On the Downlink Spectral Efficiency of DS-CDMA Systems

Using MMSE Detectors

Oliver Prator,¨ Ingmar Blau and Gerhard P. Fettweis Vodafone Chair Mobile Communications Systems, Dresden University of Technology D-01062 Dresden, Germany e-mail: {praetor,blau,fettweis}@ifn.et.tu-dresden.de

Abstract— The spectral efficiency of single-cell DS- orthogonal codes as well. For the AA this can be CDMA systems deploying higher order sche- achieved by introducing the class of Haar distributed mes is investigated for the downlink in presence of fa- codes [3], which were shown to behave very similar ding channels. Therefore, the asymptotic analysis (AA) is evaluated for random and orthogonal Haar distributed to the Walsh-Hadamard codes used within the 3GPP spreading codes to obtain the signal-to-interference-and- UMTS specification. noise ratio (SINR) after the optimal and a suboptimal From the AA the signal-to-interference-and-noise ratio linear MMSE detector. An extension to the AA is proposed (SINR) after the linear detector is obtained. This value for both receiver structures to include the intersymbol is independent of the actual spreading code realization interference, which is shown by means of simulations to significantly improve the results for multipath and thus allows for a general statistical interpretation. channels. Based on the extension, the SINR can be used As it describes the quality of the symbol estimates in conjunction with measures of the information theory to before demodulation and decoding, it can be used in determine the spectral efficiency of the system. For modu- conjunction with measures of the information theory to lation and coding at the cut-off rate the tradeoff between determine the overall system spectral efficiency. multi-code transmission and higher order modulation is in- vestigated. Furthermore, an efficiency comparison between In this work first the accuracy of the AA theory is the suboptimal and the optimal LMMSE is conducted. compared to simulation results. For multipath channels Key Words: DS-CDMA, spectral efficiency, MMSE, an extension is presented that improves accuracy. asymptotic analysis Based on the SINR the spectral efficiency is evaluated for modulation and coding at the cut-off rate. A 1. INTRODUCTION comparison between multi-level modulation and multi- Code division multiple access (CDMA) based on code transmission is shown, which is of high interest the direct sequence (DS) technology for system design, especially when the orthogonality was chosen for most third generation between the codes is destroyed by multipath fading. communications systems. Therefore, extensive research Within the 3GPP specification, higher order modulation was conducted to find DS-CDMA implementations up to 64-QAM has already been specified within that provide both high peak data rates and low error the advanced modulation and coding schemes for rates. Advanced receiver concepts were proposed, the High Speed Downlink Packet Access (HSDPA). among which the linear minimum mean-square-error However, results are usually obtained by simulation. (LMMSE) detector is very promising w.r.t. the bit error The approach presented allows to determine the system rate performance [1]. In contrast to the conventional efficiency for different modulation schemes without matched filter detector it deploys the knowledge of the simulation, and to check whether HSDPA parameters spreading sequences and channel impulse responses of are well-chosen. all users to suppress interference. As a second step, the achievable efficiencies of One of the most important aspects for system design the optimal and suboptimal LMMSE detectors is the spectral efficiency. Therefore, it is desirable are compared. The suboptimal implementation is to determine this measure for different parameters important as it covers the situation where the codes without the need of extensive system simulations. In of the interfering users are not known. Furthermore, this work the efficiency of the DS-CDMA downlink its complexity is reduced. The analysis procedure is investigated on a analytical basis. For detection the presented can also be extended to other receivers and optimal and a suboptimal LMMSE implementation are to multi-cell scenarios. considered. The results are based on the asymptotic The structure of this paper is as follows. First, in analysis (AA), which was introduced for the uplink and section 2 the system model and