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Orthogonal Space-Time Block Coding over Time-Selective FadingChannels: a PIC Detector for the Gi Systems

F.-C. ZhengSchool of Electrical EngineeringVictoria U niversity of Techn ologyMelboume, Australia(Email: fzheng@ ieee.org)Abstract' - All the orthogonal space-time block coding(0-STBC) schemes are based on the followingassumption: the channel remains static over the entirelength of the codeword. However, time selective fadingchannels do exist, and in such case the conventional 0-STBC detectors can suffer from a large error floor in thehigh signal-to-noise ratio (SNR) cases. This paperaddresses such an issue by introducing a parallelinterference cancellation (PIC) based detector for thecoded systems (i= 3 an d 4).

I. I ntroduct ionOver the past five years, orthogonal space-time blockcoding (0 -STBC) technology has attracted enormousinterest due to its high diversity order and low decodingcomplexity [1]-[4]. The low decoding complexity of 0-STB C is directly due to the linear maxim um likelihood(ML) decoder at the receiver. The linear ML decoder,however, relies on the so-called "quasi-static channel"assumption: the channel remains static over the lengthof the entire codeword: 2Ts for the two transmitantenna (2-Tx) STB C (i.e., G? th e Alamouti code [ I ] ) ,4T, for the Hi systems, and 8T, for the systems( i = 3 for 3-Tx, i = 4 for 4-Tx, and T, is the symbolperiod). While such an assumption is reasonable inmost cases, time selective or fast fading channels doexist in practice, even for the 2-Tx case (see [4][5] an dthe references therein). In these scenarios, the channelstate varies from symbol to symbol. Clearly, the 3- an d4-Tx 0-STBC cases (especially the Gi systems) aremuch more vulnerable to channel variation than the 2-Tx case due to the much longer STBC codewo rd.

The above channel variation will destroy theorthogonality of the channel matrix and therefore cause*Thiswork was supported by th e UK Engineering an d PhysicalSciences Research Council (EPSRC) under Grant GRR94428.

A . G. BurrDepartment of ElectronicsThe University of YorkYork, UK(Email: alister@ohm.york.ac.uk)inter element interference (IEI). The end effect of allthis is an irreducible error floor in the bit error rate(BER) curves in the high signal-to-noise ratio (SNR)region. To suppress such an error floor in the 2-Txcase, an elegant decoder was presented in 141. Since thedecoder structure in [4] cannot be used for the 3- or 4-Tx case, an effective yet simple zero forcing (ZF)scheme for the 8, nd G4 systems was proposed in [6]and 171.This paper proposes an altemative approach to the ZFdetector in [6] and [7]. Based on the principle ofparallel interference cancellation (PIC), the newdetector (termed "PIC detector") offers an even betterperformance than the ZF detector. The computationalcomplexity of the PIC de tecto r is higher than that of theconventional and the ZF detector, but is still veryaffordable. Only the 4, oded syste ms are considered inthis paper (the H , oded systems will be addressed in[SI).

11.The S y s t e m M o d e l fo r t h e G4Sys temConsider a typical G4 encoded 4-Tx 0-STBC systemwith 4 transmit (4-Tx) and 1 receive (1-Rx)ntennas. Agroup of four complex symbols, sl, s2, : ~ , nd s4, ar epassed through a G4 encoder before transmitted overST,. The encoder output is therefore a 8 x 4 matrixC = [ c~ , ] ,where c,, is & s ~ , o r S E (conjugate of s k ) ,and is transmitted by Tx j at t ime i. Also, by letting thechannel gain from Tx j to the Rx at time ib e h,(z), thereceived signal at time i is

,=1where n.,s a complex additive white Gaussian noise(AWGN) with zero mean and a variance of U:(therefore 0 . 5 ~ : er dimension). Also, h,(i) is subject

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to Rayleigh fading but is normalised, i.e., U: = 1, orRe[hj(i )], Im[hj(i)] - N(0, 0.5).The "quasi-static channel" assumption in all 0-STBCschemes requires that hj(i)= constant over the entirecodeword length. This, in the case of G4 an d G3systems, means that the channel remains static overXr9.t has been shown in [6][7][14] that even undernormal vehicle speeds, the assumption of "quasi-staticchannel" may not hold f or the 4-Tx STBC (e.g., '& and8 ,) systems. As such, this paper assumes that thechannel is static ove r T, only and from one rs o th enext, it is time variant.Clearly, this is a much moregeneral and realistic model with the quasi-staticchannel now becoming a special case. Perfect channelstate information (CSI) is assumed in this paper. Forthe estimation o f CSI, please see [ 5 ] [ 9 ] - [ 21.

