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: fpraetor,blau,[email protected] Abstract— The spectral efficiency of single-cell DS- orthogonal codes as well. For the AA this can be CDMA systems deploying higher order modulation 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 fading 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) spread spectrum technology for system design, especially when the orthogonality was chosen for most third generation wireless 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 MODEL AND RECEIVER CONCEPTS are Throughout this work, the downlink of a synchronous, x^ = 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 chip matched filter output of the received wmf = Hc1; (4) signal is given by p 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 ] comprisesp 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 p 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 ¯1 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 lengthp normalized to unity spreading p 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 p¹ 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 p K ¡1 w = p p¹HHH + 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 ´ 2¼i 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 ¯ 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 ¯1;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 ¯ = p cH HH p¹HHH +σ2I Hc ; 1;subopt 1 1 N N 1 18 (8) 16 where p¹ 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.