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Effective capacity of cognitive radio systems in asymmetric fading channels
The Journal of China Universities of Posts and Telecommunications June 2015, 22(3): 18–25 www.sciencedirect.com/science/journal/10058885
http://jcupt.xsw.bupt.cn
Effective capacity of cognitive radio systems in asymmetric fading channels Shi Xiangqun1 (
), Chu Qingxin2
1. School of Electronic and Information Engineering, University of Electronic Science and Technology of China, Zhongshan Institute, Zhongshan 528402, China 2. School of Electronic and Information Engineering, South China University of Technology, Guangzhou 510641, China
Abstract As known that the effective capacity theory offers a methodology for exploring the performance limits in delay constrained wireless networks, this article considered a spectrum sharing cognitive radio (CR) system in which CR users may access the spectrum allocated to primary users (PUs). Particularly, the channel between the CR transmitter (CR-T) and the primary receiver and the channel between the CR-T and the CR receiver (CR-R) may undergo different fading types and arbitrary link power gains. This is referred to as asymmetric fading. The authors investigated the capacity gains achievable under a given delay quality-of-service (QoS) constraint in asymmetric fading channels. The closed-form expression for the effective capacity under an average received interference power constraint is obtained. The main results indicate that the effective capacity is sensitive to the fading types and link power gains. The fading parameters of the interference channel play a vital role in effective capacity for the looser delay constraints. However, the fading parameters of the CR channel play a decisive role in effective capacity for the more stringent delay constraints. Also, the impact of multiple PUs on the capacity gains under delay constraints has also been explored. Keywords
effective capacity, delay quality-of-service constraint, asymmetric fading channels, cognitive radio systems
1 Introduction In future wireless communication system, the higher data rates, efficient uses of spectrum and the scarcities are still major design challenges. CR is a promising technique to improve the utilization of radio spectrum [1]. There are different spectrum sharing approaches for CRs, e.g., interweave, overlay, and underlay. The underlay approach allows for spectrum sharing between PUs and CR users, under the constraint that the interference caused by the CR users does not degrade the performance of the PUs. This underlay approach is adopted in this article. The information theory provides ultimate performance limits and identifies most efficient use of resource. Some representative work on CR systems from an information-theoretic perspective with the goal of Received date: 29-07-2014 Corresponding author: Shi Xiangqun, E-mail: [email protected] DOI: 10.1016/S1005-8885(15)60648-3
identifying the fundamental capacity limits shows available in Refs. [2–5]. Ref. [2] investigated the capacity of fading channels with the average interference power (AIP) constraint at primary receiver, while the fading conditions for interference channel (CR-T to PU receiver (PU-R)) and the CR channel (CR-T to CR-R) are the same, and showed significant capacity gains may be achieved if the channels are varying due to fading. Similar results were also extended to ergodic, delay-limited and outage capacities of CR networks under combinations of AIP/peak interference power (PIP) constraints in Ref. [3]. Ref. [4] explored the impact of asymmetric fading (e.g., the interference channel and the CR channel experience different fading conditions) on the capacity gains under the same constraint with Ref. [2]. In Ref. [5], the impact of the channel knowledge on capacity was examined. However, information-theoretic studies generally aim to maximize ergodic capacity without taking into account delay constraints. In recent years, there has been a dramatic
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Shi Xiangqun, et al. / Effective capacity of cognitive radio systems in asymmetric fading channels
increase for the delay-sensitive wireless services such as voice over Internet and video streaming delivery etc. These services are required to satisfy a delay QoS. Hence, it is essential to characterize the performance of CR systems in the presence of delay limitations. An effective capacity model proposed by Wu and Negi has been used as a delay sensitive performance metric for analysis of the properties of time varying wireless network [6]. The application and analysis of effective capacity in various wireless systems has attracted much interest [7–9]. In Refs. [7–8], the maximum capacity gains of the CR channel with delay constraint was investigated and the effective capacity under the AIP constraint in Nakagami-m fading channels was derived. In Ref. [9], the work of Ref. [7] was extended to the low power regime of Nakagami-m fading channels. Motivated by the results of Ref. [4], we extend the work of Ref. [7] and consider fading conditions of the interference channel and the CR channel. In practice, these two paths could experience different fading types and different link power gains, due to various factors such as different path length or shadowing. We obtain an arrival rate of the CR channel subject to delay constraints and analyze the capacity performances with diverse statistical delay provisioning under different fading environments. The main contributions of this article are summarized as follows. 1) The closed-form expressions for the effective capacity under AIP constraints are obtained for asymmetric fading environments. It is shown that the effective capacity is more sensitive to the fading types of the interference channel for the looser delay constraints. The effective capacity is more sensitive to the fading types of the CR channel for the stringent delay constraints. 2) The effective capacity depends on the average power ratio of the CR channel to the interference channel. When power ratio increases, the effective capacity increases accordingly. 3) The impact of multiple PU-Rs in symmetric Rayleigh fading was studied. It is shown that the effective capacity reduces significantly as the number of primary receivers gets larger for the looser delay constraints, while the influences decrease gradually as the delay constraints become the more stringent. The rest of the article was organized as follows. The system and channel model is described in Sect. 2. The effective capacity and an efficient power allocation policy
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for asymmetric fading conditions are investigated in Sect. 3. Extensions of these results to multiple PUs are presented in Sect. 4, followed by a conclusion in Sect. 5.
2 System model The system model was borrowed from Ref. [2]. Consider an underlay cognitive spectrum-sharing system in which the CR user shares spectrum with a PU as depicted in Fig. 1. Single PU assumption is just for the sake of clarity of final results, and the extensions to multiple PUs are investigated at the next section. The asymmetric fading scenarios were considered. The interference channel between the CR-T and PU-R, and the CR channel between the CR-T and CR-R have different fading parameters, including fading types ( Rayleigh and Nakagami-m) and link power gains. Although Rayleigh fading can be obtained as a special case of Nakagami-m (m=1), we explicitly left the Rayleigh as a separate entry since it is the most popular communication scenario. All channels are assumed to be ergodic and independent block-fading channels. Let gss and gsp denote the instantaneous channel power gains of the interference path and the CR path, respectively. All channel envelopes are modeled as Nakagami-m fading with fading parameter mss and msp, respectively. The probability density function (PDF) of channel power gains was expressed as m
f ( x) =
1 m mx x m −1 exp − ; x≥0 Γ ( m) Ω Ω
(1)
where, Γ(m) is gamma function, and m denotes the fading parameter of Nakagami-m distribution, Ω is average power gain.
