Pre-equalization techniques for MC-CDMA systemswith reduced cyclic prefix redundancy

Nevio Benvenutot, Paola Bisagliat, Federico BoccardittDipartimento di Ingegneria dell'Informazione

University of Padova, Via Gradenigo 6/B 35131 Padova, ItalytDora Spa, ST Microelectronics Group, Via Lavoratori Vittime del Col du Mont 24, 1 1 100 Aosta, Italy

Email: nb@dei.unipd.it, paola-dora-spa.bisaglia@st.com, federico.boccardi@dei.unipd.it

Abstract- For an increased bandwidth efficiency, a TDD MC-CDMA system with cyclic prefix inserted every K OFDMsymbols is utilized. In this context, to combat inter-carrier andinter-symbol interference along with multiple access interference,new pre-equalization schemes for both downlink and uplinkare introduced; for downlink, a multiple transmit antennaconfiguration is also considered. Results of computer simulationsshow that the proposed pre-equalization techniques are veryeffective with respect to other pre or post-equalization solutionsand furthermore they can trade performance and bandwidthefficiency against complexity.

Index Terms-MC-CDMA, pre-equalization, cyclic prefix,downlink, uplink.

I. INTRODUCTIONM/[Ulticarrier-code division multiple access (MC-CDMA)

systems are being considered as potential candidatesfor next generation (4G) technology. MC-CDMA inherits fromCDMA the multiple access flexibility, while from orthogonalfrequency division multiplexing (OFDM) characteristics suchas robustness against multi-path propagation [1], [2]. In OFDMsystems, the usual approach for combating the multi-path na-ture of wireless channels is by insertion of a cyclic prefix (CP)to each transmitted OFDM symbol, where the length of theCP exceeds the maximum channel impulse response length. Ifon the one hand the use of a CP simplifies the equalizationproblem, on the other hand the bandwidth efficiency is signif-icantly reduced. Recently, in order to increase the efficiency,OFDM systems with CP inserted every K symbols have beenconsidered and correspondingly post-detection methods havebeen proposed [3].

In this context, the aim of this paper is to introduce new pre-equalization schemes, for both downlink and uplink, in orderto combat inter-carrier and inter-symbol interference alongwith multiple access interference (MAI). We note that, in thelast few years, pre-equalization techniques at the transmitterhave been proposed for downlink [4], [5], [6] and uplink[7], [8] conventional MC-CDMA systems, in conjunctionwith time division duplex (TDD), where the same frequencychannel is used for both uplink and downlink transmissions.The principle behind pre-equalization at the transmitter is tovary the gain assigned to each sub-carrier on the basis of the

This work was supported in part by the Italian Ministry of University andResearch (MIUR), under the FIRB project. The work was initiated when P.Bisaglia was with the University of Padova.

0-7803-9206-X/05/$20.00 ©2005 IEEE

channel state information (CSI) which can be obtained fromthe received signal using the TDD channel reciprocity. Theconsequence is that at the receiver no equalization is requiredand only de-spreading may be performed.As regards downlink transmissions, we present a novel pre-

equalizer scheme based on the total interference removal (TIR)approach [5]. In our proposed algorithm the problem of perfectcancellation of the inter-carrier and inter-symbol interferenceas well as multiple-access interference, is tackled allowing alower redundancy due to the cyclic prefix. To increase thediversity gain of the system, a multiple antenna configurationhas been considered. As regards uplink transmissions, inanalogy with detection techniques at the receiver, we proposepre-equalization techniques based on zero forcing (ZF) orminimum mean square error (MMSE) criteria. Moreover, forboth downlink and uplink transmissions, the key issue of pre-equalizing the signal at the transmitter side without increasingthe transmit power is considered and criteria are proposed.

The remainder of this paper is organized as follows. InSection II the structure of the MC-CDMA system is given.In Section III pre-equalizer techniques, for both downlinkand uplink transmissions are presented. Numerical results arereported in Section IV and conclusions are in Section V.

