Performance Enhancement of Cognitive Relay Networks using Regression Functions Kamel Tourki
Mazen O. Hasna
Mathematical and Algorithmic Sciences Lab, France Research Center Huawei Technologies Co. Ltd, France Email:
[email protected]
College of Engineering Qatar University, Qatar Email:
[email protected]
Abstract—In a spectrum sharing relay-aided secondary network, the design objective shifts from minimizing the violation of the primary system constraints when using the outdated channel state information (CSI), to making the best use of the available (imperfect) channel gain while maintaining the primary system Quality of Service (QoS). To address this trend, we propose a new power allocation based on a linear regression model and we carry out the outage analysis as well as the asymptotical analysis where we deduce the diversity order of the secondary network. The performance results show that the proposed power allocation strategy outperforms the existing schemes in the literature and provides an accurate tight approximation of the ideal case under perfect CSI while respecting the primary system QoS and using outdated CSI.
I.
I NTRODUCTION
Cognitive radio (CR) emerged as a promising technique to overcome the spectrum scarcity for wireless applications. Associated to relaying techniques, this mixture inherits promising gains [1]–[6]. Involving a single relay, the impact of the primary network (PN) interference on the secondary network (SN) was investigated in [3], [4]. In particular, the authors in [4] considered the maximum transmit power and several primary transmitters (PTs) and receivers (PRs). This work has been extended to multihop (more than two hops) scheme over Nakagami-m fading channels [5]. On the other hand, the authors in [6] considered incremental opportunistic relaying scheme for the SN and proposed power allocation schemes depending on the amount of the channel state information (CSI) at the secondary transmitter (ST). However, all the aforementioned works assumed perfect CSI for all involved links. Recently, by neglecting the direct link and the transmit power limitation at the ST, the authors in [7] and [8] carried out the outage probability of the SN when the CSI of the ST-PR and the selected relay-PR interference channel links are assumed to be imperfect. In [9], the authors introduced the direct link to the aforementioned relay selection scheme. Being constrained by the maximum transmit power, the secondary transmit power allocation is based on the instantaneous outdated CSI, where a power margin has been introduced to limit the violation of the interference constraint set by the PN [10]. However, besides harming the PRs’ interference constraint even at low probability, the SN performance seems to be severely affected by using imperfect CSI. The design objective requests to overcome this performance degradation while maintaining the PN QoS where statistically speaking, the focus is on the relationship between a dependent random variable (RV) (actual interference channel gains) and another independent variable
(outdated interference channel gain). Furthermore, to model a fading process, linear models1 have been widely used in the literature [11]. Henceforth, we aim to propose a linear model which accurately predicts the channel gains’ evolution based on the knowledge of the outdated CSI to the extent possible. In this paper, we propose a new power allocation strategy based on linear regression model, where to the best knowledge of the authors, none of the appeared works in the literature have investigated. Considering the same system architecture in [9], we derive the outage performance as well as the diversity order of the SN assuming outdated CSI and considering the maximum transmit power constraint at each secondary transmitter (ST or the selected relay) as well as the peak interference power at PRs. We show that the SN performance could be recovered while using outdated CSI and respecting the PN QoS. We confirm the results via simulations, and show that our proposed scheme results corroborate the perfect CSIbased scheme, and keeps a least implementation complexity. II.
