Subcarrier excluion technique for coded OFDM ytem Kai-Uwe Schmidt, Jochen Ertel, Michael Benedix, and Adolf Finger Communication Laboratory, Dreden Univerity of Technology, 62 Dreden, Germany email: {chmidtk, ertel, benedix, finger}@ifn.et.tu-dreden.de Abtract In OFDM baed wirele LAN an efficient coding cheme, e.g. turbo coding, can improve the performance coniderably. However, the bit error rate i mainly affected by trongly attenuated ubcarrier. In thi paper we propoe two algorithm to adapt the tranmiion to the channel condition by leaving out weak ubcarrier when neceary. Thi way, the performance i le dependent on the fading channel, though the data rate i degraded lightly. In a firt approach the average channel capacity i increaed. The econd propoal provide a olution that bound the bit error rate even under very poor channel condition. The algorithm are derived and verified for an indoor propagation channel at 7 GHz. Index Term ubcarrier excluion, adaptive tranmiion, information theory, turbo coding, OFDM I. INTRODUCTION IN an OFDM ytem the total bandwidth i plitted into everal equally paced ubchannel. Each ubchannel operate with a particular ubcarrier frequency, where the ubcarrier frequencie are orthogonal to each other. Since the channel ha uually a wideband characteritic, the channel tranfer function i frequency elective. Each ubchannel can be aumed to be non-frequency elective, if the number of ubchannel i large enough for the occupied bandwidth. Thu, every ubchannel can be conidered a an AWGN channel. Though the average received power i roughly equal, ome ubcarrier may be ignificantly attenuated due to deep fade. Thi reult in a high bit error rate BER) on thoe ubcarrier. Even though mot ubcarrier may be received without error, the overall bit error rate i mainly dominated by the few ubcarrier with the mallet magnitude. That mean, the performance of the ytem trongly depend on the channel fading characteritic. In order to achieve the bet performance for a given or etimated) channel tranfer function, tranmiion parameter have to be elected carefully. There are everal propoal to adapt the tranmiion to the channel condition. Depending on the channel fading characteritic the allocated power and the modulation cheme for each carrier can be varied. An optimal power allocation i alo known a Water-Filling []. The aim of thi technique i to maximize the channel capacity under the contraint of a certain total power budget. The adaption of the modulation cheme for each carrier i uually calculated by bit loading algorithm [2], [3]. A in wirele network the channel fading characteritic can change quickly, uing thee technique reult in a large ignaling effort. Thu, an amount of the gained performance can get lot. A implification, ome ubcarrier can be witched off, intead of uing an adaptive modulation and/or adaptive power cheme. Thi way, the ignaling effort can be trongly reduced. Some baic of thi technique were invetigated in [4]. There, the idea wa to leave out a contant number of ubcarrier, where alway the weaket carrier are elected. Thi method allow only little adaption, ince the number of excluded ubcarrier i fixed. In thi paper we propoe technique that elect the number of excluded ubcarrier adaptively. Thu, the number of ued ubcarrier i reduced only for bad channel condition and in many cae all ubcarrier are ued and no bandwidth i wated. Thi paper i organized a follow. Section II give a brief decription of the employed ytem model. In ection III we analyze the effect of the fading channel on the ytem performance. Two different cheme for adaptive ubcarrier excluion are derived in ection IV. The derived technique are analyzed and compared with repect to bit error rate performance and data rate degradation in ection V. Section VI conclude the paper. II. SYSTEM MODEL The conidered OFDM ytem employ N = 28 ubcarrier in a MHz channel. Standard cyclic prefix CP) OFDM i ued with a guard interval long enough to prevent inter-ymbol interference. Perfect ynchronization in time and frequency i aumed. Conequently, the relationhip between the tranmitted ubcarrier ymbol X k and the received ubcarrier ymbol Y k can be decribed a Y k = H k X k + V k, ) where H k denote the complex fading coefficient of the k th ubcarrier and V k the Gauian noie tranformed into frequency domain. For the channel coding a turbo code with variable code rate achievable by puncturing ha been elected. In the encoder the code polynomial 3, ) octal are ued to generate the redundant bit. Here, we ue a block length of 24 bit. The decoder i baed on a Max-LogMAP algorithm, where four iteration are carried out. Soft deciion decoding i performed.
