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[10] K. C. Toh, M. J. Todd, and R. H. Tutuncu, “SDPT3: A Matlab software package for semidefinite programming,” Optim. Methods Softw., vol. 11, no. 1–4, pp. 545–581, Jan. 1999. [11] A. F. Molisch and F. Tufvesson, “Multipath propagation models for broadband wireless systems,” in CRC Handbook of Signal Processing for Wireless Communications. Boca Raton, FL, USA: CRC, 2004. [12] Technical Specification Group Radio Access Networks; Deployment Aspects, Third Generation Partnership Project TR 25.943 V10.0.0, 2011.
Partial Relay Selection With Fixed-Gain Relays and Outdated CSI in Underlay Cognitive Networks Bin Zhong, Zhongshan Zhang, Member, IEEE, Xu Zhang, Jun Wang, and Keping Long, Senior Member, IEEE
Abstract—The impact of an imperfect channel estimation on the amplify-and-forward (AF) mode cooperative communications systems is studied, with some important factors, including the probability characteristic of the secondary user’s end-to-end signal-to-noise ratio (SNR), the outage probability, the symbol error probability (SEP), and the channel capacity, being analyzed. Different from the conventional relay selection schemes, we assume that the primary users share their bandwidth with the secondary users to enable a secondary relay-aided communication if the interference added to the primary users is kept below a certain threshold in an underlay cognitive network. In particular, both the feedback delay and Doppler frequency shift are assumed to be within a tolerable range, and as compared with the conventional methods, less channel state information (CSI) feedback is required in the proposed method due to partial relay selection being performed in the latter. The proposed scheme is validated by carrying out both theoretical analysis and numerical simulation, and the theoretical approximations of closed-form expressions for some figures of merit, e.g., the outage probability, the SEP, and the channel capacity, are all consistent with the numerical results. The simulations also prove that the performance of the proposed scheme is considerably affected by some other critical parameters, such as the number of relays, the channel correlation coefficient, and the interference threshold. In the presence of multiple candidate relays, an optimum solution in terms of either the outage probability or the SEP performance can always be found within the SNR range of (0, 10 dB). Index Terms—Amplify-and-forward (AF), cognitive radio, cooperative networks, outage probability, outdated channel state information (CSI), partial relaying.
Manuscript received September 17, 2012; revised April 27, 2013; accepted May 25, 2013. Date of publication May 31, 2013; date of current version November 6, 2013. This work was supported in part by the National Natural Science Foundation of China under Grant 61172050, by the National Basic Research Program of China (973 Program) under Grant 2012CB315905, by the Program for New Century Excellent Talents in University under Grant NECT-12-0774, by the Beijing Science and Technology Program under Grant Z111100054011078, by the Foundation of Beijing Engineering and Technology Center for Convergence Networks and Ubiquitous Services, and by the National Key Projects under Grant 2012ZX03001029-005 and Grant 2012ZX03001032003. This paper was presented in part at the IEEE Wireless Communications and Networking Conference, Shanghai, China, April 7–10, 2013. The review of this paper was coordinated by Prof. C. P. Oestges. The authors are with the Institute of Advanced Network Technology and New Services, and Beijing Engineering and Technology Research Center for Convergence Networks and Ubiquitous Services, University of Science and Technology Beijing, Beijing 100083, China (e-mail: zhongbin-1982@163. com;
[email protected];
[email protected];
[email protected];
[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.2013.2265280