Fig. 1
System model for spectrum sharing
Assume that in the CR network, the data-link layer packets are organized into frames with the time duration, Tf. These frames are divided into bit streams to be transmitted through the channel. Let us denote the systems total spectral bandwidth by B. The additive noises at CR-R and PU-R are all assumed to be independent additive white
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Gaussian noise (AWGN), which have zero mean and power spectral density by N0. The perfect channel state information (CSI) on gss and gsp are assumed to be available at CR-T. We further assume that the transmission of CR user must satisfy statistical QoS constraints. As stated earlier, the effective capacity was proposed as a link layer channel model supporting statistical delay QoS guarantees. Let the sequence {R(i), i=1,2,…} is a discrete-time stationary and ergodic stochastic service process and denotes the t
allocated rate to user. S (t ) = ∑ R(i ) is the partial sum of i =1
the service process. The effective capacity is formulated as follows [6] t Ec (θ ) = − lim 1 log E exp −θ ∑ R(i ) (2) t →∞ θ t i =1 When the sequence {R(i), i=1,2,…} is an uncorrelated process. Ec(θ) satisfies Ec (θ ) = − 1 log{E[e −θ R ( i ) ]} (3)
θ
Note that the effective capacity describes the maximum arrival rate that can be supported under guaranteed delay QoS requirement. The parameter θ, a certain positive constant called QoS exponent, signifies the exponential decay rate of the QoS violation probabilities. A more stringent QoS requirement corresponds to a larger QoS exponent θ. On the other hand, a looser QoS requirement corresponds to a smaller QoS exponent θ. E[⋅] is the expectation operator.
3 Effective capacity under single PU 3.1
Effective capacity
There are two different networks in CR systems referred as power-interference limited (PIL) networks and interference limited (IL) networks. We focus on the IL networks. So, the effective capacities derived serve as upper-bounds for the capacity under interference power constraints. It is assumed that the cognitive user protects the PU transmission via transmit power control by applying the so-called interference temperature constraint at PU-R in form of either the AIP constraint or the PIP constraint over all different fading environments. From the perspective of protecting the PU, the PIP constraint is more restrictive, on the contrary, from the CR’s perspective, AIP constraints is more favorable [10]. In this section, we will
2015
use AIP constraint. Thereafter, the extension of its results to the PIP constraint is straightforward. Then, we consider the following optimal power allocation problem that maximizes the effective capacity of the CR user under an AIP constraint at the PU-R, given by 1 max Ec (θ ) = − log{E[e −θ R (i ) ]} θ (4) s.t. E[ gsp P( g ss , g sp , θ )]≤I th P( gss , gsp , θ )≥0; ∀( gss , gsp ,θ ) where Ith denotes the AIP constraint that the PU can tolerate at its receiver. The instantaneous service rate R(i) of the frame i can be express as gss P( gss , gsp , θ ) R(i ) = Tf B log 1 + (5) N0 B where P(gss, gsp, θ) denotes the allocated power for CR-T. In Eq. (5), we assume that the effect of the PU transmitter (PU-T) on the CR-R is neglected. This assumption is valid when either the CR-R is outside the PU’s transmission range or the CR-R can decode the primary’s data, particularly when the PU signal is strong, and cancel the resulting interference. The solution for Eq. (4) can be obtained by the Lagrangian method, and the optimal power allocation P(gss, gsp, θ) given by [7] 1 gss β +1 N 0 B ; g ≥γ g ss sp − 1 g P( gss , g sp ,θ ) = γ gsp (6) ss 0; gss < γ gsp where β = θTf B as the normalized QoS exponent. γ =λN0B /β as the nonnegative variable can be numerically obtained by substituting Eq. (6) into the constraint E[gssP(gss, gsp, θ)] = Ith, i.e. 1 g gss β +1 − 1 sp f ( gss ) f ( gsp )dg ss dg sp = α (7) ∫∫ gss gss ≥γ gsp γ g sp where α = I th / ( N 0 B ) , is the signal-to-interference plus noise ratio (SINR) at the primary receiver. f ( gss ) and
f ( gsp ) denote the PDF of gss and gsp. Two special cases were analyzed for power allocation policy. As the θ → 0, P(gss,gsp,θ) reduces to that in Ref. [2], i.e. 1 1 γ g − g N 0 B; gss ≥γ gsp lim P( gss , g sp ,θ ) == sp (8) ss θ →0 0; g < γ g ss sp The power control policy in Eq. (8) is in the form of
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Shi Xiangqun, et al. / Effective capacity of cognitive radio systems in asymmetric fading channels
water-filling [2], in which, the system can tolerate an arbitrarily long delay. However, the water level in Eq. (8) is in general time-varying and is inversely proportional to gsp, that is, more transmission power is used when either gss increases or gsp decreases. On the other hand, as θ → ∞ , P(gss,gsp,θ) will converge to [11]
lim P( gss , g sp , θ ) =
θ →∞
σ
(9)
gss
In this case, σ can be found by substituting Eq. (9) into the constraint E[gssP(gss, gsp, θ)] = Ith, i.e. mss − 1 I th ; mss ≥1 (10) σ = mss τ 0; otherwise where, we define τ =Ωss/Ωsp, in which, Ωss and Ωsp are average channel power gains of the CR channel and interference channel respectively. Eq. (9) shows that as θ → ∞ , the power allocation policy becomes the policy of zero-outage channel inversion, and P(gss, gsp, θ) only relies on the fading type of CR channel, the power ratio τ and interference power constraint Ith that the PU can tolerate at its receiver, and is unrelated to msp. Give the optimal power allocation policy, we can derive the expression for the effective capacity Ec(θ) as follows 1 Ec (θ ) = − log ∫∫ f gss ( gss ) f gsp ( gsp )dgss dgsp + θ gss <γ gsp
f gss ( gss ) f gsp ( gsp )dgss dgsp = β 1 ∞ 1 γ β +1 − log (11 ) + γ f ( g )d g ( g ) f ( g )dg 1 ∫ ∫ 0 θ γ where g and f(g) denote the random variable gsp/gss and its PDF, respectively. With gsp and gss being distributed according to Eq. (1) with their m parameter being msp and mss, Ω being Ωss and Ωsp, respectively, gsp/gss may be shown to have the following probability distribution [2]
gss γ g ∫∫ sp gss ≥γ gsp
mss Ω f gsp ( x) = ss msp gss Ωsp
−
2
F1 (a, b; c; z ) =
x
msp −1
; x≥0 (12) m +m mss ss sp Ω Β (msp , mss ) x + mss sp Ω sp By using the change of variable y = 1–γx, and the Gauss hypergeometric function 2F1(a,b;c,z) [12], which can be expressed in terms of the following integration