We ue a channel model for an indoor environment at 7 GHz. Thi model ha a non line of ight NLOS) characteritic and conider 7 path. A complete decription of the channel i given in []. an ALA larger than 4. db form the th group. Figure 2 how the probability that an arbitrary channel realization belong to one of the ten group..3 replacement III. FADING CHANNEL EFFECTS Uually, in OFDM ytem we have to deal with wideband frequency elective channel. It i well known that the performance decreae for thoe fading channel compared to flat AWGN channel, although the mean power gain of both channel i rather equal. That mean, PSfrag the performance replacement highly depend on the frequency electivity of the channel. Even in a particular environment e.g. an office) an infinite number of channel realization with different fading characteritic can occur. In order to enure tranmiion at a target BER, it i important to ae the channel condition. Thu, we have to find a meaure to indicate the frequency electivity of the channel. We define the average logarithmic attenuation ALA) D = N N k= log H k 2 ) [db]. 2) An ALA equal to zero mean the whole channel ha a flat fading characteritic, i.e. the channel i an AWGN channel. The higher the ALA the larger i the performance degradation of the ytem. In figure the tranfer function of two channel realization are plotted. Note, the overall ubcarrier attenuation [db] Fig.. - - - -2-2 D=.46 db D=.8 db -3 6 32 48 64 8 96 2 ubcarrier number Channel realization with different ALA gain of the two realization i normalized to one. The ALA of one realization i cloe to zero, which mean the ytem performance i cloe to the performance of an AWGN ytem. The ALA of the econd realization i much higher, which correpond to a large performance degradation. In order to how the dependence between the ytem performance and the ALA of the according channel tranfer function, we ubdivide all poible channel realization into ten group. The firt nine group contain the channel realization with an ALA from to 4. db. The limit are equally paced by. db. The channel realization with Of coure it would be the bet to chooe an infinite number of channel group, i.e. each channel realization form one group. Thi caue a very large effort in imulation and viualization. A a trade-off between imulation effort and realitic conideration we choe ten group. So, we aume that all channel realization of one group have a imilar behavior with repect to ytem performance and ALA. P D In) Fig. 2..2.2....9e-3.8e-3 7.6e-4.. 2 2. 3 3. 4 4. Probabilitie of the group of channel realization The BER performance correponding to the defined group i preented in figure 3. In addition the BER for an AWGN channel and the average BER for the invetigated channel model are plotted a dahed line. It can eaily be log BER - -2-3 -4 - D AWGN D=dB) average BER -6 2 4 6 8 2 E b /N [db] Fig. 3. BER for channel group with different ALA, 6-QAM, code rate /2 een that the BER performance lo i proportional to the ALA. Furthermore, at a BER of the required E b /N can vary from about 4 db to more than +4 db compared to the average required E b /N. The relationhip between the average BER BER) and the BER for the channel realization group BERI n )) can be calculated with the Baye theorem BER = n BERI n ) P D I n ). 3) According to figure 3 we can aume for a mall average BER: BERI n n =..9) BERI ). For thi condition we can approximate equation 3 BER BERI ) P D I ). 4) That mean, for a mall average BER the BER mainly depend on the BER for the wort channel realization. Equation 4 can be eaily verified in figure 3. The graph for the average BER and the BER for the wort channel realization are nearly parallel and differ by a factor of about P D I ) for lower BER.
From thee conideration we can conclude: when electing the tranmiion parameter, it i not optimal to focu on the average performance of the ytem. Since the performance can vary ignificantly, in many cae a large amount of power i wated or the achieved BER i very poor. In order to allow efficient tranmiion, parameter have to be adapted to the channel condition. IV. SUBCARRIER EXCLUSION TECHNIQUES A. Maximizing the average channel capacity Since a powerful error correction coding allow tranmiion near the Shannon limit, it i ueful to conider the channel capacity a a meaure for the ytem performance. The channel capacity i an upper bound for the data rate of a communication ytem. The capacity of one ubchannel can be calculated by the theorem of Shannon [6]. In an OFDM ytem the overall capacity i the um of all ubchannel capacitie. Since the carrier attenuation are time-variant, the overall channel capacity i time-variant, too. Thu, it i common to indicate the average channel capacity for an infinite number of OFDM ymbol. If the probability denity function PDF) of the channel coefficient ph) i known, the average ubchannel capacity i given by ) E C Fading = N C AWGN H 2 E ) ph)dh, ) N where C AWGN.) i the capacity of one ingle AWGN channel and C Fading.) i the average capacity. The ratio E /N denote the SNR of one ubcarrier ymbol. According to equation the channel capacity i given in bit/ymbol. One technique to increae the channel capacity i the well known Water-Filling. The goal of thi method i to maximize the channel capacity for a given