I. I NTRODUCTION Cooperative relays forward copies of the message of a source to a destination using relaying channels, in addition to the directly sourceto-destination (S → D) link, and the diversity gain of the relaying system is consequently improved. By using multiple relays, both the diversity gain and radio coverage of a cooperative communications system can be thus improved by combatting the severe multipath fading [1]–[3]. However, in the presence of multiple relays, a smart resource-allocation algorithm is required to guarantee an orthogonal channel (in either the time or frequency domain) being allocated to each relay to effectively mitigate the interrelay interference [4]. As the number of relays increases, the cost of an orthogonal resource allocation in terms of spectral efficiency may deteriorate the overall performance of the cooperative system [5], [6]. In consideration of the aforementioned challenge, relay selection can be regarded as one of the most attractive methods to solve the complicated interference mitigation issue met in the multirelay network systems [7]. In general, the existing relay selection methods can be classified into two categories, i.e., the opportunistic relay selection and the partial relay selection [8]–[10], where in the former, the SNR of both the source-to-relay (S → R) and relay-to-destination (R → D) links is required by the central unit [8], but in the latter, the SNR of only the S → R or R → D link is necessarily considered [9]. Considering imperfect channel state information (CSI) with a high feedback rate and a sufficiently high maximum Doppler shift, the partial relay selection has advantages over the opportunistic relay selection in term of both outage and symbol error probabilities (SEPs) [10]. Moreover, in a cognitive network, the selected relays for the secondary users should maintain a strict interference threshold in their signal transmission to avoid the primary users being interfered by secondary users [11]–[13]. Currently, most of the studies about relay selection have been focused on the scenarios of a constant channel condition [14], [15]. A partial relay selection method for the underlay cognitive networks is studied in [14], where the fixed-gain relays are assumed to operate in the amplify-and-forward (AF) mode. However, the practical channel condition is usually time variant, particularly in high-mobility scenarios (a large Doppler shift is observed) with an accurate CSI feedback being infeasible. In this case, performing relay selection by considering an outdated CSI feedback would be a suitable method to optimize the performance of the cooperative systems at a reasonable CSI feedback cost [10]. To the best of the authors’ knowledge, partial relay selection with an outdated CSI feedback in underlay cognitive networks has not been considerably studied in prior works. In this paper, partial relay selection in underlay cognitive networks is studied, with fixed-gain AF relaying mode and an outdated CSI feedback being considered. The CSI feedback burden can be greatly reduced in a partial relay selection scheme due to an outdated CSI of the S → R link being required by the source node. Moreover, utilizing fixed-gain relays rather than adaptively variant gain relays can further simplify the relaying operation and without sacrificing too much performance. As compared with the existing works, the main contributions of this paper are exhibited as follows: 1) The impact of the interference on the primary user during the relay selection process of AF-based underlay cognitive radio networks is analyzed; 2) relay selection with imperfect CSI is studied; 3) partial relay selection with fixed-gain relays instead of full opportunistic relaying is studied; and 4) approximated closed-form expressions of some figures of merit, including the outage probability, the SEP, and the channel capacity, have been considered for the proposed scheme.
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In the following, without loss of generality, the transmission signal power in each node is assumed to be Es . Considering cognitive network systems (as shown in Fig. 1), S is communicating with D through relay Rn . s(t) is assumed to be the signal transmitted by S. The received interference signal at node p through interference channel S → Rn → p can be written as rp (t) = hRn P · g · hSRn · s(t).
(1)
Furthermore, the interference signal power received at the primary user through the interference channel S → Rn → p can be derived as InP = Es g 2 |hSRn |2 |hRn P |2 .
(2)
The cdf of interference signal power received at the primary user can be given by
FInP (λ) = 1 − 2
λ 2 2 σSR σ Es g 2 n Rn P
× K1
Fig. 1. System model of a cognitive network, with cooperative relays being closed to a primary user.
