A Novel MIMO SC-DS-CDMA HARQ scheme for Frequency ...

achieve the maximum frequency diversity, the symbols ofTXjl ,repeated Q -times in the frequency domain mustexperience independent fading. For that to happen, thefrequency spacing between the repeated symbols must behigher than the coherence bandwidth of the channelBcoh≈ 1 τmwhere τmthe delay spread of the channel is. Theabove condition is satisfied if NTc≥ 1 τmthat is N ≥ Tc τm. Alsoin order to guarantee that each subcarrier experience flatfading, we must have Tc ≥ τm. Therefore the chip duration T cmust satisfy the conditionτ ≤T≤ Nτ(8)m c mThe total system bandwidth B= ( NQ+Δ) Tcmust also be lessthan the available bandwidth B o, i.e.( NQ +Δ) Tc ≤ Bo(9)So the frequency domain spreading is limited by the availablebandwidth B .i.e.oQ≤( T B −Δ ) N(10)coLet Yr ⎡Yr,0 , , Yr, G−1vector arerl ,= ⎣ ⋯ ⎤ ⎦ be the NQ× G matrix whose columnY of (12) ; it is given by:T ∗ T T ∗ TYr = H ⎡r1⎣S1Qc1 − S2Q c ⎤ ⎡ ⎤2⎦ + Hr2 ⎣S2Qc1 + S1 Qc2⎦+ W (14)rwhere Wr = ⎡ ⎣ Wr,0 , ⋯ , Wr, G−1⎤ ⎦ . By multiplying (14) by c 1and thenby c2following conjugation, we get respectively:Z = H S + H S + Wc (15)r1 r1 1Q r2 2Q r 1Z = H S − H S + W c (16)∗ * ∗*r2 r2 1Q r1 2Q r 2where Zr1 = Ycr 1and Zr2 = Ycr 2(noting that c 2is real). Eq (15)and (16) can be rewritten as:Z 1r1 = ( Δr1S1+ Δr2S2)+ Wrc 1(17)Q∗ 1 ∗ ∗ ∗Zr2 = ( Δr1S1− Δr2S2)+ Wrc 2(18)QAt the receive antenna r , the CP are first removed (Fig. 2) andthe remaining received chip-blocks corresponding to theTtransmitted sub-blocks xjl ,are applied to an NQ -PointFFT block to yield:⌢ ⌢Yrl ,= Hr1X1, l+ Hr2 X2, l+ Wrl ,, l = 0, ⋯ , G−1(11)where Hrjis a NQ × NQ diagonal matrix whose NQ diagonalelements (the n th and ( n+ mN) th elements where n= 1, ⋯ , N andm= 1, ⋯ , Q are uncorrelated due to the condition in (8) ) are thefrequency response of the channel gain from the j th transmitthantenna to the r receive , X ⌢ jl ,is an NQ × 1 vector given in (5) ,Wrl ,is an NQ × 1 vector of the NQ -Point FFT of the channelnoise at the receive antenna r front end and Y rl ,is anNQ × 1 vector of the frequency domain received signal. Using(1) and (2) , (11) can be written as:Y = H ⎡⎣c () l S − c () l S ⎤⎦ + H ⎡⎣c () l S + c () l S ⎤⎦ + W(12)where∗∗rl , r1 1 1Q 2 2Q r2 1 2Q 2 1 Q rl ,1 T TTjQ= ⎡j jQ⎣ ⋯ ⎤ ⎦Q−timesS S S(13)Tis a NQ × 1 vector and Sj= ⎡ ⎣ Sj1⋯ S ⎤jN ⎦ is a N × 1 vector of N -TPoint FFT of sj= ⎡ ⎣ sj,1 ⋯ s ⎤j,N ⎦ . and cj() l is a scalar.(1) (2) ( Q)Twhere Δrj = [ Hrj Hrj ⋯ Hrj] is of size NQ× N and H( q)rjis aN× N diagonal matrix with diagonal elements the ( q− 1) N + 1 tothe qN diagonal elements of Hrj. The factor 1 Q in (17)and (18) come from (13).Combining (17) and (18) into one column vector we get:where1 ⎛Δr1 Δ ⎞r2Δr= * *Q ⎜⎟⎝Δr2 − Δr1⎠TTT⎡∗⎡ ⎤r r1 1, ⎡ ⎤ ⎤r2 2Zr = Δr S + Wr(19), ⎡1;2T ∗TS = ⎣ S S ⎤ ⎦ , Zr = ⎡ ⎣ Zr1,Z ⎤r2⎦W = ⎢ ⎣ ⎣ W c ⎦ ⎣ W c ⎦ ⎥ ⎦and Z is of size 2NQ × 1rConsidering theMRreceive antennas we get:Z = ΔS + W (20)Twhere , , TTTZ= ⎡ ⎣Z ⎤1⋯ ZMR ⎦ , , , TTW= ⎡ ⎣W ⎤1⋯ WMR ⎦ are of size 2MRNQ×1Tand , , TTΔ= ⎡ ⎣Δ ⎤1⋯ ΔMR ⎦HIt is easily shown that Δ Δ is diagonal matrix whichcompletely decouple the transmitted vectors S1and S2.T299

The minimum mean square estimation (MMSE) of S is givenby:ˆ 1( )SNRH−1HSmmse= Δ Δ + I2NΔ Z (21)We then take the N -Point IFFT of (21) to obtain the decisionT Tstatistics sˆ [ ˆ ˆmmse= s1, s2]mmse. Note that zero-forcing detection canHalso be used which gives similar results as the MMSE as Δ Δ isa diagonal matrix.The two received blocks s ˆT1and sˆT2are multiplexed to formone block and fed to the viterbi decoder; the CRC is checkedin the packet for error. If the is no error the packet is acceptedand a positive Acknowledgement (ACK) is sent to thetransmitter otherwise the receiver send a negativeAcknowledgement (NACK). The transmitter then resends thepacket exactly as described above. The received data