1 Γ (c ) t b −1 (1 − t ) − b + c −1 (1 − tz ) − a dt ; ∫ 0 Γ (b )Γ ( c − b ) (13) Re c > Re b > 0, z < 1
We can derive the closed-form expression of Ec (θ ) as shown in Eq. (14) (see Appendix A). It indicates in Eq. (14) that Ec (θ ) is not only associated with the fading types and power ratio of the CR path and the interference path, but also related to the delay constraint of the CR user and the interference constraint at the PU-R. m (γ mss )mss (τ msp ) sp 1 ⋅ Ec (θ ) = − log 1 + mss + msp θ Β (msp , mss )(γ mss + τ msp )
τ msp 2β + 1 2 F1 mss + msp ,1; msp + β + 1 ; γ m + τ m ss sp β msp + β +1 τ msp 2 F1 mss + msp ,1; msp + 1; γ mss + τ msp msp
−
(14)
We also obtain a solution for γ by evaluating the integration in Eq. (7) as follows (see Appendix A)
γm
ss
α=
−1
(mss ) mss (τ msp )
msp
Β (msp , mss )(γ mss + τ msp )
mss + msp
⋅
τ msp 2β + 1 2 F1 mss + msp ,1; msp + β + 1 ; γ m + τ m ss sp β msp + β +1 τ msp 2 F1 mss + msp ,1; msp + 2; γ mss + τ msp msp + 1
β β +1
mss
21
3.2
−
(15)
Simulation results and discussion
In this subsection, we illustrate and discuss the simulation results for the effective capacity of the cognitive channel under AIP constraints for different scenarios. For example, the CR channel may experience Rayleigh fadings whereas the interference channel may undergo Nakagami-m fadings with various fading parameters and different link power gains. We assume TfB= 1. The channel models are presented in Table 1. Table 1 Scenario 1 2 3
Cognitive Channel and Interference channel model Channel model Cognitive channel model Interference channel model Rayleigh Nakagami-m Nakagami-m Rayleigh Nakagami-m Nakagami-m
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2015
Fig. 2 to Fig. 4 shows the normalized effective capacity versus θ under AIP constraints for different fading conditions. In these figures, we set α = 0 dB, τ = 0 dB. It is observed that the effective capacity decreases with increasing θ. This indicates that effective capacity degrades as a result of a more stringent delay requirement. This reveals an important fact that there exists a fundamental tradeoff between effective capacity and delay QoS provisioning.
Fig. 3
Normalized effective capacity vs. θ for scenario 2
In Fig. 4, we plot the normalized effective capacity versus θ in scenario 3, which the interference path and the CR path experiences symmetric fading conditions with mss = msp =1 or 2 or 5.
Fig. 2
Normalized effective capacity vs. θ for scenario 1
In Fig. 2, we plot the normalized effective capacity versus θ in scenario 1. The CR path experiences Rayleigh fading and the interference path experiences Nakagami-m fading with msp = 1 or 2 or 5. As θ is smaller, the effective capacity when msp = 1 is higher than that when msp = 5. This indicates that fading of the interference channel is beneficial to the effective capacity for the looser QoS guarantees. With the increase of θ, corresponding to the more stringent QoS guarantees, the effective capacity is not sensitive to the fading type of the interference channel. This is because that the power allocation policies varies from the water-filling formula with a variable water level (which is relevant to msp) to zero-outage channel inversion with the increase of θ. Hence, the capacity gain of the CR channel is uncorrelated to msp gradually. This proves the correctness of the theory analysis of Eq. (10) in subsection 3.1. In Fig. 3, we plot the normalized effective capacity versus θ in scenario 2, which the interference path experiences Rayleigh fading and the CR path experiences Nakagami-m fading with mss = 1 or 2 or 5. It is observed that the effective capacity when mss = 1 is slight lower than that when mss = 5. This indicates that fading of the cognitive path is harmful to the capacity gain of the CR channel regardless of θ, however, is not significant.
Fig. 4
Normalized effective capacity vs. θ for scenario 3
It is observed that as θ is below some fixed level, the fading of symmetric channels is beneficial to the effective capacity of the CR channel. However, there exists opposite results when θ exceeds some fixed level, that is, the fading of the symmetric path is harmful to the effective capacity of the CR channel. Compared to Fig. 2 and Fig. 3, it shows that the fading parameters of interference channel have a significant effect on capacity gains of CR channel for the looser delay constraints, while the fading parameters of CR channel plays a key role to the capacity gains of CR channel for the more stringent delay constraints. Fig. 5 shows the normalized effective capacity versus α in scenario.3 when θ = 10 −2 1/nat and τ = 10 dB. It is observed that the effective capacity increases if the PU-R can tolerate more interference. This is evident that PU-T is able to transmit with higher power. The case of interest in engineering practice is for a low value of α. When α is lower, the effective capacity is sensitive to the fading type, however, for large α, the impact of fading types on the
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Shi Xiangqun, et al. / Effective capacity of cognitive radio systems in asymmetric fading channels
effective capacity is reduced gradually.
23
constraint is formulated by the following n constraints gsp,i P( gss , gsp,1 ,..., g sp, n ,θ )≤I th ,i ; i = 1, 2,..., n (16) Eq. (16) equivalently
P( gss , g sp,1 ,..., g sp, n ,θ )≤ min i
I th,i g sp,i
; i = 1, 2,..., n
For ease of exposition, we assume
(17)
I th,i = I th ,
i = 1, 2,..., n . With a slight abuse of the notation, I th denotes the PIP constraint at PU-R this time. The effective capacity is achieved when the maximum allowed power according to Eq. (17) is transmitted. Thus, the effective capacity Ec (θ ) is given by Fig. 5
Normalized effective capacity vs. α for scenario 3
In a practical environment, due to path length different or shadowing of the CR path and interference path, link power gains, Ωss and Ωsp, may be different. It can be shown from Eq. (14) the effective capacity depends on the average power ratio τ. Hence, Fig. 6 show the impact of power ratio τ on the effective capacity of the CR channel in scenario.3 when θ = 10 −2 1/nat and mss = msp = 1.
Fig. 6
Normalized effective capacity vs. α for τ = ±10 dB
We assume that the power ratios between both links are τ = 10 dB and τ = − 10 dB. It is observed that the effect on effective capacity is a simple scaling by the power ratio τ.