The remainder of this paper is organized as follows. Section II introduces the system model of partial relay selection with fixed-gain relays and outdated CSI feedback in underlay cognitive networks. The approximate closed-form expressions of some critical parameters, including the outage probability, the SEP, and the channel capacity, are derived in Section III. Section IV gives out the numerical results. Finally, Section V concludes this paper. Notation: {x} and {x} are the real and imaginary parts of x, respectively. A circularly symmetric complex Gaussian variable with mean a and variance σ 2 is denoted by z ∼ CN (a, σ 2 ). γab represents the SNR of link a → b by considering outdated CSI feedback. γ¯ab and γˆab stand for the mean and estimated SNR of link a → b, respectively. fX (·) and FX (·) represent the probability density function (pdf) and the cumulative distribution function (cdf) of random variable (RV) X, respectively. MX (s) denotes the moment-generating function (mgf) of RV X. II. S YSTEM M ODEL Here, an underlay cognitive network, which consists of primary-user terminal P , secondary source terminal S, and N half-duplex AF fixedgain relays (as denoted by the set of Ω = {Rn , n = 1, 2, . . . N }) with each relay having a gain of g and destination terminal D, is considered, as shown in Fig. 1. Each link in the cognitive network system is subjected to zero-mean additive white Gaussian noise (AWGN) with variance N0 . The fading in all S → Rn (n = 1, . . . , N ) links is assumed to be independent and identically distributed (i.i.d.) Rayleigh-distributed 2 . Similarly, the fading in all RVs with mean 0 and variance σSR Rn → D and Rn → P links is assumed to be zero-mean i.i.d. 2 2 and σRP , respectively. Rayleigh-distributed RVs with variance σRD The channel gain between terminals a and b is denoted by hab , where 2 ), where a, b ∈ {S, P, D} ∪ Ω. Therefore, we have hab ∼ CN (0, σab 2 2 2 2 2 2 σSRi = σSR , σRi D = σRD , and σRi P = σRP ∀1 ≤ i ≤ N . Moreover, |hab |2 = ({hab })2 + ({hab })2 , where {hab } and {hab } 2 /2. are i.i.d. zero-mean Gaussian RVs with common variance σab 2 2 , |hab | can be thus formulated as an exponential RV with mean σab 2 −(x/σab ) 2 . and its pdf is given by f|hab |2 (x) = (1/σab )e
2
λ 2 σSR σ 2 Es g 2 n Rn P
(3)
where K1 (·) is the first-order modified Bessel functions of the second kind, and λ represents a preset interference threshold signal power received at the primary user. In the proposed partial relay selection scheme, each candidate relay may be activated with the probability of Pλ = FInP (λ) or, otherwise, be dropped from the selection pool with a probability of P¯λ = 1 − FInP (λ). The set of relays (without loss of generality, the cardinality of this set is assumed to be l, where l ≤ N ) out of N candidates satisfying the aforementioned interference constraint can be therefore denoted by Ω = {Rn |InP ≤ λ, n = 1, 2, . . . N }. For each i.i.d. link S → Rn , n = 1, . . . , N , since the received SNR at Rn can be written as γSRn = Es (|h2SRn |/N0 ) = (Es /N0 )({hSRn })2 + ({hSRn })2 , the pdf of γSRn is γSRn )e−(x/¯γSRn ) , where thus given by fγSRn (x) = (1/¯ 2 )/N denotes the mean of γ . Likewise, γ¯SRn = (Es σSR 0 SRn n γSD and γRn D are exponentially distributed with parameters 2 2 )/N0 and γ¯Rn D = (Es σR )/N0 , respectively. For γ¯SD = (Es σSD nD the i.i.d. case, we can easily derive γ¯SRi = γ¯SR and γ¯Ri D = γ¯RD , where ∀1 ≤ i ≤ N . III. A PPROXIMATED C LOSED -F ORM A NALYSIS A. PDF and CDF of the Relay-Selection Channel Here, two scenarios, i.e., l ≥ 1 and l = 0, will be separately analyzed as follows. 1) l ≥ 1: The secondary user selects the optimal relay according to the following rule, i.e., γSRi ) k = arg max (ˆ i:Ri ∈Ω
(4)
where γˆSRi is the estimated SNR at time t using the feedback CSI, which is not the concurrent information due to the existence of the feedback delay τ . For each a → b link, using the Jakes’ autocorrelation ˆ ab can be repmodel, the correlation coefficient between hab and h resented as ρab = J0 (2πτ fab ), where J0 (·) denotes the zeroth-order Bessel function of the first kind, fab is the maximum Doppler Shift ˆ ab represent the channel coefficient on the a → b links, and hab and h ˆ and its estimation, respectively. The relationship between hab and hab ˆ ab = ρab hab + 1 − ρ2 · wab , with wab representing is given by h ab a circularly symmetric complex Gaussian RV, whose distribution is identical to hab .
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The conditional pdf of γSRk for a given l can be derived as [10]
(−1) l−1
fγSRk |l (γ|l) = l
m l−1 m
Am e−Am γ
(5)
(m + 1)
m=0
derived as Fγtotal (γT ) =
N N
l
l=1
l−1 (−1)m
Pλl P¯λN −l l
l−1 Am
m
m+1
m=0
γT
− × Qm (γT ) + P¯λN 1 − e γ¯SD
where Am =
(m + 1) . m(1 − ρ2SRk ) + 1 γ¯SR
(−1)
FγSRk |l (γ|l) = l
m l−1 m
(1 − e−Am γ )
Using [17, Eq. (2.3–10)] and [18, Eq. (4)], the SEP of the proposed method is approximated as .
