Distributed joint power control and beamforming algorithms

Int. J. Electron. Commun. (AEÜ) 65 (2011) 235–238

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International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.de/aeue

LETTER

Distributed joint power control and beamforming algorithms Jui Teng Wang ∗ Graduate Institute of Communication Engineering, National Chi Nan University, 1 University Road, Puli, Nantou 545, Taiwan

a r t i c l e

i n f o

Article history: Received 24 April 2009 Accepted 15 February 2010 Keywords: Power control Beamforming Feedback Broadcast Overhead

a b s t r a c t Two distributed joint power control and beamforming algorithms, namely, the power feedback based algorithm and the broadcast based algorithm, are proposed. The power feedback based algorithm can dispense with the redundancy of power computation that occurs in the algorithm in Rashid et al. (1998) and Olfat et al. (2005) [1,2]. Also, the power feedback based algorithm and the broadcast based algorithm can both consume less overhead of feedback than the algorithm in Rashid et al. (1998) and Olfat et al. (2005) [1,2]. © 2009 Elsevier GmbH. All rights reserved.

1. Introduction Cochannel interference, which is inevitable in wireless networks due to the frequency reuse, can decrease the received signal to interference and noise ratio (SINR). Two techniques are commonly used for enhancing the SINR in wireless networks with cochannel interference: beamforming and power control. Beamforming acts as a spatial filter that aims to maintain constant gain at the direction of the target signal and minimizes the cochannel interferences that come from other directions. On the other hand, power control is a technique that aims to appropriately adjust the power levels for all users so that the cochannel interference can be suppressed and the SINR requirement can be satisfied. Beamforming and power control can be jointly operated to further improve the performance as in [1–3]. An algorithm, namely, Algorithm A, was proposed in [1] to find the optimal feasible power and weight for joint power control and beamforming. To make Algorithm A work in practice, the authors of [1] further suggest a distributed algorithm to implement Algorithm A. This distributed algorithm was also used in [2] for the OFDM system. In the distributed algorithm in [1,2], each user computes the updated power according to the interference received at the base station, however, the base station also needs to compute the power of the user to determine the interference, as a result, there is a redundancy of power computation. In this paper, we first propose a power feed-

∗ Tel.: +886 49 2910960. E-mail address: [email protected]. 1434-8411/$ – see front matter © 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2010.02.011

back based algorithm in which the updated power of the user is determined by the base station so that the base station always knows the power of the user. Compared to the distributed algorithm in [1,2], the power feedback based algorithm can dispense with the redundancy of power computation and reduce the overhead (in terms of the number of parameters to be sent). In addition, we further propose a broadcast based algorithm that broadcasts a set of parameters to all users in the same cell so that each user can use these parameters to determine its updated power. Numerical results show that the broadcast based algorithm can outperform the distributed algorithm in [1,2] in the saving of the overhead for the CDMA networks.

2. System model Assume that each user i is equipped with transmitter i and receiver i on opposite sides of the link, each transmitter has one antenna element, and each receiver has M antenna elements. Let Pi and si represent the transmitting power and the message signal of j j user i, respectively. Also, let xi and wi represent the total received signal and the weight at the jth antenna element of receiver i. Then the output of the combiner for receiver i is given by

yi =

M  j=1

j

∗

j

(w) x i

i

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J.T. Wang / Int. J. Electron. Commun. (AEÜ) 65 (2011) 235–238

Note that



j

xi =

j

j

li

i

Pl Gli asl + n

l

where Gli denotes the link gain between transmitter l and receiver j

i, ali denotes the array response between transmitter l and receiver j

i at the jth antenna element, and ni denotes the noise at the jth antenna element of receiver i. Furthermore, the received signal of receiver i at the jth antenna element that comes from transmitter i j is denoted by di , which can be expressed as j

di =



Step 1: Let Pi0 = P0 . Step 2: Measure xm and calculate ˚m by i i

The SINR for receiver i is given by E(wH d dH w i ) i i i H H [E(wi xi xi wi ) − E(wH d dH wi )] i i i j

j

=

wH ˝i wi i H wi ˚i wi − wH ˝i wi i

H

= E[xm (xm ) ] ˚m i i i Step 3: Calculate the optimal weight vector by

j

where wi = {wi }, xi = {xi }, di = {di }, ˚i = E(xi xH ) and ˝i = E(di dH i ). i Note that ˚i and ˝i are the correlation matrixes for the total received signal and the received signal of interest, respectively. Assume that the message signals are uncorrelated with zero mean and E(|si |2 ) = 1, then we have ˝i = Pi Gii aii aH ii and ˚i =



H

j

where ali = {ali } and Ni denotes the noise power at receiver i. Therefore, we have Pi Gii wH a aH w i ii ii i H

H

H

i

li

i

(1)