receiver concepts random codes in [2]. Recently, the AA was conducted are introduced. Next, the concept of the asymptotic for the downlink as well [3]. Due to the inherent analysis for LMMSE detectors is explained in section synchronism in the downlink it is necessary to consider 3. Furthermore, in this section an extension is presented r Linear xˆ 1 for multipath fading channels. Section 4 includes Demodulation Decoding Sink Detector β the evaluation of the spectral efficiency based on the 1 AA SINR results. Therefore, the trade-off between multi-code transmission and multi-level modulation is Fig. 2. DS-CDMA downlink: System model of receiver determined, and a comparison between the optimal and suboptimal LMMSE detectors is conducted. Conclusions end the paper. seen through the same channel from the perspective of an individual user. The detection is done using a linear filter w, such that the data estimates of desired user 1 2. SYSTEM MODELAND RECEIVER CONCEPTS are Throughout this work, the downlink of a synchronous, xˆ = wH r. (3) single-cell DS-CDMA system with K active users and 1 spreading factor N is assumed. Furthermore, multipath A block diagram of the receiver is shown in Fig. 2. Rayleigh fading channels are considered. Then, with The conventional receiver for CDMA systems is the MF channel matrix H, code matrix C and transmit power pk receiver with of user k, the matched filter output of the received wmf = Hc1, (4) signal is given by √ usually implemented as Rake receiver. However, it is r = HC Px + n, (1) optimal only in presence of white interference, which T where vector x = [x1, x2, . . . , xK ] comprises√ the is generally not true. The linear MMSE detector de- complex data symbols of the active users, matrix P ploys knowledge of the signatures of all active users to √ is a diagonal matrix with entries pk and vector n suppress the multiple access interference (MAI), thus is the complex additive white Gaussian noise with maximizing the SINR β1 of the symbol estimates after variance σ2/2 per component. The code matrix in- the filter. For the desired user, its filter coefficients N cludes the length√ normalized to unity spreading √ H H 2 −1 T wopt = p1(HCPC H + σ IN ) Hc1 (5) codes ck = 1/ N[ck,1, ck,2, . . . , ck,N ] of the users, i.e. C = [c , c ,..., c ], and channel matrix H is 1 2 K can be easily derived from standard MMSE equations a circulant Toeplitz matrix consisting of the length L as e.g. in [1]. Especially for orthogonal codes also an channel impulse response h = [h , h , . . . , h ]T in 0 1 L−1 suboptimal LMMSE receiver is of interest. In contrast each column. To obtain a circular structure of H the to the optimal LMMSE it has knowledge of the average intersymbol interference (ISI) of the previous symbol is power p¯ of the interferers only, and not of the spreading interpreted as part of the desired signal of the present codes. Thus, it coefficients can be determined to symbol. If the spreading factor approaches infinity the µ ¶ impact of this simplification can be neglected. The √ K −1 w = p p¯HHH + σ2I Hc . (6) advantage of considering a ciculant Toeplitz channel subopt 1 N N 1 matrix is that its eigenvalues can be determined to ³ ´ 2πi l eig(H) = h e N , l = 0,...,N − 1. (2) 3. ASYMPTOTIC ANALYSIS FOR MULTIPATH This property is used within the derivation of the asym- CHANNELS ptotic analysis [3] for simplification. The impact of this The SINR β describes the quality of the symbol assumption is considered in detail in section 3. 1 estimates before demodulation and decoding, as indi- A schematic of the transmitter and channel of such a cated in Fig. 2. To obtain general conclusions about the system is shown in Fig. 1. The source bits of each system behavior and optimal parameter sets it would user are encoded and modulated, before they are spread, be desirable to obtain an expression of the SINR that is resulting in the symbols x . It should be noted that in k independent of the signature realization but incorporates the downlink, the superposition of all user’s signals is the statistics of it. In this section, analytical SINR expressions will be shown at the examples of the optimal p x 1 1 and a suboptimal LMMSE detector, where it is assumed AWGN Source Encoder Modulation Spreading n that all users transmit with equal power pk = p1. The User 1 c1 + Channel instantaneous SINR after the optimal LMMSE is given p x r K K h by [3] Source Encoder Modulation Spreading

