- Email: [email protected]
Precoder design in cognitive radio networks with channel covariance information
Signal Processing 92 (2012) 3056–3061
Contents lists available at SciVerse ScienceDirect
Signal Processing journal homepage: www.elsevier.com/locate/sigpro
Fast communication
Precoder design in cognitive radio networks with channel covariance information$ Bangwon Seo n Electronics and Telecommunications Research Institute (ETRI), Daejeon, Republic of Korea
a r t i c l e in f o
abstract
Article history: Received 1 June 2011 Received in revised form 20 April 2012 Accepted 11 May 2012 Available online 31 May 2012
In cognitive radio communications, the coexistence of a secondary network is allowed as long as the interference power from the secondary network measured at the primary user (PU) is kept below a given threshold. In this paper, we consider a communication scenario for cognitive radio networks where the channel state information (CSI) between the secondary base station (SBS) and the PU is perfectly known at the SBS transmitter whereas only the covariance information is known for the channel of the SBS-secondary user (SU) link at the SBS transmitter. A closed-form precoder design method is proposed for the purpose of maximizing the upper bound of the expected achievable rate of the SU. Even though the proposed scheme uses only partial CSI of the SBS-SU link, the simulation results show that the expected achievable rate of the proposed scheme approaches to that of a conventional scheme with full CSI as the channel correlation increases. & 2012 Elsevier B.V. All rights reserved.
Keywords: Cognitive radio (CR) Precoder design Channel covariance information Achievable rate
1. Introduction As the demand for wireless services and applications requiring high data rate transmission are explosively growing, it becomes a very important issue to develop the technologies for efficient spectrum usage due to the limited available frequency resources. Recent Federal Communications Commission (FCC) measurement has shown that more than 70% of the allocated spectrum is not utilized in United States [1]. Therefore, cognitive radio (CR) technology has been considered as one of the attractive candidates to tackle such a challenge [2–5]. In a spectrum sharing based CR network, the secondary users (SUs) are allowed to utilize the frequency band licensed to the primary user (PU) if the interference
$ This work was supported by the IT R&D program of MKE/KEIT in Republic of Korea (10038765 Development of beyond 4G technologies for smart mobile services). n Tel.: þ 82 42 860 5903; fax: þ82 42 860 6789. E-mail address: [email protected]
0165-1684/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2012.05.011
power received at the primary user due to secondary transmission is less than a given threshold [6–9]. In cognitive radio (CR) communications, precoding matrix design problems of maximizing an achievable rate of secondary user (SU) have been studied for situations where instantaneous channel state information (CSI) for both secondary base station (SBS)–SU link and SBSprimary user (PU) link is available at the SBS transmitter [6,8]. Recently, a more realistic scenario has been treated in [7] where the SBS transmitter has instantaneous CSI for the SBS-SU link, but no information on SBS-PU link channel. On the contrary, in this paper, we consider the case where the SBS transmitter has instantaneous CSI for SBS-PU link, but only channel covariance information for SBS-SU link. This situation can occur when both primary and secondary systems are deployed by time-division duplex (TDD) technology in which the same frequency band is used for uplink and downlink transmissions. In this case, a PU periodically transmits its pilot symbol to the primary BS in the uplink. Then, SBS can estimate the instantaneous CSI for the SBS-PU link by overhearing the
B. Seo / Signal Processing 92 (2012) 3056–3061
PU’s pilot signal and using the uplink-downlink channel reciprocity. However, an SU is allowed to transmit its pilot symbol to the SBS only when its transmission will hardly influence the primary network communications. Therefore, the SBS can estimate only the channel covariance of the SBS-SU link instead of its instantaneous CSI because the pilot signals are transmitted aperiodically and sparsely from the SU. In the past two decades, precoding matrix design problems have received considerable attention in nonCR settings for situations where only partial CSI such as mean or covariance information of the channel is available at the transmitter [10,11]. However, the literature is generally not applicable to CR networks because of the additional constraint on PU interference in CR settings. In this paper, we propose a closed-form precoding matrix design method for a CR network. The aim of the method is to maximize the upper bound of the expected achievable rate of the SU in situations where the CSI for the SBS-PU link is perfectly known at the SBS transmitter, but only the covariance information of the SBS-SU link channel is available at the SBS transmitter. The rest of this paper is organized as follows. In Section 2, we present the system model and problem formulation. In Section 3, we explain our proposed precoding matrix design method. In Section 4, we present the simulation results. Finally, in Section 5, we present our conclusion.
