Transmit Power and Bit Allocations for OFDM Systems in a Fading Channel

Transcription

1 Transit Power and Bit Allocations for OFD Systes in a Fading Channel Jiho Jang *, Kwang Bok Lee, and Yong-Hwan Lee * Sasung Electronics Co. Ltd., Suwon P.O.Box, Suwon-si, Gyeonggi-do 44-74, Korea School of Electrical Engineering, Seoul National University, Seoul -74, Korea E-ail: jiho@obile.snu.ac.kr, {klee, ylee}@snu.ac.kr Abstract -In this paper, we propose a coputationally efficient algorith for transit power and bit allocations in wireless OFD systes aiing at axiization of data rate under the constraint of total transit power and bit error rate. Although the proposed algorith is based on the water-filling approach, the coputational coplexity is only O ( k ), where k denotes the nuber of iterations required to eet the constraints and is the nuber of subchannels. By coputer siulations, we verify that iterations are sufficient to satisfy the constraints, and the decrease of data rate for the proposed algorith is shown to be less than one percent relative to the optial ethod at the average SNR above 4dB. The proposed algorith is appropriate for OFD systes in a tie varying wireless channel for its low coputational coplexity and good perforance. I. INTRODUCTI The growing deand for ultiedia services requires reliable and high rate data counications, but the high rate data transission is significantly liited by intersybol interference (ISI). Recently, a ulticarrier syste has been taken a great interest as a solution to the proble of transitting data over channels with severe ISI [~]. The principle of the ulticarrier syste is to split a high rate data strea into a nuber of lower rate parallel streas that are transitted siultaneously on a nuber of orthogonal subchannels. In the ulticarrier syste, when the channel state inforation (CSI) is available at the transitter, transit power and nuber of bits to be transitted for each subchannel can be adapted according to the CSI in order to reduce transit power consuption or to increase data rate [6~]. The algorith to adaptively allocate transit power and bits for each subchannel in the ulticarrier syste is known as a loading algorith for the application of Asyetric Digital Subscriber Lines (ADSL), where a discrete ultitone (DT) technology is generally used. Existing loading algoriths in [6], [7], [8], and [9], which are developed for DT systes, inherently assue that the channel is quasistatic. In the quasi-static channel, the channel state can be assued not to vary with tie. Therefore, the transitter allocates transit power and bits once at the beginning, and the allocation ay not be changed for a long tie. The coputational coplexity of the loading algorith in this case ay not be a proble, if the transit power and bit allocations can be done within the tie period given for the initial loading. However, in Orthogonal Frequency Division ultiplexing (OFD) systes, which are popular in these days for the wireless application of the ulticarrier syste, the channel state ay vary with tie. Thus low coplexity and efficient coputation are iportant for the loading algorith applied to the OFD systes in order to adapt transit power and nuber of bits for each subchannel within the coherence tie of the wireless channel. Leke and Cioffi developed a loading algorith for a wireless ulticarrier syste in [], but the coputational coplexity of the algorith is still high. In this paper, we propose a coputationally efficient loading algorith for OFD systes in a tie varying wireless channel. We ai at axiization of data rate under the constraint of total transit power and bit error rate (BER). The proposed loading algorith is based on the water-filling approach [], which is known as optial to axiize data rate under the constraint of total transit power. In the proposed algorith, however, the water-filling power allocation is not fully perfored. Instead, by adjusting only the water-filling level needed in the water-filling power allocation, we ay adapt transit power and nuber of bits for each subchannel with low coputational coplexity, and the total nuber of loaded bits in an OFD sybol can be axiized while satisfying the constraints. This paper is organized as follows. In next section, existing loading algoriths are briefly reviewed. In Section III, we forulate the proble of transit power and bit allocations for OFD systes considered in this paper, and propose a loading algorith appropriate for the wireless OFD syste. Siulation results are shown in Section IV, and we conclude this paper in Section V. II. EXISTING ALGORITHS In [6], Hughes-Hartogs developed a loading algorith to iniize transit power consuption under the requireents of data rate and BER. This algorith achieves optial perforance using a greedy search ethod, where one additional bit is allocated to the subchannel that requires the least increental power at each step until the target data rate is reached. Since the Hughes-Hartogs algorith requires

