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Code Division Multiple Access
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Published Date:25-10-2017
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4 Code Division Multiple Access In Section 1.1.2 different multiple access techniques were introduced. Contrary to time and (FDMA) frequency division multiple access schemes, each user occupies the whole time-frequency domain in (CDMA) code division multiple access systems. The signals are separated with spreading codes that are used for artificially increasing the signal bandwidth beyond the necessary value. Despreading can only be performed with knowledge of the employed spreading code. For a long time, CDMA or spread spectrum techniques were restricted to military appli- cations. Meanwhile, they found their way into mobile radio communications and have been established in several standards. The IS95 standard (Gilhousen et al. 1991; Salmasi and Gilhousen 1991) as a representative of the second generation mobile radio system in the United States employs CDMA as well as the third generation Universal Mobile Telecom- munication System (UMTS) (Holma and Toskala 2004; Toskala et al. 1998) and IMT2000 (Dahlman et al. 1998; Ojanpera ¨ and Prasad 1998a,b) standards. Many reasons exist for using CDMA, for example, spread spectrum signals show a high robustness against multipath propagation. Further advantages are more related to the cellular aspects of communication systems. In this chapter, the general concept of CDMA systems is described. Section 4.1 explains the way of spreading, discusses the correlation properties of spreading codes, and demon- strates the limited performance of a single-user matched filter (MF). Moreover, the differ- ences between principles of uplink and downlink transmissions are described. In Section 4.2, the combination of OFDM (Orthogonal Frequency Division Multiplexing) and CDMA as an example of multicarrier (MC) CDMA is compared to the classical single-carrier CDMA. A limiting factor in CDMA systems is multiuser interference (MUI). Treated as addi- tional white Gaussian noise, interference is mitigated by strong error correction codes in Section 4.3 (Dekorsy 2000; Kuhn ¨ et al. 2000b). On the contrary, multiuser detection strate- gies that will be discussed in Chapter 5 cancel or suppress the interference (Alexander et al. 1999; Honig and Tsatsanis 2000; Klein 1996; Moshavi 1996; Schramm and Muller ¨ 1999; Tse and Hanly 1999; Verdu 1998; Verdu and Shamai 1999). Finally, Section 4.4 presents some information on the theoretical results of CDMA systems. ¨174 CODE DIVISION MULTIPLE ACCESS 4.1 Fundamentals 4.1.1 Direct-Sequence Spread Spectrum The spectral spreading inherent in all CDMA systems can be performed in several ways, for example, frequency hopping and chirp techniques. The focus here is on the widely used direct-sequence (DS) spreading where the information bearing signal is directly multiplied with the spreading code. Further information can be found in Cooper and McGillem (1988), Glisic and Vucetic (1997), Pickholtz et al. (1982), Pickholtz et al. (1991), Proakis (2001), Steele and Hanzo (1999), Viterbi (1995), Ziemer and Peterson (1985). For notational simplicity, the explanation is restricted to a chip-level–based system model as illustrated in Figure 4.1. The whole system works at the discrete chip rate 1/T and c the channel model from Figure 1.12 includes the impulse-shaping filters at the transmitter and the receiver. Certainly, this implies a perfect synchronization at the receiver. For the moment, though restricted to an uncoded system the description can be easily extended to coded systems as is done in Section 4.2. The generally complex-valued symbols a at the output of the signal mapper are multiplied with a spreading code c, k. The resulting signal 1  √ ± for N ≤k(+ 1)N s s N s xk= a· c, k with c, k= (4.1) 0else  has a chip index k that runs N times faster than the symbol index .Since c, kis s nonzero only in the interval N ,(+ 1)N , spreading codes of consecutive symbols do s s not overlap. The spreading factor N is often termed processing gain G and denotes s p the number of chips c, k multiplied with a single symbol a. In coded systems, G p also includes the code rate R and, hence, describes the ratio between the durations of an c information bit (T ) and a chip (T ) b c T T N b s s G = = = . (4.2) p T R · T R c c c c This definition is of special interest in systems with varying code rates and spreading factors, as discussed in Section 4.3. The processing gain describes the ability to suppress interfering signals. The larger the G , the higher is the suppression. p c, k nk k yk r matched hk,κ filter a xk  k Figure 4.1 Structure of direct-sequence spread spectrum systemCODE DIVISION MULTIPLE ACCESS 175 Owing to their importance in practical systems, the following description