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channel distribution, can be written as h i C exp ðFÞ ¼ EH log2 9IN þ R1=2 HFFH HH R1=2 9 z z
ð3Þ
where EH ½U is the statistical expectation operator with respect to H. If we denote the maximum transmit power at the SBS by Pth , the SBS transmit power constraint can be written as Es ½sH FH Fs ¼ trfFEs ½ssH FH g ¼ trfFFH g r Pth
ð4Þ
Since we consider the case the PU has one receive antenna, the channel response from the SBS to the PU is represented by an M 1 vector g and the interference signal from the SBS to the PU is given by gH Fs. Let us define an interference power threshold by Gth , then the interference power constraint at the PU is given by E½gH FssH FH g ¼ gH FE½ssH FH g ¼ gH FFH g r Gth
ð5Þ
Therefore, the maximization problem of C exp ðFÞ in a CR network can be formulated as h i max EH log2 9IN þ R1=2 HFFH HH R1=2 9 z z F
subject to trfFFH g r Pth ,
gH FFH g r Gth
ð6Þ
3. Proposed precoder design method 2. System model and problem formulation In this paper, we consider a point-to-point single-user communication system from an SBS to an SU in a CR network. We assume the spectrum is licensed to only one PU. We also assume that the SBS transmitter has M transmit antennas, the SU receiver has N receive antennas, and the PU has a single receive antenna. In addition, we assume that the full CSI of the SBS-PU link is known at the SBS transmitter, but only the covariance information of the SBS-SU link channel is known at the SBS transmitter. The channel between the SBS transmitter and the SU receiver is expressed by an N M matrix H. The input signal at the SBS transmitter is represented by a complex Gaussian distributed vector, s, of size J 1 with mean E½s ¼ 0 and covariance E½ssH ¼ IJ , where IJ is a J J identity matrix. The selection of the parameter J should be based on consideration of the rank of the SBS-SU link channel matrix, i.e., J ¼ rankðHÞ. An M J precoding matrix, F, is applied to the input signal vector, s. Then, the received signal at the SU receiver is represented by y ¼ ½y1 ,y2 ,. . .,yN T ¼ HFs þ z
ð1Þ
where z is the sum of the signal from the primary network and the additive white Gaussian noise (AWGN). We assume that z has a colored Gaussian distribution with mean E½z ¼ 0 and covariance E½zzH ¼ Rz . The achievable rate of the SU is given by [12] 9 CðFÞ ¼ log2 9IN þRz1=2 HFFH HH R1=2 z
ð2Þ
where 9U9 is the determinant operator. The expected value of CðFÞ for the random matrix H, averaged over the
The channel between the SBS transmitter and the SU receiver is assumed to be correlated at the transmit antennas, but uncorrelated at the receiver antennas. This model is appropriate for an environment where the SBS transmitter is not impeded by local scatterers and the SU receiver is surrounded by local scatterers [13]. The SBS-SU link channel is therefore modeled as [13] H ¼ Hw R1=2
ð7Þ
where Hw is a complex Gaussian random matrix of size N M with mean 0 and variance s2h IN and R is an M M transmit antenna covariance matrix. We assume that the SBS can know R by using the pilot signals aperiodically and sparsely transmitted from the SU. If we substitute (7) into (6), the constrained optimization problem can be rewritten as h i 1=2 max EHw log2 9IN þ R1=2 Hw R1=2 FFH R1=2 HH 9 z w Rz F
subject to trfFFH g r Pth ,
gH FFH g r Gth
ð8Þ
It is impossible to find the closed-form optimal solution of this original problem Hw. Therefore, in order to obtain the closed-form solution, we resort to the upper bound optimization solution of the expected achievable rate. From Jensen’s inequality [14] and the determinant identity 9IN þ AB9 ¼ 9IM þBA9 for the N M matrix A and M N matrix B, the upper bound C exp ðFÞ of the expected achievable rate is given by 1=2 1 FFH R1=2 9 C exp ðFÞ r log2 9IM þEHw ½HH w Rz Hw R
¼ log2 9IM þ s2h bR1=2 FFH R1=2 9 ¼ C exp ðFÞ
ð9Þ
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B. Seo / Signal Processing 92 (2012) 3056–3061
where b ¼ trfR1 z g and the following relation are used: 1 EHw ½HH w Rz Hw ¼
2 h bUIM
s
ð10Þ
The derivation of (10) is given in Appendix A. We assume that the SU reports b to the SBS through the feedback channel and therefore the SBS knows b. Now, the upper bound maximization problem of the expected achievable rate can be simplified as max log2 9IM þ s
subject to trfFFH g r P th ,
s2h blm @ 2 Lðq1 ,. . .,qM , m, rÞ ¼ 0 ) mr9wm 9 ¼ 0, @qm 1 þ s2h blm qm m ¼ 1,2,. . .,M, M X
ð17Þ
qm Þ ¼ 0,
rðGth
m¼1
gH FFH g r Gth
ð11Þ
The PU interference power constraint in (11) complicates finding an optimal closed-form solution for this optimization problem [6]. Thus, we propose a suboptimal solution with the closed-form in this paper. Let us define the eigenvalue decomposition of R as R ¼ UKUH , where the columns of U are eigenvectors of R with the property U UH ¼ UH U ¼ IM and K ¼ diagðl1 , l2 ,. . ., lM Þ is an eigenvalue matrix with l1 Z l2 Z Z lM . Also, without loss of generality, the precoding matrix can be written as F ¼ U U for any M J matrix U. The upper bound of the expected achievable rate can then be rewritten as C~ exp ðUÞ ¼ log2 9IM þ s2h bU K1=2 UUH K1=2 UH 9 i ¼ log2 9IM þ s2h bK1=2 UUH K1=2 9 ,
