COMPARISON OF MULTI-CARRIER MODULATION TECHNIQUES

Andrew S. Ling and Laurence B. MilsteinUniversity of California, San Diego

e-mail: asling@ucsd.edu,

ABSTRACTThis work presents a side-by-side comparison between

two multi-carrier schemes which have equal bandwidth,information rate, and transmit power One system usesmany narrow sub-bands, while the other is made up offewer relatively-wide sub-bands. Closed-form expressionsfor the average bit error probability (BEP) are derived forboth systems under two different cases for the coherencebandwidth of the channel, where estimates of the channelare obtained at the receiver via pilot symbols insertedamong the data. These results are compared with thosewhere the availability ofperfect channel state information(CSI) at the receiver is assumed. Finally, the trade-offsbetween the two systems are investigated under differentvalues for the information rate, number of users in thesystem, and number of pilot symbols used in the estimate.

INTRODUCTIONOver the past decade, extensive research has been carried

out on multi-carrier systems [1]-[4] because of their capa-bility of supporting high data rates and their effectivenessin combating a frequency-selective multipath fading chan-nel. The majority of this research, however, has focused ononly one system at a time, and of the papers which havelooked at two or more multi-carrier schemes [5], most ofthe results given are based on computer simulations.

In this paper, we address the question of whether direct-sequence spreading should be used at each sub-carrierin a multi-carrier system by comparing side-by-side twosystems which have equal bandwidth, information rate,and transmit power. The first scheme, which uses manynarrow sub-bands, will be referred to as multi-carrierCDMA (MC-CDMA). The second scheme, which will belabeled as multi-carrier direct-sequence CDMA (MC-DS-CDMA), performs direct-sequence spreading at each sub-carrier and, hence, uses fewer relatively-wide sub-bands.We will analyze the two systems under two separate casesfor the coherence bandwidth of the channel. Estimates ofthe fade amplitudes and phases introduced by the channel

This work was partially supported by the Center for Wireless Commu-nications at UCSD, the UC Discovery Program of the State of California,the US Army Research Office under the MURI grant number W91 1LNF-04 1 0224, and KDDI R&D Laboratories, Inc.

- La Jolla, CA USA 92093-0407milstein@ece.ucsd.edu

will be made using pilot symbols sent along with the data.We will present closed-form expressions for the averagebit error probability (BEP) for both systems under eachscenario analyzed, and the trade-off between frequencydiversity and channel estimation errors will be investigatedfor different values of the information rate, number of usersin the system, and number of pilot symbols used in theestimate.

OVERVIEW

The MC-DS-CDMA system used in this paper is similarto the one proposed in [4] and investigated further in [6]-[8]. The MC-CDMA system is set up to match this MC-DS-CDMA system as closely as possible. As a result, itwill deviate slightly from the conventional MC-CDMA set-up that is normally seen throughout the literature. In bothsystems, waveform shaping is used to band-limit the signalat each sub-carrier, and the sub-carriers are spaced in sucha way that adjacent sub-bands do not overlap. In MC-DS-CDMA, M symbols are transmitted in parallel, with eachsymbol repeated over R1 different sub-carrier frequencies.The processing gain in each sub-band is defined to be N.Thus, the overall bandwidth is divided into MR1 disjointfrequency sub-bands of equal width. In MC-CDMA, wetransmit the same M symbols in parallel, but each symbolis repeated over R2 different sub-carrier frequencies insteadof R1. Here, the overall bandwidth is divided into MR2disjoint frequency sub-bands of equal width. If we defineW1 and W2 to be the bandwidth of a sub-band in MC-DS-CDMA and MC-CDMA, respectively, then

1

Since both systems have the same overall bandwidth, thisimplies that

R2= N*R. (2)

Fig. 1 shows an example of how the two systems comparein the frequency domain. As for the time domain, in bothsystems, the data being transmitted at each sub-band isdivided into frames, each with Q time slots, as shown inFig. 2. Qe of the Q slots are used to transmit pilot symbols,while the remaining Q-Qe slots are used to transmit actualdata. Although the pilot symbols are free to be distributed

I of 9

(1)

, MC-DS-CDMA

MC-CDMA

Fig. 1. Spectra of the MC-DS-CDMA and MC-CDMA signals withN = 4 and MR1 = 4.

throughout the frame, in this paper, we will confine themto the first Qe slots of each frame.

I_ frame: Q time slotsl- o-l

MR teststatistics

Fig. 4. Receiver block diagram.

