H. Zamiri-Jafarian1 and M. Abbasi Jannat-Abad1,2. 1 Electrical Engineering Department, Ferdowsi University of Mashhad. 2 Communications and Computer ...
Cooperative Beamforming and Power Allocation in the Downlink of MIMO Cognitive Radio Systems H. Zamiri-Jafarian1 and M. Abbasi Jannat-Abad1,2 1
2
Electrical Engineering Department, Ferdowsi University of Mashhad Communications and Computer Research Center, Ferdowsi University of Mashhad
Abstract—In this paper, a new algorithm is proposed for cooperative beamforming and power allocation (CBPA) in the downlink of multi-input multi-output (MIMO) cognitive radio systems. Increasing the efficiency of spectrum usage is the main goal of the cognitive radio systems by servicing a number of secondary users in addition to the primary user. The proposed CBPA algorithm is developed based on a criterion in which the total signal to interference plus noise ratio (SINR) of the secondary users is maximized subject to guarantee the primary user's quality of service and a maximum transmit power. In addition, the CBPA algorithm provides the same quality of service for all secondary users. The transmitter and receiver beamforming vectors along with allocated powers to users are estimated by the CBPA algorithm in an iterative manner. The performance of the CBPA algorithm is evaluated by computer simulations and compared with that of the uplink-downlink duality method. Index Terms— Cognitive radio, MIMO systems, cooperative beamforming, power allocation
I.
INTRODUCTION
To increase spectrum usage efficiency, two kinds of users are serviced in cognitive radio (CR) systems, primary user (PU) and secondary user (SU). Although the bandwidth originally licensed to the PU, when the PU does not occupy the bandwidth, it can be allocated to the SUs. By making the use of licensed bandwidth possible for SUs, the efficiency of spectrum usage is increased in the CR systems [1]. Due to PU’s priority in bandwidth usage, the main challenge is to guarantee the PU’s quality of service (QoS), subject to efficient use of spectrum. Joint beamforming and power allocation methods have been proposed under different criteria in recent years to mitigate this challenge. In [2], a joint beamforming and power allocation method was proposed for uplink of single input multi-output (SIMO) CR system under two criteria. In the first one, the sum of SUs' capacity is maximized subject to a threshold for PU's interference temperature and a maximum allowable transmit power for each SU. In the second one, the minimum SU’s signal to interference plus noise ratio (SINR) is maximized under constraints similar to the first criterion. In [3], another joint beamforming and power allocation algorithm has been proposed for downlink of multi-input single output (MISO) CR system, in which transmit power is minimized subject to a PU's
interference temperature limit and SUs' SINR thresholds. In [4], the authors have studied joint beamforming and power allocation under two different scenarios, in which the sum of SUs' bit rates is maximized under a limit of interference temperature for the PU and a threshold for SUs' SINR constraints. In the first scenario, each user is supported by a required QoS. In this case, a solution has been given based on nonlinear programming method. In second scenario, each SU whose SINR is below a threshold is not serviced. In this context, by defining SUs' priority classes, the SUs, which maximize system revenue function, are selected to be serviced In this paper, we propose a new algorithm for cooperative beamforming and power allocation (CBPA) in the downlink of MIMO CR systems. The proposed CBPA algorithm is developed under a criterion with three constraints. In the criterion of the CBPA algorithm, total SINR of all SUs is maximized subject to 1) a maximum allowable transmit power, 2) a minimum threshold for PU’s SINR in order to guarantee the required PU’s QoS and 3) providing the same SINR for all SUs. A two-step iterative CBPA algorithm estimates the transmitter and receiver beamforming vectors of the PU and SUs by maximizing the total SINR and also computes allocated powers to the PU and SUs by applying the given three constraints. In the following, a MIMO CR system is modeled based on one PU and K SUs in Section II. The CBPA algorithm is developed in Section III. Computer simulation results are presented in Section IV and Section V concludes the paper.
II.