ð12Þ
Furthermore, the transmit power and PU interference constraints can be written as trfU UUH UH g ¼ trfUUH g rP th ,
listed as follows:
mðPth
1=2 2 FFH R1=2 9 h bR
F
where m and r are Lagrangian coefficients with m Z 0 and r Z 0. The Karush–Kuhn–Tucker (KKT) conditions are
wH UUH wr Gth
M X
2
9wm 9 qm Þ ¼ 0
ð18Þ
m¼1
From (17), the solution for qm is given by " #þ 1 1 qopt,m ¼ , m ¼ 1,2,. . .,M m þ r9wm 92 s2h blm
ð19Þ
where ½x þ ¼ maxðx,0Þ. The Lagrangian coefficients m and r should be chosen to satisfy the following constraints: " #þ L X 1 1 rP th , ð20Þ 2 s2h blm m ¼ 1 m þ r9wm 9 M X m¼1
9wm 9
2
"
1
1 m þ r9wm 92 s2h blm
#þ r Gth
ð21Þ
One simple method of selecting m and r is to use the bisection method for the two parameters [6]. After Q opt ¼ diagðqopt,1 ,qopt,2 ,. . .,qopt,M Þ has been obtained, the expected achievable rate of the proposed scheme is given by h i 1=2 9 ð22Þ C pro ¼ EHw log2 9IN þRz1=2 Hw R1=2 Q opt R1=2 HH w Rz
ð13Þ
H
respectively, where w ¼ U g. After Q ¼ UUH is defined and the determinant propQ erty 9A9 r m ½Am,m is used for any non-negative definite matrix A [15], where [A]m,n is the (m,m) th element of A , the following inequality is satisfied: 9IM þ s2h bK1=2 Q K1=2 9 r
M Y
ð1 þ s2h blm qm Þ
ð14Þ
m¼1
where qm ¼ ½Q m,m and the equality is satisfied when Q is diagonal, i.e., Q ¼ diagðq1 ,q2 ,. . .,qM Þ. Because C~ exp ðQ Þ has its maximum value when Q is diagonal, we confine Q to the diagonal matrix in order to find the closed-form solution. The problem formulation for the suboptimal closed solution can now be written as max
q1, ...,qM
M X
log2 ð1 þ s2h blm qm Þ
m¼1
subject to
M X
qm rP th ,
m¼1
M X
2
9wm 9 qm r Gth
ð15Þ
m¼1
A Lagrangian function of this optimization problem is given by Lðq1 ,. . .,qM , m, rÞ ¼
M X
log2 ð1 þ s2h blm qm Þ
m¼1
þ mðP th
M X m¼1
qm Þ þ rðGth
M X m¼1
2
9wm 9 qm Þ
ð16Þ
4. Simulation results The simulations are performed for the antenna combinations M ¼ N ¼ 2 and 4. Each element of Hw and g is generated by an independent and circularly symmetric complex Gaussian random variable distributed as CNð0,1Þ. For the transmit antenna covariance matrix, R, of the SBSSU link channel, the exponential correlation model [16] is used, i.e., ½Rm,n ¼ a9mn9 , 0 o a o1, where a means the channel correlation coefficient. The covariance matrix, Rz , of the additive noise, z, is set to IN for simplicity, and J is chosen as J ¼ rankðHÞ. The SBS transmit power constraint, P th , sweeps from 1 to 100, which is equivalent to a range of signal-to-noise ratios (SNRs) per receive antenna from 0 dB to 20 dB. The simulation results are implemented over 2000 randomly generated channel pairs ðHw ,gÞ, and the expected achievable rates are plotted against the SNR values. Fig. 1 compares the expected achievable rates of the conventional schemes and the proposed scheme for when M ¼N ¼2, Gth ¼ 0:5, and a ¼ 0:1,0:9. Fig. 2 represents the expected achievable rates for M ¼N ¼4. In the figures, the term ‘Full CSI’ represents the conventional scheme with full CSI [6], the term ‘Partial CSI (Proposed)’ represents the proposed scheme with partial CSI. Furthermore, the term ‘W/o PU constraint’ denotes the non-CR scheme without consideration of the PU interference constraint [15]. The figures show that the performance of the proposed scheme is about 0.5 dB worse than that of the conventional scheme
B. Seo / Signal Processing 92 (2012) 3056–3061
3059
Expected value of the achievable rate (bps/Hz)
11 (α=0.1) W/o PU constraint (α=0.1) Full CSI (α=0.1) Partial CSI (Proposed) (α=0.9) W/o PU constraint (α=0.9) Full CSI (α=0.9) Partial CSI (Proposed)
10 9 8 7 6 5 4 3 2 1
0
5
10 SNR (dB)
15
20
Fig. 1. Expected achievable rate comparison when M ¼ N ¼2 and Gth ¼ 0:5.
Expected value of the achievable rate (bps/Hz)
22 (α=0.1) W/o PU constraint (α=0.1) Full CSI (α=0.1) Partial CSI (Proposed) (α=0.9) W/o PU constraint (α=0.9) Full CSI (α=0.9) Partial CSI (Proposed)
20 18 16 14 12 10 8 6 4 2
0
5
10 SNR (dB)
15
20
Fig. 2. Expected achievable rate comparison when M ¼ N ¼4 and Gth ¼ 0:5.
with full CSI for the same expected achievable rate when the channel correlation coefficient, a, is 0:1 and M ¼ N ¼ 4. However, by comparing the curves for a ¼ 0:1 and a ¼ 0:9, it can be observed that the performance gap between the two schemes decreases as the channel correlation coefficient increases. Fig. 3 shows the expected achievable rate for the primary base station (PBS)-PU link. The received signal
model at the PU and an achievable rate analysis for the PBS-PU link is given in Appendix B. From the figure, it can be observed that the achievable rate for the PBS-PU link is degraded due to the presence of interference from the SU compared to the case without the interference and this degradation is natural in cognitive radio networks. However, the performance degradation for the PBS-PU link gets smaller as Gth decreases.