1 Ec (θ ) = − log ∫ (1 + α x) − β f X ( x)dx
θ
X
(18)
where X = gss / max i gsp,i . The PDF of X is given by [2]
n − 1 k f X ( x) = n∑ (−1) k (1 + x + k )2 k =0 n −1
(19)
Substituting Eq. (19) into Eq. (18) yields Ec (θ ) under PIP constraint with n PUs as n − 1 ∞ (1 + α x) − β 1 n −1 Ec (θ ) = − log n∑ (−1) k dx ( 20 ) ∫ 2 θ k 0 (1 + x + k ) k = 0 In Fig. 7, we plot the normalized effective capacity versus θ for different numbers of PUs when all the interference channels and the CR channel experience Rayleigh fading. We observe that the effective capacity degrades with increasing number of the PUs for the looser delay constraints, while this influence disappears gradually for the more stringent delay constraints. In fact, the increasing n means that the probability of having the larger channel gain from CR-T to PU-R increases. Hence, the CR-T has to send less power to meet the interference power constraints.
4 Effective capacity under multiple PUs The previous results were derived under the assumption that the CR user shares spectrum with one PU. When more PUs are present, the transmit power of the CR user would be subject to additional constraints. This leads to a capacity reduction [2]. In this section, we focus on the effect of the scaling of the PUs on the effective capacity of the CR channel. For the sake of simplicity, we consider the scenario 3 in symmetric Rayleigh fading. Let gsp,i denote the channel power gain of the CR- T to the ith PU- R, PIP
Fig. 7 Normalized effective capacity vs. θ for difference numbers of PUs, τ = 0 dB
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The Journal of China Universities of Posts and Telecommunications
As mentioned above, when the delay constraints are more stringent, the capacity gains of CR channel become less and less relevant to the interference channel, so this influence decreases gradually.
5 Conclusions
β 1 ∞ log f ( x )dx + γ (γ x) β +1 f ( x)dx = 1 ∫ ∫ 0 θ γ mss m ss Ωss m −1 ∞ x sp m dx + sp m +m ∫1 1 mss ss sp − log Ω γ sp θ Ω x + ss Β ( m sp , mss ) msp Ωsp
x = gsp gss
Ec (θ )
=
2015
−
1
The authors in this article paid focus on the impact of asymmetric fading channels on the effective capacity of the CR channel under the delay QoS constraints over spectrum sharing systems. The theory analysis and β m −1 1 x sp γ β +1 numerical evaluations indicate that the fading parameter γ dx ( ) x mss + msp ∫0 mss τy==1Ω−ssγ xΩsp msp has a significant effect on effective capacity of the CR = Ω channel for the looser delay constraints. However the x + ss msp fading parameters mss plays an important role to the Ω effective capacity for the more stringent delay constraints. sp Furthermore, the impact of multiple PUs on the capacity m mss (γ mss ) (τ msp ) sp 1 gains under delay constraints has also been investigated. It − log 1 + ⋅ mss + msp θ indicates that the effective capacity reduces as the number Β (msp , mss )(γ mss + τ msp ) m − 1 of primary receivers increases for the looser delay 1 (1 − y ) sp β +1 dy − ∫ constraints, while the influences decrease gradually for the mss + msp τ msp 0 more stringent delay constraints. From a practical point of 1 − γ m + τ m y ) ss sp view, these results provide benchmarks for performance msp −1 comparison and crucial insights into achievable capacity 1 (1 − y ) dy gains under delay constraints in next generation wireless mss + msp ∫0 (A.1) τ msp cognitive systems. 1 − γ m + τ m y ss sp Altogether, the effect of interference from the PU-T to the CR-R was ignored. However, the interference from the We set a = mss + msp , b = 1, c = 2 + msp − 1 ( β + 1) , PU-T may also limit the performance of the CR systems. z = τ msp ( γ mss + τ msp ) for the first integral in Eq. (A.1), Thus, the effect of channel parameters from PU-T to CR-R and set a = mss + msp , b = 1, c = msp − 1, z = τ msp ( γ mss + τ m on the effective capacity remains to be analyzed. The research topics will allow us to develop completelyγ m + τ msp ) for the second integral in Eq. (A.1). Using the equality Γ(m + 1) = mΓ(m) and Eq. (13), we can obtain analytical approaches to research effective capacity of CR channel. Eq. (14). In the same way, we derive expression (15) from Eq. (7) Acknowledgements as 1 gss β +1 gsp f ( g ) ⋅ α = This work was supported by the National Natural Science ss − 1 ∫∫ gss gss ≥γ gsp γ g sp Foundation of China (61171029). τ =Ω ss Ω sp x = g sp g ss
Appendix A
f ( gsp )dgss dgsp mss
Here, we derive closed-form expression (14) from Eq. (11) as
=
mss m τ 1 1 − γ sp β +1 [( ) − 1] ⋅ x γ Β(msp , mss ) ∫ 0
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Shi Xiangqun, et al. / Effective capacity of cognitive radio systems in asymmetric fading channels
x
msp
y =1− γ x
mss x+ m τ sp
γm
ss
−1
mss + msp
dx =
(mss ) mss (τ msp )
msp
Β(msp , mss )(γ mss + τ msp )
mss + msp
⋅
m − 1 (1 − y ) sp β +1 dy − ∫ mss + msp 0 τ msp 1 − γ m + τ m y r sp 1
(A.2)
dy τ msp 1 − γ m + τ m y ss sp We set a = mss + msp , b = 1, c = 2 + msp − 1 ( β + 1) , z = τ msp ( γ mss + τ msp ) for the first integral in Eq. (A.2),
∫
1
0
(1 − y )
msp
mss + msp
and set a = mss + msp , b = 1, c = msp + 2, z = τ msp
( γ mss + τ m
γ m + τ msp ) for the second integral in Eq. (A.2). Using the equality Γ(m + 1) = mΓ(m) and Eq. (13), we can obtain Eq. (15).