P | P | P D D ...D P |

H Qe pilotsymbols

Q-Qe datasymbols

Fig. 2. Data frame.

The general transmitter block diagram for both systemsis shown in Fig. 3. The input data, d,(k) is assumed tobe a random binary sequence of Al's, and the data forthe K users in the system are assumed to be independent.QeM pilot symbols (+l's) are inserted into the input databefore each group of (Q- Q)M bits, so that at everysub-carrier frequency, the pilot symbols will be located atthe beginning of each data frame, as described earlier. Theresulting sequence is then serial-to-parallel-converted intogroups of M symbols each, {ap:),m}m4M, where p is theframe index and q is the time slot index. Note that for any pand m, ap,),m = 1 for q = 1, . . ., Qe. A rate 1/R repetitioncode is applied to each of the M symbols at the outputof the serial-to-parallel converter, and the resulting MRsymbols are mapped to the MR sub-carrier frequenciesin such a way that the distance in the frequency domainbetween any two repetitions of a symbol is maximized,with the ith repetition of the mth symbol being mapped tosub-carrier frequency fm,i (m = 1, ... , M; i = 1,.. , R).The modulators for the two systems are different, and theywill be described in more detail in their respective sections.The general receiver block diagram for both systems is

shown in Fig. 4. The demodulators, which will also be

Rate 1 Rrepetition code

k) Pilot Symbol Bank of MR + R' ()tdh k~ ~ S - - 7 < (k) (t)Insertion Mapper ulators

M parallel MR symbolsdata symbols

Fig. 3. Transmitter block diagram.

described later in more detail, output test statistics whichare associated with either pilot symbols (q = 1, ..., Qe)or actual data (q = Qe + 1, . . . , Q). At each sub-carrierfrequency, the channel estimate is formed from the nor-malized sample average of the Qe pilot test statistics, andthese estimates become the weighting coefficients in themaximal-ratio combiner. As for the MR test statisticsassociated with actual data, they are de-mapped into Mgroups, with each group corresponding to one of the Msymbols. Maximal-ratio combining is then performed onthe R test statistics in each group, and a hard decisionis made on the output of each maximal-ratio combiner.Finally, parallel-to-serial conversion is performed to gived(k) the estimate of the transmitted data sequence.

In both the transmitter and receiver block diagrams, forthe MC-DS-CDMA system, we have R = R1, and forMC-CDMA, R= R2.We will analyze the two systems for two different cases

of the coherence bandwidth of the channel, (Af),. In whatwill be referred to as Scenario #1, we assume (Af)c = WI,while in Scenario #2, we assume (Af)c= W2.

MC-DS-CDMA

A. Transmitter

The MC-DS-CDMA modulator for the kth user at sub-carrier frequency fm,i (m = 1,...,M;i = 1,...,R1)is shown in Fig. 5. The input symbol, ap,),T, is modu-lated by N chips from the kth user's spreading sequence,{C>}k fL. It is assumed that long spreading sequences

Impulse period T,

(k x Impulse H(f x ()pq,m Vi(t)ModulatorC(Fki)M DS DA(27f,t+dat r ier.

Fig. 5. M[C-DS-CDMIA modulator at sub-carrierfmi

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y y y y y y y y y y y y y y y

f

-0 - .-I

are used. After passing through an impulse modulatorand a chip wave-shaping filter with excess bandwidth 3(O < 3 < 1), the signal is then modulated onto thecorresponding sub-carrier. Finally, the signals at all MR1frequencies are summed together to give the transmittedsignal

M R1

s(k) ()= EERefs(k)i (t)Cj2-Ff, it}, (3)m=l i=l

where0 Q N

<'t:)i(t) = A1 E3 ZZap, mCp neJ«P(4)p=-00 q=I n=l

*hi{t-[(pQ- q -1)N + n-1T}

In (4), A1 is the transmit amplitude, h, (t) is the impulseresponse of the chip wave-shaping filter, j(k) is a randomphase uniformly distributed over [0, 2wF), and T? = T810t/Nis the chip duration, where

Tslot = MT (5)is the duration of a time slot, and T is the duration of abit in the input data sequence.

to the desired user, are i.i.d. uniform random variables over[0, Ta). We assume that the fading characteristics of thechannel remain constant over the duration of a frame, suchthat for any two time slots qi and q2 within the same frame,

(k) (k)gTm,i,q, 9mgT,i,q2' (8)