'

C =

111. The C onve n t iona l 0 - S T B C D e te cto r f o r G*For symb ol group s = [SI, 2, s j , sd]'. the code matrixfor 94 [2] is

SI 32 s:%-s 2 SI - 1-s:$ 81 s1 - S A

(2)-s* - s s2 sisf s; s s '-s; s; -s; s;- s j s s; -s;- -s; -s; s; s;

From Eq.( I ) , the "manipulated received signal vector"r = T I , ..., r:,; rf , ..,

r = H s + n , (3)can then be written as

where the channel matrix (ChM )

At the receiver end, the conventional 0-STBC detectorassumes that the channel is time invariant over theentire S l ; period. Regardless of any channel variation,the following expressio n is effectively employed:( 5 )= Hs + n ,

where H is the estimated channel m atrix for the "quasi-static channel":

390

It can be proved that the "linear maximum likelihood(ML)" G, detector in [3] is equivalent to the followingtwo-step procedure [6][7]:[Step C l ] Apply linear transform Q = H to thereceived signal r:

-H

r' = Qr = QHs + v ' (7 )[Step CZ] Carry out the "linear ML" detection:

X7= arg{ m inl[r']; - j s rn l*} , i = 1,2,3,4, (8 )Here, v = a n , S s the symbol alphabet, [r 'Ii the ithelement of r', and A; = 2xlhj/2i.e., the ith elementj= 1of diagonal QH).In reality, however, th e true physical process in Step C1is

r' = Qr = QH s + v . (9 )=QH = &]4x4. (10)

rmtS

4 -

Note that in general XVH in Eq.(9) is non-diagonal:

The conventional 2-step linear detection procedure istruly ML if and only if the channel is truly static over8Th, n which case H =H and thus $ = 0 fo r i # j ,and 4;t = X i .

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For a time selective fading channel, however, H # 6and thus 4 i j # 0 fo r i # j . Physically, this leads tointer element interference (IEI). The value o f the non-diagonal 4ij depends upon the time-selectivity of thechannel. When using the above conventional detector(thus the 84 decoder in [3]) for a time selective fadingchannel, these non-zero &'s are effectively ignored,resulting in extra detection errors in addition to thosecaused by the AWGN. These extra errors will form anirreducible error floor in thc BER curves in the highSNR region.IV . T h e PI C Detec tor for G4Although & # 0 (for i # j) in Eq.(lO), it is also truethat normally 1dijI

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[32 + A 4 + (3 + A . l ) I ] complex number (CN)multiplications/symhol and [28 + A[ + 311 CNadditions/symbol, where M is the modulation level andI is the number of iterations. Therefore, thecom putatio nal complexity of the PIC detector is higherthan that of both the conventional and the ZF decoders[6][7]. As I is normally small ( 5 3) , however, theextra computation in the PIC detector is still moderate(compared with the full ML search over al l 4 symbols,whose complexity is O ( M ' ) ) , and can well he justifiedby the enorm ous performance improvem ent.[The Gj encoded systems] By setting I L ~ ( Z )= 0 inE q . ( l ) a nd al l the other related equations, the abovePIC detector can directly be used in the E, encodedsystems.V. S i m u l a t i o n sT he G4 system under 16-QAM (Gray encoded)modulation and the time selective fading channel inEq.(lI) are employed. The signal to noise ratio (SNR)at the receiver is defined asSN R = (MT~zE.9 ) /~ :MTES/u: (since U%= l),where E , is the Tx power at eac h antenna, and A& = 4(for the '& system). Also, the UMTS symbol ra tes arec o n s i d e r e d : 7: = SF/(3.84 x 10') seconds wi thf = 2 GHz.Five vehicle speeds are simulated V = 0 (quasi-staticchannels) , 70, 100, 130, and 160 km/h (leading todifferent cy, values) for SF = 128 (or equivalentlyV = 0, 35, 50, 65, and 80 km/h for SF=256). Thesecorrespond to the f d T s values of 0, 0.0043, 0.0062,0.0080, and 0.0099 (see Fig. 1for the p values). For thecase o f f & = 0.0099, the BER details of the PIC forI = 0 (i.e., the conventional decoder), 1, 2, and 3 ar cshown in Fig. 2. Clearly, two or three iterations delivermost of the performance gain even for such a highspeed case. For al l the other speeds, only the BERresults for I = 3 are plotted in Fig.3. For comparison,the re sult s of the conventional G4 detector are shown inFig. 4. It is easy to see that the P IC detector exhibits n oerror floor while the conventional G4 detector does.Most importantly, the BER degradation of the PICcaused by channel variation is very small indeed withinthe considered f,,T, range. The penalty, however, is arelatively higher computational complexity. In addition,our simulations (not shown here) have indicated that

the sam e observation also applies to the GJ coded 3-Txsystems.VI. Conclus ionsThis paper has presented a PIC based detec tor structurefor the '& 0-STBC systems over time-selective fadingchannels. While the conventional '& detectors undersuch conditions tend to suffer from a considerableirreducible error floor in the high SNR cases, the PICdetector shows no error floor at all. The PIC'Srelativelyhigher computational co st can he justified by itsenormous performance gain.

. ..08 I

fdTe' lOOFig. 1 The value of printed as "mu" here) withrespect to f,jT6.

5 10 15 20 25 30 35 40 ' 45SNR

Fig. 2 Th e BE R performance of the PIC '& detectorwith respect to the number of iterations.

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5 10 15 20 25 30 35 40 45SNR .Fig. 3 Th e BER performance of the PI C G d detectorfor I=3.

~

5 10 15 XI 25 30 35 40 45SNRFig. 4 Th e BER performance of the conventional

E d detector.

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0-7803-8255-2/04/$20.00 I EEE.

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