References 1. Haykin S. Cognitive radio: Brain-empowered wireless communications. IEEE Journal on Selected Areas in Communications, 2005, 23(2): 201−220 2. Ghasemi A, Sousa E S. Fundamental limits of spectrum-sharing in fading environment. IEEE Transactions on Wireless Communications, 2007, 6(2): 649−658
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3. Kang X, Liang Y C, Nallanathan A, et al. Optimal power allocation for fading channels in cognitive radio networks: ergodic capacity and outage capacity. IEEE Transactions on Wireless Communications, 2009, 8(2): 940−950 4. Suraweera H A, Gao J, Smith P J, et al. Channel capacity limits of cognitive radio in asymmetric fading environments. Proceedings of the International Conference on Communications (ICC’08), May 19−23, 2008, Beijing, China. Piscataway, NJ, USA: IEEE, 2008: 4048−4053 5. Rezki Z, Alouini M S. Ergodic capacity of cognitive radio under imperfect channel-state information. IEEE Transactions on Vehicular Technology, 2012, 61(5): 2108−2119 6. Wu D P, Negi R. Effective capacity: A wireless link model for support of quality of service. IEEE Transactions on Wireless Communications, 2003, 2(4): 630−643 7. Musavian L, Aissa S. Effective capacity of delay-constrained cognitive radio in Nakagami fading channels. IEEE Transactions on Wireless Communications, 2010, 2(3): 1054−1062 8. Elalem M, Zhao L. Effective capacity optimization for cognitive radio network based on underlay scheme in gamma fading channels. Proceedings of the International Conference on Computing, Networking and Communications (ICNC’13), Jan 28−31, 2013, San Diego, CA, USA. Piscataway, NJ, USA: IEEE, 2013: 714−718 9. Benkhelifa F, Rezki Z, Alouini M S. Effective capacity of Nakagami-m fading channels with full channel state information in the low power regime. Proceedings of the 24th International Symposium on Personal Indoor and Mobile Radio Communications (PIMRC’13), Sept 8−11, 2013, London, UK. Piscataway, NJ, USA: IEEE, 2013:1883−1887 10. Zhang R. On peak versus average interference power constraints for protecting primary users in cognitive radio networks. IEEE Transactions on Wireless Communications, 2009, 8(4): 2112−2120 11. Tang J, Zhang X. Quality-of-service driven power and rate adaptation over wireless links. IEEE Transactions on Wireless Communications, 2007, 6(8): 3058−3068 12. Gradshteyn I S, Ryzhik I M. Table of integrals, series, and products. Boston, MA, USA: Academic Press, 1992
(Editor: Zhang Kexin)
http://jcupt.xsw.bupt.cn
Effective capacity of cognitive radio systems in asymmetric fading channels Shi Xiangqun1 (
), Chu Qingxin2
1. School of Electronic and Information Engineering, University of Electronic Science and Technology of China, Zhongshan Institute, Zhongshan 528402, China 2. School of Electronic and Information Engineering, South China University of Technology, Guangzhou 510641, China
Abstract As known that the effective capacity theory offers a methodology for exploring the performance limits in delay constrained wireless networks, this article considered a spectrum sharing cognitive radio (CR) system in which CR users may access the spectrum allocated to primary users (PUs). Particularly, the channel between the CR transmitter (CR-T) and the primary receiver and the channel between the CR-T and the CR receiver (CR-R) may undergo different fading types and arbitrary link power gains. This is referred to as asymmetric fading. The authors investigated the capacity gains achievable under a given delay quality-of-service (QoS) constraint in asymmetric fading channels. The closed-form expression for the effective capacity under an average received interference power constraint is obtained. The main results indicate that the effective capacity is sensitive to the fading types and link power gains. The fading parameters of the interference channel play a vital role in effective capacity for the looser delay constraints. However, the fading parameters of the CR channel play a decisive role in effective capacity for the more stringent delay constraints. Also, the impact of multiple PUs on the capacity gains under delay constraints has also been explored. Keywords
effective capacity, delay quality-of-service constraint, asymmetric fading channels, cognitive radio systems
1 Introduction In future wireless communication system, the higher data rates, efficient uses of spectrum and the scarcities are still major design challenges. CR is a promising technique to improve the utilization of radio spectrum [1]. There are different spectrum sharing approaches for CRs, e.g., interweave, overlay, and underlay. The underlay approach allows for spectrum sharing between PUs and CR users, under the constraint that the interference caused by the CR users does not degrade the performance of the PUs. This underlay approach is adopted in this article. The information theory provides ultimate performance limits and identifies most efficient use of resource. Some representative work on CR systems from an information-theoretic perspective with the goal of Received date: 29-07-2014 Corresponding author: Shi Xiangqun, E-mail: [email protected] DOI: 10.1016/S1005-8885(15)60648-3
identifying the fundamental capacity limits shows available in Refs. [2–5]. Ref. [2] investigated the capacity of fading channels with the average interference power (AIP) constraint at primary receiver, while the fading conditions for interference channel (CR-T to PU receiver (PU-R)) and the CR channel (CR-T to CR-R) are the same, and showed significant capacity gains may be achieved if the channels are varying due to fading. Similar results were also extended to ergodic, delay-limited and outage capacities of CR networks under combinations of AIP/peak interference power (PIP) constraints in Ref. [3]. Ref. [4] explored the impact of asymmetric fading (e.g., the interference channel and the CR channel experience different fading conditions) on the capacity gains under the same constraint with Ref. [2]. In Ref. [5], the impact of the channel knowledge on capacity was examined. However, information-theoretic studies generally aim to maximize ergodic capacity without taking into account delay constraints. In recent years, there has been a dramatic
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Shi Xiangqun, et al. / Effective capacity of cognitive radio systems in asymmetric fading channels
increase for the delay-sensitive wireless services such as voice over Internet and video streaming delivery etc. These services are required to satisfy a delay QoS. Hence, it is essential to characterize the performance of CR systems in the presence of delay limitations. An effective capacity model proposed by Wu and Negi has been used as a delay sensitive performance metric for analysis of the properties of time varying wireless network [6]. The application and analysis of effective capacity in various wireless systems has attracted much interest [7–9]. In Refs. [7–8], the maximum capacity gains of the CR channel with delay constraint was investigated and the effective capacity under the AIP constraint in Nakagami-m fading channels was derived. In Ref. [9], the work of Ref. [7] was extended to the low power regime of Nakagami-m fading channels. Motivated by the results of Ref. [4], we extend the work of Ref. [7] and consider fading conditions of the interference channel and the CR channel. In practice, these two paths could experience different fading types and different link power gains, due to various factors such as different path length or shadowing. We obtain an arrival rate of the CR channel subject to delay constraints and analyze the capacity performances with diverse statistical delay provisioning under different fading environments. The main contributions of this article are summarized as follows. 1) The closed-form expressions for the effective capacity under AIP constraints are obtained for asymmetric fading environments. It is shown that the effective capacity is more sensitive to the fading types of the interference channel for the looser delay constraints. The effective capacity is more sensitive to the fading types of the CR channel for the stringent delay constraints. 2) The effective capacity depends on the average power ratio of the CR channel to the interference channel. When power ratio increases, the effective capacity increases accordingly. 3) The impact of multiple PU-Rs in symmetric Rayleigh fading was studied. It is shown that the effective capacity reduces significantly as the number of primary receivers gets larger for the looser delay constraints, while the influences decrease gradually as the delay constraints become the more stringent. The rest of the article was organized as follows. The system and channel model is described in Sect. 2. The effective capacity and an efficient power allocation policy
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for asymmetric fading conditions are investigated in Sect. 3. Extensions of these results to multiple PUs are presented in Sect. 4, followed by a conclusion in Sect. 5.