2 exhaustive sorting and adds one ore bit at a tie, the order of operations is expected to be O( Bˆ log ), where Bˆ denotes the total nuber of loaded bits and is the nuber of subchannels. Krongold et al. treated the data rate axiization proble under the constraint of total transit power and BER in [7]. They used a table lookup ethod with precoputed rate-snr tables, and a bisection ethod to obtain an optial Lagrange ultiplier solution. The aount of transit power and nuber of bits to be allocated for each subchannel are obtained fro the value of Lagrange ultiplier and the corresponding operating point in the lookup table. This algorith requires large size of eory to keep all the values of operating points on rate-snr curve. Also, since the Krongold s algorith needs table lookups at the initialization stage and at each iteration, and each table lookup requires the operations of O ( D), where D denotes the axiu nuber of bits apped for a sybol in the constellation, the order of operations for the algorith is O (( k + ) D), where k denotes the required nuber of iterations. Chow and Cioffi proposed a suboptial loading algorith in [8] and [9], where they optiized transission bandwidth to axiize data rate under the constraint of flat on/off power allocation and BER. In this algorith, total transit power is equally distributed over turned-on subchannels, which are allowed to transit data bits, and then bits are allocated for each subchannel with the strategy that large nuber of bits for subchannels with high gain, and sall nuber of bits for subchannels with low gain. Also in [], Leke and Cioffi developed an efficient ipleentation ethod of the water-filling power allocation by decoupling the transission bandwidth identification fro the transit power allocation as in [8] and [9]. The Leke s water-filling power allocation ethod yields higher data rate than the Chow s flat power allocation schee. When the restriction of integer bit constellation is given as in [9] and [], the nuber of loaded bits is rounded off to be an integer nuber, and the transit power for each subchannel is rescaled accordingly to eet the BER constraint. However, since the rescaled transit power ay not satisfy the total transit power constraint, the nuber of loaded bits for each subchannel should be adjusted at each iteration and the procedure should be repeated until the total transit power constraint can be et. Unfortunately however, the iteration procedures are not fully described in [9] and []. Since the Chow s aglorith and Leke s algorith require sorting of the channel power gains at the transission bandwidth identification stage, the order of operations for both algoriths is expected to be O ( log + ( k + ) ), where k denotes again the required nuber of iterations. III. PROPOSED ALGORITH In this section, we propose a siple and efficient loading algorith that axiizes data rate of the OFD syste under the constraints of total transit power and BER. A. Proble Forulation In OFD systes, since the total bandwidth is divided into subchannels and the loading algorith assigns a certain nuber of bits, b for the th subchannel, the total nuber of loaded bits in one OFD sybol is written as B. () The nuber of loaded bits, be obtained by [9] b b for the th subchannel ay sg, () b log + σ Γ where s and g denote the transit power allocated to the th subchannel, and the power gain of the th subchannel, respectively. σ is the variance of the additive white Gaussian noise. Γ represents the SNR gap which is a function of the target BER and a channel coding schee. Note that b in () ay have a real nuber in this for, and hence we round off to ake b be an integer nuber for practical odulation/deodulation as in [9] and [] such that b ˆ round[ b ], for,,. (3) Then, the total nuber of loaded bits in an OFD sybol can be rewritten as Bˆ ˆ. (4) b Consequently, the proble considered in this paper ay be forulated as ax Bˆ, () BER BER, subject to and S where BERt arg et t arg et represents the axiu allowable BER, ŝ denotes the allocated transit power corresponding to the integer nuber of bits, for the th subchannel, and S is the total available transit power budget. B. Algorith Under the assuption that the nuber of loaded bits can have infinite granularity in constellation size, the optial transit power allocation policy that axiizes the total nuber of loaded bits () can be easily derived using a Lagrange ultiplier technique. The solution is the wellknown water-filling [] over the subchannels, and the transit power allocated to the th subchannel can be written as +, (6) s σ Γ λ g where [ x] + ax{ x,}, and λ denotes a threshold representing the water-filling level. By replacing s in () with (6), the nuber of bits b for the th subchannel can be reexpressed as

