Exploring the UMTS WCDMA-Receiver DesignSpace Using a Semianalytical ApproachGunnar Fock, Jens Baltersee, Peter Schulz-Rittich, and Heinrich Meyr

Abstract—A fast simulation technique for the analysis of the sy-

stem performance of digital receivers is presented and ap-plied to the UMTS terrestrial radio access (UTRA). This se-mianalytical method comprises the computation of receiveroutput statistics, conditioned on a static channel model, agiven spreading sequence and a given transmitted symbolsequence. Averaging over the known channel tap probabi-lity density function and over all symbol sequences yieldsthe system performance, conditioned on the selected sprea-ding sequence. This method permits the qualitative classifi-cation and comparison of different detection algorithms forWCDMA in the performance-complexity design space. Inthe UTRA environment, the performance is shown to va-ry significantly with the choice of the OVSF spreading se-quence. Simulation results for a Rake receiver in indoor andoutdoor scenarios are compared to lower and upper boundsof the probability of error of a maximum likelihood sequenceestimation (MLSE) algorithm. It is shown that a Rake re-ceiver provides sufficient detection performance in medium-rate outdoor scenarios, whereas in high-rate indoor scena-rios, especially in the uplink, alternative receiver structuresmust be considered.

I. INTRODUCTION

A great problem in the numerical simulation of digitalreceiver performance is the large number of trials neces-sary to gain bit error rates with sufficiently small varian-ce. For receivers with relatively low complexity, such asthe well-known Rake receiver [3], receiver output statisticscan be computed analytically for a simplified system mo-del. In order to make the results representative, averagingover several channel tap realizations and over the transmit-ted symbols is then performed. In this paper we employthe technique to a transmission system according to theUMTS terrestrial radio access standard draft [5]. It utilizeswideband code division multiple access (WCDMA). TheRake receiver is well known to exploit the signal energyof several multipath components arriving at the receiver,a scenario common in mobile transmission environments.It is suggested in [5] as a low-complexity solution for fast

The authors are with the Integrated Systems for Signal ProcessingLaboratory, Aachen University of Technology (RWTH), Aachen, Ger-many. Telephone: +49 (241) 80 7632. Fax: +49 (241) 8888 195. E-mail:ffock, balterse, rittich, meyrg@ert.rwth-aachen.de.

time-to-market and will be the receiver of choice for a gre-at part of the first UMTS applications. In terms of detec-tion performance it is suboptimal, especially if comparedto the MLSE, which is known to be optimal in the sen-se of minimal probability of error; its complexity, growingexponentially with the number of users and the channelinfluence length, is prohibitive for the implementation inmost cases. In the scope of this paper the MLSE is usedas a best-case reference. The channel models proposed in[5] are chip-tap models where each channel tap is assumedto be Rayleigh faded. Furthermore, the transmission chainfrom the spreading to the channel convolution can be mo-deled in a simplified way to facilitate the computation ofthe aforementioned output statistics.

II. SYSTEM MODEL

In a CDMA transmission system the user data sym-bols fakg are oversampled by the spreading factorN = T=Tc and then multiplied by a user spreading se-quencec= (c0 : : : cN�1), T andTc being the symbol andchip duration, respectively. For the UTRA,Tc = 244:14 ns.If a chip-tap model is available for a multipath fading chan-nel of delay spreadLc �Tc, meaning that each path delay isan integer multiple of the chip duration, then the operati-ons spreading and channel convolution can be describedjointly in matrix notation and the received signal samplesare

wi = 0B@ z0;0 � � � z0;Ls

.... . .