2 System model The system model was borrowed from Ref. [2]. Consider an underlay cognitive spectrum-sharing system in which the CR user shares spectrum with a PU as depicted in Fig. 1. Single PU assumption is just for the sake of clarity of final results, and the extensions to multiple PUs are investigated at the next section. The asymmetric fading scenarios were considered. The interference channel between the CR-T and PU-R, and the CR channel between the CR-T and CR-R have different fading parameters, including fading types ( Rayleigh and Nakagami-m) and link power gains. Although Rayleigh fading can be obtained as a special case of Nakagami-m (m=1), we explicitly left the Rayleigh as a separate entry since it is the most popular communication scenario. All channels are assumed to be ergodic and independent block-fading channels. Let gss and gsp denote the instantaneous channel power gains of the interference path and the CR path, respectively. All channel envelopes are modeled as Nakagami-m fading with fading parameter mss and msp, respectively. The probability density function (PDF) of channel power gains was expressed as m
f ( x) =
1 m mx x m −1 exp − ; x≥0 Γ ( m) Ω Ω
(1)
where, Γ(m) is gamma function, and m denotes the fading parameter of Nakagami-m distribution, Ω is average power gain.
Fig. 1
System model for spectrum sharing
Assume that in the CR network, the data-link layer packets are organized into frames with the time duration, Tf. These frames are divided into bit streams to be transmitted through the channel. Let us denote the systems total spectral bandwidth by B. The additive noises at CR-R and PU-R are all assumed to be independent additive white
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Gaussian noise (AWGN), which have zero mean and power spectral density by N0. The perfect channel state information (CSI) on gss and gsp are assumed to be available at CR-T. We further assume that the transmission of CR user must satisfy statistical QoS constraints. As stated earlier, the effective capacity was proposed as a link layer channel model supporting statistical delay QoS guarantees. Let the sequence {R(i), i=1,2,…} is a discrete-time stationary and ergodic stochastic service process and denotes the t
allocated rate to user. S (t ) = ∑ R(i ) is the partial sum of i =1
the service process. The effective capacity is formulated as follows [6] t Ec (θ ) = − lim 1 log E exp −θ ∑ R(i ) (2) t →∞ θ t i =1 When the sequence {R(i), i=1,2,…} is an uncorrelated process. Ec(θ) satisfies Ec (θ ) = − 1 log{E[e −θ R ( i ) ]} (3)
θ
Note that the effective capacity describes the maximum arrival rate that can be supported under guaranteed delay QoS requirement. The parameter θ, a certain positive constant called QoS exponent, signifies the exponential decay rate of the QoS violation probabilities. A more stringent QoS requirement corresponds to a larger QoS exponent θ. On the other hand, a looser QoS requirement corresponds to a smaller QoS exponent θ. E[⋅] is the expectation operator.
3 Effective capacity under single PU 3.1
Effective capacity
There are two different networks in CR systems referred as power-interference limited (PIL) networks and interference limited (IL) networks. We focus on the IL networks. So, the effective capacities derived serve as upper-bounds for the capacity under interference power constraints. It is assumed that the cognitive user protects the PU transmission via transmit power control by applying the so-called interference temperature constraint at PU-R in form of either the AIP constraint or the PIP constraint over all different fading environments. From the perspective of protecting the PU, the PIP constraint is more restrictive, on the contrary, from the CR’s perspective, AIP constraints is more favorable [10]. In this section, we will
2015
use AIP constraint. Thereafter, the extension of its results to the PIP constraint is straightforward. Then, we consider the following optimal power allocation problem that maximizes the effective capacity of the CR user under an AIP constraint at the PU-R, given by 1 max Ec (θ ) = − log{E[e −θ R (i ) ]} θ (4) s.t. E[ gsp P( g ss , g sp , θ )]≤I th P( gss , gsp , θ )≥0; ∀( gss , gsp ,θ ) where Ith denotes the AIP constraint that the PU can tolerate at its receiver. The instantaneous service rate R(i) of the frame i can be express as gss P( gss , gsp , θ ) R(i ) = Tf B log 1 + (5) N0 B where P(gss, gsp, θ) denotes the allocated power for CR-T. In Eq. (5), we assume that the effect of the PU transmitter (PU-T) on the CR-R is neglected. This assumption is valid when either the CR-R is outside the PU’s transmission range or the CR-R can decode the primary’s data, particularly when the PU signal is strong, and cancel the resulting interference. The solution for Eq. (4) can be obtained by the Lagrangian method, and the optimal power allocation P(gss, gsp, θ) given by [7] 1 gss β +1 N 0 B ; g ≥γ g ss sp − 1 g P( gss , g sp ,θ ) = γ gsp (6) ss 0; gss < γ gsp where β = θTf B as the normalized QoS exponent. γ =λN0B /β as the nonnegative variable can be numerically obtained by substituting Eq. (6) into the constraint E[gssP(gss, gsp, θ)] = Ith, i.e. 1 g gss β +1 − 1 sp f ( gss ) f ( gsp )dg ss dg sp = α (7) ∫∫ gss gss ≥γ gsp γ g sp where α = I th / ( N 0 B ) , is the signal-to-interference plus noise ratio (SINR) at the primary receiver. f ( gss ) and
f ( gsp ) denote the PDF of gss and gsp. Two special cases were analyzed for power allocation policy. As the θ → 0, P(gss,gsp,θ) reduces to that in Ref. [2], i.e. 1 1 γ g − g N 0 B; gss ≥γ gsp lim P( gss , g sp ,θ ) == sp (8) ss θ →0 0; g < γ g ss sp The power control policy in Eq. (8) is in the form of
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Shi Xiangqun, et al. / Effective capacity of cognitive radio systems in asymmetric fading channels
water-filling [2], in which, the system can tolerate an arbitrarily long delay. However, the water level in Eq. (8) is in general time-varying and is inversely proportional to gsp, that is, more transmission power is used when either gss increases or gsp decreases. On the other hand, as θ → ∞ , P(gss,gsp,θ) will converge to [11]
lim P( gss , g sp , θ ) =
θ →∞
σ
(9)
gss