to binary √ spreading sequences is restricted, that is, the chips take the values ±1/ N . Hence, the s signal-to-noise ratio (SNR) per chip is N times smaller than for a symbol aand E /N = s s 0 N · E /N holds. Since the local generation of spreading codes at the transmitter and the s c 0 receiver has to be easily established, feedback shift registers providing periodical sequences are often used (see Section 4.1.4). Short codes and long codes are distinguished. The period of short codes equals exactly the spreading factor N , that is, each symbol a is multiplied s with the same code. On the contrary, the period of long codes exceeds the duration of one symbol a so that different symbols are multiplied with different segments of a long sequence. For notational simplicity, short codes are referred to only unless otherwise stated. In Figure 4.1, spreading with short codes for N = 7 is illustrated by showing the signals s a, c, k, and xk. Figure 4.2 shows the power spectral densities of aand xk for a spreading factor N = 4, an oversampling factor of w= 8, and rectangular pulses of the chips. Obviously, s 2 the densities have a (sin(x)/x) shape and the main lobe of xk is four times broader than that of a. However, the total power of both signals is still the same, that is spreading does not affect the signal’s power. Hence, the power spectrum density around the origin is larger for a. As we know from Section 1.2, the output of a generally frequency-selective channel is obtained by the convolution of the transmitted signal xk with the channel impulse response hk,κ and an additional noise term L−1 t  yk= xk∗ hk,κ+ nk= hk,κ· xk− κ+ nk. (4.3) κ=0 Generally, it can be assumed that the channel remains constant during one symbol duration. In this case, the channel impulse response hk,κ can be denoted by h, κ which will be used in the following derivation. Inserting the structure of the spread spectrum signal given 0 unspread spread −10 −20 −30 −40 −50 −60 0 0.125 0.25 0.375 0.5 f · T → sampl Figure 4.2 Power spectral densities of original and spread signal for N = 4 s (f · T )→ sampl176 CODE DIVISION MULTIPLE ACCESS in (4.1) and exchanging the order of the two sums delivers L−1 t   yk= h, κ· a· c, k− κ+ nk κ=0  L−1 t   = a· h, κ· c, k− κ+ nk  κ=0  = a· s, k+ nk with s, k= c, k∗ h, k. (4.4)  The convolution between the spreading code c, k and the channel impulse response is termed signature s, k and describes the effective channel including the spreading. Hence, the receive filter maximizing the SNR at its output has to be matched to the signature s, k also and not only to the physical channel impulse response. It inherently performs the despreading also. Next, the specific structures of the MF for frequency-selective and nonselective channels are explained in more detail. Matched Filter for Frequency-Nonselective Fading For the sake of simplicity, the discussion starts with the MF for frequency-nonselective channels represented by a signal coefficient h. Therefore, the signature reduces to s, k= h· c, k and the received signal becomes   yk= a· hc, k+ nk= a· s, k+ nk. (4.5)   ∗ 1 The MF that maximizes the SNR has the form g , k= s , (+ 1)N − k. The MF s convolution of yk with g k now yields MF (+1)N −1 s    r k= yk− k · g , k T MF c  k=N s (+1)N −1 s     ∗  = a· s, k− k + nk− k · s , (+ 1)N − k . s  k=N  s  Exchanging the order of the two sums and locating all terms independent of k in front of this sum leads to the chip rate filter output (+1)N−1 s    ∗  r k= a· s, k− k · s , (+ 1)N − k + n k T s T c c   k=N s  = a· φ (+ 1)N − k+ n k. (4.6) SS s T c  1 For simplicity, the normalization of g , k to unit energy has been dropped. MFCODE DIVISION MULTIPLE ACCESS 177 In (4.6), n k denotes the noise contribution at the MF output and φ k denotes the T SS c autocorrelation of the signature s, k which is defined by (+1)N −1 s   ∗  φ k= s, k+ k · s , k SS  k=N s (+1)N −1 s  2   =h · c, k+ k · c, k  k=N s 2 =h · φ k. (4.7) CC For frequency-nonselective channels, φ k simply consists of the product of the channel SS coefficient’s squared magnitude and the spreading code’s autocorrelation function φ k. CC Hence, the output of the MF is simply the correlation between ykand s, k. Naturally, the autocorrelation function has its maximum at the origin implying that the optimum sampling time with the maximum SNR for r kis k= (+ 1)N . According to (4.1), T s c φ 0= 1 holds. Furthermore, the spreading code is restricted to one symbol duration T CC s resulting in φ k=0fork≥ N . Hence, only one term of the outer sum contributes to CC s the results and we obtain (+1)N−1 s  .  ∗  2 . r= r k = yk · s , k =h · a+˜ n. (4.8) T c k=(+1)N s  k=N s 2 The MF delivers the original symbol a weighted with the squared magnitudeh of the channel coefficient and disturbed by white Gaussian noise with zero mean and variance 2 2 2 σ =h σ . Since the signal-to-noise ratio ˜ N N 2 σ E s A 2 SNR= =h · 2 N σ 0 ˜ N is the same as that for narrow-band transmission, spread spectrum gives no advantage in single-user systems with flat fading channels. Matched Filter for Frequency-Selective Fading The broadened spectrum leads in many cases to a frequency-selective behavior of the mobile radio channel. For appropriately chosen spreading codes, no equalization is necessary and the MF is still a suited mean. The signature cannot be simplified as for flat fading chan- nels so that the length of the signature now exceeds N samples, and successive symbols s interfere. Correlating the received signal yk with the signature s, k yields after some manipulations (+1)N +L−1 s t  ∗ r= s, k · yk k=N s L−1 (+1)N +L−2 t s t   ∗ = h, L − 1− κ · yk− κ· c, k− L + 1. (4.9) t t κ=0 k=N +L−1 s t178 CODE DIVISION MULTIPLE ACCESS yk T T T c c c c, k− L + 1 c, k− L + 1 c, k− L + 1 t t t (+1)N+L−2 (+1)N+L−2 (+1)N+L−2 s t s t s t k=N +L−1 k=N +L−1 k=N +L−1 s t s t s t ∗ ∗ ∗ h , L − 1 h , L − 2 h , 0 t t L−1 t r κ=0 Figure 4.3 Structure of Rake receiver as parallel concatenation of several correlators Implementing (4.9) directly leads to the well-known Rake receiver that was originally introduced by Price and Greene (1958). It represents the matched receiver for spread spec- trum communications over frequency-selective channels. From Figure 4.3 we recognize that the Rake receiver basically consists of a parallel concatenation of several correlators also called fingers, each synchronized to a dedicated propagation path. The received signal yk is first delayed in each finger by 0≤κL , then weighted with the spreading code t (with a constant delay L − 1), and integrated over a spreading period. Notice that integra- t tion starts after L − 1 samples have been received, that is, even the most delayed replica t h, L − 1· xk− L + 1 is going to be sampled. Next, the branch signals are weighted t t with the complex conjugated channel coefficients and summed up. Therefore, the Rake receiver maximum ratio combines the propagation paths and fully exploits the diversity (see Section 1.5) provided by the frequency-selective channel. All components of the Rake receiver perform linear operations and their succession can be changed. This may reduce the computational costs of an implementation that depends on the specific hardware and the system parameters such as spreading factor, maximum delay, and number of Rake fingers. A possible structure is shown in Figure 4.4. The tapped delay line represents a filter matched only to the channel impulse response and not to the whole signature. We need only a single correlator at the filter output to perform the despreading. Next, we have to consider the output signal rk in more detail. Inserting  yk= a· s, k+ nk  into (4.9) yields L−1 t    ∗   r= h, L − 1− κ · a · s,k− κ+ nk− κ · c, k− L + 1. t t  κ=0 k CODE DIVISION MULTIPLE ACCESS 179 yk T T T c c c ∗ ∗ ∗ h , L − 1 h , L − 2 h , 0 t t c, k− L + 1 t r L−1 (+1)N +L−2 t s t κ=0 k=N+L−1 s t Figure 4.4 Structure of Rake receiver as serial concatenation of channel matched filter and correlator  Since the signatures s, k exceed the duration of one symbol, symbols at  = ± 1 overlap with a and cause intersymbol interference (ISI). These signal parts are comprised ISI  in a term n  so that in the following derivation we can focus on  = . Moreover, the noise contribution at the Rake output is denoted by n˜. We obtain with s, k= h, κ· c, k− κ κ L−1 L−1 t t   ISI ∗  r= n +˜ n+ a· h, L − 1− κ · h, κ (4.10) t  κ=0 κ=0 (+1)N+L−2 s t   · c, k− κ− κ · c, k− L + 1. t k=N +L−1 s t  The last sum in 4.10 represents again the autocorrelation φ , κ+ κ − (L − 1)of the CC t spreading code c, k. The substitution κ→ L − 1− κ finally results in t L−1 L−1 t t   ∗   ISI r= a· h, κ · h, κ · φ , κ − κ+ n +˜ n (4.11a) CC  κ=0 κ=0 a PCT ISI = r + r + n +˜ n. (4.11b) We see from (4.11a) that the autocorrelation function of spreading codes influences the output of the Rake receiver. If it is impulse-like, that is, φ , κ≈0for κ= 0, each CC branch of the Rake receiver extracts exactly one propagation path and suppresses the other interfering signal components. More precisely, the first (left) finger extracts the path with the largest delay (h, L − 1) because we start integrating at k= N + L − 1 while the t s t last (right) finger detects the path with the smallest delay corresponding to h, 0. Owing to this temporal reversion, all signal components are summed synchronously and the output of the Rake receiver consists of four parts as stated in (4.11b). The first term L−1 t  a 2 r = h, κ · a (4.12) κ=0  obtained for κ = κ combines the desired signal parts transmitted over different propagation 2 paths according to the maximum ratio combining (MRC) principle. This maximizes the L−1 2 t 2 Compared to (1.104), the normalization with h, κ was neglected. κ=0180 CODE DIVISION MULTIPLE ACCESS SNR and delivers an L -fold