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B. Seo / Signal Processing 92 (2012) 3056–3061
Expected achievable rate for PU (bps/Hz)
3.5 Without interference from SU With interference from SU (Γth=0.1) 3
With interference from SU (Γ =0.2) th
With interference from SU (Γth=0.5) 2.5
2
1.5
1
0.5
0
5
10 SNR (dB)
15
20
Fig. 3. Comparison of the expected achievable rates for the PBS-PU link.
(
5. Conclusion
¼
In this paper, we propose a closed-form precoding matrix design method for maximizing the upper bound of the expected achievable rate of the SU for when the CSI of the SBS-PU link is perfectly known at the SBS transmitter, but only the covariance information of the channel of the SBS-SU link is known at the SBS transmitter. Even though the proposed scheme uses only partial CSI of the SBS-SU link, its expected achievable rate approaches to that of a conventional scheme requiring full CSI of the SBS-SU link as the channel correlation increases. Appendix A. Derivation of (10) If we define ðm,nÞth element of Hw by hm,n , we know n that Eh ½hm,n hm0 ,n0 ¼ s2h dðmm0 Þdðnn0 Þ where dðmÞ is the Kronecker delta function because Hw is a complex Gaussian random matrix of size N M with mean 0 and variance s2h IN . We define ðm,nÞth element of R1 z by qm,n and partition Hw into its subvectors, i.e., Hw ¼ ½h1 ,h2 ,. . .,hM where hm ¼ ½hm,1 ,hm,2 ,. . .,hm,N T . Then ðm,nÞth element of EHw 1 ½HH w Rz Hw , dm,n , is given by ( ) " # N N X X H 1 n dm,n ¼ Eh ½hm Rz hn ¼ Eh ðhl,m vl,k Þ hk,n k¼1
¼
N X N X
n
vl,k Eh ½hl,m hk,n ¼
k¼1l¼1
¼
N X k¼1
s2h b if m ¼ n 0
otherwise
where vl,k ¼ ½R1 z l,k
1 we obtain EHw ½HH w R z Hw ¼ s
Appendix B. Signal model at PU and an expected achievable rate for PBS-PU link Let us define the input signal at PBS transmitter by a complex Gaussian distributed random variable sPU with zero mean and unit variance. Then, the received signal at the PU can be written as: yPU ¼ hPU sPU þ gH F s þ ZPU
N X N X
vl,k s2h dðlkÞdðmnÞ
1 2 h trðR z ÞdðmnÞ ¼
s
ð23Þ
where hPU represents a channel response for the PBS-PU link and ZPU is AWGN with zero mean and variance s2Z . In (23), gH F s means the interference from the SU. The achievable rate of the PU is given by [12] C PU ¼ log2 1þ
9hPU 9
!
2
ð24Þ
gH F FH g þ s2Z
The expected value of C PU for the random vector g and random variable hPU , averaged over the channel distribution, can be written as "
vk,k s
vk,k ¼ trðR1 z Þ ¼ b. Finally, k¼1 2 h bUIM for any distribution of z.
l¼1
k¼1l¼1
2 h dðmnÞ ¼
N P
and
2 h bdðmnÞ
s
C PU,exp ¼ Eg,hPU log2 ð1 þ
#
2
9hPU 9
gH F FH g þ s2Z
Þ
ð25Þ
B. Seo / Signal Processing 92 (2012) 3056–3061
References [1] Facilitating Opportunities for Flexible, Efficient, and Reliable Spectrum Use Employing Cognitive Radio Technologies, Notice of Proposed Rule Making and Order, FCC 03-322 Federal Communications Commission, 2003. [2] F.-X. Socheleau, S. Houcke, P. Ciblat, A. Aissa-El-Bey, Cognitive OFDM system detection using pilot tones second and third-order cyclostationarity, Signal Processing 91 (2) (2011) 252–268. [3] S.V. Nagaraj, Entropy-based spectrum sensing in cognitive radio, Signal Processing 89 (2) (2009) 174–180. [4] J. Chen, X. Zhang, Y. Kuo, Adaptive cooperative spectrum sharing based on fairness and total profit in cognitive radio networks, ETRI Journal 32 (4) (2010) 512–519. [5] B. Seo, Joint design of precoder and receiver in cognitive radio networks using an MSE criterion, Signal Processing 91 (11) (2011) 2623–2629. [6] R. Zhang, Y.-C. Liang, Exploiting multi-antennas for opportunistic spectrum sharing in cognitive radio networks, IEEE Journal of Selected Topics in Signal Processing 2 (1) (2008) 88–102. [7] L. Zhang, Y.-C. Liang, Y. Xin, H.V. Poor, Robust cognitive beamforming with partial channel state information, IEEE Transactions on Wireless Communications 8 (8) (2009) 4143–4153.