In this case, σ can be found by substituting Eq. (9) into the constraint E[gssP(gss, gsp, θ)] = Ith, i.e. mss − 1 I th ; mss ≥1 (10) σ = mss τ 0; otherwise where, we define τ =Ωss/Ωsp, in which, Ωss and Ωsp are average channel power gains of the CR channel and interference channel respectively. Eq. (9) shows that as θ → ∞ , the power allocation policy becomes the policy of zero-outage channel inversion, and P(gss, gsp, θ) only relies on the fading type of CR channel, the power ratio τ and interference power constraint Ith that the PU can tolerate at its receiver, and is unrelated to msp. Give the optimal power allocation policy, we can derive the expression for the effective capacity Ec(θ) as follows 1 Ec (θ ) = − log ∫∫ f gss ( gss ) f gsp ( gsp )dgss dgsp + θ gss <γ gsp
f gss ( gss ) f gsp ( gsp )dgss dgsp = β 1 ∞ 1 γ β +1 − log (11 ) + γ f ( g )d g ( g ) f ( g )dg 1 ∫ ∫ 0 θ γ where g and f(g) denote the random variable gsp/gss and its PDF, respectively. With gsp and gss being distributed according to Eq. (1) with their m parameter being msp and mss, Ω being Ωss and Ωsp, respectively, gsp/gss may be shown to have the following probability distribution [2]
gss γ g ∫∫ sp gss ≥γ gsp
mss Ω f gsp ( x) = ss msp gss Ωsp
−
2
F1 (a, b; c; z ) =
x
msp −1
; x≥0 (12) m +m mss ss sp Ω Β (msp , mss ) x + mss sp Ω sp By using the change of variable y = 1–γx, and the Gauss hypergeometric function 2F1(a,b;c,z) [12], which can be expressed in terms of the following integration
1 Γ (c ) t b −1 (1 − t ) − b + c −1 (1 − tz ) − a dt ; ∫ 0 Γ (b )Γ ( c − b ) (13) Re c > Re b > 0, z < 1
We can derive the closed-form expression of Ec (θ ) as shown in Eq. (14) (see Appendix A). It indicates in Eq. (14) that Ec (θ ) is not only associated with the fading types and power ratio of the CR path and the interference path, but also related to the delay constraint of the CR user and the interference constraint at the PU-R. m (γ mss )mss (τ msp ) sp 1 ⋅ Ec (θ ) = − log 1 + mss + msp θ Β (msp , mss )(γ mss + τ msp )
τ msp 2β + 1 2 F1 mss + msp ,1; msp + β + 1 ; γ m + τ m ss sp β msp + β +1 τ msp 2 F1 mss + msp ,1; msp + 1; γ mss + τ msp msp
−
(14)
We also obtain a solution for γ by evaluating the integration in Eq. (7) as follows (see Appendix A)
γm
ss
α=
−1
(mss ) mss (τ msp )
msp
Β (msp , mss )(γ mss + τ msp )
mss + msp
⋅
τ msp 2β + 1 2 F1 mss + msp ,1; msp + β + 1 ; γ m + τ m ss sp β msp + β +1 τ msp 2 F1 mss + msp ,1; msp + 2; γ mss + τ msp msp + 1
β β +1
mss
21
3.2
−
(15)
Simulation results and discussion
In this subsection, we illustrate and discuss the simulation results for the effective capacity of the cognitive channel under AIP constraints for different scenarios. For example, the CR channel may experience Rayleigh fadings whereas the interference channel may undergo Nakagami-m fadings with various fading parameters and different link power gains. We assume TfB= 1. The channel models are presented in Table 1. Table 1 Scenario 1 2 3
Cognitive Channel and Interference channel model Channel model Cognitive channel model Interference channel model Rayleigh Nakagami-m Nakagami-m Rayleigh Nakagami-m Nakagami-m
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2015
Fig. 2 to Fig. 4 shows the normalized effective capacity versus θ under AIP constraints for different fading conditions. In these figures, we set α = 0 dB, τ = 0 dB. It is observed that the effective capacity decreases with increasing θ. This indicates that effective capacity degrades as a result of a more stringent delay requirement. This reveals an important fact that there exists a fundamental tradeoff between effective capacity and delay QoS provisioning.
Fig. 3
Normalized effective capacity vs. θ for scenario 2
In Fig. 4, we plot the normalized effective capacity versus θ in scenario 3, which the interference path and the CR path experiences symmetric fading conditions with mss = msp =1 or 2 or 5.
Fig. 2
Normalized effective capacity vs. θ for scenario 1
In Fig. 2, we plot the normalized effective capacity versus θ in scenario 1. The CR path experiences Rayleigh fading and the interference path experiences Nakagami-m fading with msp = 1 or 2 or 5. As θ is smaller, the effective capacity when msp = 1 is higher than that when msp = 5. This indicates that fading of the interference channel is beneficial to the effective capacity for the looser QoS guarantees. With the increase of θ, corresponding to the more stringent QoS guarantees, the effective capacity is not sensitive to the fading type of the interference channel. This is because that the power allocation policies varies from the water-filling formula with a variable water level (which is relevant to msp) to zero-outage channel inversion with the increase of θ. Hence, the capacity gain of the CR channel is uncorrelated to msp gradually. This proves the correctness of the theory analysis of Eq. (10) in subsection 3.1. In Fig. 3, we plot the normalized effective capacity versus θ in scenario 2, which the interference path experiences Rayleigh fading and the CR path experiences Nakagami-m fading with mss = 1 or 2 or 5. It is observed that the effective capacity when mss = 1 is slight lower than that when mss = 5. This indicates that fading of the cognitive path is harmful to the capacity gain of the CR channel regardless of θ, however, is not significant.
Fig. 4
Normalized effective capacity vs. θ for scenario 3
It is observed that as θ is below some fixed level, the fading of symmetric channels is beneficial to the effective capacity of the CR channel. However, there exists opposite results when θ exceeds some fixed level, that is, the fading of the symmetric path is harmful to the effective capacity of the CR channel. Compared to Fig. 2 and Fig. 3, it shows that the fading parameters of interference channel have a significant effect on capacity gains of CR channel for the looser delay constraints, while the fading parameters of CR channel plays a key role to the capacity gains of CR channel for the more stringent delay constraints. Fig. 5 shows the normalized effective capacity versus α in scenario.3 when θ = 10 −2 1/nat and τ = 10 dB. It is observed that the effective capacity increases if the PU-R can tolerate more interference. This is evident that PU-T is able to transmit with higher power. The case of interest in engineering practice is for a low value of α. When α is lower, the effective capacity is sensitive to the fading type, however, for large α, the impact of fading types on the
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Shi Xiangqun, et al. / Effective capacity of cognitive radio systems in asymmetric fading channels
effective capacity is reduced gradually.