C. Channel: Scenario #2

In Scenario #2, the channel is assumed to be a slowly-varying, frequency-selective Rayleigh fading channel with(Af)c = W2. From (1), W2 < W1 , so each MC-DS-CDMA sub-band will also see frequency-selective fading,where the number of independently fading paths arrivingat the receiver at each sub-carrier frequency is given by [9]

LfW1(/Af)cN. (9)

With a combination of both path and frequency diversity,each symbol will therefore experience an overall diversityof order R1L = R2The complex low-pass equivalent impulse response of

the channel for the sub-band associated with sub-carrierfrequency fm,i for user k during the qth time slot can bewritten as

B. Channel: Scenario #1

The channel is assumed to be a slowly-varying,frequency-selective Rayleigh fading channel with (Af)c =Wi. We assume each sub-band experiences flat fading,and the fade amplitudes between different sub-bands areindependent. Therefore, the order of frequency diversityfor each symbol is R1. With perfect power control, thereceived signal is given by

K M RIr(t) zE ERe.{,{(k).)(t)ej2wf, i} + nwL(t), (6)

k:=l m= I i= Iwhere

00 Q N

r$i(t) A1 > >3>3 p,q,m pq,n9gmi,q(kp=-oo q=1 n=l

*hi{t -[(pQ + q -1)N + n- 1]Tc- (k)} (7)In (7), the zero-mean complex Gaussian random variable

(k) a- (k)e () i q is the transfer function for the sub-9m,i,'q e' is

band corresponding to sub-carrier frequency fm,i duringthe qth time slot, and nw(t) is additive white Gaussiannoise (AWGN) with two-sided power spectral densityNo/2. The {j)jq)i'} are i.i.d. Rayleigh random variableswith unit second moment, and the {/(ki q} are i.i.d. uni-form random variables over [0, 2wF). Also, the {T(k)}, whichrepresent the relative time delays of each user with respect

Lc(k) ~(k) 6[t (I(k) q(t) = >gm,i,q,lL

1=11)TC] (10)

where (k) a (k) ,Ji<m,q,i The {aemiqi} are in-gm,i,q,l m,i,'q,1l' ,II ae i

dependent Rayleigh random variables whose distributionsdepend on the multipath intensity profile (MIP) of the chan-nel, and the {/3ik)iq } are i.i.d. uniform random variablesover [0, 2wF). Assuming perfect power control, the receivedsignal is then given by

K M R1 L

r(t) > >3 Re{r1(k) (t)ej2wfm it} + nw(t),k=1 m=l i=1 1=1

(1w1)where

00 Q N

< I(t)= Al E >3 >3 ci<im2Zp=-00 q= I n=l

* hi{t- [(pQ +q-l1)N +n-1]T,-(1- 1)T,- T(k)}(12)

We assume that for a given path index I and any two timeslots qi and q2 within the same frame,

(k)=

, (k)gm,i,qj,l 9m,i,q2,U1 (13)

We will analyze the system for the case of both a rectan-gular and an exponential MIP for the channel.

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D. Receiver: Scenario #1

For Scenario #1, the complex low-pass equivalent repre-sentation of the demodulator for the kth user at sub-carrierfrequency fm,i is shown in Fig. 6. The received signalpasses through a band-pass matched filter with a complexlow-pass equivalent impulse response given by

(14)

after which the signal is sampled, de-spread, and summed.We assume the chip wave-shaping filter, H (f), satisfiesthe Nyquist criterion and is of unit energy. We define

xi(t) X Xl(f) _ IHI(f)l2 (15)

kr7 t_ho ) hX(-t)

t = [(pQ+q- 1)N+n- 1]T,

C(k)

t {[(pQ+q 1)N+n- 1] + (L- 1)}T

1N 1 Wrn,i,q,Lor

Ym,i,q,LyC(k)P,q,n

Fig. 7. Complex low-pass equivalent representation of the MC-DS-CDMA demodulator at sub-carrier fm,,i for Scenario #2.

and assume a raised-cosine frequency characteristic forXl (f). Since each sub-band is band-limited and thereis no overlap between adjacent sub-bands, inter-carrierinterference (ICI) may be ignored. Also, since XI (f)satisfies the Nyquist criterion, we can ignore inter-chipinterference as well.

t = [(pQ+ q-I)N+ - 1]T

1Ni1 Wrn,i,qZr$,i(t) xhn,i(t) h= ( t) 1 ° or

k N ~~~~~~~~~~n=OYm,-,qt,1c(k)p,q,n

Fig. 6. Complex low-pass equivalent representation of the MC-DS-CDMA demodulator at sub-carrier m,i for Scenario #1.