User K c H H H H 2 −1 K β1,opt = p1c1 H (HUPU H +σ IN ) Hc1, (7) Fig. 1. DS-CDMA downlink: System model of transmitter and where matrix U is equal to the code matrix C without channel the column for the desired first user. After the subopti- 22 mal LMMSE the SINR can be determined to E /N =20dB µ ¶ b 0 K − 1 −1 20 β = p cH HH p¯HHH +σ2I Hc , 1,subopt 1 1 N N 1 18 (8) 16 where p¯ denotes the average of the powers of the 14

interfering users. in dB 1 In this work, we consider two classes of spreading β 12 codes, random codes and orthogonal Haar distributed 10 AA ext. (11), β , Haar codes. For both a general expression for the SINR can be 1,opt 8 Sim., β , Haar obtained for the multipath fading channel based on the 1,opt AA (9), β , Haar AA [3]. However, the resulting implicit expressions can 6 1,opt 0.5 1 1.5 be solved only for a particular channel realization. To α obtain the general behavior w.r.t. the fading statistic, the AA has to be evaluated for a sufficiently large number of Fig. 3. SINR vs. load for optimal LMMSE, Haar codes and 3-tap channel realizations. In this paper analytical expressions fading channel (N = 64, Eb/N0 = 20dB and QPSK modulation) will be shown for Haar codes only, details on the expres- sion for random codes can be found in [3]. Incorporating 14 E /N =10dB b 0 the statistics of orthogonal Haar distributed codes, the 12 AA leads to the instantaneous SINR ¡ ¢ 10 Z 1/2 −j2πf 2 h e p1df 8 β1,opt = ¡ ¢2 , ¡ ¢ −j2πf −1/2 β1,opt αh e p1 Sim., β , Haar σ2 1 − α + 6 1,opt 1+β 1+β in dB 1,opt 1,opt 1 AA ext. (11), β , Haar β 1,opt (9) 4 Sim., β , Haar for the optimal LMMSE detector, and for the suboptimal 1,subopt AA ext. (14), β , Haar LMMSE to 2 1,subopt Sim., β , random Z ¡ ¢2 0 1,opt 1/2 −j2πf AA, β , random h e p1df 1,opt β1,subopt = ¡ ¢2 , (10) −2 −j2πf −1/2 2 αh e p1 0 0.5 1 1.5 σ + α 1+αβ1,subopt 1+αβ1,subopt The AA is derived for large systems, i.e. N → ∞. Fig. 4. Comparison between optimal and suboptimal LMMSE Therefore, the impact of the simplification of a circulant detectors for the 3-tap fading channel (N = 64, Eb/N0 = 10dB and QPSK modulation) matrix H, as already mentioned in section 2, is neglected in (9) and (10). Thus, system simulations with finite spreading factors over multipath channels yield a gap of several dB in comparison to theory, especially around instead of α in the orthogonal gain of (11). As shown full system load as shown in Fig. 3. However, the in Fig. 3 for simulations with spreading factor N = 64, accuracy can be improved by applying two extensions Eb/N0 = 20dB and QPSK modulation, with (11) the to (9), i.e. AA accuracy is very high for all system loads. It was ¡ ¢ shown by means of simulation in [4] that the results Z 1/2 −j2πf 2 cISI h e p1df for the orthogonal Haar codes represent the results for β1,opt = ¡ ¢2 . ¡ 0 ¢ αh e−j2πf p Walsh-Hadamard codes very closely as well. −1/2 σ2 1 − α cISI β1,opt + 1 1+β1,opt 1+β1,opt The correction terms (12) and (13) have to be applied (11) to the suboptimal LMMSE detector as well, leading to Because over one symbol period a certain amount of ISI ¡ ¢ Z 1/2 −j2πf 2 is treated as desired signal power, we need to scale the cISI h e p1df β1,subopt = ¡ ¢2 . power of the desired signal in the numerator by using −j2πf −1/2 σ2 αh e p1cISI 1+α0β + 1+α0β 1 LX−1 1,subopt 1,subopt c = 1 − lh2. (12) (14) ISI NL l l=1 In Fig. 4 a comparison between the SINR after the Furthermore, as the ISI is not orthogonal to the user optimal and suboptimal LMMSE detectors is shown for of interest, the orthogonal gain in the denominator is the 3-tap multipath channel, E /N = 10dB and QPSK reduced by c . Also, because orthogonality can be b 0 ISI modulation. Interestingly, for α > 1 the performance achieved only for a maximum load of one, we include ½ of the suboptimal LMMSE is as good as the one of α α ≤ 1 the optimal LMMSE. The reason is that for higher α0 = (13) 1 α > 1 loads the interference suppression capability is similar, 0.9 because there are no further degrees of freedom that E /N =5dB can be exploited by the optimal LMMSE based on the 0.8 b 0 knowledge of the codes. Furthermore, the SINR after the 0.7 optimal LMMSE detector for random codes is shown, 0.6 which is significantly lower than for orthogonal codes 0.5 also in the multipath scenario. 0.4