S YSTEM MODEL
Consider a SN consisting of a transmitter (ST), a receiver (SR), and a cluster of K potential relays, coexisting with a PN consisting of L PTs and L PRs, where each node is equipped with a single antenna and each relay operates in half-duplex decode-and-forward (DF) mode. We neglect the effect of the PTs on the SRs, and we assume that the L PRs are close to each other and form a cluster2 . We denote hij the coefficients of the channels between a transmitter i and a receiver j, modeled as flat fading and Rayleigh distributed with variances λij 3 . Thereby, we assume that λsr = λsk , λrs = λks and λs(r)p = λs(k)l for all k, l. Being a dual hopbased transmission, the ST first broadcasts its information to the SR and the relays where the successful decoding relays form a decoding set referred to as C. Then, a single relay r is selected according to r = arg maxk∈C |hks |2 to assist the ST transmission. Both received replicas at the SR are combined using maximum ratio combining technique. Under spectrum sharing constraints, the secondary node (ST and 1 The advantages of a linear model is twofold: (1)The models which depend linearly on their unknown parameters are easier to implement than models which are non-linearly related to their parameters, (2)The statistical properties of the resulting estimators are easier to determine. 2 These assumptions have been widely considered in the literature for the sake of the presentation clarity 3 i ∈ {s, k, r} and j ∈ {s, k, r, l}. When i = s (j = s) it refers to the ST (SR) and l stands for the lth PR. However, k and r stand for the kth relay and the selected relay, respectively.
978-1-4799-8091-8/15/$31.00 ©2015 IEEE
r) transmission is constrained by the interference threshold I at the PRs as well as the maximum transmit power Pm . Accordingly, the secondary transmit power should follow IN0 , P Ps(r) = min (1) m . maxl=1,...,L (|hs(r)l |2 ) However, such power allocation could be inefficient since the channel coefficients change due to several factors such as the mobility and feedback delay. Therefore, the outdated CSI (denoted as hij ) used for the power allocation in (1) may differ from the actual values referred to as gij , giving gs(r)l = ρs(r) hs(r)l + 1 − ρ2s(r) ws(r) , gsr(s) = ρ1 hsr(s) + 1 − ρ21 w1 , and grs = ρ2 hrs + 1 − ρ22 w2 , where ws(r) and w1(2) are circularly symmetric complex Gaussian random variables (RV) having the same variance as RVs hs(r)l and hss(rs) , respectively; ρs(r) 4 (ρ1(2) ) is the correlation coefficient between gs(r)l and hs(r)l (gsj(rs) and hsj(rs) , j ∈ {s, r}). We can determine the probability density function (PDF) of |gij |2 where i ∈ {s, r} and j ∈ {r, s, l} as (2) f|gij |2 (y) = E|hij |2 f|gij |2 ||hij |2 , where f|gij |2 ||hij |2 is the conditional PDF of |gij |2 conditioned on |hij |2 given by [14] −
y+ρ2 x 2
e (1−ρ )λij I0 p|gij |2 ||hij |2 (y|x) = (1 − ρ2 )λij
2
ρ2 yx (1 − ρ2 )2 λ2ij
probability of occurrence ε, which we shall henceforth refer to as CSI-based scheme, such that I 2 |gip | > I ≤ ε, i = s, r. (7) P r ηi |hip |2 However, besides harming the PR’ interference constraint even at low probability, the SN performance seems to be severely affected (as it will be shown in figures 1 and 3). Our main focus is how to find the best approximation of the (actual) interference channel gains without degrading the SN performance nor the PN QoS when imperfect CSI is used. To this end, we propose the use of regression functions of ˜ 1 = |hsp1 |2 and Z1 |gsp1 |2 and Z2 |grp2 |2 based on Z 2 ˜ Z2 = |hrp2 | , respectively, and the new power allocation, referred to as LR-based scheme5 , is summarized in the following lemma: Lemma 1: By approximating Z1 and Z2 using their regres˜ 2 , respectively, the ST and ˜ 1 and Z sion functions based on Z r transmit powers are given by IN0 P˜s = min , P (8) m , ρ2s |hsp1 |2 + λsp (1 − ρ2s ) and P˜r = min
,
˜1 Let p1 and p2 be the indices of the PRs that satisfy Z 2 2 2 ˜ |hsp1 | = maxl (|hsl | ) and Z2 |hrp2 | = maxl (|hrl |2 ), respectively. Ideally, the secondary power allocation would be defined using gsp1 and grp2 , and given by
IN0 Ps(r) = min (4) , Pm , |gs(r)p1(2) |2
1(2)
(y) =
L
L−1
λs(r)p
l=0
L−1 (−1)l − Day e s(r) l Ds(r)
III.