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[8] L. Zhang, Y.-C. Liang, Y. Xin, Joint beamforming and power allocation for multiple access channels in cognitive radio networks, IEEE Journal of Selected Areas in Communications 26 (1) (2008) 38–51. [9] M.H. Islam, Y.-C. Liang, A.T. Hoang, Joint power control and beamforming for cognitive radio networks, IEEE Transactions on Wireless Communications 7 (7) (2008) 2415–2419. [10] E. Visotsky, U. Madhow, Space–time transmit precoding with imperfect feedback, IEEE Transactions on Information Theory 47 (6) (2001) 2632–2639. [11] S. Zhou, G.B. Giannakis, Optimal transmitter eigen-beamforming and space-time block coding based on channel mean feedback, IEEE Transactions on Signal Processing 50 (10) (2002) 2599–2613. [12] D. Tse, P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2004. [13] D. Shiu, G.J. Foschini, M.J. Gans, J.M. Kahn, Fading correlation and its effect on the capacity of multielement antenna systems, IEEE Transactions on Communications 48 (3) (2000) 502–513. [14] T.M. Cover, J.A. Thomas, Elements of Information Theory, Wiley, New York, 1991. [15] I.E. Telatar, Capacity of multi-antenna Gaussian channels, European Transactions on Telecommunications 10 (6) (1999) 585–595. [16] V.A. Aalo, Performance of maximal-ratio diversity systems in a correlated Nakagami-fading environment, IEEE Transactions on Communications 43 (8) (1995) 2360–2369.
Contents lists available at SciVerse ScienceDirect
Signal Processing journal homepage: www.elsevier.com/locate/sigpro
Fast communication
Precoder design in cognitive radio networks with channel covariance information$ Bangwon Seo n Electronics and Telecommunications Research Institute (ETRI), Daejeon, Republic of Korea
a r t i c l e in f o
abstract
Article history: Received 1 June 2011 Received in revised form 20 April 2012 Accepted 11 May 2012 Available online 31 May 2012
In cognitive radio communications, the coexistence of a secondary network is allowed as long as the interference power from the secondary network measured at the primary user (PU) is kept below a given threshold. In this paper, we consider a communication scenario for cognitive radio networks where the channel state information (CSI) between the secondary base station (SBS) and the PU is perfectly known at the SBS transmitter whereas only the covariance information is known for the channel of the SBS-secondary user (SU) link at the SBS transmitter. A closed-form precoder design method is proposed for the purpose of maximizing the upper bound of the expected achievable rate of the SU. Even though the proposed scheme uses only partial CSI of the SBS-SU link, the simulation results show that the expected achievable rate of the proposed scheme approaches to that of a conventional scheme with full CSI as the channel correlation increases. & 2012 Elsevier B.V. All rights reserved.
Keywords: Cognitive radio (CR) Precoder design Channel covariance information Achievable rate
1. Introduction As the demand for wireless services and applications requiring high data rate transmission are explosively growing, it becomes a very important issue to develop the technologies for efficient spectrum usage due to the limited available frequency resources. Recent Federal Communications Commission (FCC) measurement has shown that more than 70% of the allocated spectrum is not utilized in United States [1]. Therefore, cognitive radio (CR) technology has been considered as one of the attractive candidates to tackle such a challenge [2–5]. In a spectrum sharing based CR network, the secondary users (SUs) are allowed to utilize the frequency band licensed to the primary user (PU) if the interference
$ This work was supported by the IT R&D program of MKE/KEIT in Republic of Korea (10038765 Development of beyond 4G technologies for smart mobile services). n Tel.: þ 82 42 860 5903; fax: þ82 42 860 6789. E-mail address: [email protected]
0165-1684/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2012.05.011
power received at the primary user due to secondary transmission is less than a given threshold [6–9]. In cognitive radio (CR) communications, precoding matrix design problems of maximizing an achievable rate of secondary user (SU) have been studied for situations where instantaneous channel state information (CSI) for both secondary base station (SBS)–SU link and SBSprimary user (PU) link is available at the SBS transmitter [6,8]. Recently, a more realistic scenario has been treated in [7] where the SBS transmitter has instantaneous CSI for the SBS-SU link, but no information on SBS-PU link channel. On the contrary, in this paper, we consider the case where the SBS transmitter has instantaneous CSI for SBS-PU link, but only channel covariance information for SBS-SU link. This situation can occur when both primary and secondary systems are deployed by time-division duplex (TDD) technology in which the same frequency band is used for uplink and downlink transmissions. In this case, a PU periodically transmits its pilot symbol to the primary BS in the uplink. Then, SBS can estimate the instantaneous CSI for the SBS-PU link by overhearing the
B. Seo / Signal Processing 92 (2012) 3056–3061
PU’s pilot signal and using the uplink-downlink channel reciprocity. However, an SU is allowed to transmit its pilot symbol to the SBS only when its transmission will hardly influence the primary network communications. Therefore, the SBS can estimate only the channel covariance of the SBS-SU link instead of its instantaneous CSI because the pilot signals are transmitted aperiodically and sparsely from the SU. In the past two decades, precoding matrix design problems have received considerable attention in nonCR settings for situations where only partial CSI such as mean or covariance information of the channel is available at the transmitter [10,11]. However, the literature is generally not applicable to CR networks because of the additional constraint on PU interference in CR settings. In this paper, we propose a closed-form precoding matrix design method for a CR network. The aim of the method is to maximize the upper bound of the expected achievable rate of the SU in situations where the CSI for the SBS-PU link is perfectly known at the SBS transmitter, but only the covariance information of the SBS-SU link channel is available at the SBS transmitter. The rest of this paper is organized as follows. In Section 2, we present the system model and problem formulation. In Section 3, we explain our proposed precoding matrix design method. In Section 4, we present the simulation results. Finally, in Section 5, we present our conclusion.