(7)
β Pe = √ 2π
fγtotal , l=0 (γ, l = 0) =
l=1
×
(−1) m N Pλl P¯λN −l l m+1 l
=
Am e−γ/¯γSD − e−Am Am γ¯SD − 1
(8)
+
β ¯N P 2 λ
1−
l=1
l ×
Pλl P¯λN −l l
l−1 (−1)m m=0
Gm =
Am · Qm (γ)
m
m+1
(9)
1 1/¯ γSD × + Am Am (Am − 1/¯ γSD )
fγtotal , l=0 (γ, l = 0)
1
(10)
γ¯SD
e
γ SD
−γ ¯
Fγtotal , l=0 (γ, l = 0) = P¯λN 1 − e
B C= 2
l=1
Pλl P¯λN −l l
l−1 (−1)m m=0
·Qm (γ) +
η η + 2Am
1 (Am − 1/¯ γSD )
η η + 2/¯ γSD
(16)
γ −γ ¯SD
(12)
m
P¯λN
log2 (1+γ)fγtotal ,l=0 (γ, l = 0)dγ
B + 2
∞ log2 (1+γ)fγtotal ,l=0 (γ, l = 0)dγ 0
=
B P¯λN e1/¯γSD E1 2 ln 2
1−e
1
+D(N, l, γ¯SR , γ¯SD )
γ¯SD
N N l=1
Am γ −γ ¯SD
l
Pλl P¯λN −l l ·
(13)
B. Outage Probability Analysis For a preset threshold γT , from (13), the outage probability of the partial relay selection scheme with an outdated CSI feedback can be
B stands for the signal bandwidth θ(m, γ¯SR , γ¯SD ) = γ¯SD e1/¯γSD E1
and E1 (x) =
∞ x
(e−t /t)dt.
l−1 (−1)m m=0
×
.
(17)
where D(N, l, γ¯SR , γ¯SD ) =
l−1
m+1
∞ 0
(11)
respectively. Hence, the unconditional cdf of the received SNR is derived as
l
(15)
In the presence of outdated CSI feedback, the approximate closedform expression for the channel capacity of the proposed partial relay selection can be derived as
N N
D. Channel Capacity Analysis
2) l = 0: In this case, relays will be dropped from the selection pool with probability Pr (l = 0) = P¯λN , if the selection threshold is not met. The joint pdf and cdf can be derived as
Fγtotal (γ) =
η¯ γSD 2 + η¯ γSD
l−1
e−x/¯γSD 1 1/¯ γSD e−Am x − + . Am Am (Am −1/¯ γSD ) Am −1/¯ γSD
= P¯λN
and β and η are modulation-specified constants determined by modulation format.
where Qm (x) =
m=0
− N N
dγ
where
γ
t2 2
which leads to Fγtotal ,l=0 (γ, l = 0) =
e−
l−1 (−1)m l−1 β N m Am · G m Pλl P¯λN −l l 2 m+1 l l=1
m=0
t η
Fγtotal
N
m l−1
l−1
2
∞ 0
In the presence of cooperative relays, some combining method, e.g., maximal-ratio combining, can be employed to optimize the effective SNR as γtotal = γeq + γSD , where γeq = (γSRk γRk D )/(G + γRk D ) represents the SNR of the S → Rk → D link, and G = Es /(g 2 N0 ), as defined by [16]. From the Appendix, the joint pdf of γtotal and l is derived as N
(14)
C. SEP Analysis
m+1
m=0
.
(6)
Evidently, (5) leads to l−1
l−1 m
m+1
θ(m, γ¯SR , γ¯SD ) · Am Am γ¯SD − 1
1
γ¯SD
−
(18)
1 · eAm ·E1(Am ) Am (19)
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 62, NO. 9, NOVEMBER 2013
Fig. 2. Outage probability versus the average SNR of the S → D links with λ = 10 and ρSRk = 0.707.
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Fig. 3. Outage probability versus the average SNR of the S → D links with λ = 10 and N = 3.
