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Transmit Power Allocation for BER Performance Improvement in Multicarrier Systems

Transmit Power Allocation for BER Performance Improvement in Multicarrier Systems
Chang Soon Park and Kwang Bok (Ed) Lee School of Electrical Engineering and Computer Science, Seoul National University Kwanak P.O. Box 34, Seoul 151-742, Korea E-mail: parkcs@mobile.snu.ac.kr, klee@snu.ac.kr

Abstract? In a multicarrier system, transmit power allocation over different subchannels is an effective means of improving the performance. In this paper, the optimal transmit power allocation scheme is developed to improve bit error rate (BER) performance in a multicarrier system with diversity reception. A simple suboptimal scheme is also derived from the optimal one, and an asymptotic case referred to as equal SNR scheme is discussed. Numerical results show that the optimal and suboptimal power allocation schemes significantly outperform the equal power allocation scheme. The effects of the modulation level, the number of receiving antennas, and the number of subchannels on the BER performance are also investigated. I. INTRODUCTION

Multicarrier communication system is promising for future wideband wireless communications, and recently the system is being applied to several fixed and mobile radio systems, such as digital audio and video broadcasting, and wireless LAN [1], [2]. In a multicarrier system, a wideband channel is divided into multiple narrowband subchannels using orthogonal subcarriers. Multiple data substreams are transmitted in parallel through subchannels, and the total transmit power should be distributed to these subchannels. It may be natural to allocate equal transmit power to multiple subchannels, when the channel state information is not available at the transmitter. When the channel state information is available at the transmitter, however, effective transmit power allocation may improve error rate performance or increase the capacity. Transmit power allocation based on the water-filling solution has been studied in [3] and [4], and it has been combined with adaptive modulation to maximize the capacity [5], [6]. These schemes of increasing capacity may be suitable for variable rate services such as e-mail and web browsing. On the contrary, delay-sensitive services such as voice or video are usually provided at a fixed rate [7]. In these applications, it is desirable to design a transmit power allocation scheme that improves error rate performance for a given rate. In this paper, we develop the optimal transmit power allocation scheme to improve bit error rate (BER) performance in a multicarrier system with receive antenna diversity. Receive antenna diversity is used to mitigate the effects of fading as in [2]. Based on the optimal scheme, a computationally efficient suboptimal scheme is also derived. Furthermore, it is shown that the equal signal-to-noise ratio (SNR) scheme, which makes the received SNR become the same for all subchannels, corresponds to an asymptotic case of the suboptimal scheme. The performance of the proposed power allocation schemes is evaluated, and compared with

that of the equal power allocation scheme. The effects of the modulation level, the number of receiving antennas, and the number of subchannels on the BER performance are also investigated. Numerical results show that the use of the optimal or suboptimal power allocation scheme significantly improves the BER performance of a multicarrier system. The equal SNR scheme is found to perform well in the presence of large number of receiving antennas. The remainder of this paper is organized as follows. Section II describes the system and channel models. In Section III, the optimal transmit power allocation scheme is developed, and a simple suboptimal scheme is derived from the optimal one. An asymptotic case of the suboptimal scheme is also investigated in Section III. Numerical results are presented in Section IV, and conclusions are drawn in Section V.

II.

SYSTEM AND CHANNEL MODELS

A multicarrier communication system considered in this paper is depicted in Fig. 1. An input data stream is divided into K parallel substreams through a serial-to-parallel converter. The transmit power pk is assigned to the kth substream dk (k = 1, 2, , K). In the multicarrier modulator, the substreams are modulated on orthogonal subcarriers to form a transmit signal. The receiver is equipped with N antennas, from which N replicas of the transmit signal are received. The multicarrier demodulator at each antenna separates the received signal into K subcarrier signals. The signals received from different antennas are assumed to undergo independent fading. We also assume that the K subcarrier signals for each antenna experience frequencynonselective and slow fading independently. Correspondingly, the output of a multicarrier demodulator for the kth subcarrier at the nth receive antenna may be expressed as yk , n = pk hk , n d k + nk ,n , k = 1, 2, ???, K , n = 1, 2, ???, N (1) where hk,n denotes the multiplicative fading coefficient for the kth subcarrier at the nth antenna, and they are assumed to be independent and identically distributed (i.i.d.) complex Gaussian random variables with zero mean and unit variance. nk,n’s represent the additive white Gaussian noise (AWGN), and they are assumed to be i.i.d. complex Gaussian random variables with zero mean and variance of 2. dk is the encoded data symbol with unit average power, and pk is the transmit power for the kth subcarrier with the total power constraint given as
K