zero-mean complex Gaussian. Therefore, we can approx-imate Wm,i,q and Ym,i,q as zero-mean complex Gaussianrandom variables as well. This approximation is exact forK = 1.The output of the maximal-ratio combiner gives the final

test statistic for the mth data symbol during the qth timeslot (q = Qe + 1,.. .,Q),

RIZm,q =E Re{W;m,iym,i,q}

i=l

E { WW,iym,i,q + 2Wm,i,iY q}iil

(16)

(17)

E. Receiver: Scenario #2

For Scenario #2, the complex low-pass equivalent repre-sentation of the demodulator for the kth user at sub-carrierfrequency fm,i is shown in Fig. 7. A RAKE demodulatorwith L fingers is used to take advantage of the pathdiversity present in each sub-band. It is assumed that eachfinger in the RAKE is perfectly synchronized to one ofthe L paths. As a result, in this scenario, there are L teststatistics associated with each sub-band, and, for each datasymbol, maximal-ratio combining is performed on R1Ltest statistics, instead of just R1 as in Scenario #1.

F Analysis: Scenario #1

We evaluate the performance of the mth symbol forthe first user (k = 1), assuming perfect carrier, chip, andbit synchronization. The spreading sequences of the inter-fering users are modeled as independent random binarysequences, while the spreading sequence of the desireduser is considered to be deterministic. We label the teststatistics associated with the pilot symbols as {Wm,i,ql}Qand the test statistics associated with the actual data as{Ym,i,q}=Q-+1. For a large number of users, K, we ap-proximate the multiple-access interference (MAI) as being

where

1 (1 Q 1

Wm,i m,, (18)

represents the estimate of the channel at sub-carrier fre-quency fm,i for the duration of the frame. Wm,i andYm,i,q are assumed to be correlated, but the R1 pairs{Wm,i, Ym,i,q}IR,I are mutually statistically independentand identically distributed. Using the results in Appendix Bof [9] and after some algebraic manipulation [10], we canshow that the average BEP associated with the test statisticZm,q, for the case 3 = 0, is given by

Pe = (2 y kk=0

k 2)(l+,u) (19)

where

1 /(1 + QeC)(1+ y)_K1 EbA 1

7 RI No) 1 + K-(1 NoEl)

(20)

(21)

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hm,i (t) = h* (-t) .

,u is the cross-correlation coefficient between Wm,i andY.,i,q, and a is the average signal-to-interference-plus-noise ratio (SINR) per sub-carrier, where the energy-per-bitcan be shown to equal Eb = NR1A2/2.

G. Analysis: Scenario #2

We evaluate the performance of the mth symbol for thefirst user (k = 1), assuming perfect carrier, chip, bit, andpath synchronization. We make the same assumptions asin Scenario #1 regarding the spreading sequences of thedesired and interfering users, and, for large K, we alsoapproximate the MAI in each finger of the RAKE to bezero-mean complex Gaussian. Furthermore, we considerthe self-interference in each RAKE finger (due to the otherL -1 paths) to be negligible when either (1) there are asmall number of users but N is large, or (2) the number ofusers is large such that the MAI will dominate. Therefore,the test statistic at each RAKE finger output, Wm,i,qil orYm,i,q,il can also be approximated as a zero-mean complexGaussian random variable. This approximation, however,is not exact for K = 1, since we are ignoring the self-interference.The final test statistic for the mth data symbol during

the qth time slot (q = Qe + 1, ... Q) is given byRI L 1 1

Zm,q >3 EWmijYm,i,q,l + 2TWm4i,lYr,i,q,ljl(22)

where

Wm,i,i E Wm,iql (23)

represents the channel estimate of the Ith path at sub-carrierfrequency fm,i.

For the case of a rectangular MIP, we have

At is the cross-correlation coefficient between Wm,i,j andY,,i,q,l, and is the average SINR per path per sub-carrier.The energy-per-bit, Eb, is the same as in Scenario #1.