IV. N UMERICAL R ESULTS Here, without loss of generality, the relaying channels and their interference on the primary user are assumed to satisfy the following γSD , γ¯RD = 1.8¯ γSD , and γ¯RP = 0.8¯ γSD . constraints, i.e., γ¯SR = 1.5¯ Binary phase-shift keying modulation is considered in this paper, and this implies that β = 1 and that η = 2. In particular, to simplify the analysis, Es = 1, N0 = 1, and g = 1 are assumed. The outage probability as a function of γSD for different N is shown in Fig. 2. Although using more relays implies obtaining a higher probability of selecting the optimum relay at low SNR, further increasing SNR may degrade the selectivity of the optimum relay due to severer interference being added to the primary user. After the SNR approaching a certain level (15 dB in this simulation), further increasing SNR may cause all the relays being dropped from the candidate pool. By keeping N = 3 and λ = 10 unchanged, for different ρSRk , a smaller ρSRk implies faster changes in channel state, and the outage probability is therefore a monotonically decreasing function of ρSRk , as shown in Fig. 3. Note that the selected relay may no longer be optimum due to imperfect CSI feedback (i.e., with outdated CSI). Outage probability as a function of an SNR for different λ is also shown in Fig. 4, with N = 3 and ρSRk = 0.707 being considered. As λ → 0, all relays are dropped from the selection pool, and only the S → D link is used for data transmission. In this case, increasing λ implies more relays could become a candidate of the best relay, and the outage probability is reduced accordingly. SEP as a function of SNR is shown in Fig. 5, where an optimum SNR can always be found to obtain the lowest SEP for each N . For a specific SNR level in the S → D link, more candidate relays implies better SEP performance due to an improved spatial diversity gain. However, increasing SNR beyond a certain threshold may also degrade the SEP of the proposed relay selection. The effect of the channel correlation coefficient ρSRk on SEP performance of the partial relay selection with outdated CSI feedback is also shown in Fig. 6. Similar to the outage probability performance, SEP is also a monotonically decreasing function of ρSRk . Similarly, the SEP performance is also a monotonically decreasing function of λ. The channel capacity as a function of SNR for the proposed relay selection scheme is described in Fig. 7. When SNR is smaller than 10 dB, the channel capacity is a monotonically increasing function of the number of available relays. However, the number of candidate
Fig. 4. Outage probability versus the average SNR of the S → D links with N = 3 and ρSRk = 0.707.
relays may decrease due to the interference from the secondary user, and this effect will consequently degrade the channel capacity. Basically, using an accurate CSI feedback is beneficial to improving the channel capacity as compared with that with outdated CSI, however, at the cost of a heavier CSI feedback burden. From this point of view, partial relay selection with outdated CSI feedback is a good choice to optimize the tradeoff between the CSI feedback and the channel capacity improvement. The effect of ρSRk on the channel capacity is also analyzed in Fig. 7, with N = 3 and λ = 10 being assumed in this simulation. The capacity with an imperfect CSI feedback is only slightly worse than that with perfect channel estimates in a low-SNR regime, but this performance gap diminishes as the SNR increases. Similarly, the capacity is affected by λ, as shown in Fig. 7, where a larger λ implies a higher capacity in the low-SNR regime but with the performance gain diminishing as the SNR increases.
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Fig. 5. SEP versus the average SNR of the S → D links with λ = 10 and ρSRk = 0.707.
Fig. 7.
Channel capacity versus the average SNR of the S → D links.
A PPENDIX A PPROXIMATION OF fγtotal ,l=0 (γ, l = 0) Using (7) and [19, Eq. (3.324.1)], the conditional cdf is derived as Fγeq |l (γ|l) = Pr [γeq < γ|l] =l
l−1 (−1)m m=0
l−1 m
m+1
× 1−2e
−Am γ
Am γG K1 2 γ¯RD
Am γG γ¯RD
(20)
where Pr {·} stands for the probability distribution, and fγRk D (γ) = (1/¯ γRD )e−(γ/¯γRD ) . The joint pdf of γeq and l can be derived as Fγeq ,l=0 (γ, l = 0) =
N N l=1
Fig. 6. SEP versus the average SNR of the S → D links with λ = 10 and N = 3.