3 [ ( λg )] + b log. (7) After the nuber of bits b for,, is calculated fro (7) for a specific λ, integer nuber of bits is obtained by rounding off it to be as (3). Then, corresponding to the obtained integer nuber of bits, the transit power for each subchannel should be rescaled to satisfy the BER constraint [9][]. Assuing that QA odulation and ideal phase detection are used as in [], the BER for the th subcarrier signal is bounded by γ, (8) BER exp. b ˆ where γ denotes the signal to noise ratio (SNR) for the th subchannel which is defined by γ ˆ s g σ. Under the requireent of BER BER t, the transit power for the arget th subchannel corresponding to ay be calculated as ( ) σ Γ, (9) g where the SNR gap, Γ in this case is written as ln( BER t arget ) Γ.. Note fro (7) and (3) that b and are deterined by only the value of the water-filling level, λ, when the channel power gain g is given. Therefore, we need not perfor the whole water-filling power allocation in [], but only adjust the value of λ to satisfy the total transit power constraint given as in (). Note that the nuber of bits S b for each subchannel increases with λ as seen fro (7). The increase of b with λ indicates the increase of and ŝ as well fro (3) and (9). Therefore, to axiize the data rate of the syste, the water-filling level λ should be kept as large as possible while satisfying the total transit power constraint. Consequently, by adjusting λ iteratively, we can adapt transit power and nuber of bits for each subchannel, and the nuber of loaded bits in an OFD sybol can be axiized while satisfying the total transit power constraint and BER requireent. At each iteration, the water-filling level λ is adjusted as + S, () λ λ µ σ Γ where < µ < is a step size, and denotes the nuber of turned-on subchannels which are allowed to transit data bits. For the initial value of λ, we ay set S, () λ + σ Γ g which can be obtained fro (6) and the condition of teporarily assuing that all the subchannels are s S turned on and relaxing the restriction of the integer bit constellation. Table. Coplexity coparisons Loading Algoriths Order of Operations Hughes-Hartogs algorith [6] O( Bˆ log ) Krongold s algorith [7] O (( k + ) D) Chow s flat power allocation [9] O ( log + ( k + ) ) Leke s water-filling ethod [] O ( log + ( k + ) ) Proposed O ( k ) ( Bˆ : total nuber of loaded bits, : nuber of subchannels, k: required nuber of iterations, D: axiu nuber of bits apped for a sybol in the constellation) b if START Initialize: S λ + g, σ Γ for [ log ( λg )] round [ b ],, then σ Γ g,, + ( ) END, S? Y, λ λ + µ N S σ Γ Fig.. Flow chart of the proposed loading algorith The followings are the suary of the proposed loading algorith, and the flow chart is depicted in Fig.. (Step ) Initialize λ as (), set, and give a specific value for the step size µ. (Step ) For,,, calculate b,, and ŝ fro (7), (3) and (9), respectively. If b ˆ (or s ˆ ), then set. (Step 3) If, then STOP. Allocate the integer S

4 nuber of bits, and the transit power ŝ for each subchannel. Otherwise, adjust λ as () and GOTO (Step ). Note that the proposed algorith does not require sorting nor table lookup, and thus the algorith requires the operations of O ( k ) only. Consequently, the proposed algorith s coputational coplexity is low, and the algorith can be applied to the wireless OFD systes. The order of operations for the proposed algorith and those for other algoriths are tabulated in Table. The required nuber of iterations and data rate perforance will be shown in next section. IV. SIULATI RESULTS In this section, we evaluate the perforance of the proposed loading algorith in ters of data rate and required nuber of iterations by coputer siulations. In siulations, the nuber of subchannels, is set to be 6, and we assue that OFD sybols undergo Rayleigh fading. We consider a wideband ultipath fading in this paper, but the bandwidth of each subchannel is sall enough to be assued flat fading for each subcarrier signal. The average channel power gain for each subcarrier signal is assued to be one, and the average received SNR can be written as S ( σ ). Also we assue that no channel coding schee is used, and the target BER is set to be 3 BER arg. Note that in this t et case, the SNR gap Γ is.48db. In figures and 3, to show the effects of the step size µ in the proposed algorith, we depict the Data-Rate-Decrease relative to the optial ethod, and the nuber of iterations required to eet the constraints, respectively, for various values of µ. The optial ethod is the odified version of the Hughes-Hartogs algorith [6], where a greedy search ethod is used to axiize data rate under the constraints of total transit power and BER. The Data-Rate-Decrease relative to the optial ethod is defined in this paper by B ˆ ˆ B, () Data - Rate- Decrease [%] Bˆ where Bˆ represents the total integer nuber of bits loaded in an OFD sybol when the Hughes-Hartogs algorith is used. In the figures, the results of, independent trials are averaged to get the Data-Rate-Decrease and the required nuber of iterations. Note fro Fig. that the Data-Rate-Decrease decreases for all the values of µ as the average SNR increases. At a fixed average SNR value, however, the Data-Rate-Decrease increases with µ. Fro Fig. 3, we ay observe that the required nuber of iterations decreases as µ increases at a fixed average SNR value. The required nuber of iterations is less than at all the values of average SNR for the cases of µ. 7, µ. 8, and µ. 9. Considering the tradeoff between the