Knopp, Raymond

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Institut Eure´comMobile Communications Department2229, route des CreˆtesB.P. 19306904 Sophia AntipolisFRANCETechnical Report no. 36Simple Power–Controlled Modulation Schemesfor Fixed–Rate Transmission over Fading Channels(submitted to IEEE Communication Letters)October 22, 1997R. Knopp1, G. Caire2Tel: (+33) 4 93 00 26 26Fax: (+33) 4 93 00 26 27Email: knopp@eurecom.fr1Eurecom’s research is partially supported by its industrial partners: Ascom, Cegetel, Hitachi, IBM France, Motorola,Swisscom, and Thomson CSF2G. Caire is with the Politecnico di Torino,Corso Duca degli Abruzzi 24, I-10129 Torino, ItalySimple Power–Controlled Modulation Schemes for Fixed–RateTransmission over Fading ChannelsR. Knopp, G. CaireAbstractThis paper examines power–controlled modulation schemes for fixed–rate transmission over fading radio channels.We are primarily interested in transmitter diversity schemes for multiple–block (slot) transmission on narrowbandtime–division duplex systems such as DECT and PHS. We consider transmitter power control over one and twoindependently–faded blocks using variable (coded) modulation schemes. The main conclusion is that simple andde facto standard coded–modulation schemes can yield performance comparable to a non–fading AWGN channelat a fixed data rate.21 IntroductionThe purpose of this paper is to show the benefits of exploiting channel state feedback for fixed–ratedata transmission over fading radio channels. We are mostly interested in illustrating this with somepractical examples. The work of Goldsmith [1] has shown that channel state feedback can be used bythe transmitter to reliably transmit at a high spectral efficiency under long–term power constraints on anergodic fading channel. This can be achieved in practice by a combined variable–rate, variable–powercoded–modulation scheme [2]. Equivalently, one may opt to spread the coded bits across a time interval(for example with the use of a bit–interleaver) over which the channel has varied greatly. Caire et al.have shown in [3] that traditional error control codes can be used effectively with large symbol alphabetsto achieve low bit error rates (BER).Roughly speaking these two techniques are equivalent, in the sense that they both exploit the ergod-icity of the channel process. Both allow for information rates approaching channel capacity, dependingon the desired BER. The amount of time needed for the average rate to be attained in the variable–ratescheme will be on the order of the interleaving delay required to achieve a high diversity order in thefixed–rate scheme.In many systems (e.g. voice telephony), fixed rate transmission with small decoding delay is re-quired. Here, we are not at liberty to exploit the ergodicity of the channel. Furthermore, in wireless loopapplications [4], the channels are often static which renders ergodic channel assumptions completelymeaningless. We may, however, perform partial averaging by coding across variations in the frequency-domain using either a slow-frequency hopping mechanism, spread-spectrum or multitone signalling. Insuch instances, Ozarow et al. have demonstrated that the fundamental performance is properly definedby an information outage probability, Pout(R) which is a function of the fixed code rate R, and that thechannel capacity is zero, when no channel state feedback is available at the transmitter. For static fadingsystems where coded data is interleaved over a small number of independent blocks, Knopp and Hum-blet have shown in [5] that practical coding schemes can achieve block error rates approaching Pout(R),which can be seen as a practical lower–bound on the achievable probability of block error.In a recent study by Caire et al.[6] it was shown that channel state feedback can be used effectivelyto reduce or drive Pout(R) to zero. The latter is typically the case when the fading process has more thanone degree of freedom which allows for a non–zero channel capacity. It is shown that for transmissionacross two static faded blocks with independent fading levels, a loss in signal–to–noise ratio (SNR)on the order of 5dB in Rayleigh fading can be expected with respect to a non–fading AWGN channel.Here we build upon this work and give a few examples of simple power–controlled modulation schemesfor transmission across one and two independently–faded blocks which essentially remove the effect ofsignal fading, under the assumption of a long–term power constraint.2 System modelConsider the transmission of F complex sequences fcf;ig; f = 1;    ; F across F independent staticfading channels, with channel strengths ffg; f = 1;    ; F . The fare assumed independent andidentically distributed with probability density and distribution functions denoted by f(u) and F(u)respectively. Each cf;iis assumed to be drawn independently from some finite variable-size constellationwithMf() signal