II. SYSTEM MODEL

We consider the MC-CDMA system depicted in Figure 1,from the transmitter to the generic receiver u. The transmitteris equipped with M antennas, while the receiver with oneantenna only. Let a(U) denote the transmitted data symbolsof user u (belonging to a unit power constellation), NSF thespreading factor and c(U)-= (): (UFl_]T the spread-ing sequence, selected from a set of orthogonal codes. Thedata symbols are first serial-to parallel converted to produceK symbols a(U),k = 0, 1..., K - 1, then each symbol isspread by repeating a(u) NSF times (upsampler and holderin Figure 1) and multiplied by the user specific spreadingsequence. A serial-to-parallel conversion then follows to loadthe chips into the OFDM modulator. The chip period Thip isrelated to the symbol period T by Thip = T/NsF. In thiswork to simplify notation the number of OFDM sub-carriersM equals the spreading factor, i.e. M = NSF. OFDM mod-ulation is implemented by using the inverse discrete Fouriertransform (IDFT). Next, pre-equalization is performed after

103

Fig. 1. Block diagram of the MC-CDMA system, between the transmitter and the generic receiver u.

OFDM modulation by multiplying the signal by a complexmatrix K(') that will be introduced later. Last, the CP isinserted every K OFDM symbols. Referring to the genericantenna m, the resulting signal ymu), received over the radiochannel h(u), from antenna m of the transmitter to the receiveru, is corrupted by the contribution of other antennas and otherusers. Furthermore, we assume that the channels are time-invariant during at least K OFDM symbols.

Let h$u) (1) be the i-th tap of the channel impulse response,of length L, between the m, m = 0, ..., M - 1, transmitantenna and the receive antenna, associated to user u.Moreover, referring to a MC-CDMA system with CP insertedevery K OFDM symbols, let

C(u) -Cs=

We introduce theexpressed as

C(U)C(U)

C(U)

C(U)C(U)

C(U)

(3)

KM x KMM system matrix H(U),

a(u) - [a(u). a(u) ...I a(U) ]T (1)

denote the vector of symbols transmitted over K OFDMsymbols by user u, and d(u) be the KMM chip vector,given by

d(u)- [d(u)T d(u)T d(u)T d(u)T]T-[O ** K-1' 0I* K-1

antenna 0 antenna M-1

ra(u) C(U)T. a(u) c(u)Tantenna 0

au) c(U)T, . a(u) c(U) ]T

antenna M-1

= C(U)a(u) (2)

where C(U) is the KMM x K spreading matrix given by

H(u) - [H(U) H(u) ...HM (4)

where H$u), mz=0, ..., M-l represents the KM x KMsingle-input single-output system matrix, after the CP hasbeen removed, found to be

h(u) (O h(u) (1)

0h(m)(L 1) h(u)(L -1)

H _u) 0 h(u (0) . 0 (5)

00 h(u) (L-1) h(u) (0)

If FM is the M x M discrete Fourier transform (DFT)matrix, with elements [FM]i,,. = W)7, where WM = e_j I,and introduced the KMM x KMM matrix

FM O MxM

OMXM FM

FM (KMI) =OMxM

*. OMXMA... OMXM

... FM I (6)

104

the KMAI received signal, when pre-equalization is applied, and applying the total interference removal (TIR) criterioncan be expressed as [4] in the time-domain, we obtain the following conditions I

U-i

Y H(V)K(V)EF (KM) d(v) +nV=0

where FM1 (KM) is the inverse of FM (KM), K(denotes the KMM x KMM pre-equalizer matrix anmis the additive white Gaussian noise (AWGN) vector, ecomponent having a variance o2.