diversity gain. The second term t L−1 L−1 t t   PCT ∗   r = a· h , κh, κ · φ , κ− κ (4.13) CC  κ=0 κ=0  κ=κ represents path crosstalk between different Rake fingers caused by imperfect autocorrela- 3 tion properties of the spreading code. For random spreading codes and rectangular chip √ impulses, φ , κ≈ 1/N holds for κ 0 and a large spreading factor N . Hence, the CC s s power of asynchronous signal components is attenuated by the factor 1/N . Path crosstalk s can be best suppressed for spreading codes with impulse-like autocorrelation functions. It has to be mentioned that the Rake fingers need not be separated by fixed time delays as depicted in Figure 4.3. Since they have to be synchronized onto the channel taps – which are not likely to be spaced equidistantly – the Rake fingers are individually delayed. This requires appropriate synchronization and tracking units at the receiver. Nevertheless, the Rake receiver collects the whole signal energy of all multipath components and maximizes the SNR. Figure 4.5 shows the bit error rates (BERs) versus E /N for an uncoded single-user DS b 0 spread spectrum system with random spreading codes of length N= 16. The mobile radio channel was assumed to be perfectly interleaved, that is, successive channel coefficients are independent of each other. The number of channel taps varies between L =1and L = 8 t t and their average power is uniformly distributed. Obviously, the performance becomes better with increasing diversity degree D= L . However, for growing L , the difference t t between the theoretical diversity curves from (1.118) and the true BER curves increases as well. This effect is caused by the growing path crosstalk between the Rake fingers due to imperfect autocorrelation properties of the employed spreading codes. 0 10 L = 1 t L = 2 t −1 L = 4 10 t L = 8 t theory −2 10 AWGN −3 10 −4 10 −5 10 0 5 10 15 20 E /N in dB→ b 0 Figure 4.5 Illustration of path crosstalk and diversity gain of Rake receiver 3 The exact expression should consider the case that the data symbol may change during the correlation due  to the relative delay κ− κ . In this case, the even autocorrelation function (ACF) has to be replaced by the odd ACF defined in (4.37) on page 191. BER→CODE DIVISION MULTIPLE ACCESS 181 N s L − 1 t s0 s1 s2 Figure 4.6 Structure of system matrix S for frequency-selective fading Channel and Rake receiver outputs can also be expressed in vector notations. We com- bine all received samples yk into a single vector y and all transmitted symbols a into a vector a. Furthermore, s contains all N + L − 1 samples of the signature s, kfor s t k= N , ... , (+ 1)N + L − 2. Then, we obtain s s t y= S· a+ n, (4.14) where the system matrix S contains the signatures s as depicted in Figure 4.6. Each signature is positioned in an individual column but shifted by N samples. Therefore, L − 1 s t samples overlap leading to interfering consecutive symbols. For N  L , this interference s t can be neglected. With vector notations and neglecting the normalization to unit energy, the Rake’s output signal in (4.9) becomes H H H r= S · y= S S· a+ S n. (4.15) 4.1.2 Direct-Sequence CDMA In CDMA schemes, spread spectrum is used for separating the signals of different sub- scribers. This is accomplished by assigning each user u a unique spreading code c , k u with 1≤ u≤ N . The ratio between the number of active users N and the spreading factor u u N is denoted as the load s N u β= (4.16) N s of the system. For β= 1, the system is said to be fully loaded. Assuming an error-free transmission, the spectral efficiency η of a system is defined as the average number of information bits transmitted per chip mN N u u η= = mR · = mR · β (4.17) c c G N p s and is averaged over all active users. In (4.17), m= log (M) denotes the number of bits 2 per symbol afor M-ary modulation schemes. Obviously, spectral efficiency and system load are identical for systems with mR = 1. c Mathematically, the received signal can be conveniently described by using vector notations. Therefore, the system matrix S in (4.14) has to be extended so that it contains the signatures of all users as illustrated in Figure 4.7. Each block of the matrix corresponds182 CODE DIVISION MULTIPLE ACCESS b) a) u u = 2 = 2 = 0 = 0 = 1 = 1 Figure 4.7 Structure of system matrix S for direct-sequence CDMA a) synchronous down- link, b) asynchronous uplink to a certain time index  and contains the signatures s  of all users. Owing to this u arrangement, the vector T a= a 0 a 0··· a 0 a 1 a 1··· (4.18) 1 2 N 1 2 u consists of all the data symbols of all users in temporal order. Downlink Transmission At this point, we have to distinguish between uplink and downlink transmissions. In the downlink depicted in Figure 4.8, a central base station or access point transmits the user signals x k synchronously to the mobile units. Hence, looking at the link between the u base station and one specific mobile unit u, all signals are affected by the same chan- nel h , κ. Consequently, the signatures of different users v vary only in the spread- u ing code, that is, s , κ= c , κ∗ h , κ holds, and the received signal for user u v v u becomes y = S· a+ n = T C· a+ n . (4.19) u u h ,κ u u √ c , k 1 P 1 a  x k 1 1 n k u y k u 8 h , κ u c , k N P u N u a  x k N N u u Figure 4.8 Structure of downlink for direct-sequence CDMA systemCODE DIVISION MULTIPLE ACCESS 183 In (4.19), T denotes the convolutional matrix of the time varying channel impulse h ,κ u response h , κand C is a block diagonal matrix u   C0   C1   C= (4.20)   C2   . . .   