1(2)
|2
(y) = L
L−1
l=0
(5)
L−1 (−1)l − ay 1 − e Ds(r) , (6) l l+1
respectively. Considering a single PR (p1 = p2 = p), recent works [9], [10] proposed to jointly use the outdated channel gains |hsp |2 and |hrp |2 and insert a power margin ηs(r) to limit the violation of the interference constraint I to a small loss of generalities, ρs = J0 (2πfd,sp Td ) where J0 (.) denotes the zeroth order Bessel function of the rst kind [12, Eq. 8.411], fd,sp is the maximum Doppler frequency on the ST-PR link, and Td is the time difference between the actual channel value and its estimate [13]. 4 Without
P ERFORMANCE A NALYSIS
In this section, we derive the outage probability of the scheme under consideration. To this end, we derive our analysis using (4) that we aim to compare with the performance results using the LR-based power allocation scheme in lemma 1. Thus, the outage probability is given by Pout
= P r[γss < Φ, |C| = 0] + P1
K
K i
i=1
where a = (l + 1)/λs(r)p , Ds(r) = ρ2s(r) + a(1 − ρ2s(r) ), and F|gs(r)p
(9)
We can note that the ST (relay) will use |hsp1 |2 (|hrp2 |2 ), ρs(r) , and λs(r)p to compute its transmit power and did not need additional feedback. Therefore, we keep the system design complexity to the minimum6 .
where the PDF and the cumulative distribution function (CDF) of |gs(r)p1(2) |2 are given by |2
IN0 , Pm , ρ2r |hrp2 |2 + λrp (1 − ρ2r )
respectively.
(3) where I0 (.) stands for the zero-order modified Bessel function of the first kind and ρ = ρs,r,1,2 depending on the (i, j) transceivers.
f|gs(r)p
P r[γss + γrs < Φ, |C| = i],
(10)
P2i
where γss = Ps |gss |2 /N0 and γrs = Pr |grs |2 /N0 are the consecutively received and combined SNRs at the SR, and Φ = 22R − 1 where R is the transmission rate. Using (2) and (3), we can easily prove that f|gss |2 (y) = e−y/λss /λss and f|gsr |2 (y) = e−y/λsr /λsr . Thus, we can write P1 as K Φ Φ I 1 − e− λsr γ¯ F|gsp |2 P1 = 1 − e− λss γ¯ + γ¯ ∞ K Φz Φz f|gsp |2 (z)dz, (11) 1 − e− λss I 1 − e− λsr I I γ ¯
5 LR stands for linear regression since the regression function is a linear function of |hsp1 |2 (|hrp2 |2 ). 6 At least, we keep the same system design complexity as [9], [10], since there we need ρs(r) and the probability of harming the interference power constraint, denoted as ε, to compute the power margin ηs(r) .
⎛ jΦ ⎞ L−1 (l+1)I (l+1)I jΦ Φ K L−1 − λsr γ¯ + λsp γ¯ Ds − λss j+l K γ ¯ + λsr γ ¯ + λsp γ ¯ Ds
L K (−1) Φ Φ I e e j l ⎝ ⎠, 1 − e− λsr γ¯ P1 = 1 − e− λss γ¯ F|gsp |2 − + jΦ jΦ l+1 Φ l+1 γ¯ λ D + + + sp s λ I λ D λ I λ I λ D j=0 l=0