III. PRE-EQUALIZATION SCHEMES

A. Downlink transmission1) Pre-equalization technique: In the downlink c

concentrating on the u-th MT, since the channel is the sa(7) becomes

U-1

Y-S H(U)Kv)EF-(KM) d(v) + nv=O

The KMM x KMMI matrix K(vRE can be written as

(7) A(U)K(u) F-4(KMI) C = IK

A(V)KP FM1(KM) C(u) = OK Vv uUu)IRE Defined the KU x KMM matrix

(14)

A()= [,(u)T, \(O)T >, (u-l)T, A(u+l)T (U-1)T]T(15)

the KMM x K matrix

(16)KPU)/ K()E FM (KM)CSand the KU x K matrix

b = [IK, OK, ..., OK]T (17)

it follows that (14) and (15) yield the following system of(8) KU equations and KMM unknowns

A(U)KpuE -b (18)

K(v) -K(v) EK(v) (9)KPRE = PREKN(

where the scalar K(v) is introduced to meet the transmit powerconstraints. We now concentrate on evaluating KPRE, whiledetails on K(v) will be deferred to the next section.

Introduced the K x KM despreading matrix

C(u)H

C(u) =

C(U)HI (10)

C(u)H_and the K x KMM matrix

A(u) = C(u)FM(K) H(u)

Now (18) can be decomposed into K independent sub-systems2

00

AF -&PRE(,i)- l i=0,..., K-1 (19)

0

where in the RHS column vector of (19), the only 1 is at thei-th row. The solution of each sub-system can be obtained bythe LS criterion. Since rank(A(U)) = KU, it follows that thepseudo-inverse of A(U) is given by

(11)

the signal at the decision point, after DFT and despreading,can be written as

a(u) - c(u)FM(K) y

-A(U)K(UEFj1 (KM) C(Uu)a(u)U-1

+ Z A(u)K V) EFM1(KM) C(V)a(v) (12)v=O, v$u

+ N

The signal in (12) involves the following terms: the desiredsignal, the multiple access interference caused by the lossof orthogonality among users, and the noise. From (12),considering the interference that the signal of user u producesat an other MT v

MAI(u')=VV)-A(VKPE)FM1(KM) C(u)a(u) (13)

A(u)# = A(u)H (A(u)A(u)H>-

and KP )E can be determined as

KR)E = A(u)#b-(u) ~ PR

Then KPRE follows from (16).2) Power limiting issues: Like in [5], we consider a per

user power constraint

where the notation III_, denotes the Frobenius norm. Weobserve that imposing different power constraints for each ac-

tive user in downlink transmission, leads to a suboptimum total

Here IK is the K x K identity matrix and OK the all zero K x K matrix.21f A is a matrix, A(:, i) denotes its i-th column.

105

(20)

(21)

K(V) =-' KiN -2

PRE HII(22)

transmit power allocation. In [9]-[l 11 precoding techniqueswith a more efficient power constraint on the overall transmitpower are considered. We stress that the scheme proposed inthis work as modification of [5] is also applicable to [91-[r 1].

3) Computational complexity: The computational complex-ity of (20) mainly comes from the inversion of the KU x KUmatrix A(u)A(u)H, which does not depend on M and M butonly on K and U.

B. Uplink transmission

1) Pre-equalization techniques: In the uplink case, weassume that the MT transmitters are equipped with oneantenna only, i.e. M - 1. As in the downlink case, thereceiver, i.e now the BS, is equipped with one antenna.Hence, the received signal in (7), reduces to

U-iY- Z H(v)K(v),EF7 (K)d(v) + n (23)

v=O

where H(v) = H(v) is a circulant matrix. Equivalently to(9), the KM x KM matrix K'(vE is determined as

K(v) -= KRKN (24)PRE PRE N

Depending on the pre-equalization criterion, namely zeroforcing (ZF) or minimum mean square error (MMSE), KPREcan be evaluated as

ZF: kv) H(v)H(H(v)H(v)H)-1 (25)

