) = lim
q5
× 2 F1 (q3 , q1 , q2 + q3 ; Ξ)
αg2 αh2
mg1
1) SNR Gap in Coding Gain: As can be explicitly observed from (18) and (29), the diversity order of the secondary network does not depend on the fading parameters pertaining to the interference links, i.e., mh1 and mh2 . Indeed, the diversity order is limited by the more severe fading channel between the two secondary hops (data links), regardless of the transmit power strategy employed. However, the same is not true when it comes to coding gains, in which an SNR gap (in decibels) between the two strategies is observed. This coding gap is defined by the ratio G = 10 log10
Γ(mh2 )Γ(mg2 + k + 1)
k=0
αg1 αh1
C. Comparison Between the Two Transmit Power Strategies
where p1 = mh1 + u + w − v + 1, p2 = mh1 + u, p3 = mg2 + mh2 , p4 = p3 + v − w − 1, p5 = mg2 + u + w − v + 1, Ξ = ((αg2 αh1 + γQ )+αh1 αh2 )/(αh1 +(αg1 γth /¯ γQ ))(αh2 +(αg2 γth / αg1 αh2 )(γth /¯ γ¯Q )), and 2 F1 (·, ·, ·; ·) is the Gauss hypergeometric function [13, Eq. (9.111)]. For the case of arbitrary fading parameter, the OP can be given by mg +k mh αg2 2 αh2 2 Γ(q3
(30)
respectively.
× 2 F1 (p3 , p1 , p2 + p3 ; Ξ)
∞
if mg1 < mg2 if mg1 = mg2 if mg1 > mg2
β3 =
w−v+1 αgu+ αg2 2 αh1 1 αh2 2 Γ(p1)Γ(p2)Γ(p3)Γ(p4) 1
×
β3 , β3 + β4 , β4 ,
and β3 and β4 are given by
v=0 w=0 mg
min(mg1 ,mg2 ) (29)
log [(1 + γ¯ )2r − 1] − 1 . (33) d(r) = − min(mg1 , mg2 ) lim γ ¯ →∞ log γ¯ Finally, by applying the l’Hospital rule to (33), the diversity– multiplexing tradeoff can be expressed as d(r) = min(mg1 , mg2 )(1 − 2r).
(34)
From (34), we conclude the following: 1) The maximum diversity order of min(mg1 , mg2 ) is achieved as r → 0, and 2) the maximum normalized spectral efficiency of 1/2 is achieved as d → 0 due to the half-duplex communication modes. IV. N UMERICAL R ESULTS AND S IMULATIONS In this section, representative numerical results are provided to validate the proposed analysis and illustrate the effects of the channel parameters, e.g., mg1 , mg2 , mh1 , mh2 and Ωg1 , Ωg2 , Ωh1 , Ωh2 , on
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Fig. 2. OP for spectrum-sharing AF relay networks with combined power constraints at the PU and SU transmitters over Nakagami-m fading.
Fig. 3. OP for spectrum-sharing AF relay networks with the power constraint at the PU only over Nakagami-m fading.
the network performance. Without loss of generality, the outage threshold γth is set to 3 dB, and the channel mean powers of the secondary network are given by Ωg1 = Ωg2 = 1. A good agreement between analysis (both exact and lower bound) and simulations is observed. In addition, it can be seen that the asymptotic curves tightly converge to the simulations in the high-SNR regime, which corroborates our analysis. For the plots, we define Network A as the CRN that employs the combined power constraints (see Section III-A) and Network B as the CRN that employs the single power constraint (see Section III-B). Figs. 2 and 3 show the OP versus the average SNR γ¯ for Networks A and B, respectively. The parameters of the primary network, i.e., those pertaining to the interference links, are fixed to mh1 = 2, mh2 = 3, and Ωh1 = Ωh2 = 7.3. With the aim to highlight the impact of the fading parameters mg1 and mg2 on the diversity gain, three different representative examples are considered: 1) Network A: {mg1 = 2, mg2 = 3}, {mg1 = 4, mg2 = 3}, and {mg1 = 4, mg2 = 4}.