total power budget. For each carrier an optimal power i determined according to it attenuation. If the attenuation of everal ubcarrier i too high, the aigned power i zero. That mean, thoe ubcarrier are not ued to carry information. From thi algorithm a carrier excluion cheme can be derived. Then, the ubcarrier whoe aigned power i zero are left out and the total power i ditributed equally over the remaining ubcarrier. Thi olution contain ome ignificant diadvantage: it i valid only for a continuou Gauian modulation cheme, rather than for dicrete input ignal et like M-QAM it i not proofed, that the number of excluded ubcarrier i optimal to maximize the channel capacity Fig. 4. complex calculation are neceary to elect the ubcarrier to be excluded. Now, an algorithm will be derived that conider the diadvantage mentioned above. In the following, we call thi algorithm Capacity Maximizing Subcarrier Excluion). We define the average portion of ued ubcarrier a the average ratio of the number of ued ubcarrier to the total number of ubcarrier. If i a degree of freedom, there i an optimal value that maximize the average channel capacity for a certain modulation cheme. Thi way, the average channel capacity i a function of the average SNR and CFading SS, E ) = N H T ) H CAWGN SS 2 E ph)dh, N 6) where the upercript mean the channel capacity i calculated for a certain input ignal et, which i different from the Water-Filling olution. The average aigned energy E correpond to the allocated energy per carrier when all ubcarrier are ued. The integration limit H T i a threhold for the channel coefficient. If a channel coefficient i maller than thi threhold, the according carrier will be excluded. The relationhip between the threhold and the average portion of ued ubcarrier i given by = H T ph)dh. 7) The maximum of the average channel capacity i defined by C SS Fading E N ) = max [ CFading SS, E )]. 8) N Uually, thi can only be olved by numerical evaluation, ince the capacity formula for an arbitrary ignal et ha high complexity. Beide, the PDF of the channel coefficient i uually given a a numerical etimation. The maximal average channel capacity for everal modulation cheme i preented in figure 4. It can be concluded C [bit/ymbol].4.2.8.6 with 6-QAM QPSK BPSK -2-2 3 4 E /N [db] Average channel capacity with that a ignificant gain i only poible for low SNR. That mean, if the SNR increae to a certain value, almot no ubcarrier will be excluded. In figure the average portion of ued ubcarrier i hown. Although the rate of excluded ubcarrier i relatively mall for typical SNR value, the portion can coniderably vary for ingle channel realization. In ubection V we how the dependence between the ALA of a channel tranfer function and the portion of excluded carrier.
replacement replacement Fig...98.96.94.92.9.88.86 BPSK.84 QPSK 6-QAM.82 64-QAM.8 continuou Gauian modulation 2 E /N [db] Average portion of ued ubcarrier ) B. Bounding the bit error rate The nature of the technique invetigated above i rather theoretical, ince the channel capacity a an upper bound i only reachable by uing an ideal coding cheme and an infinite time diverity. From the practical point of view, the achievable bit error rate i more intereting than the channel capacity. The BER performance doe not only depend on the channel capacity, but alo on the particular error correction code. In order to focu more on the BER performance, the coding impact on the bit error rate ha to be conidered. A mentioned in ection III, the performance for a particular channel realization can be predicted roughly by calculating the average logarithmic attenuation. Auming the bit error rate increae for larger ALA, a bounding of the ALA lead to a bounding of the bit error rate. For thi reaon we define a cut-off value D co that bound the ALA. If the ALA of a particular channel realization i equal to or maller than the cut-off value, all ubcarrier are ued to carry information. In the other cae the weaket ubcarrier are elected and left out until the ALA i equal to or maller than the cut-off value. Then, only the ued ubcarrier are conidered to recalculate the ALA. We name thi technique BBSE Bit error rate Bounding Subcarrier Excluion). The reulting average portion of ued ubcarrier i plotted over the cut-off value in figure 6. Note, one can vary Fig. 6..9.9.8.8.7.7.. 2 2. 3 3. 4 D co [db] Average portion of ued ubcarrier BBSE) the average portion of excluded ubcarrier by adjuting the cut-off value. When the cut-off value decreae, the performance i expected to improve. Hence, for a particular cenario a trade-off between data rate reduction and performance gain ha to be made. V. SIMULATED PERFORMANCE AND COMPARISONS In the previou ection two different carrier excluion cheme have been derived. In the following, the BER performance of thee technique baed on imulation i preented. All imulation have been carried out under the following condition. For each data block of 24 bit an independent channel realization i generated. The overall channel gain i normalized to one for each channel realization. During the tranmiion of one data block the channel i uppoed to be tationary. For all imulation we employ 6-QAM and a code rate of /2. Similar reult can be achieved with other modulation cheme BPSK, QPSK, 64-QAM) and code rate /3, 3/4). In figure 7 the average bit error rate i preented. The performance for BBSE i hown for different cut-off value. If the derived carrier excluion cheme are applied, log BER Fig. 7. 