Antenna Diversity Techniques for a Single Carrier System with Frequency Domain Equalization An Overview

Transcription

1 Atea Diversity Techiques for a Sigle Carrier System with Frequecy Domai Equalizatio A Overview. Witschig, G. Strasser,. Weigel, A. Spriger Istitute for Commuicatios ad Istitute for Techical Electroics Iformatio Egieerig, Uiversity of Erlage, Germay Uiversity of Liz, Austria Abstract: I this work we ivestigate the possibilities of combiig diversity techiques with a Sigle Carrier System with Frequecy Domai Equalizatio (SC/FDE, as future trasmissio cocepts will also be judged by their possibilities to be combied with multiple ateas. Cocepts of multiple ateas allow either a sigificat performace gai through diversity or sigificat higher capacity based o spacial multiplexig. I this cotributio we will poit out the excellet possibilities of SC/FDE to be combied with diversity techiques, as the process of the equalizatio itself ca be combied with receive diversity as well as with space time block (STC decodig. This combiatio gives a reduced sigal processig complexity as well as the expected eormous performace gai. Especially the cocept of STC allows a advatageous combiatio with SC/FDE as both are based o a blockwise processig that ca be formulated advatageously i the frequecy domai. Our simulatio results are based o the parameters of the IEEE 80.a stadard.. Itroductio The ambitio for extreme high data rates as well as high quality of service make ew cocepts for commuicatio systems idispesable. Future physical layer cocepts will have to esure better performace, higher data rates as well as a high spectral efficiecy. Aother decisive factor is the sigal processig complexity, especially for mobile applicatios. Cocepts of multiple ateas allow to fulfill several of the above metioed criteria i a superior way. While o the oe had multiple atea systems allow to ehace the spectral efficiecy tremedously by spacial multiplexig, they allow to improve the performace by diversity o the other had. From that poit of view, the questio is obvious, if there are trasmissio cocepts (modulatios schemes/physical layer cocepts that ca be combied i a more profitable way with cocepts of multiple ateas tha others - cocepts that combie the best performace with a low sigal processig complexity. Especially the wireless idoor radio commuicatio has to face problems, as the time dispersio caused by multipath propagatio makes powerful equalizatio cocepts ecessary to guaratee a acceptable performace, at the cost of a high sigal processig complexity. Cocepts that implemet this equalizatio i the frequecy domai are able to fulfill both criteria, good performace ad reduced implemetatio effort. The mai advatage over a time domai equalizatio is that the complexity for a FFT based frequecy domai equalizatio grows oly slightly more tha liear with the bit rate, while the complexity grows at least quadratically with the bit rate for traditioal time domai equalizers []. Cocepts that implemet a efficiet FFT based frequecy domai equalizatio are OFDM (Orthogoal Frequecy Divisio Multiplexig ad SC/FDE (Sigle Carrier System with Frequecy Domai Equalizatio. While the cocept of OFDM is used i differet stadards, the cocept of SC/FDE is rarely ivestigated, although this approach combies the advatages of OFDM ad sigle carrier trasmissio [][3][4]. This paper is orgaized as follows: I the ext sectio we will itroduce the cocept of the ivestigated SC/FDE system. The followig chapter deals with receive diversity ad how this ca be combied with the SC/FDE cocept. Possible performace gais ad the required implemetatio effort will be discussed. The same discussio will be carried out for trasmit diversity (STC. Fially a overall perceptio that evaluates the applicability of SC/FDE for diversity techiques will coclude this work.. The Cocept of SC/FDE I a SC/FDE system the received sigal must be trasferred to the frequecy domai (by meas of a efficiet FFT operatio, where the equalizatio takes place. Similar to OFDM, a cyclic prefix is iserted betwee successive blocks i the trasmitter i order to mitigate iterblock iterferece (II [5]. To prevet this II the duratio of the guard period T G must be loger tha the duratio of the chael impulse respose T h. Due to the cyclic extesio, the liear covolutio of oe cyclically exteded trasmitted block ad the chael impulse respose h(t appears as a circular covolutio correspodig to the frequecy domai relatio ( f0 ( f0 S( f0 + ( f0 ( for Z ad f0/tfft. ere the fuctios (f, S(f ad (f are related to the time domai sigals r(t (oe period of the received data block, s(t (the origial, o cyclically exteded block ad h(t by the