SYSTEM MODEL
A downlink of MIMO CR system with one primary user (PU) and K secondary users (SU) is shown in Fig. 1 where the base station (BS) , PU and ith SU are equipped by antenna arrays with N , M p and M s i elements, respectively. When
s p and s si are transmitted signals for the PU and the ith SU, respectively, the transmitted signal from the BS is given as K
x = v p s p + ¦ v si ssi i =1
(1)
where s p and s s i are independent and identically distributions (i.i.d.) signals with normalized energy. Also, v p and v s i are
978-1-4244-3574-6/10/$25.00 ©2010 IEEE
§ K G3i = u sHi H si ¨¨ ¦ v s j v sHj © j =1, j ≠ i =
¦
H sj
transmitted powers of the PU and the ith SU, respectively. After applying u p ( PU’s receiver beamforming vector) and
usi ( ith SU’s receiver beamforming vector), the received signals of the PU, y p , and the ith SU, y si , are given as
j =1
ysi = u sHi H si v p s p + u sHi H si v si ssi +
K
¦
j =1, j ≠ i
(3)
u sHi H si v s j ss j + u sHi n si ,
COOPERATIVE BEAMFORMING AND POWER ALLOCATION ALGORITHM
K
TSINR =
¦G i =1
K
¦ (G i =1
(2)
where H p is the channel between the BS and the PU and H s i is the channel between the BS and the ith SU. Also, n p and
H si
Increasing the efficiency of spectrum usage is the main goal of the cognitive radio system. In this section, a new cooperative beamforming and power allocation (CBPA) algorithm is proposed to improve spectrum usage efficiency. The CBPA algorithm is developed based on a criterion in which total SINR (TSINR) of the SUs is maximized in order to service more users. Also, PU’s QoS is guaranteed by means of defining a threshold for the SINR PU . The TSINR is defined as
K
y p = u Hp H p v p s p + ¦ u Hp H p v s j ss j + u Hp n p ,
H si
v H u si u H si v s j
III.
the PU's and ith SU's transmit beamforming vectors, respectively, such that p p = v Hp v p and p s i = v sHi v s i are
2i
1i
+ G3i + N si
SINR PU =
,
K
¦ v sHj H Hp u p u Hp H p v s j + N p
(4)
j =1
SINR SU i = G1i
CCBPA =
arg max Vs , Us , v p , u p
{TSINR} ,
2i
+ G 3i + N s i
)
for i = 1," , K
(5)
where N p = E ª¬u Hp n p n Hp u p º¼ and N si = E ª¬u sHi n si n sHi u si º¼ . Also, G1i , G2i and G3i are defined as G1i = v H u si u H si v si
(6)
G2i = v Hp H sHi u si u sHi H si v p
(7)
H si
H si
H si
K ° p p + ¦ p s i = p max ° i =1 ° s .t . ®SINR PU ≥ γ p ° °SINR SUi = SINR SUj ° for i = j = 1," , K ¯
(10)
where VS = ª¬ v s1 , v s2 ,", v sk º¼ and U S = ª¬u s1 , u s2 ,", u sk º¼ . We employ a two-step algorithm in order to maximize the TSINR. At the first step, it is assumed that U S matrix and u p vector are known and VS matrix and v p vector are estimated by maximizing the TSINR. After doing some manipulations, from (8) we have K
(G
)
the same SINR for all SUs.
as v Hp H Hp u p u Hp H p v p
(9)
The criterion of the CBPA algorithm, CCBPA , is defined based on maximizing the TSINR subject to a maximum transmit power, Pmax , a threshold of the SINR PU , γ p , and providing
nsi are zero mean additive white Gaussian noise (AWGN) vectors. Meanwhile, E ª¬n p n Hp º¼ = N p I M p and E ª¬n si n sHi º¼ = N si I M s , i where Ι N is N × N identity matrix and (.)H represents complex conjugate operation. Based on (2) and (3), the SINR of the PU, SINR PU , and the SINR of the ith SU, SINR SUi , are defined
(8)
K
j =1, j ≠ i
Fig. 1. The model of cognitive radio system.
· H ¸¸ H si u si ¹
¦G i =1
3i
K § K + N si = ¦ u sHi H si ¨¨ ¦ v s j v sHj i =1 © j =1, j ≠ i
· H ¸¸ H si u si + N si ¹
K § K Ns = ¦ v Hsi ¨ ¦ H sHm u sm u sHm H sm + i I M s i ¨ psi i =1 © m =1, m ≠ i
· ¸v s j ¸ ¹
(11)
By using the following decomposition K
¦
m =1, m ≠ i
H sHm u sm u Hsm H sm +
Ns i ps i
I M s = ī iH ī i i
(12)
we have
psi Ai
K
K
G3i + N si = ¦ v sHi ΓiH Γi ¦ i =1 i =1
K
v si = ¦ fiH fi
(13)
i =1
p p Bi +
K
ps j Ci , j + N si
¦
j =1, j ≠ i
psi Ai +1
= p p Bi +1 +
K
¦
j =1, j ≠ i +1
ps j Ci +1, j + + N si +1
(21)
i = 1," , K − 1
for
where fi = Γi v si for i = 1," , K . By substituting v si = īi−1fi in (6), G1i becomes G1i = fi
(ī
H
−H i
H si
H si
H u si u H si ī
−1 i
)f
(14)
i
Meanwhile, by using the eigenvalue decomposition of K
H ¦ i 1 =
H si
u si u H si = QȁQ H si
H
and also substituting (13) and (14)
in the TSINR, we have K
¦1 f ( ī H i
TSINR =
−H i
i=
)
H sHi u s i usHi H si īi−1 fi K
(15)
i =1
To maximize the TSINR, fi should be proportional to eigenvector of Φ i = īi− H H sHi u s i u sHi H s i īi−1 corresponding to the maximum eigenvalue of Φ i and v p should be proportional to the eigenvector of QȁQ H corresponding to the minimum eigenvalue of QȁQ H that we note it q min [5]. Since the Φ i matrix has only one non-zero eigenvalue, we have fi = ī i− H H sHi u s i
psi
vp =
H i
īi
)
−1
H sHi us i
(ī
H i
īi
)
−1
Bi = w Hp H sHi u si
,
2
and
2
. Note that if the minimum value of
p p obtained from the above equations becomes negative value, the BS sets p p = pmax and only services the PU.