pk = KP

(2)

k =1

0-7803-7589-0/02/$17.00 ?2002 IEEE

PIMRC 2002

where P denotes the average transmit power per subcarrier, when the total transmit power is equally distributed into the K subcarriers. As shown in Fig. 1, signals received from N antennas are combined for each subcarrier to achieve antenna gain and diversity gain. It is assumed that the maximal ratio combining (MRC) is employed to maximize the SNR [1]. The transmit symbol for each subcarrier is estimated based on the output of each MRC combiner. After MRC, the SNR k for the kth subcarrier may be calculated as γ k = α k pk (3) where α k =
N n =1

hk ,n

2

σ 2 is the ratio of the combined

gain of the kth subchannel to the noise power, representing the overall channel state for the kth subchannel. The channel states k (k = 1, 2, , K) required to determine the transmit power pk (k = 1, 2, , K) are assumed to be perfectly known to the transmitter. These channel states can be obtained by feedback from the receiver in a frequency division duplex (FDD) system, or can be estimated at the transmitter in a time division duplex (TDD) system.

III.

TRANSMIT POWER ALLOCATION

In this section, several transmit power allocation schemes are described. The equal power allocation scheme is briefly discussed in Section III-A, and power allocation scheme that is optimal in terms of the BER is developed in Section III-B. A suboptimal scheme is derived as a simplified version of the optimal scheme in Section III-C, and the equal SNR scheme is discussed as an asymptotic case of the suboptimal scheme in Section III-D.

A. Equal Power Allocation When the channel state information is not available at the transmitter, it is natural to allocate equal transmit power to K subcarriers. This scheme is referred to as the equal power allocation scheme. In this scheme, the transmit power for each subcarrier is P in (2), and the corresponding SNR γ k for the kth subcarrier is α k P . When the channel state information is available at the transmitter, the total transmit power may be allocated in a more effective way to achieve better BER performance, as described in Section III-B D.
B. Optimal Power Allocation To derive the optimal power allocation scheme, we first express the overall BER as a function of the transmit power for K subcarriers, { pk k = 1, 2, ???, K } , and then find { pk } that minimizes the overall BER. The BER for the kth subcarrier is generally a function of the SNR γ k , and thus the BER Pb (e α k ) for a given channel state α k may be expressed as Pb (e α k ) = f (γ k ) = f (α k pk ) , k = 1, 2, ???, K (4) where f ( ? ) is a function determined by a specific modulation scheme. Since data streams are transmitted over independent subchannels, the overall BER for given channel states of {α k k = 1, 2, ???, K } can be calculated as an arithmetic mean of Pb (e α k ) in (4):