For the case of an exponential MIP, we have

Qm,i,q,l E[(aj()jq1)2] = Qoe-r(l-1), I 1, ,(28)

where Qo i1seL the average power of the initialpath, and r is the decay rate of the average power. Wecan apply the results given in Appendix B of [9] to findthe characteristic function of Zm,q, take its inverse Fouriertransform, and then integrate the resulting pdf to show that,for Qe = 1 and 3= 0, the average BEP is given by [10]

L R1 + (

PeC 1 bl,(R-k12)/=1 b=1 _t=>(+=)

RIL-1 k -1+ n

n=O

(29)

( 2 )

where

t RI No )1 + K-1 I Ebl)

(30)

(31)

Similarly, for an integer Qe > 1 and /3 = 0, we can showthat [10]

L L R1 R1

Pe 3EE E E Cll,(Rf-kl+1)dl2,(R-lk2+1)11=1 12= 1k=l k2=1

(1 I )kljk (k-Ii -+n)(1n2+ > (32

where

I = 1, ..., L. (24)

The R1L pairs {Wm,i,i,Ym,i,q,i}i 1Ril 1L are theni.i.d., which implies that the double summation in (22) canbe treated as a single summation of i.i.d. terms. We followthe same steps as in Scenario #1 to get, for 3 = 0,

Fe( R)L RL(R1L -k) (1 + t)k

(25)where

t (1+ QeC)(1+ y)_1 K1 EbA 1

L tRI No) 1 + K-(1 INo)1 IN

Qoe-r(i-1) (1 + I)

Qoc-r(ii-1) (1 + I

+ Qoe-r(12 -1) (1

-Qoe-r(12-1) (1

12)

(1)(33)

ti /(1+ QeI Y)(l +i) (34)

In the above equations, {bl,k}, {cl,k}, and {dl,k} representcoefficients associated with three different partial-fractionexpansions, and closed-form expressions for these coeffi-cients are given in [10].

(26) MC-CDMA

A. Transmitter(27) The MC-CDMA modulator at sub-carrier frequency fm,i

(m= 1,...,M;i= 1,...,R2) is shown in Fig. 8. The

5 ot 9

Q =- E[(a (k) )2] 1

M,i.q.l M,i q,l L

Impulse period = T,

a(km Impulse H2(f) (t)Aq,m ~Modulator M

CC ,q,i A26j(fM ,i+()

Fig. 8. MC-CDMA modulator at sub-carrier fm,i.

input symbol, ap,kq,m, is modulated by only one chip inthe kth user's spreading sequence, as opposed to N chipsin the case of MC-DS-CDMA. The R2 chips in the kthuser's spreading sequence, {C ii}f21, each modulate adifferent repetition of a given data symbol, and the samespreading sequence is used for all M data symbols. As inthe MC-DS-CDMA system, long spreading sequences areassumed. The rest of the modulator is similar to the one forMC-DS-CDMA, except that the wave-shaping performedhere is done at the symbol level as opposed to the chiplevel. The transmitted signal is given by

M R2

s(k)(t) >E ERe{s(k) (t)ej2wfT it}, (35)m=l i=1

where

00 Q

Smi(t) A2 apZaqpmC(q).eJi« (3ip=-oo q=

*h2[t -(pQ- q -1)Ts].

In (36), A2 is the transmit amplitude and Ts = Ti0t= NTJis the symbol duration, where T8i0t was given in (5).

B. Channel: Scenario #1

With (Af), = W1 = N W2, the coherence band-width spans several MC-CDMA sub-bands. We assume acorrelated block fading model (correlation in frequency),where the MR2 sub-bands are partitioned into MR2 blocksNof N sub-bands each, and each block occupies the samefrequency range as one of the MC-DS-CDMA sub-bands.We assume the fading between different blocks to beindependent, while the fading within each block is perfectlycorrelated; that is, the fade amplitudes associated with theN sub-bands within each block are identical. Even thougheach symbol is transmitted over R2 different frequencies,the effective order of frequency diversity per symbol inthis scenario is only MN 2. With perfect power control, thereceived signal is given by

K M R2

r(t) =EE E Re{r$()i(t)eJ2wfit} + n,(t), (317)k =lm=l i=l

where0o Q

r$)i(t) = A2 E p,q mCp,q,igmj,qei «p=-0o q=l (38)

. h2[t -(pQ + q- 1)T- 7(k)]The {T(k) } are i.i.d. uniform random variables over [0, Ts),the {(1.1i q} are identically distributed Rayleigh randomvariables with unit second moment, and the {i(k)i Iare identically distributed uniform random variables over[0, 2wF). As before, we assume

(k) (k)g9m,i,q, 9mgT,i,q2 (39)

for any two time slots ql and q2 within the same frame.