...zN�1;0 � � � zN�1;Ls

1CA0B@ ai�Ls

...ai

1CA +ni= Zai +ni (1)

The vectorwi contains exactlyN chips belonging tothe transmitted data symbolai . The matrixZ has the di-mensionN� (Ls+ 1), Ls = b(Lc � 1)=Nc+ 1 being thesymbol equivalent influence length of the chip-tap chan-nel model. The matrixZ is computed by multiplying the(Ls+1)N� (Ls+1)N Toeplitz matricesC andH, descri-bing the spreading and channel convolution, respectively,and extracting everyNth column and the lastN rows. A

......Σ

1T

c

1T

c

1T

c

g* 2

ΣN

1

ΣN

1

ΣN

1

h^

q*

h^

2*

h^

1*

^ak

δ( )Tcτ1k+

δ( )Tcτqk+

δ( )Tcτ2k+

n

2 gZw

ak

Rake receiver

Fig. 1. Transmission model with Rake reception

prerequisite for this model is the equivalence of analogand digital signal processing (i.e. [1]), which can easily beshown if the transmit filter proposed in [5] - a root-raisedcosine filter withα = 0:22 - is also used as a receive filter.An additional advantage when using this receive filter isthe whiteness of the additive white gaussian noise processafter chip rate sampling.The power delay profiles (PDP) of the chip-tap channelmodels used in this paper are shown in Figure 2.

0 2 4 6 8 10 12

−30

−20

−10

0

indoor A

t / Tc

Tap

Pow

er [d

B]

0 2 4 6 8 10 12

−30

−20

−10

0

vehicular A

t / Tc

Tap

Pow

er [d

B]

Fig. 2. UTRA channel models

III. PROBABILITY OF ERROR FORRAKE RECEPTION

The transmission model according to section II with aRake receiver is depicted in Figure 1. The received datachips are distributed to a total ofq Rake fingers. In eachfinger, the estimated path delayτ is compensated, desprea-ding and symbol rate sampling are performed, followed bymaximum ratio combining. Throughout this paper we shallassume perfect timing and channel estimation, i.e. the pathdelaysτ and the channel tap weightsh are assumed to beperfectly known. Also, the multipath channel is assumedto be constant for the duration of one symbol interval.The conditional symbol error probability at the output ofthe Rake receiver can be expressed as

Ps( ak 6= ak jh;a;c) = Q

�E[ ak jh;a;c] sgn(ak)p

Var[ ak jh;a;c] �(2)

E[ ak jh;a;c] is the expected value of the data sym-bol at the receiver output, conditioned on the know-ledge of the instantaneous static channel tap vectorh = (h0 � � � hLc), the sequence of transmitted data symbolsa= (ak�Ls � � � ak+Lr ), and on the spreading sequencec. Lr

is the noncausal part of the data symbol sequence influ-ence length, caused by the property of the Rake receiver toassign the greatest delay to the multipath component arri-ving first.Var[ ak jh;a;c] is the corresponding conditionalvariance. It was shown in [2] that the expected value canbe written as

E[ ak jh;a;c] (3)= q

∑n=1

h�n (k+1)N�1

∑j=kN

Ls

∑l=0

z( j+τn=Tc)mod N;l ab j+τn=TcN c�Ls+l

cjmod N

The variance is given by

Var[ ak jh;a;c] (4)= σ2n

q

∑n=1

q

∑m=1

h�n hm

(k+1)L�1

∑i=kL

cimod Nc(i+(τn�τm)=Tc)mod N

and can thus be easily computed by multiplying thechannel noise powerσ2

n with a factor dependent onh, c andq. Monte-Carlo averaging over several channel tap realiza-tions and all QPSK modulated data sequences of lengthLs+Lr +1, which are assumed to be uniformly distribu-ted, leads to an error probability which is only dependenton the chosen spreading sequence:

Ps(c) = ∑a

∑h

p(a) p(h)Ps( ak 6= ak jh;a;c) (5)

With the assumed independence of the data, I- and Q-components of the QPSK symbols can be treated separate-ly by computing (5) for real and imaginary parts and sub-sequent averaging.

IV. ERROR BOUNDS FORMLSE

Forney proposed upper and lower bounds for the proba-bility of error of maximum likelihood sequence detectionin a PAM-system in [4]. These bounds can be employed fora CDMA system by using the received data chips to com-pute the euclidean distance metric. In a Viterbi implemen-tation of the MLSE, these would constitute the path metricincrements. The conditional upper bound can be expressedas

Pu( ak 6= ak jh;c) = ∑d2D

Q

�d( ak jh;c)