2 cotiuous Fourier trasform. (f is the Fourier trasform of the additive oise. Figure shows the trasmitted data structure, which cosists of the data sequece cotaiig symbols ad the sequece of the cyclic prefix with G symbols. The duratio of a processed block is T FFT T ad the duratio of the cyclic prefix is T G G T. T G I this cotributio a fractioally spaced equalizer is implemeted that allows a low complexity implemetatio as well as a compact mathematical descriptio that icludes multiple ateas. The received sigal is sampled with twice the symbol rate, which is sufficiet for bad-limited pulses ad the sampled sequece of each received ad processed block is split ito its polyphases r (k ad r (k, were both sequeces cotai samples that correspod to symbol rate sequeces. The -poit FFT trasformed sequeces (f 0 ad (f 0 are the multiplied (equalized with the equalizer phases E (f 0 ad E (f 0.The equalized sigal Y T D ( f0 E ( f0 X ( f0 + E ( f0 X( f0 ( is fially trasformed back to the time domai by a - poit IFFT. The equivalece of classical structure ad the polyphase structure is documeted i Figure. Two equalizatio criteria are discussed i this cotributio - the Zero Forcig (ZF criterio ad the Miimum Mea Square Error (MMSE criterio. The polyphase represetatio of the ZF equalizer has bee developed i [6] ad is give i equatio (3 ˆ ˆ F ( f F ( f (3 E ZF ( f Fˆ ( f Fˆ ( f, i i where the superscript deotes ermitia traspose. I this equatio [ Fˆ ( f, Fˆ ( f ] t F ˆ ( f (4 T FFT CP Data CP Data CP y(k y(k T G T FFT CP CP 3... Figure Trasmitted data structure i a SC/FDE. f ˆ( t ˆf ˆf (t t 0 + m T x ( k x ( k ( k x(m e e e(m yˆ yˆ Figure : Fractioally spaced equalizatio. xˆ ( mt / T/ xˆ T x ˆ T describes the fourier trasformed, chael distorted trasmitted pulse that has to be derived by pilot aided chael estimatio procedures. It is essetial to metio that i the case of a blockwise trasmissio based o cyclic prefix, a optimal equalizer of ifiite legth is implemeted. From the aalytical expressio of the equalizer it gets obvious that the omiator of equatio (3 characterizes a matched filter that maximizes the S for the two polyphases, while the deomiator describes the ZF equalizer itself that forces a ISI free detectio. It is well kow that the ZF equalizer suffers from oise amplificatio if the chael trasfer fuctio shows deep spectral fades. A equalizer that avoids this problem by compromisig the oise amplificatio ad the ISI reductio is the Miimum Mea Square Error equalizer (MMSE. The optimum ifiite-legth MMSE equalizer is give by ˆ FD ( f E D, MMSE ( f. (5 ˆ T F ( f + σ / σ i i, D ere it is assumed that the iput data sequece d(k is ucorrelated with variace σ d ad the oise i the two braches is white with variace σ. Compared to the ZF equalizer the additive term σ / σ d i the deomiator protects agaist ifiite oise amplificatio sice the deomiator is always strictly positive. The ZF equalizer is a sub-optimum equalizer i the case of strog multipath propagatio but it will be show that for multiple ateas this is sufficiet. 3. eceive Diversity for SC/FDE Differet cocepts as switched atea diversity, selectio diversity, equal gai combiig, or maximum ratio combiig are well kow. The most powerful but also the cocept with the highest implemetatio effort is maximum ratio combiig. There, the sigal that is received due to multiple ateas (M is added ad additioally weighted i a optimal way to ehace the actual S. The optimal equalizer for multiple ateas ca be defied as follows Fˆ ( f Fˆ ( f E ZF ( f M, (6 Fˆ ( f Fˆ ( f were Froted Froted FFT FFT FFT FFT j i i, j d Equalizer Figure 3: Implemetatio of the receiver. IFFT yˆ