C. Channel: Scenario #2

With (Af)c = W2, we assume each sub-band experi-ences flat fading, and the fade amplitudes between differentsub-bands are independent, as was the case with MC-DS-CDMA in Scenario #1. The order of frequency diversityper symbol, therefore, is equal to R2. With perfect powercontrol, the expressions for the kth user's received signalare also given by (37) and (38). For this scenario, however,the {aE(qq} and {k(i3 q} are i.i.d. Rayleigh and uniformrandom variables, respectively.

D. Receiver: Scenarios #1 and #2

The demodulator for the kth user at sub-carrier fre-quency fm,i, as shown in Fig. 9, is the same for bothscenarios, since each sub-band is assumed to have flatfading in both cases. After band-pass matched filtering,the signal is sampled and "de-spread" (multiplication bya single chip). Once again, we assume a raised-cosinefrequency characteristic for X2 (f) H2 (f) 2, and weignore both inter-carrier and inter-symbol interference.

t=(pQ+q-)T

, r($) (t)X hm,i (t) = h2(-t) -S x ork ym,,q

ff(k)Apq,i

Fig. 9. M-CDMA demodulator at sub-carrier fm,i for both scenarios.

E. Analysis: Scenario #1

We evaluate the performance of the mth symbol for thefirst user (k = 1), assuming perfect carrier and bit syn-chronization. Making the same assumptions as in the MC-DS-CDMA analysis regarding the spreading sequences ofthe desired and interfering users and the MAI, we canonce again approximate Wm,i,q and Yn,i,q, the demodulator

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outputs, as zero-mean complex Gaussian random variables.This approximation is exact for K = 1.The final test statistic for the mth data symbol during

the qth time slot (q = Qe + 1, . .. , Q) is given by

R2

Zm,q = { -W?~ iym,i,q + 2W,Y qJiil

(40)

where Wm,i is as defined in (18), but with A1 replacedby A2. Wm,i and Ym,i,q are assumed to be correlated.Because of the correlated block fading, the R2 pairs{Wm,i, Ym,i,q}fi2 are identically distributed but not nec-essarily independent.We derive the average BEP using an approach similar to

the one described in Scenario #2 for the MC-DS-CDMAsystem with an exponential MIP. For Qe = 1 and 3 = 0,we can show that [10]

Fe (21) sQ(bnl=O

a b-0-nl

2 (a(1/1) E20an2w=e

where

a-1 +n,n1i

(1

1 +n2 1 +p,2n2n2 J 2J

where

( Qe7-1) +Q(7 + 1) +

(1 Qe?1)+19

( 4Q--7 + 1))(1 /---e7) +

13 (1 + Qe7)-(1 + Qe7)

II A=

- (1 +Qe?)(1+?)-/(1 +Qe?Y)(1+?)

/(1 +Qe?)(1 +?)

/(1 +Qe?)(1 +?)

/(1 +Qe?)(1 +?)/(1 +Qe?)(1 +?)

(1- Qe)+ (1+Qe\)(1+)

/15(1 +Qe?y)(1+?)

(46)

(47)

(48)

(49)

(50)

For the MC-CDMA system, the energy-per-bit can beshown to equal Eb = R2A2/2.

F Analysis: Scenario #2

Since the same receiver is used for both scenarios, thefinal test statistic is also given by (40). Wm,i and Ym,i,qare assumed to be correlated, but in this scenario, the

(41) R2 pairs { Wm,i, Ym,i,q }i=21 are both mutually statisticallyindependent and identically distributed. Following the samesteps as in the analysis for the MC-DS-CDMA system inScenario #1, for 3 = 0, we get

(42)

1

1)I(Eb)

Pe = (

(43)

(44)

For an integer Qe > 1 and 3= 0, we have [10]

e 21) 1a0 (b- a-1 +nt) 1(1) n

a -1 +n2) (I+/11)1n( 2 I\2J

1-122 a-1 +n3) 1 +1 2

Vn3 JV2

(b-a-1+nl) 1 +13'

b-(a -1 +n23 (1 + P)V n2 JV2

n2 (a-1 +n3) ( +/515)

2

1n3

2 )R2R2 l(R

2 ykk=0

k ) ( 2 )' ( 1)

where

1 /(1+ Qe )(1+?y)_K/ 1 Eb

a R2No 1+(K-

(52)

1

1) (I2E) (53)

,u is the cross-correlation coefficient between Wm,i andYm,i,q, and is the average SINR per sub-carrier. Theenergy-per-bit, Eb, is the same as in Scenario #1.