3057
channel distribution, can be written as h i C exp ðFÞ ¼ EH log2 9IN þ R1=2 HFFH HH R1=2 9 z z
ð3Þ
where EH ½U is the statistical expectation operator with respect to H. If we denote the maximum transmit power at the SBS by Pth , the SBS transmit power constraint can be written as Es ½sH FH Fs ¼ trfFEs ½ssH FH g ¼ trfFFH g r Pth
ð4Þ
Since we consider the case the PU has one receive antenna, the channel response from the SBS to the PU is represented by an M 1 vector g and the interference signal from the SBS to the PU is given by gH Fs. Let us define an interference power threshold by Gth , then the interference power constraint at the PU is given by E½gH FssH FH g ¼ gH FE½ssH FH g ¼ gH FFH g r Gth
ð5Þ
Therefore, the maximization problem of C exp ðFÞ in a CR network can be formulated as h i max EH log2 9IN þ R1=2 HFFH HH R1=2 9 z z F
subject to trfFFH g r Pth ,
gH FFH g r Gth
ð6Þ
3. Proposed precoder design method 2. System model and problem formulation In this paper, we consider a point-to-point single-user communication system from an SBS to an SU in a CR network. We assume the spectrum is licensed to only one PU. We also assume that the SBS transmitter has M transmit antennas, the SU receiver has N receive antennas, and the PU has a single receive antenna. In addition, we assume that the full CSI of the SBS-PU link is known at the SBS transmitter, but only the covariance information of the SBS-SU link channel is known at the SBS transmitter. The channel between the SBS transmitter and the SU receiver is expressed by an N M matrix H. The input signal at the SBS transmitter is represented by a complex Gaussian distributed vector, s, of size J 1 with mean E½s ¼ 0 and covariance E½ssH ¼ IJ , where IJ is a J J identity matrix. The selection of the parameter J should be based on consideration of the rank of the SBS-SU link channel matrix, i.e., J ¼ rankðHÞ. An M J precoding matrix, F, is applied to the input signal vector, s. Then, the received signal at the SU receiver is represented by y ¼ ½y1 ,y2 ,. . .,yN T ¼ HFs þ z
ð1Þ
where z is the sum of the signal from the primary network and the additive white Gaussian noise (AWGN). We assume that z has a colored Gaussian distribution with mean E½z ¼ 0 and covariance E½zzH ¼ Rz . The achievable rate of the SU is given by [12] 9 CðFÞ ¼ log2 9IN þRz1=2 HFFH HH R1=2 z
ð2Þ
where 9U9 is the determinant operator. The expected value of CðFÞ for the random matrix H, averaged over the
The channel between the SBS transmitter and the SU receiver is assumed to be correlated at the transmit antennas, but uncorrelated at the receiver antennas. This model is appropriate for an environment where the SBS transmitter is not impeded by local scatterers and the SU receiver is surrounded by local scatterers [13]. The SBS-SU link channel is therefore modeled as [13] H ¼ Hw R1=2
ð7Þ
where Hw is a complex Gaussian random matrix of size N M with mean 0 and variance s2h IN and R is an M M transmit antenna covariance matrix. We assume that the SBS can know R by using the pilot signals aperiodically and sparsely transmitted from the SU. If we substitute (7) into (6), the constrained optimization problem can be rewritten as h i 1=2 max EHw log2 9IN þ R1=2 Hw R1=2 FFH R1=2 HH 9 z w Rz F
subject to trfFFH g r Pth ,
gH FFH g r Gth
ð8Þ
It is impossible to find the closed-form optimal solution of this original problem Hw. Therefore, in order to obtain the closed-form solution, we resort to the upper bound optimization solution of the expected achievable rate. From Jensen’s inequality [14] and the determinant identity 9IN þ AB9 ¼ 9IM þBA9 for the N M matrix A and M N matrix B, the upper bound C exp ðFÞ of the expected achievable rate is given by 1=2 1 FFH R1=2 9 C exp ðFÞ r log2 9IM þEHw ½HH w Rz Hw R
¼ log2 9IM þ s2h bR1=2 FFH R1=2 9 ¼ C exp ðFÞ
ð9Þ
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where b ¼ trfR1 z g and the following relation are used: 1 EHw ½HH w Rz Hw ¼
2 h bUIM
s
ð10Þ
The derivation of (10) is given in Appendix A. We assume that the SU reports b to the SBS through the feedback channel and therefore the SBS knows b. Now, the upper bound maximization problem of the expected achievable rate can be simplified as max log2 9IM þ s
subject to trfFFH g r P th ,
s2h blm @ 2 Lðq1 ,. . .,qM , m, rÞ ¼ 0 ) mr9wm 9 ¼ 0, @qm 1 þ s2h blm qm m ¼ 1,2,. . .,M, M X
ð17Þ
qm Þ ¼ 0,
rðGth
m¼1
gH FFH g r Gth
ð11Þ