estimating U S matrix and u p vector. By assuming to know VS and v p , from (6), (7) and (8) the G1i , G2i and G3i can be written as G1i = u sHi H si v si v sHi H sHi u si
(22)
G2i = u sHi H si v p v Hp H sHi u si
(23)
G3i =
K
u Hsi H si v s j v sHj H sHi u si ¦ j 1, j i =
(24)
≠
By following a procedure similar to (11)-(14) and doing some manipulations, the TSINR becomes K
(16) TSINR =
Therefore, v s i and v p become
(ī
Ci , j = w sHj H sHi u si
2
At the second step, we assume that VS matrix and v p vector are known and the TSINR is maximized by
v Hp QȁQ H v p + ¦ fiH fi
vs i =
Ai = w sHi H sHi u si
where
g iH ( ϒ i− H H si v s i vsHi H sHi ϒ i−1 ) g i ¦ i =1
(25)
K
¦ g Hj g j i =1
H sHi us i
for i = 1," , K
(17) (18)
p p q min
where ϒ i and g i for i = 1," , K are computed from the following relations. H s i v p v Hp H sHi +
K
¦
m =1, m ≠ i
H si v s m v sHm H sHi + N s i I M s = ϒ iH ϒ i i
(26)
By defining vsi = p si wsi and v p = p p w p , the estimations of
g i = ϒ i u si
normalized w si and w p vectors can be obtained from (17) and (18), respectively. By substituting (17) and (18) in the constraints of (10), K + 1 unknown parameters p p and p s i (for
By maximizing the TSINR, the normalized receiver beamforming vectors, u si for i = 1," , K , are given as
i = 1," , K ) can be obtained by solving the following
u si =
equations.
(27)
1
(
)
−1 ϒ iH ϒ i H si v si
(ϒ
)
−1 H H si v si i ϒi
for i = 1," , K
(28)
K
p p + ¦ p s i = p max
(19)
i =1
To estimate u p , from (4) and (10) we can write the PU’s SINR constraint as
2
K
p p w Hp H Hp u p − ¦ psi γ p w sHi H Hp u p i =1
2
= N pγ p
(20)
ª §K ·º u Hp « H p v p v Hp H Hp − γ p ¨ ¦ H p v si v sHi H Hp +N p I M p ¸ » u p ≥ 0 © i =1 ¹¼ ¬
By using the following decomposition
(29)
§K · H p v p v Hp H Hp − γ p ¨ ¦ H p vs i v sHi H Hp +N p I M p ¸ = DǻDH © i =1 ¹
(30)
0
10
-1
10
We can obtain u p as
-2
10
u p = d min
(31)
-3
bit error rate
10
where d min is the eigenvector of DΔD H corresponding to the minimum eigenvalue of DΔD . Note that p p and p s i (for H
-5
10
i = 1," , K ) have been chosen in (20) such that the right hand
-6
side of (29) does not become negative value.
10
The CBPA algorithm is estimated transmitter and receiver beamforming vectors of the PU and SUs based on the proposed two-step algorithm in an iterative manner. It should be noted that in the second step of the CBPA algorithm, the receiver beamforming vectors, u si and u p are estimated directly by
10
maximizing the TSINR. Another approach that can be used for the second step is to estimate the u si and u p by employing
-7
-8
10
format. Meanwhile, the number of SUs is K = 2 and the threshold value of the SINR PU is γ p = 20dB in simulations.
5
10
15
20 SNR (dB)
25
30
35
PU , SU , PU , SU , PU , SU , PU , SU , PU , SU ,
3
SINR
10
2
10
4-1-1 4-1-1 6-1-1 6-1-1 8-1-1 8-1-1 4-2-2 4-2-2 6-2-2 6-2-2
1
10
0
10
0
5
10
15
20 SNR (dB)
25
30
35
Fig. 3. The SINRs of the PU and SU in MISO and MIMO systems with different number of antennas.