MMSE:KPRE = H(v)H (H(v)H()H+ nIKM)

where we recall c2 is the noise variance.For the considered MC-CDMA system with CP inserted

every K OFDM symbols, the system matrix in (5), in thecase of M = 1, can be diagonalized as

H(v) = F-1 D(v)FKM (26)

where D(V) is a diagonal KM x KM matrix. Using (26),(25) can therefore be rewritten as

ZF:K(v) F-1 D(v)-FK (27)

MMSE: KRE= FKM D(v)H(D(v)D(v)H +±o21) -F

allowing to reduce the computational complexity.2) Power limiting issues: Equivalently to (22), the criterion

to determine K(v) is now

Kv K EF (K)C v)I (28)

3) Computational complexity: We consider separately thecomputational complexity (in terms of complex multiplica-tions) due to the derivation of the filter coefficients (designstage) and of the processing of the received signal (processingstage). Referring to (27), with regard to the ZF scheme thecomputational complexity is KM +KM log2(KM) for thedesign stage and KM +2KM lg2(KM) for the processingstage. With regard to the MMSE scheme the computationalcomplexity is 2KM + KM log2(KM) for the design stageand KM + 2KM log2(KM) for the processing stage.

IV. NUMERICAL RESULTS

In order to evaluate the performance of the proposed pre-equalizer techniques, a common scenario for both downlinkand uplink transmissions is envisaged. In particular, theOFDM system consists of 16 sub-carriers, each using QPSK,with a CP equal to 4 every K symbols. Each user has adifferent Walsh-Hadamard sequence and the same powerconstraint. If B is the system bandwidth, we performedsimulations with a multi-path Rayleigh fading channelwith an exponential power delay profile and average rmsdelay spread, normalized to 1/B, equal to 1. In the casemultiple antennas are used at the BS, they are assumed to besufficiently separated in space to consider independent fadingprocesses. The results are given in terms of average bit errorrate (BER) versus Eb/No, the ratio of the average energyper information bit to the noise power spectral density at thereceiver input. It is useful to express the relationship betweenEb/No and the system parameters namely the spreadingfactor NSF, the number of OFDM subcarriers M, the CPlength LCp, the number of symbols in the constellation NS,the system bandwidth B and the average received power PRas follows

Eb _ PR 1NO No B .10g2(NVS) R

where R is the information rate

(29)

(30)-= M/NSFM + LCp/K

Note that an increase in K results in increase of R and hencein a a decrease in the required Eb/No.

A. Downlink

Figure 2 illustrates performance of the pre-equalizationstructures described in Section 111-A, for both K = 1 andK = 4. There are 8 active users (half-load) and the numberof transmit antennas is M = 1 or 2. We note that, for K = 4performance is almost the same with respect to the case ofK = 1. More importantly, we highlight that the informationrate, from K = 1 to 4, increases of almost 15%. Furthermore,using 2 antennas at the BS entails a gain of approximately6.5 dB at a BER = 10-3 with respect to the single antennacase.

106

100

1 01

1o-2

m-310

10

-4

10 -015 2f 5 10 15 20

Eb/N0 [dB]

Fig. 2. Downlink transmission. Average BER versus Eb/NO with U = 8,K = 1 and K = 4, using Al = 1 and M = 2 antennas.

B. Uplink

Figure 3 reports performance of the ZF and MMSE pre-equalization techniques, again for both K = 1 and K = 4. Thenumber of active users has been fixed equal to 8. We observethat in the case of the MMSE pre-equalization technique, forK 4 performance is slightly improved with respect to thecase of K = 1. For the pre-equalizer ZF technique, instead,performance is almost invariant as K increases. However, inboth cases again, the information rate, from K = 1 to K =

4, increases of almost 15%. Finally, we note that at a BER= 10-3 the MMSE pre-equalizer outperforms the ZF pre-

equalizer by 10 dB. For comparison, in Figure 3 performanceof a conventional single user ZF at the receiver (POST-ZF)[1] is also reported. Indeed, results indicate a considerableperformance improvement of pre-equalization with respect topost-equalization.