The PU interference power constraint in (11) complicates finding an optimal closed-form solution for this optimization problem [6]. Thus, we propose a suboptimal solution with the closed-form in this paper. Let us define the eigenvalue decomposition of R as R ¼ UKUH , where the columns of U are eigenvectors of R with the property U UH ¼ UH U ¼ IM and K ¼ diagðl1 , l2 ,. . ., lM Þ is an eigenvalue matrix with l1 Z l2 Z Z lM . Also, without loss of generality, the precoding matrix can be written as F ¼ U U for any M J matrix U. The upper bound of the expected achievable rate can then be rewritten as C~ exp ðUÞ ¼ log2 9IM þ s2h bU K1=2 UUH K1=2 UH 9 i ¼ log2 9IM þ s2h bK1=2 UUH K1=2 9 ,
ð12Þ
Furthermore, the transmit power and PU interference constraints can be written as trfU UUH UH g ¼ trfUUH g rP th ,
listed as follows:
mðPth
1=2 2 FFH R1=2 9 h bR
F
where m and r are Lagrangian coefficients with m Z 0 and r Z 0. The Karush–Kuhn–Tucker (KKT) conditions are
wH UUH wr Gth
M X
2
9wm 9 qm Þ ¼ 0
ð18Þ
m¼1
From (17), the solution for qm is given by " #þ 1 1 qopt,m ¼ , m ¼ 1,2,. . .,M m þ r9wm 92 s2h blm
ð19Þ
where ½x þ ¼ maxðx,0Þ. The Lagrangian coefficients m and r should be chosen to satisfy the following constraints: " #þ L X 1 1 rP th , ð20Þ 2 s2h blm m ¼ 1 m þ r9wm 9 M X m¼1
9wm 9
2
"
1
1 m þ r9wm 92 s2h blm
#þ r Gth
ð21Þ
One simple method of selecting m and r is to use the bisection method for the two parameters [6]. After Q opt ¼ diagðqopt,1 ,qopt,2 ,. . .,qopt,M Þ has been obtained, the expected achievable rate of the proposed scheme is given by h i 1=2 9 ð22Þ C pro ¼ EHw log2 9IN þRz1=2 Hw R1=2 Q opt R1=2 HH w Rz
ð13Þ
H
respectively, where w ¼ U g. After Q ¼ UUH is defined and the determinant propQ erty 9A9 r m ½Am,m is used for any non-negative definite matrix A [15], where [A]m,n is the (m,m) th element of A , the following inequality is satisfied: 9IM þ s2h bK1=2 Q K1=2 9 r
M Y
ð1 þ s2h blm qm Þ
ð14Þ
m¼1
where qm ¼ ½Q m,m and the equality is satisfied when Q is diagonal, i.e., Q ¼ diagðq1 ,q2 ,. . .,qM Þ. Because C~ exp ðQ Þ has its maximum value when Q is diagonal, we confine Q to the diagonal matrix in order to find the closed-form solution. The problem formulation for the suboptimal closed solution can now be written as max
q1, ...,qM
M X
log2 ð1 þ s2h blm qm Þ
m¼1
subject to
M X
qm rP th ,
m¼1
M X
2
9wm 9 qm r Gth
ð15Þ
m¼1
A Lagrangian function of this optimization problem is given by Lðq1 ,. . .,qM , m, rÞ ¼
M X
log2 ð1 þ s2h blm qm Þ
m¼1
þ mðP th
M X m¼1
qm Þ þ rðGth
M X m¼1
2
9wm 9 qm Þ
ð16Þ
4. Simulation results The simulations are performed for the antenna combinations M ¼ N ¼ 2 and 4. Each element of Hw and g is generated by an independent and circularly symmetric complex Gaussian random variable distributed as CNð0,1Þ. For the transmit antenna covariance matrix, R, of the SBSSU link channel, the exponential correlation model [16] is used, i.e., ½Rm,n ¼ a9mn9 , 0 o a o1, where a means the channel correlation coefficient. The covariance matrix, Rz , of the additive noise, z, is set to IN for simplicity, and J is chosen as J ¼ rankðHÞ. The SBS transmit power constraint, P th , sweeps from 1 to 100, which is equivalent to a range of signal-to-noise ratios (SNRs) per receive antenna from 0 dB to 20 dB. The simulation results are implemented over 2000 randomly generated channel pairs ðHw ,gÞ, and the expected achievable rates are plotted against the SNR values. Fig. 1 compares the expected achievable rates of the conventional schemes and the proposed scheme for when M ¼N ¼2, Gth ¼ 0:5, and a ¼ 0:1,0:9. Fig. 2 represents the expected achievable rates for M ¼N ¼4. In the figures, the term ‘Full CSI’ represents the conventional scheme with full CSI [6], the term ‘Partial CSI (Proposed)’ represents the proposed scheme with partial CSI. Furthermore, the term ‘W/o PU constraint’ denotes the non-CR scheme without consideration of the PU interference constraint [15]. The figures show that the performance of the proposed scheme is about 0.5 dB worse than that of the conventional scheme
B. Seo / Signal Processing 92 (2012) 3056–3061
3059
Expected value of the achievable rate (bps/Hz)
11 (α=0.1) W/o PU constraint (α=0.1) Full CSI (α=0.1) Partial CSI (Proposed) (α=0.9) W/o PU constraint (α=0.9) Full CSI (α=0.9) Partial CSI (Proposed)
10 9 8 7 6 5 4 3 2 1
0
5
10 SNR (dB)
15
20
Fig. 1. Expected achievable rate comparison when M ¼ N ¼2 and Gth ¼ 0:5.