containing in its blocks C= c ··· c  the spreading codes 1 N u   T c = c , N ··· c , (+ 1)N − 1 u u s u s of all users. This structure simplifies the mitigation of MUI because the equalization of the channel can restore the desired correlation properties of the spreading codes as we will see later. However, channels from the common base station to different mobile stations are differ- ent, especially the path loss may vary. To ensure the demanded Quality of Service (QoS), for example, a certain signal to interference plus noise ratio (SINR) at the receiver input, power control strategies are applied. The aim is to transmit only as much power as necessary to obtain the required SINR at the mobile receiver. Enhancing the transmit power of one user directly increases the interference of all other subscribers so that a multidimensional problem arises. In the considered downlink, the base station chooses the transmit power according to the requirements of each user and the entire network. Since each user receives the whole bundle of signals, it is likely to happen that the desired signal is disturbed by high-power signals whose associated receivers experience poor channel conditions. This imbalance of power levels termed near–far effect represents a penalty for weak users because they suffer more under the strong interference. Therefore, the dynamics of downlink power control are limited. In wideband CDMA systems like UMTS (Holma and Toskala 2004), the dynamics are restricted to 20 dB, to keep the average interference level low. Mathematically, power control can be described by introducing a diagonal matrix P into (4.14) containing the user-specific power amplification P (see Figure 4.8). u 1/2 y= SP · a+ n (4.21) Uplink Transmission Generally, the uplink signals are transmitted asynchronously, which is indicated by different starting positions of the signatures s  within each block as depicted in Figure 4.7b. u Moreover, the signals are transmitted over individual channels as shown in Figure 4.9. Hence, the spreading codes have to be convolved individually with their associated channel impulse responses and the resulting signatures s  from (4.4) are arranged in a matrix S u according to Figure 4.7b. The main difference compared to the downlink is that the signals interfering at the base station experienced different path losses because they were transmitted over differ- ent channels. Again, a power control adjusts the power levels P of each user such that u184 CODE DIVISION MULTIPLE ACCESS √ c , k P 1 1 a  x k 1 1 h , κ nk 1 yk 8 c , k N P u N u a  x k N N u u h , κ N u Figure 4.9 Structure of uplink for direct-sequence CDMA system its required SINR is obtained at the receiving base station. Contrary to the downlink, the dynamics are much larger and can amount to 70 dB in wideband CDMA systems (Holma and Toskala 2004). However, practical impairments like fast fading channels and an imper- fect power control lead to SINR imbalances also in the uplink. Additionally, identical power levels are not likely in environments supporting multiple services with different QoS con- straints. Hence, near–far effects also influence the uplink performance in a CDMA system. Receivers that care about different power levels are called near-far resistant. In the context of multiuser detectors, a near–far-resistant receiver will be introduced. Multirate CDMA Systems As mentioned in the previous paragraphs, modern communication systems like UMTS or CDMA 2000 are designed to provide a couple of different services, like speech and data transmission, as well as multimedia applications. These services require different data rates that can be supported by different means. One possibility is to adapt the spreading factor N . Since the chip duration T is a constant system parameter, decreasing N enhances the s c s data rate while keeping the overall bandwidth constant (T = T /N → B= N /T ). c s s s s However, a large spreading factor corresponds to a good interference suppression and subscribers with large N are more robust against MUI and path crosstalk. On the contrary, s users with low N become quite sensitive to interference as can be seen from (4.27) and s (4.31). These correspondences are similar to near–far