Pl Gli wali awi + Ni wwi

l= / i

Let Ii denote the total interference and noise for receiver i, then we also have Ii =

 l= / i

H

H

H

i

li

i

Pl Gli wali awi + Ni wwi

The minimum variance distortionless response (MVDR) beamforming is accomplished by minimizing the interference and noise subject to wH a = 1. It was reported in [1] that the weight for the i ii MVDR beamforming is given by ˜i = w

(˚m − ˝im ) i aH (˚m i ii

−1

aii

−1 − ˝im ) aii

where ˝im = Pim Gii aii aH . ii Step 4: Compute the total interference by H

li



= wm i

Iim = (wm ) (˚m )wm − Pim Gii i i i

Pl Gli ali a + Ni I

l

i =

3.1. DJPCB algorithm

j

Pi Gii aii si

i =

A. This distributed algorithm was also used in [2] for the OFDM system. For convenience, such a distributed algorithm is called the distributed joint power control and beamforming (DJPCB) algorithm in this paper. The DJPCB algorithm uses only local information to iteratively adjust the power and weight of each individual user. In the following, we give the description for the DJPCB algorithm , xm , ˝im , that works in the uplink. For this description, we let ˚m i i m ,  m and I m denote the m th-discrete-time value for ˚ , wm , P i i i i i xi , ˝i , wi , Pi , i and Ii , respectively, and we let i denote the SINR requirement of user i.

(˚i − ˝i )

−1

aH (˚i − ˝i ) ii

i determines Pim+1 as per the value of Iim . It is observed that the DJPCB algorithm has the problem below: the base station needs to know Pim to calculate wm and Iim . To overcome this probi lem, we can let the base station execute Step 6 to obtain the value of Pim . However, Step 6 is also executed by each user, thus, there is a redundancy of power computation in the DJPCB algorithm. To dispense with the redundancy of power computation, we propose the following power feedback based DJPCB algorithm, which is similar to the DJPCB algorithm except for Step 5 and Step 6.

(2)

aii

As a result, the received SINR with the MVDR beamforming for receiver i can be expressed as i = Pi Gii (aH (˚i − ˝i ) ii

then go to Step 2. Note that in each iteration of the above algorithm, the base station first sends the value of Iim to each user i, then each user

3.2. Power feedback based DJPCB algorithm

aii

−1

Step 5: Send the value of Iim to each user i. Step 6: Each user i updates the power according to the following iteration:  Pim+1 = i Iim Gii

−1

aii )

(3)

And the total interference with the MVDR beamforming for receiver i can be expressed as ˜H ˜ i − Pi Gii Ii = w i (˚i )w

(4)

3. Distributed joint power control and beamforming algorithms An algorithm, namely, Algorithm A, was proposed in [1] to find the optimal feasible power and weight for joint power control and beamforming. To make Algorithm A work in practice, the authors of [1] further suggest a distributed algorithm to implement Algorithm

Step 1 to Step 4: Same as Step 1 to Step 4 of the DJPCB algorithm. Step 5: Determine the updated power of each user i according to the following iteration: Pim+1 =

i m I Gii i

Step 6: Send the value of Pim+1 to each user i and go to Step 2. Note that in the power feedback based DJPCB algorithm, the updated power is computed by the base station, so the base station has learned Pim and can directly use it to calculate wm and i Iim , also, the user need not compute the updated power, thus there is no redundancy of power computation. As the DJPCB algorithm requires the base station to send the value of Iim to each user i in each iteration, the power feedback based DJPCB algorithm requires the base station to send the value of Pim+1 to each user i in each iteration, and this may consume much overhead (in terms of the number of parameters to be sent) when the number of users per

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237

cell is large. To save the overhead for the CDMA networks, we further propose in this paper a broadcast based DJPCB algorithm that sends the values of xm instead of the value of Iim or Pim+1 . Coni sider the CDMA networks where all users use the same frequency band and denote the set of users connected to base station k by Sk . Since xm is the same for all users i ∈ Sk , we let xm represent xm i k i for all i ∈ Sk . In each iteration of the broadcast based DJPCB algorithm, each base station k first broadcasts xm to all users i ∈ Sk , then, k each user i ∈ Sk can use the broadcasted xm to compute wm , Iim and k i Pim+1 .

3.3. Broadcast based DJPCB algorithm Step 1: Let Pi0 = P0 . Step 2: Choose a user i ∈ Sk for each base station k, then measure and let xm = xm . xm i k i Step 3: Each base station k broadcasts xm to all users i ∈ Sk . k by Step 4: Each user i ∈ Sk figures ˚m i H

Fig. 1. The overhead (the number of parameters to be sent per cell) against the number of users per cell for the DJPCB algorithm, the power feedback based DJPCB algorithm and the broadcast based DJPCB algorithm.