1 1 Pb (e α k ) = f (α k pk ) . (5) K k =1 K k =1 Note that the average BER becomes minimal when the BER in (5) is minimized for each given channel state. To find the optimal {pk} that minimizes (5), we use the Lagrange multiplier method with the total power constraint in (2). The Lagrangian function may be expressed as K K 1 J ( p1 , p2 , ???, pK ) = f (α k pk ) + λ pk ? KP (6) K k =1 k =1 where λ denotes the Lagrange multiplier. By differentiating (6) with respect to pk and setting it to zero, we obtain a set of K equations as 1 d f (α k pk ) + λ = 0, k = 1, 2, ???, K . (7) K dpk Solving K + 1 simultaneous equations in (2) and (7), we can calculate the optimal set of the transmit power {pk}. As mentioned above, the BER function in (4) is a function determined by a specific modulation scheme. For a binary differential phase-shift keying (DPSK) with single antenna reception, for example, the BER function may be expressed as an exponential function [1], and a closed-form solution of (2) and (7) may be easily found. For an M-ary phase-shift keying (PSK) or M-ary quadrature amplitude modulation (QAM), however, the exact or approximate BER function may be expressed as a Q-function [1], and it may be difficult to find a closed-form solution. In this case, an adaptive method such as the steepest descent algorithm [8] may be employed to find a solution in an iterative manner as follows. Stage 1) Initialization: Set an iteration number i = 0, a step size ?(0) = ?0, and an arbitrary initial positive power set {pk(0)} satisfying (2). Stage 2) Power set update: For k = 1, 2, , K, update the transmit power pk(i) as ? pk (i + 1) = pk (i ) ? ? (i ) J ( p1 (i ), p2 (i ), ???, pK (i )) ?pk (i ) (8) 1 d = pk (i ) ? ? (i ) f (α k pk (i ) ) + λ (i ) K dpk (i ) where λ(i) is determined from the power constraint in (2) and is updated as K 1 d (9) λ (i ) = ? 2 f (α k pk (i )) . K k =1 dpk (i ) Stage 3) Step size adjustment: If all components of the updated power set {pk(i +1)} in Stage 2 are positive, then go to Stage 4 with ?(i +1) = ?0. Otherwise, compute 1 d ? k (i ) = pk (i ) f (α k pk (i ) ) + λ (i ) K dpk (i ) Pb (e α1 ,α 2 , ???,α K ) =
for k’s associated with pk(i +1) 0, set the step size ?(i) to ρ ? min ? k (i ) where is a
k : pk ( i +1) ≤ 0

K

K

positive scaling factor smaller than one, and
1 This step size adjustment is performed to make all the updated power components become positive, and is set to 0.9 for numerical results in Section IV.

1

return to Stage 2. Stage 4) Repetition or termination: If more iterations are required for convergence, increase i by one and go to Stage 2. Otherwise, terminate the adaptive procedure. The adaptive algorithm described above converges to the global optimum solution for the convex BER function [9]. Note that the Q-function, which is the exact or approximate BER function for an M-ary PSK or M-ary QAM, is a convex function. C. Suboptimal Power Allocation In case that a closed-form solution cannot be found, an adaptive method may be used to find the optimal solution, as described in Section III-B. However, it generally takes a lot of iterations for an adaptive solution to be close to the optimal one. A simpler approach is to find an approximate closed-form solution using a simple approximation of the BER rather than the exact BER expression. Based on this approach, in this subsection, we derive a closed-form transmit power allocation scheme for M-ary square (M = 4m, m = 1, 2, ) QAM schemes. The derivation procedure may easily be applied to other modulation schemes with modifications for approximate BER functions. For an M-ary square QAM, the BER function in (4) may be approximated using an upper bound as [10], [11] a b f (α k pk ) ? a Q bα k pk ≤ exp ? α k pk (10) 2 2

D. Equal SNR Power Allocation The transmit power can be assigned to subcarriers so that the received SNR k in (3) and hence the BER f (γ k ) in (4) become equal for all subchannels. We call this scheme the equal SNR power allocation scheme. It can be shown that the equal SNR power allocation scheme corresponds to an asymptotic case of the suboptimal scheme in (11). From (11) and (12), it can be shown that pk becomes λ0 k asymptotically, when k is sufficiently large for all k. In this case, the power allocation in (11) may be written as KP 1 pk = λ0 α k = ? . (13) K α (1 α l ) k
l =1

The corresponding SNR is calculated as KP γ k = λ0 = , k = 1, 2, ???, K , K (1 α l )
l =1

(14)