effects – a small spreading factor is equivalent to a low transmit power and vice versa. Hence, low spreading users need either a higher power level than the interferers, a cell with only a few interferers, or sophisticated detection techniques at the receiver that are insensitive to these effects. The multicode technique offers another possibility to support multiple data rates. Instead of decreasing the spreading factor, several spreading codes are assigned to a subscriber demanding high data rates. Of course, this approach consumes resources in terms of spread- ing codes that can no longer be offered to other users. However, it does not suffer from an increased sensitivity to interference. A third approach proposed in the UMTS standard and limited to ‘hot-spot’ scenar- ios with low mobility is the HSDPA (high speed downlink packet access) channel. ItCODE DIVISION MULTIPLE ACCESS 185 employs adaptive coding and modulation schemes as well as multiple antenna techniques (cf. Chapter 6). Moreover, the connection is not circuit switched but packet oriented, that is, there exist no permanent connection between mobile and base station but data pack- ets are transmitted according to certain scheduling schemes. Owing to the variable coding and modulation schemes, an adaption to actual channel conditions is possible but requires slowly fading channels. Contrary to standard UMTS links, the spreading factor is fixed to N = 16 and no power control is applied (3GPP 2005b). s 4.1.3 Single-User Matched Filter (SUMF) The optimum single-user matched filter (SUMF) does not care about other users and treats their interference as additional white Gaussian distributed noise. In frequency-selective environments, the SUMF is simply a Rake receiver. As described earlier, its structure can be mathematically described by correlating y with the signature of the desired user. Using vector notations, the output for user u is given by H H H H r = S · y= S S · a + S S · a + S · n (4.22) u u u \u \u u u u u where S contains exactly those columns of S that correspond to user u (cf. Figure 4.6). u Consequently, S consists of the remaining columns not associated with u.The same \u H notation holds for a and a . The noise n˜= S n is now colored with the covariance u \u u H 2 H matrix  = En˜n˜ = σ S S . ˜ ˜ u u NN N H If the signatures in S are mutually orthogonal to those in S , then S S is always u \u \u u zero and r does not contain any MUI. In that case, the MF describes the optimum detector u and the performance of a CDMA system would be that of a single-user system with L -fold t diversity. However, although the spreading codes may be appropriately designed, the mobile radio channel generally destroys any orthogonality. Hence, we obtain MUI, that is, symbols of different users interfere. This MUI limits the system performance dramatically. The output of the Rake receiver for user u can be split into four parts a MUI ISI r = r + r + r +˜ n  (4.23) u u u u u Comparing (4.23) with (4.11) shows that path crosstalk, ISI, and noise are still present, but a fourth term denoting the multiple access interference stemming from other active users now additionally disturbs the transmission. This term can be quantified by L−1 N L−1 t u t    8 MUI ∗   r = P · h , κh , κ · φ , κ− κ · a a  (4.24) v v C C u v u v u u  κ=0 v=1 κ=0 v=u where the factor P adjusts the power of user v. From (4.24), we see that the crosscorrelation v  function φ , κ− κ of the spreading codes determines the influence of MUI. For C C u v MUI orthogonal sequences, r  vanishes and the MF is optimum. Moreover, the SUMF is not near–far resistant because high-power levels P of interfering users increase the v interfering power and, therefore, the error rate. Assuming a high number of active users, the interference is often modeled as additional Gaussian distributed noise due to the central limit theorem. In this case, the SNR defined186 CODE DIVISION MULTIPLE ACCESS in (1.14) has to be replaced by a (SINR). 2 2 σ σ X X SNR= −→ SINR= (4.25) 2 2 2 σ σ + σ I N N 2 The term σ denotes the interference power, that is, the denominator in (4.25) represents I the sum of interference and noise power. Generally, these powers vary in time because they depend on the instantaneous channel conditions. For simplicity, the following analysis on the additive white Gaussian noise (AWGN) channel is restricted. Assuming random spreading codes, the power of each interfering user is suppressed in the average by a factor N and s   E 1 E s s 2 2 σ = · P · φ 0= · · P (4.26) v v I u,v T N T s s s v= u v= u holds. Next, the difference between uplink and downlink is illuminated, especially for real- valued modulation schemes. For the sake of simplicity, only binary phase shift keying (BPSK) and quaternary phase shift keying (QPSK) are considered. Downlink Transmission for AWGN Channel Three cases are distinguished: 1. No power control and real symbols. If the modulation alphabet contains only real symbols, we consider only the real part of the matched filtered signal and only half of the noise power disturbs the 2 transmission. Hence, σ = N /2/T has to be inserted into (4.25) (cf. page 12). 