NUMERICAL RESULTS

7J In this section, we present numerical results basedon the closed-form expressions for the average bit error

1n2 probabilities that we have derived, and we compare theseJ results with those where perfect channel state information

n2 (CSI) is assumed [10]. We consider an MC-CDMA system48 with 512 sub-carriers and an MC-DS-CDMA system withJ N = 64. Thus, the MC-DS-CDMA system will have eight

n3 sub-carriers, i.e., MR1 = 8.In Scenario #1, we noted earlier that each symbol in

MC-DS-CDMA experiences frequency diversity of order(45) R1, while the effective order of frequency diversity in

7 ot 9

p= 1 + a7K1 EbA

? (M;o)1+(KMR2

a= N'R b= R2.

1

(1

1

(1

_,,at

2,

2)

-12/182

2- 1-t52J

b-a-I-n-

n2a=

b-a-I-n-

n3 =0

5-a a-I

ni =0

-a a-I-n_

n2 =°

a-1-nl-,

n3=0

2 4 6 8Eb/N0 (dB)

(a) K =1

100

na

10 12 14 0 2 4 6 8Eb/NO (dB)

(a) K 1

10°

-AA X.. XX-.MCD-CM: =1A

n 10-3

X04 MCDSGCDMA: 0e 16

A0 MC-GDMA:Q0 =16MC-DS-CDMACSI:rfectCCS

-6+MC-CDMA: Perfect CS and MC-DS-CDMA0 5 10

Eb/N0 (dB)15

n

20

(b) K = 16

Fig. 10. Scenario #1: = 1, I = 8 R2= 5:12. Both systems havefrequency diversity of order R1 =MR2 - 8N

10°

1O,>e~~~A XX.X

2-MG-DS-GDMA:Q0e=C1-A- MG-DS-GDMA:Qe=16 O.

-2 MC-DS-CDMA:Q =32

-I- MG-DS-GDMA: Perfect GSI ±. ±

3-X MG-GDMA:Qe=1 ±

A MC-CDMA: Q= 16

o MC-CDMA: Q= 32

4-± MC-CDMA: Perfect CSI

0 5 10Eb/N0 (dB)

15 20

(b) K =16

Fig. 11. Scenario #1: LI = 4, I = 2, 2 = 128. The MC-DS-CDMAand MC-CDMA systems have frequency diversity of orders RI = 2 andR

= 8, respectively.

MC-CDMA is equal to MNR2 . From (2), we have MN2 >NNR1, with equality when M = 1. Figs. 1O(a) and 1O(b)show the M = 1 case with K = 1 and K = 16,respectively. With perfect CSI, the two systems exhibitequivalent performance, in both the single- and multi-usercases. With noisy channel estimation, however, there is a

huge gap between the corresponding curves for the twosystems, for both Qe = 1 and Qe = 16, with MC-DS-CDMA always having a lower average BEP. The reason

is because the energy transmitted at each sub-carrier inMC-CDMA is only (1/N)th the energy transmitted ateach sub-carrier in MC-DS-CDMA. Therefore, the channelestimates made at each sub-carrier in MC-CDMA are muchnoisier than the channel estimates made in MC-DS-CDMA,resulting in the huge degradation in performance. For the

case where M = 4, the MC-CDMA system has a higherorder of frequency diversity. As a result, with perfect CSI,the average BEP is always lower in MC-CDMA for bothK = 1 and K = 16, as shown in Figs. 11(a) and11(b). With noisy channel estimation, however, we see

that there exists a trade-off when K = 1. For relativelylow values of Eb/No, the MC-DS-CDMA system givesbetter performance, again, because it is able to performchannel estimation at each sub-carrier with a higher SNR.But because the MC-CDMA curves have a steeper slope,the corresponding curves for the two systems eventuallyintersect, and to the right of this cross-over point, the MC-CDMA system always gives a lower average BEP. ForK = 16, a similar trade-off exists for Qe = 32, but not

8 of 9

100

10 -2

0-4

10o-5

1 o-60

X, X

-- MC-DS-CDMA: Qe= 16

AMC-DS-CDMA: Qe= 16

- MC-DS-CDMA: Perfect CSIx MC-CDMA: Qe = 1

AMC-CDMA: Qe= 16 Perfect CSI: MC-CDMA± MC-CDMA: Perfect CSI and MC-DS-CDMA

10 12 14

no 103

I

IV

comes large, for both K = 1 and K = 16. Since the slopesof these MC-DS-CDMA curves are less steep, having anexponential MIP basically translates into a loss of diversity.