Expected value of the achievable rate (bps/Hz)
22 (α=0.1) W/o PU constraint (α=0.1) Full CSI (α=0.1) Partial CSI (Proposed) (α=0.9) W/o PU constraint (α=0.9) Full CSI (α=0.9) Partial CSI (Proposed)
20 18 16 14 12 10 8 6 4 2
0
5
10 SNR (dB)
15
20
Fig. 2. Expected achievable rate comparison when M ¼ N ¼4 and Gth ¼ 0:5.
with full CSI for the same expected achievable rate when the channel correlation coefficient, a, is 0:1 and M ¼ N ¼ 4. However, by comparing the curves for a ¼ 0:1 and a ¼ 0:9, it can be observed that the performance gap between the two schemes decreases as the channel correlation coefficient increases. Fig. 3 shows the expected achievable rate for the primary base station (PBS)-PU link. The received signal
model at the PU and an achievable rate analysis for the PBS-PU link is given in Appendix B. From the figure, it can be observed that the achievable rate for the PBS-PU link is degraded due to the presence of interference from the SU compared to the case without the interference and this degradation is natural in cognitive radio networks. However, the performance degradation for the PBS-PU link gets smaller as Gth decreases.
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B. Seo / Signal Processing 92 (2012) 3056–3061
Expected achievable rate for PU (bps/Hz)
3.5 Without interference from SU With interference from SU (Γth=0.1) 3
With interference from SU (Γ =0.2) th
With interference from SU (Γth=0.5) 2.5
2
1.5
1
0.5
0
5
10 SNR (dB)
15
20
Fig. 3. Comparison of the expected achievable rates for the PBS-PU link.
(
5. Conclusion
¼
In this paper, we propose a closed-form precoding matrix design method for maximizing the upper bound of the expected achievable rate of the SU for when the CSI of the SBS-PU link is perfectly known at the SBS transmitter, but only the covariance information of the channel of the SBS-SU link is known at the SBS transmitter. Even though the proposed scheme uses only partial CSI of the SBS-SU link, its expected achievable rate approaches to that of a conventional scheme requiring full CSI of the SBS-SU link as the channel correlation increases. Appendix A. Derivation of (10) If we define ðm,nÞth element of Hw by hm,n , we know n that Eh ½hm,n hm0 ,n0 ¼ s2h dðmm0 Þdðnn0 Þ where dðmÞ is the Kronecker delta function because Hw is a complex Gaussian random matrix of size N M with mean 0 and variance s2h IN . We define ðm,nÞth element of R1 z by qm,n and partition Hw into its subvectors, i.e., Hw ¼ ½h1 ,h2 ,. . .,hM where hm ¼ ½hm,1 ,hm,2 ,. . .,hm,N T . Then ðm,nÞth element of EHw 1 ½HH w Rz Hw , dm,n , is given by ( ) " # N N X X H 1 n dm,n ¼ Eh ½hm Rz hn ¼ Eh ðhl,m vl,k Þ hk,n k¼1
¼
N X N X
n
vl,k Eh ½hl,m hk,n ¼
k¼1l¼1
¼
N X k¼1
s2h b if m ¼ n 0
otherwise
where vl,k ¼ ½R1 z l,k
1 we obtain EHw ½HH w R z Hw ¼ s
Appendix B. Signal model at PU and an expected achievable rate for PBS-PU link Let us define the input signal at PBS transmitter by a complex Gaussian distributed random variable sPU with zero mean and unit variance. Then, the received signal at the PU can be written as: yPU ¼ hPU sPU þ gH F s þ ZPU
N X N X
vl,k s2h dðlkÞdðmnÞ
1 2 h trðR z ÞdðmnÞ ¼
s
ð23Þ
where hPU represents a channel response for the PBS-PU link and ZPU is AWGN with zero mean and variance s2Z . In (23), gH F s means the interference from the SU. The achievable rate of the PU is given by [12] C PU ¼ log2 1þ
9hPU 9
!
2
ð24Þ
gH F FH g þ s2Z
The expected value of C PU for the random vector g and random variable hPU , averaged over the channel distribution, can be written as "
vk,k s
vk,k ¼ trðR1 z Þ ¼ b. Finally, k¼1 2 h bUIM for any distribution of z.
l¼1
k¼1l¼1
2 h dðmnÞ ¼
N P
and
2 h bdðmnÞ
s
C PU,exp ¼ Eg,hPU log2 ð1 þ
#
2
9hPU 9
gH F FH g þ s2Z
Þ
ð25Þ
B. Seo / Signal Processing 92 (2012) 3056–3061
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