0 s  N Without power control, all users experience the same channel in the downlink so that their received power levels P = 1 are identical. The resulting average SINR for v BPSK can be approximated by E E s b SINR≈ = . (4.27) N /2+ (N − 1)E /N N /2+ (N − 1)E /N 0 u s s 0 u b s Obviously, enlarging the spreading factor N results in a better suppression of inter- s fering signals for fixed N . Figure 4.10 shows the SINR versus the number of active u 4 users and SINR versus the 2E /N . We recognize that the SINR decreases dramat- b 0 ically for growing number of users. For very high loads, the SINR is dominated by the interference and the noise plays only a minor role. This directly affects the bit error probability so that the performance will degrade dramatically. According to the general result in (1.49) on page 21, the error probability amounts to " " 2 2 1 σ 1 σ X X P = · erfc = · erfc b 2 2 2 2 σ 2σ  N N for BPSK transmission over an AWGN channel. The argument of the complementary 2 2 error function is half of the effective SNR σ /σ after extracting the real part. Using  X N 4 For BPSK, E = E holds. Furthermore, we use the effective SNR 2E /N after extracting the real part since b s b 0 this determines the error rate in the single-user case.CODE DIVISION MULTIPLE ACCESS 187 20 20 15 15 E /N 10 b 0 10 5 5 0 0 N ,β u −5 −5 0 5 10 15 20 0 5 10 15 20 N = β· N → 2E /N in dB→ u s b 0 Figure 4.10 SINR for downlink of DS-CDMA system with BPSK, random spreading (N = 16) and AWGN channel, 1≤ N ≤ 20 s u this result and substituting the SNR by the SINR, we obtain for the considered CDMA system " A 1 SINR 1 E b P ≈ · erfc = · erfc . (4.28) b 2 2 2 N + 2(N − 1)E /N 0 u b s Figure 4.11 shows the corresponding results. As predicted, the bit error probability increases dramatically with growing system load β. For large β, it is totally dominated by the interference. 0 10 N ,β u −2 10 −4 10 −6 10 0 5 10 15 20 E /N in dB→ b 0 Figure 4.11 Bit error probability for downlink of DS-CDMA system with BPSK, random spreading (N = 16) and an AWGN channel, 1≤ N ≤ 20 s u SINR in dB→ BER→ SINR in dB→188 CODE DIVISION MULTIPLE ACCESS 0 10 −1 10 −2 E /N s 0 10 −3 10 0 1 2 10 10 10 P → v Figure 4.12 Bit error probability for downlink of DS-CDMA system with power control, BPSK, random spreading (N = 16) and an AWGN channel, N =3users s u 2. No power control and complex symbols. 2 2 If we use a complex QPSK symbol alphabet, the total noise power σ instead of σ  N N affects the decision and (4.27) becomes with E = 2E s b E E s b SINR≈ = . (4.29) N + (N − 1)E /N N /2+ (N − 1)E /N 0 u s s 0 u b s This is the same expression in terms of E as in (4.27). Therefore, the bit error rates b of inphase and quadrature components equal exactly those of BPSK in (4.28) when E is used. This result coincides with those presented in Section 1.4. b 3. Power control and real symbols. As a last scenario, we look at a BPSK system with power control where the received power of a single-user v is much higher than that of the other users (P  P ). v u= v The SINR results in E b SINR ≈ . (4.30) u N /2+ E /N P 0 b s v v= u Figure 4.12 shows the results obtained for N = 3 users from which one of the inter- u ferers varies its power level while the others keep their levels constant. Obviously, the performance degrades dramatically with growing power amplification P of user v. v For P →∞, the SNR has no influence anymore and the performance is dominated v by the interferer. Hence, the SUMF is not near–far resistant. Uplink Transmission The main difference between uplink and downlink transmissions is the fact that in the first case each user is affected by its individual channel, whereas the signals arriving at a certain mobile are passed through the same channel in the downlink. We now assume BER→CODE DIVISION MULTIPLE ACCESS 189 20 20 15 15 E /N b 0 10 10 5 5 0 0 N ,β u −5 −5 0 5 10 15 20 0 5 10 15 20 N = β· N → E /N /2in dB→ u s b 0 Figure 4.13 SINR for uplink of DS-CDMA system with BPSK, random spreading (N = s 16) and AWGN channels with random phases, 1≤ N ≤ 20 u a perfect power control that ensures the same power level for all users at the receiver. Note that this differs from the downlink where all users are influenced by the same chan- nel and a power control would result in different power levels. Again, we restrict to the AWGN channel but allow random phase shifts by ϕ on each channel. We distinguish two u cases: 1. Real symbols and AWGN channel with random phases. −jϕ u After coherent reception by multiplying with e , real-valued modulation schemes like BPSK benefit from the fact that the interference is distributed in the