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On the Performance of MC-CDMA Systems with Partial Combining and Multiple Antennas in Fading Channels

On the Performance of MC-CDMA Systems with Partial Combining and Multiple Antennas in Fading Channels
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  On the Performance of MC-CDMA Systems withPartial Combining and Multiple Antennas in FadingChannels Flavio Zabini DEIS, University of Bologna, ItalyEmail: fzabini@deis.unibo.it Barbara M. Masini IEIIT/CNR, University of Bologna, ItalyEmail: barbara.masini@unibo.it Andrea Conti ENDIF, University of Ferrara, ItalyEmail: a.conti@ieee.org  Abstract —In this paper we analytically evaluate the downlinkperformance of a multi-carrier code division multiple access(MC-CDMA) system by employing partial combining whenmultiple antennas at the transmitter or at the receiver areconsidered. It is known that for single reception/transmission,the partial combining technique depends on a parameter whichcan be optimized as a function of the number of subcarrier, thenumber of active users and the SNR improving the performancewith respect to other classical techniques (such as maximal ratiocombining, equal gain combining, or orthogonality restoringcombining). In this paper we extend the analysis considering alsomultiple antennas aiming at showing how the spatial diversityaffects the performance as a function of the partial combiningparameter. I. I NTRODUCTION Multi-carrier code division multiple access (MC-CDMA)consists in a combination of orthogonal frequency divisionmultiplexing (OFDM) and spreading. Systems based on MC-CDMA are widely considered to counteract the frequencyselectivity of the wireless channel, to avoid inter-symbol andinter-carrier interference and achieve high spectral efficiency.The basic principle of MC-CDMA is to spread each datasymbol over several subcarriers and, hence, to efficientlyexploit frequency diversity of the channel [1]. In this work we consider the MC-CDMA architecture presented in [2]and [3], where the spreading is performed in the frequency-domain and Walsh-Hadamard (WH) codes are used withspreading factor equal to the number of subcarriers. We focuson the downlink, hence the system is assumed synchronousand different users experience the same channel. However,in spite of this, in frequency selective fading channels, theorthogonality between spreading sequences is corrupted byfading, thus the performance at the receiver side is affectedby multi-user-interference (MUI) [2], [3]. Focusing on linear combining techniques and assuming perfectchannel state information (CSI), many techniques are knownin the literature trying to minimize the bit error probabil-ity: maximum ratio combining (MRC) reduces the effectof noise but completely destroys the orthogonality betweenthe sequences (increasing the MUI), whereas orthogonalityrestoring combining (ORC) reduces the effect of MUI butenhances that of noise. It is shown in [4] that partial combining(PC) technique represent a good trade-off to reduce both theeffect of noise and MUI. In single transmission and receptionsystems (single input single output - SISO - systems), PCconsists in choosing the complex weighs for the  m th subcarrier( m  = 0 ,...,M   − 1 ) as: G m  =  H  ∗ m | H  m | 1+ β , β   ∈ [ − 1 , 1] ,  (1)where  H  m  is the  m th complex channel coefficient and  β   is thePC parameter to be optimized in order to minimize the bit errorprobability (BEP). It is readily understandable that when  β   =0 ,  − 1  and  1 , then (1) coincides with EGC, MRC and ORC,respectively. In [4] the optimal value of   β   for SISO systemswas derived as a function of the number of subcarrier  M  , thenumber of active users  N  u  and the mean signal-to-noise-ratio(SNR) averaged over small-scale fading,  γ  . In this work weconsider multiple antennas at the transmitter or at the receiverside in order to extend the analytical framework to jointlyconsider PC on both subcarriers and branches when multipleantennas are employed and to verify the improvement achievedby adding spatial diversity. For a deep investigation on howantenna diversity affects MC-CDMA systems performance,see, at instance, [5]–[7]. II. S YSTEM  M ODEL Following the MC-CDMA architecture presented in [4],each data-symbol is copied over all sub-carriers and multipliedby the chip assigned to each particular sub-carrier. Binaryphase shift keying (BPSK) modulation is assumed (see Fig.1 for details).  