V. C ONCLUSION The impact of an imperfect channel estimation on the partial relay selection in AF relaying cooperative communications systems has been studied, with the approximate closed-form expressions for some critical figures of merit, including the probability characteristic of the secondary user’s end-to-end SNR, the outage probability, the SEP, and the channel capacity, being derived. The validity of the proposed theoretical approximation on the critical figures of merit, including the outage probability, the SEP, and the channel capacity, was proven via simulations, and the theoretical analysis matches the corresponding numerical results well. It has been also shown in the numerical results that some other parameters, including the number of relays, the channel correlation coefficient, and the interference threshold, significantly affect the system performance in the presence of multiple candidate relays, and an optimum solution in terms of both outage probability and SEP performance can always be found within the SNR range of (0, 10 dB).
l
Pλl P¯λN −l l
l−1 (−1)m l−1 m m+1
m=0
where
Zm = 1 − 2e−Am γ
Am γG K1 γ¯RD
2
Am γG γ¯RD
Zm
(21)
(22)
and K1 (x) ≈ (1/x). Hence, (21) can be approximated as Fγeq , l=0 (γ, l = 0) ≈
N N l=1
l
Pλl P¯λN −l l
l−1 (−1)m m=0
l−1 m
m+1
× [1 − e−Am γ ]
(23)
which leads to fγeq ,l=0 (γ, l = 0) =
N N l=1
l
Pλl P¯λN −l l
l−1 (−1)m m=0
l−1 m
m+1 × Am e−Am γ .
(24)
The mgf of γtotal is derived as Mγtotal (s) = Mγeq (s)MγSD (s)
(25)
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 62, NO. 9, NOVEMBER 2013
where Mγeq (s) =
N l=1
(−1) m N Pλl P¯λN −l l m+1 l l−1
m=0
∞ e−sγ fγSD (γ)dγ =
MγSD (s) =
m l−1
Am s+Am
1/¯ γSD s + 1/¯ γSD
(26)
(27)
0
γSD )e−(γ/¯γSD ) . with fγSD (γ) = (1/¯ By performing inverse Laplace transform on Mγtotal (s), the joint pdf of γtotal and l can be derived as fγtotal ,l=0 (γ, l = 0) =
N N l=1
×
l
Pλl P¯λN −l l
l−1 (−1)m m=0
l−1 m
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[14] S. I. Hussian, M.-S. Alouini, M. Hasna, and K. Qarage, “Partial relay selection in underlay cognitive networks with fixed gain relays,” in Proc. IEEE VTC, May 2012, pp. 1–5. [15] Z. Zhang, C. Tellambura, and R. Schober, “Improved OFDMA uplink transmission via cooperative relaying in the presence of frequency offsets—Part II: Outage information rate analysis,” Eur. Trans. Telecommun., vol. 21, no. 3, pp. 241–250, Apr. 2010. [16] M. O. Hasna and M.-S. Alouini, “A performance study of dual-hop transmissions with fixe gain relays,” IEEE Trans. Wireless Commun., vol. 3, no. 6, pp. 1963–1968, Nov. 2004. [17] J. G. Proakis and M. Salehi, Digital Communications. New York, NY, USA: McGraw-Hill, 2008. [18] Y. Zhao, R. Adve, and T. J. Lim, “Symbol error rate of selection amplify-and-forward relay systems,” IEEE Commun. Lett., vol. 10, no. 11, pp. 757–759, Nov. 2006. [19] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. New York, NY, USA: Academic, 2007.