Fig. 3. Uplink transmission. Average BER versus Eb/NO of ZF and MMSEpre-equalization techniques, with U = 8 and K = 1 and K = 4. Comparisonwith the classical ZF post-equalizer is also reported for K = 1.

[51 T. Salzer, A. Silva, A. Gameiro and D. Mottier, "Pre-filtering usingantenna arrays for multiple access interference mitigation in multi-carrierCDMA downlink," IST Mobile Communications Summit, June 2003.

[6] T. Salzer and D. Mottier, "Downlink strategies using antenna arraysfor interference mitigation in multi-carrier CDMA," MCSS 2003, Sept.2003.

[71 S. Nobilet and J.-F. Helard, "A pre-equalization technique for uplinkMC-CDMA systems using TDD and FDD modes," VTC-fall 2002, pp.346-350, Sept. 2002.

[81 D. Mottier and D. Castelain, "SINR-based channel pre-equalization foruplink multi-carrier CDMA systems," PIMRC 2002, pp. 1488-1492,Sept. 2002.

[91 P. Bisaglia, N. Benvenuto, S. Pupolin, L. Sanguinetti and M. Morelli,"Pre-equalization techniques for downlink and uplink TDD MC-CDMAsystems," Proc of. WPMC'04, Sept. 2004.

[10] L. Sanguinetti, M. Morelli and I. Cosovic, "MMSE pre-filtering tech-niques for TDD MC-CDMA downlink transmissions," Proc. VehicularTechn. Conf, VTC 2005-Spring, May 2005.

[11] N. Benvenuto, P. Bisaglia and F. Boccardi, "Maximum sum-rate linearbeamforming strategies for downlink TDD MC-CDMA systems," IEEEISWCS, Sept. 2005.

V. CONCLUSIONSIn this paper, pre-equalizer techniques for downlink and

uplink MC-CDMA systems with cyclic prefix inserted everyK symbols have been analyzed. The simulation results haveshown that the proposed schemes can trade performance andbandwidth efficiency against complexity.

[1] K. Fazel and S. Kaiser, Multi-carrier and spread spectrum systems,Chichester: Wiley, 2003.

[21 H. Atarashi, N. Maeda, S. Abeta and M. Sawahashi, "Broadbandpacket wireless access based on VSF-OFCDM and MC-DS-CDMA,"PIMRC'02, pp. 992-997, Sept. 2002.

[31 C. V. Sinn, J. Gotze and M. Haardt, "Efficient data detection algorithmsin single- and multi-carrier systems without the necessity of a guardperiod," Proc. IEEE ICASSP, May 2002.

[41 A. Silva, and A. Gameiro, "Pre-filtering techniques using antenna arraysfor downlink TDD MC-CDMA systems" MCSS 2003, Sept. 2003.

107

100

-1

o-2

:::: :::: ........ ::::. ......: : ... .. . . . . . . . . . 4_.......... K=1......

............................... .=

~ - >. . . . . *AWGN channeli .............

b~~~~~~~~M

~~~~~~~~.

.,.: ..>-- M-2 ~ ~ ~ ~

'\* ' >..~~.

~~~~~~~~~~~~.

LLm

10-3

................................................................ K=1..................... ........................... .

..> >. ..>_ . ............ *AWGN channel .

~~~~~~~~~~~~~~~...>-<. .......

~~~~~~~~~~~~~~~...........~~~~~~~~~~~~~~. . ... . . .\ . \ . . .s.............. ... ~.......... ........................~~~~~~~~~~~~~~~~~~~~~~~~~. . . ... . . . . \. . . .

-~ ~ ~ ~ ~ ~~~~ ~ ~ ~ ~~~~~~~~.-~~~~~~~~~~~~~~~. .. ... ,\

5 10Eb/No [dB]

25 30D

10-4

lo-5[c 1-