3 [ Fˆ ( f, Fˆ ( f, Fˆ ( f, Fˆ ( f ] t,,,, F ˆ ( f,, (7 for the case of M. The MMSE Equalizer is defied equivaletly. otice that the equalizatio term itself is idepedet of the idividual atea path or polyphase ad ca therefore be separated from the matched filter part. This allows the followig implemetatio as show i Figure 3. The advatages of this implemetatio are as follows: a The matched filter for every path implemets the optimal weightig ad guaratees a optimal performace. b The equalizatio has to be carried out oly oce for every block istead of for every path. A additioal advatage is that a simple -poit IFFT is ecessary to trasform the received ad equalized blocks back to the time domai. The performace gai of receive diversity is based o two differet aspects: a Atea Gai: Due to the receptio of the same set sequece o every atea, the received sigal power is improved. This ehaces the average output S based o the fact that the received sequeces due to differet ateas are correlated, while the oise sequeces are ucorrelated. I the AWG case this leads to gai M for M receive ateas SMx G( M AWG M (8 S x Doublig the umber of ateas leads to a gai of 3 d. Equatio (8 is still valid for multipath coditios, if the ivestigated likelihood chael sequece trasfer estimatio fuctios eedless, show uity as average the power (that m additioal implemetatio effort will ot justify the b Diversity gai: The output-sigal of every matched possible performace gai. filter ad its additio ca simply be described as the quadratical summatio of several chael trasfer E+00 fuctios as give i equatio (9. A ( f M ( f (9 (8 already. A high umber of receive ateas leads to a almost flat behaviour of the overall trasfer fuctio ad prevets additioal gai due to diversity. Figure 4 clearly documets this fact. There, the possible performace gai based o diversity that is limited by the matched filter boud is show as a fuctio of the umber of ateas (for this diagram, the costat gai of M based o the atea gai has bee subtracted. It is obvious that already for four ateas the possible additioal gai based o diversity is less tha d ad will i geeral ot justify additioal ateas. To coclude this cosideratio, the bit error behaviour for oe, two ad four receive ateas is give Figure 5. The simulatio results are based o 00 radomly chose idoor radio chael sapshots based o the IEEE stadardizatio model (tapped delay lie, each tap with radom uiformly distributed phase, ayleigh distributed magitude ad with the power delay profile decayig expoetially. The E curves prove the expected behaviour. esides the tremedous gai it gets obvious that the performace gai of a MMSE equalizer i compariso to a simple ZF equalizer is ot more tha d for two ateas ad vaishes completely for four ateas. The simulatios additioally documet that the performace loss i compariso to the matched filter boud is ot more tha d for four ateas. This makes other equalizatio cocepts as decisio feedback equalizatio or maximum E-0 Txx ZF Equalizatio MMSE Equalizatio oud Due to this additio, deep spectral fades of sigle paths are suppressed uder the assumptio of ucorrelated chael trasfer fuctios. A(f follows a chi-square distributio with a mea value of E(AM. That meas that the variace of A(f about its mea value dimiishes for icreasig M over the frequecy bad. Fially we ca coclude that for a high umber of ateas [7] G AWG ( M E[ A] M (0 This statemet was aticipated i equatio Possible remaiig gai [d] umber of receive ateas Figure 4: Possible remaiig gai at a E of 0-4. E E-0 E-03 E-04 E-05 Txx Tx4x Eb/o Figure 5: E for receive diversity. Of course it is to metio that a practical implemetatio of a higher umber of ateas tha two is critical, especially whe talkig of mobile ad hadheld applicatios. From that poit of view it is of iterest to ivestigate the possible improvemet due to trasmit diversity.

4 4. Trasmit Diversity based o Space- Time block codig. Orthogoal desigs of space time codes provide a efficiet meas to geerate codes that achieve the full diversity gai. For two trasmit ateas, the orthogoal desig is kow as the Alamouti scheme, origially formulated for the flat fadig case. I [9] this cocept was exteded for the case of frequecy selective fadig by Al-Dahir. While the cocept of Alamouti describes a symbol by symbol codig, the cocept of Al-Dahir implemets a block by block codig/processig, that is combied with the cocept of SC/FDE with less additioal effort. Figure 6 shows the block format of STC for SC/FDE ad Table gives the codig scheme. (i the followig the expressio b m (i stads for the ith of symbols of the mth block. otice that besides a simple complex cojugatio of every secod processed block, the order is chaged that allows a advatageous implemetatio of the decodig i the frequecy domai. The advatage of this chaged order gets obvious if we take a look at the Fourier trasformatio of the coded blocks. bm ( i o m. ( bm (( i o m I this equatio (-i stads for (-i modulo. ased o the blockwise processig, two received blocks ca be writte as follows. * + * + ( The factor cosiders that the set eergy over the ateas is the same as for the system with oe atea. eceivig these distorted ad coded blocks, they have to be decoded based o the followig scheme: -b *( Datablock. Atea. Atea b b b (i b (i b 3 -b *((-i b *((-i b 4 b 3 (i b 4 (i b 5 -b 4 *((-i b 3 *((-i Table : Codig scheme. -b *(- -b *(0 b (- b ( b (0 pulse shapig + ( ( * * * * (3 From equatio (3 the mai advatage of a combiatio of SC/FDE ad STS gets abvious agai. The mai part of the decodig is the multiplicatio with (k* that characterizes the matched filter that is implemeted ayway. A decoded block of symbols ca fially be calculated as follows: ˆ ˆ (4 ased o the received ad decoded block, the S at every frequecy poit ca be calculated as ρ s S( k (5 ρ were ρ defies the power of the sigal ad S ρ defies the power of the additive oise. This leads to the fial S of the etire block S E + E ρ s S( k (6 k 0 ρ where E describes the eergy of the chael trasfer fuctio. Equatio (6 demostrates that the cocept of STC based o Alamouti or Al Dahir leads to a averagig of the S of both chaels but ot to a atea gai due to multiple receptio of the same sigal as it is the case for receive diversity. The possible gai due to diversity is comparable for trasmit as well as receive diversity. Comparable codig cocepts of STC exist for four ad eight ateas [0],[]. Figure 7 documets that the icrease of trasmit ateas leads to a covergece to the matched filter boud for oe atea. Usig four trasmit ateas, the performace loss i compariso to the case of AWG is ot more tha d aymore. Agai it gets obvious that for more tha two trasmit ateas the differece betwee ZF ad MMSE equalizatio is almost egligible. evertheless, i compariso to receive diversity, the possible gai is reduced by 6 d (see Figure 5. otice that the cocept of block codig leads to a time delay of oe block usig two ateas ad four blocks usig four ateas that might be take ito accout for time critical implemetatios. Additioally it is to metio that STCs for more tha two ateas will lead to a loss of data rate, because oly three blocks of iformatio are set over four ateas i the ivestigated case. b *( b *(- b *(0 b (- b ( b (0 pulse shapig Figure 6: STC for SC/FDE.