sr
sp
s
ss
sr
sp
s
(12)
where by using (5) and (6), and after some manipulations, P1 is given by (12), in the top of the next page. On the other hand, P2i can be written as Φ P r[γss < Φ − y, |C| = i|y] pγrs (y)dy, (13) P2i = 0 T2
where pγrs (.) is the PDF of γrs given by the derivative of (14). It follows that by splitting the integration in (13) and after applying some algebraic manipulations, P2i can be given by (15) P2i = T21 + T22 , where T21 is given by T21
I 1−e = e F|gsp1 |2 γ¯ Φ (16) × Fγrs (Φ) − e− λss γ¯ I21 , − λiΦ ¯ sr γ
Φ − λsr γ ¯
K−i
where Fγrs (.) and I21 are given by (14) and (17), respectively, where 1−e−Φ(B1 −b1 ) , B1 = b1 , B1 −b1 F1 (b1 , B1 ) = (18) Φ, B1 = b1 , where B1 = j/(λrs γ¯ D2 ) and b1 = 1/(λss γ¯ ), F2 (., ., ., .) is given by θ1 −e θ2 (B1 −b1 ) θ1 F2 (b1 , B1 , θ1 , θ2 ) = Ei −(B1 − b1 ) θ2 θ2 θ1 +Ei −(B1 − b1 ) Φ + , (19) θ2 where θ1 = (l + 1)/(λrp Dr ) and θ2 = j/(Iλrs D2 ), and F3 (., ., ., .) is given by F3 (b1 , B1 , θ1 , θ2 ) = −
1 e−Φ(B1 −b1 ) − θ1 θ2 θ22 Φ + θθ12
B1 − b1 F2 (b1 , B1 , θ1 , θ2 ), θ2
(20)
A(B1 −b1 )
(23)
where A = (Φ/(λss I)) + (((i + s)Φ)/(λsr I)) + ((l + 1)/(λsp Ds )) and C = 1/(λss I), and G2 (b1 , A, C, B1 , θ1 , θ2 ) = +
C G1 (b1 , A, C, B1 ) Cθ1 + Aθ2
θ2 F2 (b1 , B1 , θ1 , θ2 ), Cθ1 + Aθ2
G3 (b1 , A, C, B1 , θ1 , θ2 ) = + +
(24)
θ2 F3 (b1 , B1 , θ1 , θ2 ) Cθ1 + Aθ2
Cθ2 F2 (b1 , B1 , θ1 , θ2 ) (Cθ1 + Aθ2 )2 C2 G1 (b1 , A, C, B1 ). (Cθ1 + Aθ2 )2
(25)
Hereafter, we provide the asymptotical analysis to get better insights on the performance results. However, the diversity gain (D) of the system under consideration would always be zero when using the traditional definition. Thus, we derive the generalized diversity gain which has been introduced in log(P f loor ) f loor [1], such that D = limλsp /λss →0 log(λspout = /λss ) , where Pout limγ¯ →∞ Pout . Lemma 2: The outage probability in high-SNR (¯ γ → ∞) and weak interference channels (λsp /λss(r) → 0 and λrp /λrs → 0) can be approximated by (26). As a consequence, we that when ρ2 = 1, we can note i i j−1 which is equal (−1) can isolate the sum term j=1 j ∞ zero [12, Eq. 0.154.2]. Hence, Pout is reduced to its first term in (26) and the diversity order is obviously equal to K + 1. ∞ On the other hand, ρ2 < 1 andi = K, Pout becomes when λsp λrp λsp λrp proportional to λss λrs + o λss λrs . Thus, we summarize this result in the following lemma. Lemma 3: The generalized diversity order of the scheme under consideration is given by K + 1, if ρ2 = 1, D= (27) 2, if ρ2 < 1. IV.