Data-Rate-Decrease and the required nuber of iterations fro the results in Fig. and Fig. 3, the value of Data-Rate-Decrease [%] µ.4 µ. µ.6 µ.7 µ.8 µ.9 Fig.. Data-Rate-Decrease for the proposed algorith for various values of µ Nuber of Iterations µ.4 µ. µ.6 µ.7 µ.8 µ Fig. 3. Required nuber of iterations for the proposed algorith for various values of µ µ is chosen to be.7 for the proposed algorith, because the Data-Rate-Decrease for the case of µ. 7 is the least aong the values of µ that require less than iterations. In figures 4 and, we depict the Data-Rate-Decrease, and the required nuber of iterations, respectively, for various loading algoriths. Note fro Fig. 4 that the Data-Rate- Decrease for the proposed algorith decreases with average SNR, and the Data-Rate-Decrease is less than one percent relative to the optial ethod at average SNR above 4dB. oreover, at the average SNR higher than db, the Data- Rate-Decrease for the proposed algorith is less than. percent and the proposed algorith shows better perforance than all other ethods. Also note fro Fig. that the proposed algorith requires less than iterations at all the values of average SNR. The required nuber of iterations for the Leke s water-filling power allocation ethod is 4~9 according to the average SNR values, and is the least aong

5 all the ethods. However, considering the coputational coplexity given in Table and the required nuber of iterations shown in Fig., we ay conclude that the proposed algorith has lower coputational coplexity than all other ethods, because the proposed algorith requires less than iterations and the order of operations for the proposed algorith is only O ( k ). V. CCLUSIS In this paper, we propose a siple and efficient loading algorith to adaptively allocate transit power and bits for each subchannel in wireless OFD systes. We ai at the axiization of data rate under the constraint of total transit power and BER. The proposed algorith is based on the well-known water-filling approach. In the proposed algorith, however, we need not perfor the whole waterfilling procedures. Instead, by adjusting only the water-filling level, which is needed in the water-filling power allocation, we adapt transit power and nuber of bits for each subchannel with low coputational coplexity, and data rate is axiized while satisfying the constraints. The proposed algorith requires the operations of O ( k ) only, where k and represent the required nuber of iterations and the nuber of subchannels, respectively. By coputer siulations, we verify that iterations are sufficient to eet the constraints for the proposed algorith. oreover, siulation results show that the Data-Rate-Decrease for the proposed algorith is negligible relative to the optial ethod, i.e. the Data-Rate- Decrease is less than one percent at the average SNR above 4dB. Consequently, the proposed loading algorith is appropriate for OFD systes in a tie varying wireless channel for its low coputational coplexity and good perforance. REFERENCES [] J. A. C. Bingha, ulticarrier odulation for data transission: an idea whose tie has coe, IEEE Coun. ag., pp. -4, ay 99. [] L. J. Ciini, Analysis and siulation of a digital obile channel using orthogonal frequency division ultiplexing, IEEE Trans. Coun., vol. CO-33, pp , July 99. [3] J. Jang, K. B. Lee, and Y. H. Lee, Frequency-tie doain transit power adaptation for a ulticarrier syste in fading channels, IEE Electronics Letters, vol. 38, No., pp. 8-, 8th Feb.. [4] J. Jang and K. B. Lee, Transit power adaptation for ultiuser OFD systes, IEEE J. Select. Areas Coun., vol., No., pp. 7-78, Feb. 3. [] J. Jang, K. B. Lee, and Y. H. Lee, Frequency-tie doain transit power adaptation for OFD systes in a ultiuser environent, IEE Electronics Letters, vol. 38, No., pp , th Dec.. [6] D. Hughes-Hartogs, Enseble ode structure for iperfect transission edia, U.S. Patents Nos. 4,679,7 (July 987), 4,73,86 (arch 988), and Data-Rate-Decrease [%] Chow [9] Krongold [7] Proposed, µ.7 Leke [] Fig. 4. Data-Rate-Decrease for various loading algoriths Nuber of Iterations Hughes-Hartogs [6] Chow [9] Krongold [7] Proposed, µ.7 Leke [] Fig.. Required nuber of iterations for various loading algoriths 4,833,76 (ay 989). [7] B. S. Krongold, K. Rachandran, and D. L. Jones, Coputationally efficient optial power allocation algoriths for ulticarrier counication systes, IEEE Trans. Coun., vol. 48, pp. 3-7, Jan.. [8] P. S. Chow and J.. Cioffi, Bandwidth optiization for high speed data transission over channels with severe intersybol interference, in proc. IEEE Globeco 9, pp. 9-63, 99. [9] P. S. Chow, Bandwidth optiized digital transission techniques for spectrally shaped channels with ipulse noise, Ph. D. Thesis, Stanford University, 993. [] A. Leke and J.. Cioffi, A axiu rate loading algorith for discrete ultitone odulation systes, in proc. IEEE Globeco 97, pp. 4-8, 997. [] T.. Cover and J. A. Thoas, Eleents of Inforation Theory, New York: John Wiley & Sons, 99. [] A. J. Goldsith and S. G. Chua, Variable-rate variable-power QA for fading channels, IEEE Trans. Coun., vol. 4, pp. 8-3, Oct. 997.