points, where = 1   F. We assume narrowband channels so that we mayuse a discrete–time channel model. This would correspond to multiple frequency-slot transmission in asystem like DECT[7] or PHS[8]. The transmitted signal energy per symbol is denoted Es. We assumea two–way system where the transmitter has perfect knowledge of the ffg, which can be achievedeither by a feedback path via the opposite link or by power measurement of the opposite link in a time–division duplex(TDD) system. This is already employed to a certain extent in DECT and PHS. Using3this information, the transmitter adjusts its power in each block by the power controller Pf(); f =1;    ; F . The signal at the receiver is given byrf;i=qfPf()Esejfcf;i+ zf;i(1)where fis the uniformly–distributed random phase induced by channel f , and zf;iis a zero–meancomplex circular–symmetric Gaussian random variable with variance N0. We approximate the BER ineach block byBERf() = Nf()Qq:5Pf()fd2f()Rb; (2)whereNf() is a factor taking into account the number of nearest neighbours for the code (constellation)used in block f and average number of bit errors for the most likely error patterns. The d2f() is theminimum square Euclidean distance for the code (constellation) used in block f . R is the informationrate per block in bits/symbol and b= Es=RN0is the signal–to–noise ratio per information bit. Note thedependence of the code parameters on the channel strengths.3 Transmission over one Block (F = 1)We begin with transmission over one block, and choose a power controller of the formP() =(K=  > T0   T(3)where K is a constant chosen such that the average transmit power is unity and is given by K =R1Tf(u)du=u 1. In unit-mean Rayleigh fading (i.e. f(u) = e u) this yields K = 1=E1(T),where E1(x) =R1xe udu=u is the first–order exponential integral [9] . We see that for non–zero K werequire T> 0. Here we have introduced an outage event which we control with the cutoff level T,which is in keeping with the results of [6]. When transmitting, the received power is kept constant, so weeffectively have a non–fading channel and any traditional coding techniques can be applied. This typeof power control was refered to as truncated channel inversion by Goldsmith [1], and is optimal in thesense of minimizing the information outage probability [6]. We may now write the BER asBER = :5F(T)+(1 F(T))NQp:5KRd2b(4)under the assumption that when   Tthe receiver chooses bits at random. This quantity can beminimized numerically by varying T. We show the minimum BER for 2 bits/symbol using uncodedQPSK transmission (N = 1; d2 = 2) and Ungerbo¨ck 8-PSK TCM schemes [10] in Figure 1. With thesesimple schemes we lose 6-8 dB (at BER=10 3) with respect to uncoded QPSK on a non-fading channel.4 Transmission over F = 2 BlocksThe power controllers and codes (constellations) are now chosen such that1. The instantaneous information rate per block is R = :5(R1+ R2) bits/symbol2. E [P1() + P2()] = 23. BERf() = BER; 8f s:t: Pf() > 040 2 4 6 8 10 12 14 16 18 2010−510−410−310−210−1100Tx SNR/bit (dB)BERQPSK − AWGN       QPSK − Rayleigh   QPSK − PC Rayleigh4−state TCM       8−state TCM       16−state TCM      32−state TCM      64−state TCM      Figure 1: Transmission schemes for F = 1 block at 2 bits/symbolThe first constraint assures that we always transmit at a fixed information rate. The second is a long–term average power constraint. By making the BER constant in each block to which non–zero power isallocated, we are again able to employ coding schemes designed for non–fading AWGN channels, ratherthan specially designed codes such as [5]. Note that we no longer have an outage region, which is nowpossible due to the diversity introduced by multiple blocks. A pragmatic power control scheme inspiredby the optimal schemes in [6] is shown for F = 2 blocks in Figure 2. We divide the (1; 2) planeinto 3 regions bordered by the horizontal and vertical axes and the lines 1= KT2and 1= 2=KT,where 0 < KT 1 is a constant to be determined. For the 2 regions where one block strength (saymax) is 1=KTtimes stronger than the other we transmit with power K1=maxin that block with a codeof rate 2R bits/symbol. In the block with the weaker channel strength, we do not transmit at all. Whenthe channel strengths of the two blocks are close to equal, we transmit with powers K2=fin both witha code of rate R bits/symbol. Note that the degerate case with KT= 1 corresponds to choosing thestrongest block (i.e. selection diversity at the transmission end.)From the third constraint above, K1and K2are related by the equationN1Qq:5K1d21Rb= N2Qq:5K2d22Rb. If we are interested only in asymptotic equality (i.e. for high SNR) this reduces to K2=d21d22K1. Thelong–term power constraint imposes that K1and K2satisfy1 =Z10"K2Z2=KTKT2f(1)d11+K1Z12=KTf(1)d11#f(2)d2(5)Noting thatR10E1(x)e xdx = log(1 + 1=) we have in unit–mean Rayleigh fading that K1log(1 +KT) K2log(KT) = 