2 3 4 6 2 3 4 6 7 8 9 E b /N [db] Average BER uing ubcarrier excluion the average BER improve ignificantly. While the BER for i fixed, the BER for BBSE highly depend on the cut-off value. If a maller cut-off value i choen, the performance get better and the throughput decreae. So, for each application a good compromie between BER performance and data rate reduction ha to be found. In order to compare the two cheme, the portion of excluded ubcarrier ha to be conidered. In Table I the average portion of excluded ubcarrier i ummarized for a BER of. Taking the BER performance and the data TABLE I AVERAGE PORTION OF EXCLUDED SUBCARRIERS AT A BER OF BBSE D co db 2 db 3 db 4.9%.7% 2.%.4% rate reduction into account, the benefit are clearly on the ide of BBSE. A we have een in ection III, the BER for a particular channel realization can differ ignificantly from the average BER. If we focu on the average BER performance ee figure 7), for one point, the BER i averaged for all channel realization at one particular SNR. In thi cae, the BER for good channel condition can be more than ufficient,
replacement replacement while the BER for bad channel condition can be very poor ee figure 3). In order to allow efficient tranmiion, it i alway deired to have a BER cloe to a target BER, no matter of the actual channel condition. Thu, it i more intereting to focu on the performance gain for particular channel realization at a target BER, rather than for an infinite number of channel realization at a fixed SNR. Therefore, imulation were done with the channel realization group, defined in ection III. For each of the ten group the required E b /N to reach a target BER of 3 and i plotted in figure 8 and 9, repectively. Eb/N [db] 8. 8 7. 7 6. 6. 4. 4 3... 2 2. 3 3. 4 4. Fig. 8. Required E b /N to reach a target BER of 3 for the channel group Eb/N [db] 2 9 8 7 6 4.. 2 2. 3 3. 4 4. Fig. 9. Required E b /N to reach a target BER of for the channel group For the performance gain at a BER of 3 i quite high, while the gain i almot vanihed at a BER of. That mean, the BER performance improve only for mall BER. It eem that thi i a contradiction to the average BER ee figure 7), where the performance gain i about 2. db for a BER of. In fact, thi gain come from the bad channel realization, which caue a high BER at the imulated E b /N. So, it i eential to ae the performance gain for particular channel realization or at leat channel group with imilar behavior), rather than for an infinite number of channel realization. When BBSE i applied, the BER fluctuation can be reduced very effectively. While the BER for channel realization with an ALA le than the cut-off value remain unaffected, the BER for wore channel realization i tabilized. That mean, one can et an upper bound for the BER by electing an appropriate cut-off value. In figure the portion of excluded ubcarrier for each channel group i preented. Uing BBSE the portion of excluded ubcarrier only depend on the cut-off value. For two graph are plotted, ince the data rate depend on the SNR and hence alo on the BER). If we look at a BER of, one can ee that the portion of excluded ubcarrier can be maller for BBSE, although the performance i better compared to. That mean, the BBSE cheme i clearly more effective than. Fig....4.4.3.3.2.2..., BER= -3, BER= -.. 2 2. 3 3. 4 4. Portion of excluded ubcarrier for the channel group VI. CONCLUSION In thi paper two different technique for ubcarrier excluion have been preented. Conidering the diverity propertie of an error correction coding, the problem of ubcarrier excluion mut be approached in a completely different way than for uncoded ytem. At firt the channel capacity, a an upper bound for the ytem performance, wa maximized by leaving out weak ubcarrier. The aim of the econd cheme wa to keep the bit error rate bounded, even under very poor channel condition. The two algorithm were compared with repect to BER performance and data rate degradation. Alo, ome invetigation on correct imulation of carrier excluion cheme from the practical point of view have been made. While the firt cheme i rather theoretical, the econd cheme allow a nearly arbitrary reduction of the BER fluctuation by electing an appropriate cut-off value. REFERENCES [] Robert G. Gallager, Information Theory and Reliable Communication, John Wiley & Son, New York, 968. [2] A.R.S. Bahai and B.R. Saltzberg, Multi-Carrier Digital Communication; Theory and Application of OFDM, Kluwer Academic/Plenum Publiher, New York, 999. [3] Andrea J. Goldmith and Soon-Ghee Chua, Variable-Rate Variable- Power M-QAM for Fading Channel, IEEE Tranaction on Communication, vol. 4, no., pp. 28 23, October 997. [4] H. Rohling and R. Grünheid, Performance of an OFDM-TDMA mobile communication ytem, Proc. VTC-96, 996. [] M.L. Rubio, A. Garcia-Armada, R.P. Torre, and J.L. Garcia, Modelling and characterization at 7 GHz for indoor broadband WLAN, IEEE Journal on Selected Area on Communication, vol. 2, no. 3, pp. 93 6, April 22. [6] C. E. Shannon, A mathematical theory of communication, Bell Syt. Tech. Journal, vol. 27, pp. 623 66, October 948.