and T22 is given by (21) where I22 is given by (22), where e− C × G1 (b1 , A, C, B1 ) = − C Φ A(B1 − b1 ) − (B1 − b1 )x Ei , C 0
and finally,
P ERFORMANCE R ESULTS
In this section, we verify our performance analysis in terms of closed-form expression of the outage probability derived in section III through comparisons with simulation results. We assume that the ST is located at the origin [0,0], the relays at [1/2,0] and the SR at [1,0]. Unless mentioned otherwise, PRs are located at [1,1]. Thus, λij of the link i → j is defined as d−4 ij where dij is the Euclidean distance between i and j nodes. We evaluate the outage probability expression of our scheme for a transmission rate R (= 1 bps/Hz), I = 5 (dB), and a number (K) of active relays in the secondary system. We will qualify the QoS of the SN as maintained if Pout ≤ 0.05. To get an equal footing comparison with CSI-based scheme, we set L = 1 and we illustrate our proposed scheme result and compare it with [9, Eq. (4)] where K = 2, 4 and ε = 0.05, 0.01 under balanced channel imperfectness (ρs = ρr = ρ2 = 0.9). Fig. 1 evaluates how our proposed scheme outperforms the CSI-based scheme. We can note that using the CSI-based power allocation scheme, the secondary system performance
(l+1)I jy i L−1 i L−1
L ij lL−1 (−1)j+l−1 − (l+1)I
L ij lL−1 (−1)j+l−1 e− γ¯ Dr + λrs γ¯ D2 y I e λrp γ¯ Dr − Fγrs (y)=F|grs |2 F|grp |2 + jy l+1 γ¯ γ¯ l + 1 λrp Dr λ D + Iλ D j=1 j=1 l=0
l=0
rp
r
rs
2
(14) i L−1
i L−1
I21 =
j
j=1 l=0 (l+1)I − λrp γ¯ Dr
e
L(−1)l+j−1 j γ¯ (l + 1)λrs D2 l
F2 (b1 , B1 , θ1 , θ2 ) +
1−e
i L−1
K−i
L−1
K−i L−1
F1 (b1 , B1 ) +
j
(i+s)Φ
(l+1)I
l
s=1 l=0
λsr I
i L−1 j
L(−1)l+j−1 j × γ¯ λrs λrp Dr D2 l
L(−1)l+j−1 j − λ(l+1)I l e rp γ¯ Dr F3 (b1 , B1 , θ1 , θ2 ) Iλrs λrp Dr D2
K−i
L−1
L(−1)l+s e− λsr γ¯ − λsp γ¯ Ds − (i+s)Φ λsp Ds + l+1 s=1
s
i L−1
j=1 l=0
i L−1
j=1 l=0
T22 =Fγrs (Φ)
(l+1)I
− λrp γ¯ Dr
l=0
λsp Ds
K−i L−1 s
L(−1)l+s − e λsp Ds
l
(i+s)Φ (l+1)I λsr γ ¯ + λsp γ ¯ Ds
(17)
Φ + λss γ ¯
I22 , (21)
i L−1
ij lL−1 L(−1)l+j−1 j (l+1)I − λrp γ¯ Dr × 1−e G1 (b1 , A, C, B1 ) + γ¯ λrs λrp Dr D2 j=1 l=0 j=1 l=0 i L−1
ij lL−1 L(−1)l+j−1 j − (l+1)I (l+1)I − λrp γ¯ Dr G2 (b1 , A, C, B1 , θ1 , θ2 ) + e λrp γ¯ Dr G3 (b1 , A, C, B1 , θ1 , θ2 ), e Iλ λ D D rs rp r 2 j=1
i L−1
I22 =
i L−1
L(−1)l+j−1 j γ¯ (l + 1)λrs D2
j
l
(22)
l=0
⎛
⎞
j (−1) λK+1 Γ(K + 2) ⎝ LΦ sp j ∞ l k 2k ⎠ (−1) (−1) ρs Pout ≈ l k IK+1 (l + 1)j+1 λss λK sr j=0 l=0 k=0
L−1 k K K−i+1 k
LΦK−i+2 Γ(K − i + 2)
s (−1)s L−1 K−i+1 K (−1)l (−1)k ρ2k + i s l k 2IK−i+1 (l + 1)s+1 s=0 i=1 l=0 k=0 ⎛ ⎞
i L 2 2
i L λK−i+1 sp j−1 l−1 jλrp ρr + l(1 − ρr ) ⎠ ⎝ × . j (−1) l (−1) 2 2 K−i lIλ ρ + j(1 − ρ ) λss λsr rs 2 2 j=1 l=1 L−1
K+1
L−1
K+1
0
10
−1
Pout
10
[9, Eq. (4), ε = 0.01] [9, Eq. (4), ε = 0.05] Eq. (10) Sim (Eqs. (8) & (9))
K=2
−2
10
K=4 −3
10
−4
10
0
5
10
15 γ [dB]
20
25
30
Fig. 1. Secondary outage probability as function of γ ¯ (dB) when K = 2, 4, L = 1, λsr = λrs = 16, and ρs = ρr = ρ2 = 0.9.
decreases when we impose a more restrictive constraint such as ε = 0.01. However, the LR-based scheme offer a very tight approximation of (10).