(

)

where Q( x) = (1 function, a=

2π )


x

exp(? t 2 2)dt , and

which indicates that the received SNR becomes equal for all subchannels by the power allocation in (13). Consequently, it has been shown that the suboptimal scheme of (11) behaves like the equal SNR scheme asymptotically, when all the subchannels are in sufficiently good conditions. The equal SNR scheme allocates transmit power inversely proportional to the channel state k, allocating the more transmit power to the more attenuated subchannel. Hence, as compared to the equal power scheme, the equal SNR scheme increases the received SNR of relatively worse subchannels, while decreases that of relatively better subchannels.

denotes the Q-

2( M ? 1)

IV.
b = 3 ( M ? 1) . By

NUMERICAL RESULTS

M log 2 M substituting the upper bound in (10) into the BER function of (7), we can find a closed-form solution of K + 1 simultaneous equations in (2) and (7). Some components of the calculated power set may be negative. In this case, we apply the Kuhn-Tucker conditions [9], from which the negative components of {pk} are set to zero and the remaining components are recalculated until all components become non-negative. Consequently, the solution is found as λ0 α k ? ( 2 b )(1 α k ) ln (1 α k ) , α k ≥ exp(?bλ0 / 2) pk = α k < exp(?bλ0 / 2) 0, (11) 2 4K λ where λ0 = ? ln is calculated to satisfy the power b ab constraint in (2) as KP + ( 2 b ) (1 α k ) ln (1 α k ) k ∈S λ0 = (12) (1 α k ) where S = {k α k ≥ exp(?bλ0 / 2)} . From (11), it can be seen that no transmit power is allocated to the kth subcarrier if the channel state k is smaller than exp(?bλ0 / 2) . Furthermore, it can be easily shown that a positive transmit power pk in (11) increases with k for k exp(1 ? bλ0 / 2) , while pk decreases with k for k > exp(1 ? bλ0 / 2) .
k ∈S

In this section, the performance of the transmit power allocation schemes described in Section III are evaluated and compared with one another. A QAM is assumed to be employed for each subcarrier, and adaptive procedure in Section III-B is used to calculate the transmit power for the optimal scheme with sufficient iterations. Equations (11) and (12) are used to determine the transmit power for the suboptimal scheme, and equation (13) is used for the equal SNR scheme. The average SNR is defined to be P / σ 2 . The average BER for each transmit power allocation scheme is calculated by averaging the BER in (5) over sufficient number of randomly generated channel states k (k = 1, 2, , K). The characteristics of power allocation schemes are illustrated in Figs. 2(a) (c), which show the transmit power pk calculated from four power allocation schemes for given values of k, when the number of subcarriers K = 20, modulation level M = 4, and P = 1. The channel states k’s are set to equi-spaced values in a given range: [ 23 dB, 4 dB] in Fig. 2(a), [ 9 dB, 10 dB] in Fig. 2(b), and [4 dB, 23 dB] in Fig. 2(c). These ranges of k in Figs. 2(a) (c), respectively, stand for relatively bad, moderate, and good channel conditions. The optimal power allocation scheme is observed to assign more transmit power to subchannel with larger k, or to less attenuated subchannel, in Fig. 2(a), while the reverse trend is observed in Fig. 2(c). This indicates that the optimal power allocation behaves like water-filling in bad channel conditions, whereas it behaves like inverse water-filling in good channel conditions. A mixture of these two behaviors can be shown in Fig. 2(b). The trends of the