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constraint is formulated by the following n constraints gsp,i P( gss , gsp,1 ,..., g sp, n ,θ )≤I th ,i ; i = 1, 2,..., n (16) Eq. (16) equivalently
P( gss , g sp,1 ,..., g sp, n ,θ )≤ min i
I th,i g sp,i
; i = 1, 2,..., n
For ease of exposition, we assume
(17)
I th,i = I th ,
i = 1, 2,..., n . With a slight abuse of the notation, I th denotes the PIP constraint at PU-R this time. The effective capacity is achieved when the maximum allowed power according to Eq. (17) is transmitted. Thus, the effective capacity Ec (θ ) is given by Fig. 5
Normalized effective capacity vs. α for scenario 3
In a practical environment, due to path length different or shadowing of the CR path and interference path, link power gains, Ωss and Ωsp, may be different. It can be shown from Eq. (14) the effective capacity depends on the average power ratio τ. Hence, Fig. 6 show the impact of power ratio τ on the effective capacity of the CR channel in scenario.3 when θ = 10 −2 1/nat and mss = msp = 1.
Fig. 6
Normalized effective capacity vs. α for τ = ±10 dB
We assume that the power ratios between both links are τ = 10 dB and τ = − 10 dB. It is observed that the effect on effective capacity is a simple scaling by the power ratio τ.
1 Ec (θ ) = − log ∫ (1 + α x) − β f X ( x)dx
θ
X
(18)
where X = gss / max i gsp,i . The PDF of X is given by [2]
n − 1 k f X ( x) = n∑ (−1) k (1 + x + k )2 k =0 n −1
(19)
Substituting Eq. (19) into Eq. (18) yields Ec (θ ) under PIP constraint with n PUs as n − 1 ∞ (1 + α x) − β 1 n −1 Ec (θ ) = − log n∑ (−1) k dx ( 20 ) ∫ 2 θ k 0 (1 + x + k ) k = 0 In Fig. 7, we plot the normalized effective capacity versus θ for different numbers of PUs when all the interference channels and the CR channel experience Rayleigh fading. We observe that the effective capacity degrades with increasing number of the PUs for the looser delay constraints, while this influence disappears gradually for the more stringent delay constraints. In fact, the increasing n means that the probability of having the larger channel gain from CR-T to PU-R increases. Hence, the CR-T has to send less power to meet the interference power constraints.
4 Effective capacity under multiple PUs The previous results were derived under the assumption that the CR user shares spectrum with one PU. When more PUs are present, the transmit power of the CR user would be subject to additional constraints. This leads to a capacity reduction [2]. In this section, we focus on the effect of the scaling of the PUs on the effective capacity of the CR channel. For the sake of simplicity, we consider the scenario 3 in symmetric Rayleigh fading. Let gsp,i denote the channel power gain of the CR- T to the ith PU- R, PIP
Fig. 7 Normalized effective capacity vs. θ for difference numbers of PUs, τ = 0 dB
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As mentioned above, when the delay constraints are more stringent, the capacity gains of CR channel become less and less relevant to the interference channel, so this influence decreases gradually.
5 Conclusions
β 1 ∞ log f ( x )dx + γ (γ x) β +1 f ( x)dx = 1 ∫ ∫ 0 θ γ mss m ss Ωss m −1 ∞ x sp m dx + sp m +m ∫1 1 mss ss sp − log Ω γ sp θ Ω x + ss Β ( m sp , mss ) msp Ωsp
x = gsp gss
Ec (θ )
=
2015
−
1
The authors in this article paid focus on the impact of asymmetric fading channels on the effective capacity of the CR channel under the delay QoS constraints over spectrum sharing systems. The theory analysis and β m −1 1 x sp γ β +1 numerical evaluations indicate that the fading parameter γ dx ( ) x mss + msp ∫0 mss τy==1Ω−ssγ xΩsp msp has a significant effect on effective capacity of the CR = Ω channel for the looser delay constraints. However the x + ss msp fading parameters mss plays an important role to the Ω effective capacity for the more stringent delay constraints. sp Furthermore, the impact of multiple PUs on the capacity m mss (γ mss ) (τ msp ) sp 1 gains under delay constraints has also been investigated. It − log 1 + ⋅ mss + msp θ indicates that the effective capacity reduces as the number Β (msp , mss )(γ mss + τ msp ) m − 1 of primary receivers increases for the looser delay 1 (1 − y ) sp β +1 dy − ∫ constraints, while the influences decrease gradually for the mss + msp τ msp 0 more stringent delay constraints. From a practical point of 1 − γ m + τ m y ) ss sp view, these results provide benchmarks for performance msp −1 comparison and crucial insights into achievable capacity 1 (1 − y ) dy gains under delay constraints in next generation wireless mss + msp ∫0 (A.1) τ msp cognitive systems. 1 − γ m + τ m y ss sp Altogether, the effect of interference from the PU-T to the CR-R was ignored. However, the interference from the We set a = mss + msp , b = 1, c = 2 + msp − 1 ( β + 1) , PU-T may also limit the performance of the CR systems. z = τ msp ( γ mss + τ msp ) for the first integral in Eq. (A.1), Thus, the effect of channel parameters from PU-T to CR-R and set a = mss + msp , b = 1, c = msp − 1, z = τ msp ( γ mss + τ m on the effective capacity remains to be analyzed. The research topics will allow us to develop completelyγ m + τ msp ) for the second integral in Eq. (A.1). Using the equality Γ(m + 1) = mΓ(m) and Eq. (13), we can obtain analytical approaches to research effective capacity of CR channel. Eq. (14). In the same way, we derive expression (15) from Eq. (7) Acknowledgements as 1 gss β +1 gsp f ( g ) ⋅ α = This work was supported by the National Natural Science ss − 1 ∫∫ gss gss ≥γ gsp γ g sp Foundation of China (61171029). τ =Ω ss Ω sp x = g sp g ss
Appendix A
f ( gsp )dgss dgsp mss
Here, we derive closed-form expression (14) from Eq. (11) as
=
mss m τ 1 1 − γ sp β +1 [( ) − 1] ⋅ x γ Β(msp , mss ) ∫ 0
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Shi Xiangqun, et al. / Effective capacity of cognitive radio systems in asymmetric fading channels
x
msp
y =1− γ x
mss x+ m τ sp
γm
ss
−1
mss + msp
dx =
(mss ) mss (τ msp )
msp
Β(msp , mss )(γ mss + τ msp )
mss + msp
⋅
m − 1 (1 − y ) sp β +1 dy − ∫ mss + msp 0 τ msp 1 − γ m + τ m y r sp 1
(A.2)
dy τ msp 1 − γ m + τ m y ss sp We set a = mss + msp , b = 1, c = 2 + msp − 1 ( β + 1) , z = τ msp ( γ mss + τ msp ) for the first integral in Eq. (A.2),
∫
1
0
(1 − y )
msp
mss + msp
and set a = mss + msp , b = 1, c = msp + 2, z = τ msp
( γ mss + τ m
γ m + τ msp ) for the second integral in Eq. (A.2). Using the equality Γ(m + 1) = mΓ(m) and Eq. (13), we can obtain Eq. (15).
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