2σn

�∑

ε2Ed

W(ε)Φ(ε);(6)

where d denotes the euclidean distance of each errorevent ε, W(ε) is the hamming weight ofε and Φ(ε) isa weighting factor proportional to the a-priori probabilitythat ε will actually occur. Accordingly,D denotes the setof all occuring euclidean distancesd(ε) andEd is the set ofall ε with d(ε) = d. The conditional lower bound is obtai-ned by looking at the term in (6) with minimum euclideandistance:

Pl ( ak 6= ak jh;c) = Q

�dmin( ak jh;c)

2σn

�∑

ε2Edmin

W(ε)Φ(ε):(7)

Then,

Pl( ak 6= ak jh;c) � Ps( ak 6= ak jh;c) < Pu( ak 6= ak jh;c):(8)

The computation of (6) requires - theoretically - theknowledge of all terms in the infinite sum. Terms withd � 2dmin were shown to have no significant contributionto the upper bound and were therefore omitted. Averagingof (6) and (7) is then performed according to (5) to yielderror bounds conditioned onc.The presented bounds are tight in many scenarios, espe-cially in the high SNR regions, as shown in the results(section V). The lower bound, equivalent to single-symbolerrors, should be representative in most scenarios, an as-sumption verified by Monte-Carlo simulation.

V. SIMULATION RESULTS

In order to reemphasize the fundamental difference ofthe two presented detection algorithms in terms of imple-mentation complexity, Figure 3 shows the number of mul-tiplications necessary for the detection of one data sym-bol. For the MLSE, a symbol equivalent channel influencelength ofLs=1 is assumed. The complexity grows linearlywith the spreading factor and exponentially with the num-ber of transmitting users. For the Rake receiver, the com-plexity grows linearly with the number of users.

105

1010

1015

1020

101

102

5 10 15 20 25 30

No. of users

No.

of m

ults

Rake, 2 fingers

MLSE, N = 256

MLSE, N = 4

Rake, 7 fingers

Fig. 3. Implementation complexity of Rake and MLSE

All simulated error probabilities are raw probabilities,i.e. no coding is included. For this reason, a target errorprobability of Ptarget = 10�2 was assumed. The simulati-ons were performed with two users sending synchronous-

ly. In the downlink, both users ”see” the same channelrealization, whereas in the uplink they differ from eachother. Also, the modulation scheme in the uplink accor-ding to [5] is dual channel QPSK, where each channel hasits own spreading sequence. The proposed scrambling isnot included in this model, underscoring the qualitativenature of the results. Upon averaging over different chan-nel tap realizations, is was shown that one ”bad” chan-nel can lead to a significant performance degradation. Inthis paper, averaging over 3000 realizations for the out-door and 5000 for the indoor scenario led to sufficientlystable results. Due to the destructive nature of unfavorablechannels, thePs-region of interest is approximately reci-procal to the number of channel realizations, in our casefor Ps� 3:33�10�4.

When referring to best-case or worst-case scenarios,the classification is done with respect to the correlationproperties of the spreading sequences of both users. Thebest-case sequence pairs have the lowest crosscorrelationenergy among all sequence pairs with the given spreadingfactor, with the worst-case pairs having the highest. Thescrambling proposed in [5] has an averaging effect on thedetection performance, making it independent of the choi-ce of the spreading sequence. A system with scramblingwill thus have a detection performance between the tworespective cases shown in this paper.

Figures 4 and 5 show simulation results for a transmis-sion scenario with the ”indoor A” PDP of Figure 2 andN = 4, corresponding to a raw bitrate of 2 MBit/s. In Figu-re 4, both users transmit with equal power, i.e. the signal tointerference ratio (SIR) is 0 dB. Even with the worst-casesequence pair selection, the 2 finger Rake receiver achievesPtarget, though with a considerable SNR loss with respectto the MLSE. In Figure 5, an uplink transmission withthe best-case sequence pair is depicted. The target errorprobability cannot be achieved by the Rake receiver. Forall indoor simulations, using more than 2 fingers does notshow a significant performance gain, in accordance withthe short channel influence length (Figure 2).