A. Multiple Transmitting Antennas In Fig. 1(a) the transmitter block scheme for the case of multiple transmitting antennas is presented: the transmittedsignal referred to the  k th user on the  l th antenna, can be writtenas: s ( l,k ) ( t ) =   2 E  b LM  + ∞  i = −∞ M  − 1  m =0 c ( k ) m  a ( k ) [ i ] g ( t − iT  b ) ×  cos(2 πf  m t  +  φ m ) ,  (2)where  E  b  is the energy per bit,  L  the number of antennas, M   the number of subcarriers. Index  i  denotes the data index, m  is the sub-carrier index,  c m  is the  m th chip of the WH 978-1-4244-1645-5/08/$25.00 ©2008 IEEE1374  code sequence,  a ( k ) [ i ]  is the data-symbol transmitted by user k  during the  i th symbol time,  g ( t )  is a rectangular pulsewaveform with duration  [0 ,T  ]  and unitary energy,  T  b  isthe bit-time,  f  m  is the  m th sub-carrier-frequency and  φ m  isthe random phase uniformly distributed within  [ − π,π [ . Inparticular,  T  b  =  T   +  T  g  is the total OFDM symbol duration,increased with respect to  T   of a time-guard  T  g  (insertedbetween consecutive multi-carrier symbols to eliminate theresidual inter symbol interference, ISI, due to the channeldelay spread).We now assume to jointly combine both antennas and subcar-riers contributions, by weighting the signal on each antennaand subcarrier with the partial combining coefficient,  G m,l ,defined as in (1) when jointly considering antenna diversity(see Fig. 1(a) as example): G m,l  = H  ∗ m,l | H  m,l | 1+ β , β   ∈ [ − 1 , 1] ,  (3)Hence, assuming uncorrelated fading channels, the total diver-sity degree is in the range  [ M,LM  ]  depending on the correla-tion among the antennas. All users exploit all the subchannels(e.g, spreading factor equal to the number of subcarriers),thus, considering perfect channel state information (CSI) atthe transmitter, the total transmitted signal on the  l th antennaresults in: s ( l ) ( t ) = N  u − 1  k =0 s ( l,k ) ( t ) =   2 E  b LM  N  u − 1  k =0+ ∞  i = −∞ M  − 1  m =0 c ( k ) m ×  a ( k ) [ i ] G m,l g ( t − iT  b )cos(2 πf  m t  +  φ m ) ,  (4)where  N  u  is the number of active users and, because of the useof orthogonal codes,  N  u  ≤  M  . Finally, the total transmittedsignal over  L  antennas is given by: s ( t ) = L − 1  l =0 s ( l ) ( t ) .  (5)  B. Multiple Receiving Antennas Following assumptions made in Section II-A (i.e., uncor-related fading among the subcarriers and perfect CSI), in theopposite case of single transmission and multiple receivingantennas (see Fig. 1(b) for details), the total transmitted signalcan be written as: s ( t ) =   2 E  b LM  N  u − 1  k =0+ ∞  i = −∞ M  − 1  m =0 c ( k ) m × a ( k ) [ i ] g ( t − iT  b )cos(2 πf  m t  +  φ m ) ,  (6)III. D ECISION  V ARIABLE We consider the downlink of a MC-CDMA system, henceusers are synchronous and different delays affecting eachsubcarriers are assumed perfectly compensated. We considera frequency-domain channel model with transfer function foreach subchannel,  H  ( f  ) , given by: H  ( f  ) ≃ H  ( f  m ) =  α m e jψ m ,  (7)where  α m  and  ψ m  are the  m th amplitude and phase coeffi-cients, respectively. Each  H  ( f  m )  is considered independentidentically distributed (i.i.d.) complex zero-mean Gaussianrandom variable (r.v.) (i.e., each subcarrier experiments flatfading, uncorrelated with the other subcarriers) with variance σ 2 H   defined as: E { α 2 m,l } = 2 σ 2 H   = 1 ,  (8)where  E {·} is the statistical expectation. Hence, we can write: M  − 1  m =0 L − 1  l =0 E { α 2 m,l } =  LM,  ∀  l,m.  (9)Each  α m,l  =  | H  m,l |  is assumed Rayleigh distributed, i.i.d.for each subcarrier and each antennas. In the following, thedecision variable for both multiple transmission and singlereception and single transmission and multiple reception isderived.  A. Multiple Transmitting Antennas In this case, the received signal can be written as: r ( t ) =   2 E  b LM  N  u − 1  k =0+ ∞  i = −∞ L − 1  l =0 M  − 1  m =0 G m,l α m,l c ( k ) m  a ( k ) [ i ] ×  g ′ ( t − iT  b )cos(2 πf  m t  + ϑ m         φ m  +  ψ m ) +  n ( t ) ,  (10)where  g ′ ( t )  is the response of the channel to  g ( t )  with unitaryenergy and duration  T  ′   T   +  T  d , being  T  d  ≤  T  g  the timedelay,  n ( t )  is the additive white Gaussian noise with two-sidepower spectral density (PSD)  N  0 / 2  and  ϑ m    φ m  +  ψ m .Note that, since  ϑ m  can be considered uniformly distributedin  [ − π,π [ , we can consider  ∠ H  ( f  m )  distributed as  ϑ m  inthe following. The receiver performs the correlation of thereceived signal with  c ( n ) m √  2cos(2 πf  m t + ϑ m ) , thus, after somealgebra, the decision variable results in: v ( n ) =   E  b δ  d LM  M  − 1  m =0 L − 1  l =0 α m,l g m,l a ( n ) +   E  b δ  d LM  M  − 1  m =0 L − 1  l =0 N  u − 1  k =0 ,k  = n α m,l g m,l c ( n ) m  c ( k ) m  a ( k ) + M  − 1  m =0 n m  (11)where  δ  d  = 1 / (1+ T  d /T  )  represents the loss of energy causedby time-spreading and the gains  g m,l  are normalized as: g m,l  =  Gα − βm,l  ,  (12)being G  =    LM   M  − 1 m =0  L − 1 l =0  α − 2 βm,l .  (13) 1375  Multiplying each term of (11) by  G , we can write: v ( n ) = U           E  b δ  d LM  M  − 1  m =0 L − 1  l =0 α 1 − βm,l  a ( n ) + I           E  b δ  d LM  M  − 1  m =0 L − 1  l =0 N  u − 1  k =0 ,k  = n α 1 − βm,l  c ( n ) m  c ( k ) m  a ( k ) + N            1 LM  M  − 1  m =0 L − 1  l =0 α − 2 βm,lM  − 1  m =0 n m  .  (14)Assuming to have a number of subcarriers sufficiently highto justify the use of the law of large numbers (LLN) andexploiting the independence among the subcarriers, by applingthe Kolmogorov’s Law, we can write: 1 M  M  − 1  m =0  L − 1  l =0 α 1 − βm,l  =  E  L − 1  l =0 α 1 − βm,l  ,  (15) 1 M  M  − 1  m =0  L − 1  l =0 α − 2 βm,l  =  E  L − 1  l =0 α − 2 βm,l  .  (16)Since the channel coefficients are Rayleigh distributed: E  α 1 − βm,l  = (2 σ 2 H  ) 1 − β 2 Γ  3 − β  2  , ∀  m, l  (17) E  α − 2 βm,l  = (2 σ 2 H  ) − β Γ(1 − β  ) , ∀  m, l,  (18)being  Γ( · )  the Euler-Gamma function. By substituting (17)and (18) in (15) and (16), respectively, and exchanging thesum operation with the statistical expectation, we have: 1 M  M  − 1  m =0  L − 1  l =0 α 1 − βm,l   =  L (2 σ 2 H  ) 1 − β 2 Γ  3 − β  2  (19) 1 M  M  − 1  m =0  L − 1  l =0 α − 2 βm,l   =  L (2 σ 2 H  ) − β Γ(1 − β  )  .  (20)Therefore, we can write: U   =   E  b δ  d LM  (2 σ 2 H  ) 1 − β 2 Γ  3 − β  2   ,  (21) N   =   (2 σ 2 H  ) − β Γ(1 − β  ) M  − 1  m =0 n m  .  (22)Since  n m  are independent Gaussian zero-mean random vari-ables with variance  N  0 / 2 , the noise term  N   is gaussiandistributed as: N   ∼N   0 ,σ 2 N    ,  (23)where σ 2 N    (2 σ 2 H  ) − β Γ(1 − β  )  MN  0 2  .  (24)For what concern the interference term  I  , we can concatenatethe sums in  m  and  l  in one vector of dimension  LM   and,by exploiting the independence between the coefficients andapplying the methodology proposed in [4], we find out: I   ∼N   0 ,σ 2 I    (25)where σ 2 I    E  b δ  d ( N  u − 1) ζ  β ( α )  (26)with ζ  β ( α ) = (2 σ 2 H  ) 1 − β  Γ(2 − β  ) − Γ 2  3 − β  2   .  (27)  B. Multiple Receiving Antennas With multiple receiving antennas, the contributions of eachantenna have to be weighted with the complex coefficients G m,l  which in multiple transmission case has been applied atthe transmitter side, as below: r ( t ) = L − 1  l =0 G l,m r ( l ) (28)being  r ( l ) ( t )  the signal received on the  l th antenna. Hence, wecan write: r ( t ) =   2 E  b LM  N  u − 1  k =0+ ∞  i = −∞ L − 1  l =0 M  − 1  m =0 G m,l α m,l c ( k ) m  a ( k ) [ i ] ×  g ′ ( t − iT  b )cos(2 πf  m t  + ϑ m             φ m  +  ψ m ) + L − 1  l =0 n l ( t ) ,  (29)being  n l ( t )  the noise at the  l -th antenna supposed to beGaussian with two-side power spectral density (PSD)  N  0 / 2 .For what concern the evaluation of the decision variable, theprocedure is similar to the one presented in [4], except for thefurther degree of freedom given by spatial diversity. Hence,the statistical distribution of   U N   and  I   results in: U   ∼ N    E  b δ  d ML E { α 1 − βl  } ,σ 2 U   N   ∼ N   0 ,σ 2 N   =  MLN  0 2 (2 σ 2 H  ) − β Γ(1 − β  )  I   ∼ N   0 ,σ 2 I   =  E  b δ  d L ( N  u − 1) ζ  β ( α )   . IV. P ERFORMANCE  E VALUATION In this section we derive the BEP and the optimum PCparameter,  β  , defined as the value of   β   minimizing the BEP.  A. Multiple Transmitting Antennas Following the previously mentioned assumptions, the BEPcan be evaluated as: P  b  = 12 erfc   U    2( σ 2 I   +  σ 2 N  )  = 12 erfc  √  L · SNIR   , whereSNIR  = γ   Γ 2  3 − β 2  2  N  u − 1 M   γ   Γ(2 − β  ) − Γ 2  3 − β 2  + Γ(1 − β  ) , 1376  represents the signal-to-noise plus interference ratio for thesingle transmission/reception scenario described in [4] and γ   = 2 σ 2 H  E  b δ  d N  0 ,  (30)is the mean SNR averaged over small-scale fading. Note that,being SNIR equal to the single transmission/reception case,the optimum value of   β   minimizing the BEP does not changewith multiple transmitting antennas, but the performance isimproved by the spatial diversity.  B. Multiple Receiving Antennas The unconditioned BEP with  L  receiving antennas is givenby: P  b  ≃  12 erfc  L − 1  l =0   E  b δ  d N  0 L (2 σ 2 l  ) 1 − β 2 Γ  3 −  β  2   (31) ×  1   L − 1 l =0  E  b δ d N  0 L  2 N  u − 1 M   σ 2 I  ( β,l ) + (2 σ 2 l  ) − β Γ[1 − β  ]  being σ 2 I  ( β,l ) = (2 σ 2 l  ) 1 − β  Γ[2 − β  ] − Γ 2  3 −  β  2   (32) + L − 1  l ′ =0 ,l ′  = l  σ 2 l,l ′ ( β  ) −  12(2 σ 2 l  ) 1 − β 2 (2 σ 2 l ′ ) 1 − β 2 Γ 2  3 −  β  2   , where, due to the independence among the subcarriers, σ 2 l,l ′ ( β  ) = 0 . 5 · E { α 1 − βm,l  α 1 − βm,l ′ } ,  ∀  m . In case of uncorrelatedantennas, the second term in (32) is zero, thus (31) reduces tothe following expression: P  b  ≃  12 erfc  √  SNIR  ,  (33)whereSNIR  =Γ 2  3 − β 2  γ  2 N  u − 1 M   Γ(2 − β  ) − Γ 2  3 − β 2   γ L  + Γ(1 − β  ) . (34)It is immediate to verify that when  L  = 1 , (33) reduces to theexpression found in [4]. Note that (32) is general, thus validalso for correlated fading among the antennas.We now aim at evaluation the optimum value of   β   whichminimizes the BEP. It can be found out that for spatialuncorrelated antennas: β  (opt) = argmin | β { P  b } = argmax | β { SNIR }  .  (35)By defining the parameter ξ  . = 2 γ N  u − 1 M  ,  (36)we find that (36) leads to: ξ   =  L  1Ψ  3 − β 2  − Ψ(1 − β  )+  β  − 1  − 1 ,  (37)where  Ψ( x )  is the logarithmic derivative of the Euler Gammafunction. Note that  ξ   is a monotonic function of   β  , thusinvertible, and it confirms that if the system is noise-limitedMRC is optimum, while if the system is interference limited,a choice close to ORC is required.Is also possible to observe that for high values of mean SNR, γ  , the second term at the denominator of (30) is negligible,thus, for interference-limited systems the SNIR evaluatedfor multiple transmissions agrees with the one of multiplereceptions.V. N UMERICAL  R ESULTS The mean BEP as a function of   β   for various SNRs,  γ  ,and diversity degrees,  L , at the transmitter and at the receiveris shown in Fig. 2 in fully loaded system conditions ( N  u  = M   = 1024 ). In Fig. 2(a) the improvement on the BEP given byspatial diversity at the transmitter can be appreciated especiallyincreasing the mean SNR,  γ  . It is also noticeable that theperformance strongly depends on  β  . Same conclusions can bedrawn by observing Fig. 2(b), where the BEP vs.  