K+1
k
k
(26)
In Fig. 2, we evaluate the outage probability expression of our scheme as function of γ¯ , under different numbers of active relays and primary receivers, (K) and (L), respectively. We reconfirm the tightness of our LR-based scheme and can note that the performance remains acceptable even when K = L = 2. Fig. 3 plots the high-SNR performance of our proposed scheme as well as of the CSI-based scheme as function of λss /λsp when L = 1 and K = 4. As before, we compare the high-SNR performance of our proposed scheme with its CSIbased counterpart, using the following benchmarks: (1) ρs = 1, ρr = 0.5, and ρ2 = 0.3, (2) ρr = 0.8 and ρs = ρ2 = 1. To get better insights on our proposed scheme achievements, we can note the performance gap between the proposed scheme and the CSI-based scheme when ρ2 < 1. This gap increases as the CSI-based scheme becomes more restrictive (ε = 0.01). On the other hand, we see that the gap shrinks when ρ2 = 1. Furthermore, we evaluate the high-SNR performance of the LR-based scheme when K = 4 and L = 3. We can note that the diversity order of K + 1 is achieved provided that ρ2 = 1. However, the generalized diversity order is reduced
expression of the outage probability as well as the diversity order of the SN, and we show that the LR-based scheme results corroborate the analysis and outperform the CSI-based scheme considered in recent works. We showed that our proposed scheme recover the SN performance degradation when using the outdated CSI while maintaining the PN QoS. Consequently, the results from the LR-based scheme are accurate, analytically tractable, and with least implementation complexity.
−1
10
Eq. (10), L = 1 Eq. (10), L = 2 Eq. (10), L = 3 Sim (Eqs. (8) & (9)) −2
10
Pout
K=2
−3
10
ACKNOWLEDGMENT K=4
This work was supported by Ooredoo under the project QUEX-Qtel-09/10-10.
−4
10
0
5
10
15 γ [dB]
20
25
30
Fig. 2. Secondary outage probability as function of γ ¯ (dB) when K = 2, 4, L = 1, 2, 3, λsr = λrs = 16, and ρs = ρr = ρ2 = 0.9.
−1
10
[9, Eq. (4), ε = 0.01] [9, Eq. (4), ε = 0.05] Eq. (10) Sim (Eqs. (8) & (9))
−2
10
−3
10
−4
Pout
10
ρs = 1; ρr = 0.5 ρ2 = 0.3
−5
10
−6
10
−7
10
ρs = ρ2 = 1 ρr = 0.8
−8
10
−9
10
0
5
10
15 λss / λsp [dB]
20
25
30
Fig. 3. Comparison of the LR-based scheme and the CSI-based scheme in high-SNR regime in terms of outage probability as function of λss /λsp (dB) when λrp = 0.01, I = 5 (dB), and L = 1.
to two once the relay selection is assumed to be carried out using outdated CSI (ρ2 < 1) as stated by lemma 3. −2
10
Eq. (10) Eq. (26) Sim (Eqs. (8) & (9))
−3
10
−4
Pout
10
−5
10
−6
ρs = 1; ρr = 0.5; ρ2 = 0.3
ρs = ρ2 = 1; ρr = 0.9
10
−7
10
ρs = 0.9; ρr = ρ2 = 0.9
−8
10
0
5
10 15 λss / λsp [dB]
20
25
Fig. 4. Asymptotical performance results of the LR-based scheme. The figure shows that ρ2 has the main impact on the diversity order as stated by lemma 3.
V.
C ONCLUSION
In this work, we proposed a new power allocation scheme based on LR model for a relay-aided SN coexisting with a PN with L primary receivers. We derived the closed-form
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