5 E E+00 E-0 E-0 E-03 E-04 E-05 Txx 4Txx ZF Equalizatio MMSE Equalizatio oud Txx Eb/o Figure 7: E for STC. 5. The combiatio of trasmit ad receive diversity Fially we wat to poit out the advatages of a combied implemetatio of trasmit ad receive diversity. Figure 8 shows the possible implemetatio of trasmit ad receive diversity i the ivestigated system. The cocept of codig is idetical to a system cotaiig several trasmit ateas ad oly oe receive atea. The differece gets obvious at the receiver, where for every receive atea a idividual decodig uit has to be implemeted ad after this the equalizatio is carried out. otice that matched filterig, optimal weightig ad decodig is implemeted i oe. The mth block of the jth receive uit ca be writte as follows t ˆ m, j m + l l k, (. (7 j m, j t I this equatio k,j describes the chael trasfer fuctio betwee trasmit atea l ad receive atea j. t describes the total umber of trasmit ateas. As every atea receives the same iformatio, a simple additio of the r receive paths implemets the receive diversity r r t m j m k j m k j l l j k, ( (, ( +. (8 t m Without goig ito details we coclude that the S iary iput data QPSK Coder QPSK IFFT STC Coder S..- ad Pulse-Shapig FE.Frot Ed iary output data Equalizer S S S STC (MF STC (MF STC (MF FE FE FE Figure 8: Cocept of trasmit ad receive diversity for SC/FDE. FFT FFT FFT of this system is r t S j l S (9 l, j t Agai it gets obvious that receive diversity results i a summatio of the average S while trasmit diversity results i a averagig of the S. Figure 9 shows the E due to trasmit ad receive diversity. For the 4Tx-4x system the performace correspods almost to the case of AWG usig four receive ateas. The additioal gai i compariso to simple receive diversity is about d at the cost of four trasmit ateas. The most iterestig cocept, focusig a competitive implemetatio, is obviously the cocept of Tx-x. This system combies high performace gai with affordable implemetatio effort. E E+00 E-0 E-0 E-03 E-04 E-05 Txx 4Tx4x Eb/o Figure 9: E for trasmit ad receive diversity. 6. Coclusio Txx ZF Equalizatio MMSE Equalizatio oud It ca be cocluded that diversity techiques allow tremedous gais i terms of E. The highest profit due to diversity is reached already whe usig two receive or trasmit ateas that combies affordable implemetatio effort with sigificat better performace. Further more it has bee demostrated that a low complexity ZF equalizer is sufficiet whe usig several ateas, although this equalizer is kow as a suboptimal solutio i the case of multipath coditios. The implemetatio effort of more powerful equalizatio cocepts will ot justify the additioal performace gai. The most importat statemet of this ivestigatios is that the cocept of SC/FDE is ot oly a powerful cadidate for future high rate commuicatio systems, but ca be combied with diversity techiques advatageously. The advatages of a SC/FDE system are its reduced implemetatio complexity combied with high performace. The combiatio of SC/FDE with diversity techiques ehaces these advatages additioally, as the cocept of the frequecy domai equalizatio ca be combied with the implemetatio of maximum ratio combiig as well as STC i a easy way.

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