CONCLUSIONS

± ± \ \ We derived closed-form expressions for the averagebit error probabilities for both the MC-DS-CDMA andMC-CDMA systems under two different cases for the

rMG-DS-GDMA = + coherence bandwidth of the channel. In Scenario #1, a-A- MG-DS-GDMA: Qe= 16 ± trade-off between the two systems may exist for M > 1

+ MC-DS-CDMA: Perfect CSIx MC-CDMA: Q 1 (depending upon the values for K and Qe), with MC-DS-A MC-CDMA: Q= 16 CDMA performing better at lower Lb/No values due to+ MC-CDMA: Perfect CSI more reliable channel estimation, and MC-CDMA faring

0 2 4 6 8 10 better at higher Eb/No values due to greater frequencyEb/NO(dB) diversity. For M = 1, the MC-DS-CDMA system always

(a) K 1 gives a lower average BEP. Finally, in Scenario #2, thetwo systems have equivalent performance when the MIPin each MC-DS-CDMA sub-band is rectangular, but theMC-DS-CDMA system loses diversity when the MIP isexponential.

10Eb/N0 (dB)

15 20

(b) K= 16

Fig. 12. Scenario #2 (Exponential MIP) Ml = 8, It = 1, R2= 64,r = 0.5. Both systems have an overall diversity of order RIL = R2 =64. The dotted curves are also the curves for MC-DS-CDMA when theMIP is rectangular.

for Qe 1 and Qe = 16. Thus, in the multi-user case

with M 4, we see that the MC-CDMA system, despitehaving a higher order of frequency diversity, must use a

larger number of pilot symbols in its estimate to overcome

the effect of estimation errors due to both noise and MAI.

For Scenario #2, the overall order of diversity is equalto R2 in both systems. We see from (25)-(27) and (51)-(53) that the two systems have equal average BEP whenthe MIP of the channel in each MC-DS-CDMA sub-bandis rectangular, since R1L = R2. When the MIP of thechannel in each MC-DS-CDMA sub-band is exponential,we see in Figs. 12(a) and 12(b) that the MC-DS-CDMAsystem always performs worse, especially as Eb/No be-

REFERENCES

[1] N. Yee, J. M. G. Linnartz, and G. Fettweis, "Multi-Carrier CDMAin Indoor Wireless Radio Networks," IEICE Trans. Commun.,vol. E77-B, pp. 900-904, July 1994.

[2] V. M. DaSilva and E. S. Sousa, "Multicarrier Orthogonal CDMASignals for Quasi-Synchronous Communication Systems," IEEEJ. Select. Areas Commun., vol. 12, pp. 842-852, June 1994.

[3] E. A. Sourour and M. Nakagawa, "Performance of OrthogonalMulticarrier CDMA in a Multipath Fading Channel," IEEE Trans.Commun., vol. 44, pp. 356-367, Mar. 1996.

[4] S. Kondo and L. B. Milstein, "Performance of Multicarrier DSCDMA Systems," IEEE Trans. Commun., vol. 44, pp. 238-246,Feb. 1996.

[5] R. Prasad and S. Hara, "An Overview of Multi-Carrier CDMA,"in Proc. IEEE ISSSTA '96, pp. 107-114, Sept. 1996.

[6] D. N. Rowitch and L. B. Milstein, "Convolutionally Coded Mul-ticarrier DS-CDMA Systems in a Multipath Fading ChannelPart I: Performance Analysis," IEEE Trans. Commun., vol. 47,pp. 1570-1582, Oct. 1999.

[7] W. Xu and L. B. Milstein, "On the Performance of MulticarrierRAKE Systems," IEEE Trans. Commun., vol. 49, pp. 1812-1823,Oct. 2001.

[8] L. L. Chong and L. B. Milstein, "Error Rate of a MulticarrierCDMA System with Imperfect Channel Estimates," in Proc. IEEEICC 2000, vol. 2, pp. 934-938.

[9] J. G. Proakis, Digital Communications, New York: McGraw-Hill,2001.

[10] A. S. Ling, Performance Analysis of Multi-Carrier ModulationSystems, Ph.D. Thesis, UCSD.

9 of 9

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>< MC-DS-CDMA: Q= 1

A MC-DS-CDMA: Qe= 16

MC-DS-CDMA: Perfect CSIx.. MC-CDMA: Qe= 1

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5

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