m+1
Am {e−γ/¯γSD − e−Am γ }. (28) Am γ¯SD − 1
A Differential Feedback Scheme Exploiting the Temporal and Spectral Correlation Mingxin Zhou, Leiming Zhang, Member, IEEE, Lingyang Song, Senior Member, IEEE, and Merouane Debbah, Senior Member, IEEE
ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their critical comments, which greatly improved this paper. R EFERENCES [1] Z. Zhang, W. Zhang, and C. Tellambura, “OFDMA uplink frequency offset estimation via cooperative relaying,” IEEE Trans. Wireless Commun., vol. 8, no. 9, pp. 4450–4456, Sep. 2009. [2] R. U. Nabar, H. Bolcskei, and F. W. Kneubuhler, “Fading relay channels: Performance limits and space-time signal design,” IEEE J. Sel. Areas Commun., vol. 22, no. 6, pp. 1099–1109, Aug. 2004. [3] W. Zhuang and M. Ismail, “Cooperation in wireless communication networks,” IEEE Wireless Commun., vol. 19, no. 2, pp. 10–20, Apr. 2012. [4] Z. Zhang, W. Zhang, and C. Tellambura, “Cooperative OFDM channel estimation in the presence of frequency offsets,” IEEE Trans. Veh. Technol., vol. 58, no. 7, pp. 3447–3459, Sep. 2009. [5] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004. [6] P. L. Yeoh, M. Elkashlan, Z. Chen, and I. B. Collings, “SER of multiple amplify-and-forward relays with selection diversity,” IEEE Trans. Commun., vol. 59, no. 8, pp. 2078–2083, Aug. 2011. [7] Z. Zhang, C. Tellambura, and R. Schober, “Improved OFDMA uplink transmission via cooperative relaying in the presence of frequency offsets—Part I: Ergodic information rate analysis,” Eur. Trans. Telecommun., vol. 21, no. 3, pp. 224–240, Apr. 2010. [8] M. Torabi, D. Haccoun, and J.-F. Frigon, “Impact of outdated relay selection on the capacity of AF opportunistic relaying systems with adaptive transmission over non-identically distributed links,” IEEE Trans. Wireless Commun., vol. 10, no. 11, pp. 3626–3631, Nov. 2011. [9] K. B. Fredj and S. Aïssa, “Performance of amplify-and-forward systems with partial relay selection under spectrum-sharing constraints,” IEEE Trans. Wireless Commun., vol. 11, no. 2, pp. 500–504, Feb. 2012. [10] D. S. Michalopoulos, H. A. Suraweera, G. K. Karagiannidis, and R. Schober, “Amplify-and-forward relay selection with outdated channel estimates,” IEEE Trans. Commun., vol. 60, no. 5, pp. 1278–1290, May 2012. [11] D. Li and A. W. Long, “Outage probability of cognitive radio networks with relay selection,” IET Commun., vol. 5, no. 18, pp. 2730–2735, Dec. 2011. [12] Z. Zhang, K. Long, and J. Wang, “Self-organization paradigms and optimization approaches for cognitive radio technologies: A survey,” IEEE Wireless Commun., vol. 20, no. 2, pp. 36–42, Apr. 2013. [13] Z. Zhang, K. Long, J. Wang, and F. Dressler, “On swarm intelligence inspired self-organized networking: Its bionic mechanisms, designing principles and optimization approaches,” IEEE Commun. Surveys Tuts., to appear.
Abstract—Channel state information (CSI) provided by a limited feedback channel can be utilized to increase system throughput. However, in multiple-input–multiple-output (MIMO) systems, the signaling overhead realizing this CSI feedback can be quite large, whereas the capacity of the uplink feedback channel is typically limited. Hence, it is crucial to reduce the amount of feedback bits. Prior work on limited feedback compression commonly adopted the block-fading channel model, where only temporal or spectral correlation in a wireless channel is considered. In this paper, we propose a differential feedback scheme with full use of the temporal and spectral correlations to reduce the feedback load. Then, the minimal differential feedback rate over a MIMO time–frequency (or doubly)-selective fading channel is investigated. Finally, the analysis is verified by simulation results. Index Terms—Correlation, multiple-output (MIMO).
differential
feedback,
multiple-input
I. I NTRODUCTION In multiple-input–multiple-output (MIMO) systems, channel adaptive techniques (e.g., water-filling, interference alignment, beamforming, etc.) can enhance the spectral efficiency or the capacity of the Manuscript received November 25, 2012; revised March 18, 2013; accepted May 3, 2013. Date of publication June 5, 2013; date of current version November 6, 2013. This work was supported in part by the National 973 project under Grant 2013CB336700, by the National Natural Science Foundation of China under Grant 61222104 and Grant 61061130561, by the Ph.D. Programs Foundation of the Ministry of Education of China under Grant 20110001110102, and by the Opening Project of the Key Laboratory of Cognitive Radio and Information Processing (Guilin University of Electronic Technology). The review of this paper was coordinated by Prof. X. Wang. M. Zhou and L. Song are with the State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China (e-mail:
[email protected];
[email protected]). L. Zhang is with Huawei Technologies Company Ltd., Beijing 100095, China (e-mail:
[email protected]). M. Debbah is with SUPELEC, Alcatel-Lucent Chair in Flexible Radio, 91192 Gif-Sur-Yvette, France (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.2013.2266379
0018-9545 © 2013 IEEE