1. The optimal value for KTwhich maximizes K1,K2(and minimizes BER) can5121= 2=KT1= KT2P1() = K1=1P2() = 0Code rate 2R bits/symbolP1() = K2=1P2() = K2=2Code rate R bits/symbolP2() = K1=2P1() = 0Code rate 2R bits/symbolFigure 2: Two–block Power Control Schemebe found numerically. For asymptotic equality of the BER in the two transmission modes we can neglectthe multiplicative terms in front of the Q-function to obtain the maximum K1,K2at KT= d21=(d22 d21),if d22 2d21. Otherwise,KT= 1, which yields K1= 1= log 2.A few examples of schemes with a transmission rate of 2 bits/symbol and minimum BER are con-sidered in Figure 3. We see that both variable modulation (using two modulation schemes) and selectiondiversity (using only one modulation scheme) effectively remove the effects of signal fading with only 2independently faded blocks (at BER=10 3 we fall 2-5 dB from QPSK on a non–fading channel.) Thisis completely in keeping with the results of [6], and, as expected, the more general variable modulationscheme significantly outperforms selection diversity. The TCM schemes are standard Ungerbo¨ck codes[10] with 32QAM and 8PSK with the number of states chosen such that the trellis complexity is the samein both transmission modes. The performance curves for the TCM examples were found by computersimulation.This two–block transmission scheme has an advantage in interference–limited (e.g. cellular) systems,since in a slotted system, some slots will remain empty a significant portion of the time, which will reduceinterference levels. Alternately, in a multiuser setting, the unused slots can be occupied by other users.5 Conclusions and Future WorkIn this letter we have presented a few examples of simple power–controlled fixed–rate modulationschemes for slowly–fading radio channels. This type of signaling could prove to be beneficial in amulti–slot (narrowband/frequency-flat) transmission mode of a TDD system such as DECT or PHS. Wehave shown that with very simple off–the–shelf error control codes and ideal channel state feedback, wecan expect error–rate performances comparable to those of non–fading AWGN channels. This is madepossible by using transmitter diversity in conjunction with power control. Near–perfect channel stateinformation should be reasonably simple to achieve in a slowly fading TDD scenario.We only considered Rayleigh distributed fading statistics, since this represents a worst case set-ting, and therefore expect even more promising results for indoor or rural communications which ex-hibit a strong deterministic component. Schemes such as this could therefore be interesting in high-speed wireless LAN and wireless loop applications. Moreover, the use of additional antenna diversity(space/polarization) with power control will provide even more performance gains. Similar ideas canalso be applied in slotted(orthogonal) multiuser systems, in conjunction with channel measurement based60 2 4 6 8 10 12 14 16 18 2010−510−410−310−210−1100Tx SNR/bit (dB)BERQPSK − AWGN                QPSK − Rayleigh            16QAM − Rayleigh Sel. Div. QPSK/16QAM − Rayleigh      4−state 32QAM TCM Sel. Div 8−state 32QAM TCM Sel. Div 16−state 32QAM TCM Sel. Div16/4−state 8PSK/32QAM TCM  32/8−state 8PSK/32QAM TCM  64/16−state 8PSK/32QAM TCM Figure 3: Transmission schemes for F = 2 blocks at 2 bits/symbolallocation schemes. These are topics of our ongoing research.References[1] A. Goldsmith, Design and Performance of High-Speed Communication Systems over Time–VaryingRadio Channels, Ph.D. thesis, University of California at Berkeley, 1994.[2] S.G. Chua and A. Goldsmith, “Variable–rate variable–power mqam for fading channels,” in Proc.VTC, April 1996, pp. 815–819.[3] G. Caire, G. Taricco, and E. Biglieri, “Bit–interleaved coded modulation,” IEEE Transactions onInformation Theory, 1997.[4] D.C. Cox, “Wireless loops: What are they?,” International Journal of Wireless Information Net-works, vol. 3, no. 3, pp. 135–138, 1997.[5] R. Knopp and P.A. Humblet, “On coding for block fading channels,” Institut Eure´com TechnicalReport no. 34, submitted to IEEE Trans. on Inform. Theory, 1997.[6] G. Caire, G. Taricco, and E. Biglieri, “Information outage rate and power control,” submitted toIEEE Trans. on Inform. Theory, 1997.[7] T. Rappaport, Wireless Communications, Prentice Hall, Englewood Cliffs, New Jersey, 1996.[8] ARIB, Personal Handy Phone System ARIB Standard, version 2, 1995.7[9] M. Abramovitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, 1965.[10] G. Ungerbo¨ck, “Channel coding with multilevel/phase signals,” IEEE Transactions on InformationTheory, vol. 28, pp. 55–67, Jan. 1982.8

Simple power-controlled modulation schemes for fixed-rate transmission over fading channels

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Simple power-controlled modulation schemes for fixed-rate transmission over fading channels

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