suboptimal power allocation scheme are shown to be similar to that of the optimal power allocation scheme, except that the power corresponding to some highly attenuated subchannels is forced to zero according to (11). As expected, the equal SNR scheme allocates the more transmit power to the more attenuated subchannel for all cases. It can be seen from Fig. 2(c) that the equal SNR scheme approaches the optimal or suboptimal scheme, when all the subchannels are in sufficiently good conditions. Fig. 3 compares the performance of the transmit power allocation schemes for various modulation levels M, when K = 8 and N = 2. As expected, the optimal power allocation scheme provides the best performance for all cases. It is noticeable that the performance of the optimal and suboptimal schemes is almost indistinguishable at high SNR range. However, the performance difference between these two schemes increases, as SNR decreases and/or M increases. The reason for this is that the approximate BER in (10) used for deriving the suboptimal scheme becomes more inaccurate, as SNR decreases and/or M increases. The performance improvement of the optimal and suboptimal schemes over the other schemes is seen to be larger for lower modulation level. In the case of QPSK modulation (M = 4), the SNR gain of the optimal scheme is about 3.8 dB over the equal power scheme, and 1.3 dB over the equal SNR scheme at BER of 10-3. Fig. 3 also shows that the equal SNR scheme is superior to the equal power scheme at high SNR range, while it is inferior to the equal power scheme at low SNR range. This may be explained using Fig. 2. As shown in Fig. 2, the equal SNR scheme is close to the optimal scheme, when all the subchannels are in good conditions. When all the subchannels are in bad conditions, however, the equal SNR scheme tends to allocate transmit power in a fashion contrary to the optimal scheme. In this case, the equal power scheme is closer to the optimal scheme than the equal SNR scheme is. The effects of the number of receiving antennas N on the BER performance are shown in Fig. 4, when M = 4 and K = 8. As shown in Fig. 3, the optimal and suboptimal schemes significantly outperform the equal power scheme, for any value of N. We can observe that the equal SNR scheme also outperforms the equal power scheme, unless the SNR is very low or the number of receiving antennas is small. Furthermore, the performance of the equal SNR scheme is found to approach that of the optimal scheme, as N increases. When N = 3, the performance difference is as small as 0.5 dB at BER of 10-3. This implies that increased diversity effects resulting from more receiving antennas provide higher probability of all the subchannels being in sufficiently good condition. Hence, simple equal SNR scheme can be an alternative to the optimal or suboptimal scheme, when the number of receiving antennas is sufficiently large. Fig. 5 shows the effects of the number of subchannels K on the BER performance, when M = 4 and N = 2. Note that the performance of the equal power scheme is independent of K. The SNR gains of the optimal and equal SNR schemes over the equal power scheme are found to increase with K increasing. Note that greater K also makes the BER curves for the optimal and equal SNR schemes decline more rapidly with SNR increasing. This phenomenon indicates that the optimal and equal SNR power allocation schemes can achieve additional diversity effects from the increased

number of subchannels.

V.

CONCLUSIONS

In this paper, we have developed the optimal transmit power allocation scheme that improves BER performance in a multicarrier system with diversity reception. A computationally efficient suboptimal scheme has also been derived for M-ary QAM, and the equal SNR scheme has been shown to be an asymptotic case of the suboptimal scheme. Numerical results have shown that the optimal power allocation scheme provides about 3.8 dB of SNR gain over the equal power scheme at BER of 10-3 for 8 subchannels, 2 receiving antennas, and QPSK modulation. The performance improvement of the optimal scheme over the equal power scheme has been found to increase, as the number of subchannels increases. The suboptimal scheme has been shown to perform as well as the optimal scheme at high SNR range. It has also been found that the performance of the equal SNR scheme approaches that of the optimal scheme, as the number of receiving antennas increases. The proposed power allocation schemes may be applied to other decoupled multi-channel systems which transmit multiple data streams through decoupled parallel channels. For example, in a multi-input multi-output (MIMO) system, the matrix channel can be converted into decoupled parallel channels using appropriate signal processing, resulting in a spatial-domain multi-channel system [12]. Therefore, proposed power allocation schemes can be used to improve the BER performance in a decoupled MIMO system.