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Fig. 4. Performance comparison for different transmission strategies with combined power constraints at the PU and SU transmitters over Nakagami-m fading.
Fig. 5. Performance comparison for different transmission strategies with the power constraint at the PU only over Nakagami-m fading.
2) Network B: {mg1 = 2.5, mg2 = 3.1}, {mg1 = 3.4, mg2 = 1.8}, and {mg1 = 3.5, mg2 = 3.5}. For Network A, the “Analysis” curves represent the lower bound of the OP, given in (13), and for Network B, the “Analysis” curves represent the exact OP, given in (27). To avoid entanglement, the lower bound of the OP for Network B is not shown in Fig. 3. However, tests were performed by the authors and reveal that the lower bound for Network B is indeed very tight to the exact result. As the diversity order is governed by the minimum of the fading severity parameters between the two hops, the higher the value of min(mg1 , mg2 ), the better the outage performance. Figs. 4 and 5 evaluate the impact of the primary network on the outage performance of Networks A and B, respectively. By keeping mg1 and mg2 fixed, three different schemes are examined for each kind of network. More specifically, for Network A, the parameters are selected as Scheme 1: mh1 = mh2 = 2 and Ωh1 = Ωh2 = 7.3,
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of the interference on the primary network and maximum transmission power at the secondary network and 2) single power constraint of the interference on the primary network. Our conclusions are at least threefold: First, the diversity order is strictly defined by the minimum fading severity between the two hops of the secondary network. Second, the secondary network achieves the full diversity order of min(mg1 , mg2 ), regardless of the transmit power constraint. Third, the diversity–multiplexing tradeoff is independent of the primary network. R EFERENCES
Fig. 6. SNR gap of the two power constraints versus the PU’s position over Nakagami-m fading.
Scheme 2: mh1 = mh2 = 3 and Ωh1 = Ωh2 = 3.3, and Scheme 3: mh1 = mh2 = 4 and Ωh1 = Ωh2 = 1. For Network B, the parameters are chosen as Scheme 4: mh1 = mh2 = 2.2 and Ωh1 = Ωh2 = 7.3, Scheme 5: mh1 = mh2 = 3.3 and Ωh1 = Ωh2 = 3.3, and Scheme 6: mh1 = mh2 = 4.4 and Ωh1 = Ωh2 = 1. It can be seen that the curves are not affected by varying the fading parameters of the primary network. Indeed, only a coding gain is observed once the curves remain parallel to each other. This validates our proposed analysis. To illustrate the effect of the two transmit power strategies on the network performance, Fig. 6 plots the SNR gain of Network B over Network A, given in (31), for three different scenarios with distinct fading severity parameters. A 2-D topology is assumed for all nodes, in which the coordinate of each terminal is defined as (x, y). Thus, the channel mean power for the fading channel from node M with coordinate (xM , yM ) to node N with coordinate (xN , yN ) −η is $ denoted as , where η is the path-loss exponent and = (xM − xN )2 + (yM − yN )2 is the distance between M and N. In this figure, S, R, and D are located at the coordinates (0,0), (1/2, 0), and (1, 0), respectively, and the curves are plotted in terms of the position of the PU, with coordinate (d, d), and by setting the average SNR γ¯ to 30 dB. Note that the gain is remarkable when the PU is located close to the secondary network. This phenomenon is explained as follows: When the PU is near S and R, i.e., h1 and h2 are high, the interference at the PU is high. In this case, the transmit powers of Networks A and B are strictly governed by the constraint Q/|h1 |2 , which leads to a similar performance for both networks. By contrast, when PU moves away from S and R, i.e., h1 and h2 are low, the interference on the PU is low. In this case, the power constraint on the PU can be neglected, and as such, Network B outperforms Network A since the latter is still controlled by the transmit power constraint of the secondary network. V. C ONCLUSION Assuming i.n.i.d. Nakagami-m fading, we have derived closed-form lower bounds and asymptotic expressions for the OP of cooperative spectrum-sharing networks with AF relaying. In doing so, two transmit power constraints have been proposed: 1) combined power constraint
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