blocksare combined to the previous one (symbol level combining)using the pre-combining scheme of [17]. After R transmissions,the decision statistics are given by:R−1R() i H () i 1 ⎞ () i H () i∑2Ni= 1 SNR⎟ ∑i=1S ˆ ⎛mmse= ⎜ +⎝Δ Δ I ⎠Δ Z (22)III. NUMERICAL RESULTSThe parameters of the simulated system are chosen asfollowed:• 2× 2 MIMO system,• Number of symbols per block per antenna : N = 128• Length of cyclic prefix : Δ= 18• Spreading Factor : 32 ( Walsh Hadamard codes)The modulation on each block is QPSK. The multipath modelis selected to be a Rayleigh fading model with anexponentially decaying power profile. The channel is specifiedby the root mean square (rms) delay spread of the tap weights.and assumed quasi-static. The impulse response of the channelis composed of complex samples with random uniformlydistributed phase and Rayleigh distributed magnitude withaverage power decaying exponentially. The k-th path of eachsub-channel is given by:α = N(0, σ ) + jN(0, σ )(23)1 2 1 2k 2 k 2 kk2 2Where σk σ0e −1= andτrms2 −rmsσ0= 1− e ; τ τrmsis the root meansquare delay spread of the channel normalized to the samplingrate. The maximal number of taps is L is chosen such that thedynamic range (the difference between the power of the lasttap and the first tap) is 20dB i.e. L ≥ 4.6τrms.In our simulation τrms= 3 and L is set to 16. As alreadydescribed above, the channel is very dispersive in the timedomain (frequency selective channel). Fig. 3 shows the BERperformance of the proposed MIMO-SC-DS-CDMA-HARQ schemefor one transmission ( R = 1)and Q = 1, 2, 3 which shows thefrequency diversity gain. Fig. 4 shows the performance afterR = 1, 2, 3 transmissions with MMSE for Q = 1, 2, 3 . We can seethe frequency diversity gain due the zeros insertion and theHARQ gain due to retransmission and symbol level combining.In Fig. 5 we compared the BER performance of the proposedscheme for one transmission ( R = 1)and Q = 1, 2 with the MIMOspace-frequency block coding (SFBC) scheme of [4] whereAlamouti space-time block coding is applied in the frequencydomain. The result shows that the proposed schemeoutperforms the conventional MIMO-OFDM SFBC. Note that inthis comparison the system bandwidth for both scheme is thesame when Q = 2 (one time repetition in the frequencydomain) in our scheme. We also note that as the number oftransmissions R increases the frequency diversity gain is lesspronounced which suggests, adaptively reducing Q at laterretransmission. The adaptive approach and the multi-user casewill be presented in another paper.IV. CONCLUSIONIn this paper, a joint FDE, antenna diversity through spacetime-blockcoding combining and employing time andfrequency domain spreading was presented for a single userMIMO SC-DS-CDMA system in a frequency selective Rayleighfading Channel. The BER performance with HARQ is presented.Frequency diversity gain is obtained though it is limited by thetotal available bandwidth. The main contribution of this workis the use of single carrier transmission combined withfrequency domain spreading (frequency diversity gain) andspace-time block coding in a MIMO DS-CDMA system. Systemperformance with HARQ and different level of frequencydiversity is examined. Performance advantage of the proposedscheme on the regular MIMO-SFBC with comparable bandwidthis also presented. This work currently is being extended tomulti-user case where the effect of the number of users and thespreading factor on the performance of the system are studied.REFERENCES[1] F. Adachi, M. Sawahashi and H. Suda, “Wideband DS-CDMA for nextgeneration Mobile Communications Systems,” IEEE Comm. Mag., Vol36, Sept 1998, pp 56-69[2] Y. Kim et al., “Beyond 3G : Vision, Requirement and EnablingTechnologies,” IEEE Comm. Mag., Vol 41, March 2003 , pp 120-124[3] Lajos Hanzo, M. Münster, B.J. 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