REFERENCES
J. G. Proakis, Digital Communications. New York: McGraw-Hill, 1995. [2] A. A. Hutter, J. S. Hammerschmidt, E. Carvalho, and J. M. Cioffi, “Receive diversity for mobile OFDM systems,” in Proc. IEEE WCNC, Chicago, USA, Sept. 2000, pp. 707-712. [3] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley, 1991. [4] J. Jang, K. B. Lee, and Y.-H. Lee, “Frequency-time domain transmit power adaptation for a multicarrier system in fading channels,” in Proc. PIMRC, San Diego, USA, Sept.-Oct. 2001, pp. D100–D103. [5] A. J. Goldsmith and S.-G. Chua, “Variable-rate variable-power MQAM for fading channels,” IEEE Trans. Commun., vol. 45, pp. 1218-1230, Oct. 1997. [6] S. T. Chung and A. J. Goldsmith, “Adaptive multicarrier modulation for wireless systems,” in Proc. 34th Asilomar Conf. Signals, Syst., Comput., 2000, vol. 2, pp. 1603 –1607. [7] “QoS concept and architecture,” 3rd Generation Partnership Project (3GPP), Tech. Spec. 23.107, V.4.0.0, Dec. 2000. [8] B. Widrow and S. D. Stearns, Adaptive Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1985. [9] D. G. Luenberger, Introduction to Linear and Nonlinear Programming. Reading, MA: Addison-Wesley, 1973. [10] M. K. Simon, S. M. Hinedi, and W. C. Lindsey, Digital Communication Techniques. Englewood Cliffs, NJ: Prentice-Hall, 1995. [11] J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering. New York: Wiley, 1965. [12] G. G. Raleigh and J. M. Cioffi, “Spatio-temporal coding for wireless communication,” IEEE Trans. Commun., vol. 46, pp. 357-366, Mar. 1998.

[1]

FIGURES
Transmitter
Power Allocation

Channel

#1

Receiver
y1,1 y2,1
? ? ?

d1 p1
Input data #2

Multicarrier demodulator

yK,1
?

MRC combiner

^ d1

d2
Serial-toparallel converter

7 7 7
dK

p2

Multicarrier modulator

7 7 7

Multicarrier demodulator

y1,2 y2,2
? ? ?

7 7 7
pK

7 7 7

7 7 7
#N

yK,2
?

MRC combiner

^ d2

7 7 7
Multicarrier demodulator

7 7 7
y1,N y2,N
? ? ?

7 7 7
^ d K

Paralleltoserial converter

Output data

yK,N
?

MRC combiner

Fig. 1.
5

Multicarrier communication system with receive antenna diversity.

Transmit Power (pk)

4 3 2 1

Equal Power Optimal Suboptimal Equal SNR

M = 64
10
-1

M = 16 M=4
10
-2

BER

10

-3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Equal Power Optimal Suboptimal Equal SNR

Subchannel Index (k)

(a)
4
Equal Power Optimal Suboptimal Equal SNR

10

-4

0

2

4

6

8

10

12

14

16

18

20

22

24

26

Average SNR (dB)

Transmit Power (pk)

Fig. 3. BER performance comparisons of power allocation schemes for K = 8 and N = 2.
N=1

3

10

-1

2
N=2

1

10

-2

BER

N=3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Subchannel Index (k)

10

-3

(b)
10
-4

Equal Power Optimal Suboptimal Equal SNR

4

-2

0

2

4

6

8

10

12

14

16

18

Transmit Power (pk)

3

Equal Power Optimal Suboptimal Equal SNR

Average SNR (dB)

Fig. 4. Effects of the number of receiving antennas N on the BER performance of power allocation schemes, when M = 4 and K = 8.
10
-1

2

Equal Power (K = 4,8,16,32) Optimal Equal SNR

1
10
-2

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20

BER

0

Subchannel Index (k)

K=4
-3

K=4 K=8 K = 16 K = 32

(c)
Fig. 2. Characteristics of power allocation schemes for M = 4, K = 20, and P = 1. (a) k= –24 + k (dB). (b) k = –10 + k (dB). (c) k = 3 + k (dB).

10

K=8 K = 16 K = 32

10

-4

-2

0

2

4

6

8

10

12

14

16

18

Average SNR (dB)

Fig. 5. Effects of the number of subchannels K on the BER performance of power allocation schemes, when M = 4 and N = 2.


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