˚m = E[xm (xm ) ] i k k Step 5: Each user i calculates the optimal weight vector by wm i

=

(˚m − ˝im ) i

−1

aH (˚m − ˝im ) ii i

aii

−1

aii

where ˝im = Pim Gii aii aH . ii Step 6: Each user i computes the total interference by H

Iim = (wm ) (˚m )wm − Pim Gii i i i Step 7: Each user i updates the power according to the following iteration:  Pim+1 = i Iim Gii then go to Step 2. Note that xm is a vector with M elements, so the broadcast i based DJPCB algorithm needs to send M parameters per iteration for each cell. In addition, each user i ∈ Sk needs to know aii and Gii in advance, thus base station k needs to send the measured values of aii and Gii to each user i ∈ Sk before executing the broadcast based DJPCB algorithm. Assume that m iterations have been executed, then the broadcast based DJPCB algorithm needs to send M · m + (M + 1) · Uk parameters for each base station k, where Uk denotes the number of users connected to base station k. On the other hand, the DJPCB algorithm requires each user i to know Gii in advance, while the power feedback based DJPCB algorithm does not. Therefore, assume that m iterations have been executed, then the DJPCB algorithm needs to send Uk · (m + 1) parameters for each base station k and the power feedback based DJPCB algorithm needs to send Uk · m parameters for each base station k. On the basis of the above analysis, it can be seen that the power feedback based DJPCB algorithm results in less overhead (in terms of the number of parameters to be sent) than the DJPCB algorithm. Also, if M < Uk and m is large, then the broadcast based DJPCB algorithm can result in less overhead than the DJPCB algorithm and the power feedback based DJPCB algorithm. However, increasing the value of Uk will increase the value of M for maintaining the performance (e.g., the probability of the success in meeting the SINR requirement), so there is a tradeoff between the performance and the overhead. 4. Numerical results In this section, we study a CDMA network that is composed of 19 cells. We assume that the locations of the users are uniformly dis-

tributed over the cell area. The link gain Gli is inversely proportional to Sli Dli˛ , where Sli is the shadowing factor between transmitter l and receiver i, Dli is the distance between transmitter l and receiver i, and ˛ is a constant that accounts for the large scale propagation loss. The shadowing factor models power variation due to shadowing, and all shadowing factors Sli are assumed to be independent, log-normal random variables with 0 dB expectation and  standard deviation. In our simulations, we choose ˛ = 4 and  = 8 dB. Also, we assume that the SINR requirement is equal to −10 dB (which is practical for CDMA networks since the SINR requirement can be multiplied by the processing gain to form the Eb /I0 requirement) for all users and 10 iterations of power control and beamforming are executed. In Fig. 1, we plot the overhead (the number of parameters to be sent per cell) against the number of users per cell for the DJPCB algorithm, the power feedback based DJPCB algorithm and the broadcast based DJPCB algorithm, where the number of antenna elements per receiver for the broadcast based DJPCB algorithm is chosen so as to maintain certain outage probability (the probability for failing to meet the SINR requirement). It can be seen from this figure that the power feedback based DJPCB algorithm results in less overhead than the DJPCB algorithm, also, when the number of iterations is large, the broadcast based DJPCB algorithm can result in less overhead than the DJPCB algorithm and the power feedback based DJPCB algorithm. Apparently, the above results coincide with the analysis of overhead in Section 3. We also found from Fig. 1 that for the broadcast based DJPCB algorithm, the overhead can be decreased with the increase in the requirement of the outage probability. The reason is that the outage probability decreases as the number of antenna elements per receiver increases, so we can relax the requirement of the outage probability to save the number of antenna elements per receiver, and in turn the overhead.

5. Conclusions We have proposed in this paper two DJPCB algorithms, which are different from the DJPCB algorithm in [1,2] in the implementing methods. Compared to the DJPCB algorithm in [1,2], the power feedback based DJPCB algorithm and the broadcast based DJPCB algorithm can both save the overhead of feedback, in addition, the power feedback based DJPCB algorithm can dispense with the redundancy of power computation.

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References [1] Rashid-Farrokhi F, Tassiulas L, Liu KJR. Joint optimal power control and beamforming in wireless networks using antenna arrays. IEEE Trans Commun 1998;46(October(10)):1313–24. [2] Olfat M, Rashid-Farrokhi F, Liu KJR. Power allocation for OFDM using adaptive beamforming over wireless networks. IEEE Trans Commun 2005;53(March(3)):505–14. [3] Wang JT. Optimal joint dual transmitter receiver diversity and power control for wireless networks. IEEE Commun Lett 2007;11(January(1)): 46–8.

Jui Teng Wang received the BS, MS and PhD degrees from National Chiao Tung University, Hsinchu, Taiwan, all in communication engineering. He is now a professor with the Graduate Institute of Communication Engineering, National Chi Nan University, Puli, Nantou, Taiwan. His research interests are in wireless communications, wireless networks and software radio.