Figures 6 through 9 show results for the ”vehicular A”PDP withN = 64, corresponding to a raw bitrate of 128KBit/s. The downlink with SIR = 0 dB is depicted in Fi-gures 6 and 7. A 7 finger Rake receiver is still able toachievePtarget even with the worst-case sequence pair. Inthe uplink with SIR = -10 dB, shown in Figures 8 and 9,the dependence of the simulated performance on the choi-ce of the sequence pair is even more obvious. A systemwith scrambling will have a performance between thesetwo scenarios, making a Rake reception feasible.

VI. CONCLUSION

A semianalytical simulation technique is presented andapplied to the simulation of the performance of two de-tection algorithms for WCDMA transmission according tothe UTRA standard draft. The method shows a simulationtime reduction by factors of several hundred when compa-red to a classical Monte-Carlo approach. It is useful for thequalitative classification of detection algorithms in the re-ceiver design space with the dimensions performance andcomplexity. One major drawback of the presented methodis the unability to include scrambling in the system mo-del. Cornerpoints of the design space have been quantifiedwith the optimal but highly complex MLSE and the low-complexity but suboptimal Rake receiver. The Rake recei-ver is shown to have sufficient detection performance inthe downlink. In the uplink however, especially in high-rate indoor environments, alternative receiver structures li-ke decision-feedback equalizers must be further analyzed.These can be classified with the presented simulation me-thod.

REFERENCES

[1] Heinrich Meyr, Marc Moeneclaey and Stefan Fechtel.DigitalCommunication Receivers: Synchronization, Channel Estimationand Signal Processing, John Wiley and Sons, New York, 1998.

[2] Peter Schulz-Rittich.Untersuchung von WCDMA Detektionsalgo-rithmen bezuglich Leistungsfahigkeit und Realisierungsaufwand,Diploma thesis, Integrated Systems for Signal Processing Labora-tory, Aachen University of Technology (RWTH), Aachen, Novem-ber 1998.

[3] R. Price and P.E. Green, Jr.A Communication Technique for Mul-tipath Channels, Proceedings of the IRE, March 1958

[4] G.D. Forney, Jr.Maximum Likelihood Sequence Detection of Di-gital Sequences in the Presence of Intersymbol Interference, IEEETransactions on Information Theory, Vol. XX, May 1972

[5] The ETSI UMTS Terrestrial Radio Access (UTRA) ITU-R RTT Candidate Submission, http://www.itu.ch/imt/2-radio-dev/proposals/etsi/utra.pdf, June 1998

worst-case

SNR per Bit0 5 10 15 3020 25

10−4

10−3

10−2

10−1

Ps

MLSE lower boundMLSE upper bound1 finger RAKE 2 finger RAKE

best-case

indoor A, N = 4

Fig. 4. Two users, downlink, SIR = 0 dB

SIR = -10 dB

SNR per Bit0 5 10 15 3020 25

10−4

10−3

10−2

10−1

Ps

MLSE lower boundMLSE upper bound1 finger RAKE 2 finger RAKE

SIR = 0 dB

indoor A, N = 4

Fig. 5. Two users, uplink, best case

0 5 10 15 3010−4

10−3

10−2

10−1

SNR per Bit

MLSE lower boundMLSE upper bound1 finger RAKE

20 25

4 finger RAKE 7 finger RAKE

Ps

vehicular A, N = 64

Fig. 6. Two users, downlink, SIR = 0 dB, best case

SNR per Bit0 5 10 15 3020 25

MLSE lower boundMLSE upper bound1 finger RAKE 4 finger RAKE 7 finger RAKE

10−4

10−3

10−2

10−1

Ps

vehicular A, N = 64

Fig. 7. Two users, downlink, SIR = 0 dB, worst case

SNR per Bit0 5 10 15 3020 25

MLSE lower boundMLSE upper bound1 finger RAKE 4 finger RAKE 7 finger RAKE

10−4

10−3

10−2

10−1

Ps

vehicular A, N = 64

Fig. 8. Two users, uplink, SIR = -10 dB, best case

SNR per Bit0 5 10 15 3020 25

MLSE lower boundMLSE upper bound1 finger RAKE 4 finger RAKE 7 finger RAKE

10−4

10−3

10−2

10−1

Ps

vehicular A, N = 64

Fig. 9. Two users, uplink, SIR = -10 dB, worst case