β   is plottedconsidering multiple receiving antennas. From a comparisonbetween the two figures, it appears that the optimum value of  β   is in the range [0.4, 0.7] and, on equal diversity degrees,multiple transmission performs better than multiple reception. 1 This is caused by summation of noise on all the antennasbranches when multiple reception is assumed.When multiple antennas at the receiver are considered, Fig.3 shows the mean BEP as a function of the mean SNR  γ   in[dB] for  β   = 0 . 5  and various diversity degrees,  L . Fully loaded( N  u  =  M   = 1024 ) and half loaded ( N  u  =  M/ 2 ) systemconditions are considered. It is possible to observe that, byfixing a target BEP,  P  b  = 10 − 3 , it can be achieved only if thesystem is half loaded with single reception, whereas a fullyloaded system can be considered by adopting two antennas atthe receiver.A CKNOWLEDGMENT The authors would like to thank Prof. Oreste Andrisano forsupporting this research activity and Prof. Larry Greensteinfor motivating this work.R EFERENCES[1] I. Cosovic amd S. Kaiser, “A Unified Analysis of Diversity Exploitation inMulticarrier CDMA”, IEEE Transactions on Vehicular Technology, Vol.56, No. 4, July 2007, Page(s): 2051 - 2062.[2] N. Yee, J.-P. Linnartz and G. Fettweis, ”Multi-Carrier-CDMA in indoorwireless networks”, in Conference Proceedings PIMRC ’93, Yokohama,Sept, 1993. p 109-113[3] N. Yee and J.-P. Linnartz, ”BER of multi-carrier CDMA in an indoorRician fading channel”, Conference Record of The Twenty-SeventhAsilomar Conference on Signals, Systems and Computers, 1993., 1-3Nov. 1993 Pages:426 - 430 vol.1[4] A. Conti, B. Masini, F. Zabini and O. Andrisano, “On the downlink Performance of Multi-Carrier CDMA Systems with Partial Equalization”,IEEE Transactions on Wireless Communications, Vol. 6, Issue 1, Jan.2007 Page(s):230 - 239. 1 Note that, being in the downlink, multiple transmission can be easilyimplemented at the base station. 1377  (a) Multiple transmission and single reception. (b) Single transmission and multiple reception.Fig. 1. Multiple transmission and multiple reception block schemes. −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 β 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 P b γ  =5dB γ  =10dB γ  =15dBL=1L=2L=4 (a) Multiple transmission and single reception. −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 β 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 P b L=1L=2L=4 γ  =5dB γ  =10dB γ  =15dB (b) Single transmission and multiple reception.Fig. 2. Mean BEP as a function of   β   for different SNRs,  γ  , and diversity degrees,  L . 0 5 10 15 γ   [dB]10 −4 10 −3 10 −2 10 −1 10 0 P b L=1, N u =ML=2, N u =ML=4, N u =ML=1, N u =M/2L=2, N u =M/2L=4, N u =M/2 Fig. 3. Mean BEP as a function of mean SNR for  β   = 0 . 5 , various diversitydegrees,  L , at the receiver side and fully or half loaded conditions.[5] J. Tang, X. Zhang, “Transmit selection diversity with maximal-ratiocombining for multicarrier DS-CDMA wireless networks over Nakagami-m fading channels”, Selected Areas in Communications, IEEE Journal on,Volume 24, Issue 1, Jan. 2006 Page(s):104 - 112 .[6] L-L. Yang, L. Hanzo, “Performance of broadband multicarrier DS-CDMA using space-time spreading-assisted transmit diversity”, WirelessCommunications, IEEE Transactions on, Volume 4, Issue 3, May 2005Page(s):885 - 894.[7] Y. Zhang, L. B. Milstein, P. H. Siegel, “Tradeoff between diversitygain and interference suppression in a MIMO MC-CDMA system”,Communications, IEEE Transactions on Volume 53, Issue 4, April 2005Page(s):623 - 631. 1378
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