Also, in all situations, the number of iterations that is used in the CBPA algorithm for obtaining the transmitter and receiver beamforming vectors is twenty.
100 90
SU percent of power allocation
80
The bit error rates (BERs) and SINRs of the PU and SUs are shown in Fig. 2 and Fig. 3 for the MISO and MIMO channels with different number of antennas. Due to having the same performance for both SUs, only the BER and SINR of one SU are indicated in Fig. 2 and Fig. 3. As it's seen, the BER of the PU is decreased when no power is allocated to the SUs; for instance in the 4-2-2 (N=4, M p = 2 and M s1 = M s 2 = 2 ) situation, no power is allocated to the SUs before SNR=10dB, thus the BER of the PU is improved by increasing the SNR. For SNR ≥ 10dB , although the average SINR of the PU is increased, the BER of the PU is increased due to errors taken place in some channel realizations in which the power is allocated to the SUs. Note that in all situations, the
40
4
10
COMPUTER SIMULATION RESULTS
In this section, simulation results are presented to evaluate the performance of the proposed CBPA algorithm. In the simulations, we use flat Rayleigh fading MISO and MIMO channels for PU and SUs such that the elements of the channel matrices are independent and have zero mean Gaussian distributions with normalized variance. The modulation is QPSK and results are obtained for 100000 realizations of channels. In the figures, the number of antennas employed in the BS, PU and SU is indicated based on “N - M p - M s i ”
0
4-1-1 4-1-1 6-1-1 6-1-1 8-1-1 8-1-1 4-2-2 4-2-2 6-2-2 6-2-2
Fig. 2. The BER performances of the PU and SU in MISO and MIMO systems with different number of antennas.
uplink-downlink duality method that has been proposed in [6] based on the duality between uplink and downlink in MIMO channels. We consider the performance of this method (called “duality”) and compare with that of our proposed approach (called “direct”) by simulations in the next section.
IV.
PU , SU , PU , SU , PU , SU , PU , SU , PU , SU ,
-4
10
70 60 50 40 30 20 10 0
0
5
10
15
20 25 SNR (dB)
30
35
40
Fig. 4. The percent of time that the BS services the SU in MISO system with N = 4 .
40
performance of the PU is equal or larger than the threshold such that at high SNRs, the BER of the PU remains constant, but the BER performances of the SUs are decreased. To highlight the behavior of the CBPA algorithm in power allocation to the SUs, the percent of time that the BS services the SUs is shown in Fig. 4 for the MISO channel with N = 4 . As it can be seen, the total transmit power is allocated to the PU at low SNR in order to satisfy the PU’s SINR constraint. However, by increasing the SNR, more frequently time the BS services the SUs. For MIMO channels, the receiver beamforming vectors are also derived based on uplink-downlink duality method [6] for the second step of the CBPA algorithm (indicated by “duality” ) in addition to the proposed direct approach in Section III (indicated by “direct”). In Fig. 5, the BER performance of the direct and duality approaches are compared with different number of antennas. As it's shown, the direct approach outperforms the duality approach.
V.
0
10
-1
10
-2
10
-3
bit error rate
10
-4
10
PU , SU , PU , SU , PU , SU , PU , SU ,
-5
10
-6
10
-7
10
-8
10
0
5
10
15
20 SNR (dB)
25
30
4-2-2 , 4-2-2 , 6-2-2 , 6-2-2 , 4-2-2 , 4-2-2 , 6-2-2 , 6-2-2 , 35
direct direct direct direct duality duality duality duality 40
Fig. 5. The BER performances of the “direct” and “duality” approaches in the MIMO channels with different number of antennas.
CONCLUSION
A cooperative beamforming and power allocation (CBPA) algorithm has been proposed in this paper for downlink of MIMO cognitive radio systems. The CBPA algorithm estimates the transmitter and receiver beamforming vectors of primary user (PU) and secondary users (SUs) along with allocated powers to them. In the CBPA algorithm, the total signal to interference plus noise ratio (SINR) of all SUs is maximized under a maximum allowable transmit power, a threshold SINR for the PU and the same SINR for all SUs constraints. The performance of the CBPA algorithm has been evaluated by computer simulations. The results indicated that the CBPA algorithm, in addition to guarantee a required performance of the PU, increases spectrum usage efficiency by servicing the SUs. Also, our proposed approach that estimates the receiver beamforming vectors directly by maximizing the total SINR outperforms the previously proposed uplinkdownlink duality method.
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ACKNOWLEDGMENT The authors would like to thank Iran Telecom Research Center (ITRC) for patialy supportting this work.
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