in bps/Hz QPSK, simulation η 4. EVALUATION OF THE SYSTEM SPECTRAL 0.3 QPSK, AA (11) 16−QAM, simulation EFFICIENCY 0.2 16−QAM, AA (11) 0.1 64−QAM, simulation Based on the SINR the system spectral efficiency 64−QAM, AA (11) can be evaluated, which is one of the most important 0 0 0.5 1 1.5 2 design goals. For multiuser systems with equal users α this efficiency is defined to Fig. 5. Spectral efficiency of multi-code transmission vs. multi-level R modulation for the optimal LMMSE, N = 64, E /N = 5dB, Haar η = K b , (15) b 0 W codes and the 3-tap fading channel where Rb is the information bit rate and W is the system . For ideal pulse shaping w.r.t. the Nyquist than single-code transmission with higher order mo- criteria N = W/Rs, with Rs denoting the symbol rate. The rate R of each user comprises both coding dulation. However, it should be noted that even for and modulation and is defined as the relation between such a multipath fading channel and low SNR the loss from using 64-QAM is very small. For a high SNR information bit rate and symbol rate by R = Rb/Rs. Thus, the spectral efficiency can be obtained to η = of Eb/N0 = 20dB as shown in Fig. 6 the 64-QAM RK/N = Rα. Rate R is also the relation between provides best performance. For the simulation results the symbol and bit energy, such that for the transmit SNR SINR was determined by means of simulation and then used with the cut-off rate definition for the maximum p1 Es Eb rate. They are very close to theory, thus showing that 2 = = R (16) σ N0 N0 this work can be used to determine optimal transmission holds. parameters for the CDMA downlink with high accuracy. The maximum spectral efficiency is achieved, when the rate is maximized w.r.t. the SINR β1 after the detector B. Comparison between Optimal and Suboptimal at the transmitter, i.e. R = Rmax(β1), and the transmit LMMSE Detectors SNR (16) is used to evaluate the implicit equation (11). For the maximum rate it is useful to deploy the measure In this subsection a comparison between the achieva- of modulation and coding at the cut-off rate as shown ble maximum efficiency for the optimal and suboptimal e.g. in [5]. LMMSE detectors will be conducted for both random and orthogonal Haar distributed codes. For the Haar A. Impact of Multi-code Transmission codes the AA extensions (11) and (14) are used to The comparison between multi-level modulation and multi-code transmission was investigated for the DS- 6 E /N =20dB QPSK, simulation CDMA uplink in [5]. In this section a similar com- b 0 QPSK, AA (11) 5 parison will be conducted for the downlink, where we 16−QAM, simulation compare the three modulation schemes QPSK, 16-QAM 16−QAM, AA (11) and 64-QAM for the optimal LMMSE detector. To be 4 64−QAM, simulation 64−QAM, AA (11) fair, the same maximum efficiency could be achieved with each scheme. This means, for each user at load 3 in bps/Hz

α = K/N we assign 3 codes for QPSK, in average 1.5 η 2 codes for 16-QAM and 1 code for 64-QAM. Depending on the SINR, the code rate R can be determined from c 1 the cut-off rate R by using Rc = R/ log2 M. In Fig. 5 the system efficiency is shown for a low SNR 0 0 0.5 1 1.5 2 of Eb/N0 = 5dB, the 3-tap channel and N = 64 α for simulations. Obviously, in this case the multi-code transmission with QPSK achieves the highest spectral Fig. 6. Spectral efficiency of multi-code transmission vs. multi-level modulation for the optimal LMMSE, N = 64, Eb/N0 = 20dB, efficiency. Thus, for this scenario users should rather Haar codes and the 3-tap fading channel deploy multi-code transmission with QPSK modulation E /N =10dB AA ext. (11), opt., Haar b 0 presence of multipath fading channels. However, the AA ext. (11), subopt., Haar 2 losses of 64-QAM are small, and for higher SNR AA, opt., random Sim., subopt., random it achieves best efficiency for the LMMSE receivers.

1.5 Finally, a comparison between optimal and suboptimal LMMSE detectors with respect to the efficiency was conducted. Interestingly, for orthogonal codes the losses

in bps/Hz 1 η of the suboptimal LMMSE are small.

0.5 REFERENCES [1] U. Madhow and M. L. Honig, “MMSE Interference Suppression 0 for Direct-Sequence Spread-Spectrum CDMA,” IEEE Transacti- 0 0.5 1 1.5 ons on Communications, vol. 42, no. 12, pp. 3178–3188, Decem- α ber 1994. [2] D. N. C. Tse and S. V. Hanly, “Linear Multiuser Receivers: Fig. 7. Comparison of the spectral efficiencies of the optimal and Effective Interference, Effective Bandwidth and User Capacity,” suboptimal LMMSE detectors at Eb/N0 = 10dB for N = 64, Haar IEEE Transactions on Information Theory, vol. 45, no. 2, pp. codes and the 3-tap fading channel 641–657, March 1999. [3] J.-M. Chaufray, W. Hachem, and P. Loubaton, “Asymptotic ana- lysis of optimum and suboptimum CDMA downlink MMSE receivers,” IEEE Transactions on Information Theory, vol. 50, obtain the SINR. The expressions for random codes no. 11, pp. 2620–2638, November 2004. [4] I. Blau, “Investigation of the Asymptotic Analysis for the Dow- can be found in [3] and [4]. This information is used nlink of Multiuser CDMA Systems,” Master’s thesis, Technische to evaluate the spectral efficiency as described in the Universitat¨ Dresden, 2005. previous section. To consider the statistics of the fading [5] O. Prator,¨ A. Lonnstrom, and G. Fettweis, “Comparison Between Multi-Code Transmission and Multi-Level Modulation in the DS- channel as well, the SINR was evaluated and averaged CDMA Uplink,” in European Wireless Conference (EW’2005), for a large number of channel realizations. This was vol. 1, April 2005, pp. 84–90. necessary as there is no closed-form solution for the SINR in multipath fading channels. The maximum system efficiencies that can be achieved at a certain load with QPSK modulation at the cut-off rate are shown in Fig. 7 for the 3-tap fading channel. Obviously, the behavior depends on the spreading code class. For random codes there is a significant loss of the suboptimal detector. This results from the non- ideal correlation properties of the random codes, which end in a strong performance advantage of the optimal LMMSE detector by considering the code structures for interference suppression. In contrast, the loss is very small for orthogonal codes as the correlation properties are much better. In this case, for loads larger than one no loss exists at all.

5. CONCLUSIONS In this work the spectral efficiency of single-cell DS- CDMA systems was investigated for the downlink and random spreading codes as well as orthogonal Haar distributed codes. For this purpose, first the asymptotic analysis known from literature was evaluated to obtain the SINR after both optimal and suboptimal LMMSE detectors. It was shown that an extension should be used to achieve reliable results for multipath channels. As expected, the SINR is much lower for random codes because they provide no orthogonal gain. It was used in combination with modulation and coding at the cut- off rate to determine the system spectral efficiency. By considering the efficiency the trade-off between multi- code transmission and higher order modulation up to 64- QAM was investigated. For low SNR and Haar codes QPSK modulation results in the highest efficiency in