complex j(ϕ −ϕ ) v u plane due to e with ϕ − ϕ = 0 while the desired signal is contained only in v u the real part. Hence, only half of the interfering power affects the real part and the average SINR becomes E 2E s b = SINR≈ . (4.31) BPSK N /2+ 1/2· (N − 1)E /N N + (N − 1)E /N 0 u s s 0 u b s Figure 4.13 shows the corresponding results for AWGN channels. A comparison with Figure 4.10 shows that the SINRs are much larger, especially for high loads, and that a gain of 3 dB is asymptotically achieved. With regard to the performance, " A 1 SINR 1 E s P ≈ · erfc = · erfc (4.32) b 2 2 2 N + (N − 1)E /N 0 u s s delivers the results depicted in Figure 4.14. A comparison with the downlink in Figure 4.11 illustrates the benefits of real-valued modulation schemes in the uplink, too. However, it has to be emphasized that complex modulation alphabets have a higher spectral efficiency, that is, more bits per symbol can be transmitted. SINR in dB SINR in dB190 CODE DIVISION MULTIPLE ACCESS 0 10 N ,β u −2 10 −4 10 −6 10 0 5 10 15 20 E /N in dB→ b 0 Figure 4.14 Bit error probability for uplink of DS-CDMA system with BPSK, random spreading codes (N = 16) and an AWGN channel with random phases, 1≤ N ≤ 20 s u 2. Complex symbols and AWGN channel with random phases. It is straightforward to recognize that the distribution of the interference in the com- plex plane provides no advantage for complex-valued modulation schemes like QPSK. 2 Therefore, the entire interfering power σ disturbs the transmission and the SINR I becomes E E s b SINR≈ = . (4.33) N + (N − 1)E /N N /2+ (N − 1)E /N 0 u s s 0 u b s Comparing (4.31) with (4.33) in terms of E , we see that BPSK and QPSK behave b differently in the uplink. It has to be mentioned that orthogonal spreading codes could be employed for syn- chronous frequency-nonselective channels. In this case, no MUI would disturb the transmission and the above discussion would be superficial. Nevertheless, the assump- tions simplified the above analysis and the principle differences between uplink and downlink still hold for frequency-selective channels. With reference to a synchronous downlink transmission, even the use of scrambled orthogonal sequences (see page 192) makes sense because synchronous signal components are perfectly suppressed and only asynchronous parts interfere. Furthermore, the equalization of the single channel could restore orthogonality. For totally asynchronous transmissions, orthog- onal codes generally do not lead to any advantage. We saw that the auto and crosscorrelation properties of spreading codes play a crucial role in CDMA systems. Therefore, the next section briefly introduces some important code families. Afterwards, the performance of a single-user MF is discussed in the context of OFDM-CDMA and in coded environments. BER→CODE DIVISION MULTIPLE ACCESS 191 4.1.4 Spreading Codes Requirements for Spreading Codes As illustrated above, the correlation properties of spreading codes have deep impact on the performance of spread spectrum and CDMA systems. To simplify the notation, we consider the spreading codes c , k only within one symbol period. Hence, the time variable  is u a constant and can be neglected in this section. The even or periodic correlation function for real signals is defined by N−1 s  even φ κ= c kc k+ κ. (4.34) u v C C u v k=0 For u= v, it represents the even (periodic) autocorrelations function, and for u= v,the even crosscorrelation function. Regarding short codes, φ κ is itself periodic with a C C u v period that equals the spreading factor N . s On the one hand, different propagation paths must be separated by the Rake receiver to exploit the diversity provided by a frequency-selective channel and to avoid path crosstalk. This requires an impulse-like shape of the autocorrelation function, that is, φ κ≈ 0 C C u u for κ= 0 is desirable. This property is also important for synchronization purposes. On the other hand, interfering users must be suppressed sufficiently by low crosscorrelations φ κ≈ 0 for arbitrary κ. Unfortunately, both conditions cannot be fulfilled simultane- C C u v= u ously as shown in Sarvate and Pursley (1980). Hence, a trade-off between autocorrelation and crosscorrelation properties is required. Moreover, a lot of spreading sequences should exist to provide many users access to the system. Regarding the uplink of a CDMA system, the transmission is generally asynchronous. Therefore, changes of the data symbols a  occur during correlation and the definition u given in (4.34) cannot be applied anymore. In fact, we need the nonperiodic correlation function  N−1−κ s   c k· c k+ κ0≤κN  u v s non k=0 φ κ= (4.35) C C N−1+κ u v s    c k− κ· c k −N κ≤ 0. u v s k=0 With (4.35), the even correlation function in (4.34) becomes even non non φ κ= φ κ+ φ N − κ. (4.36) s C C C C C C u v u v u v Equivalently, the odd correlation function odd non non φ κ= φ κ− φ N − κ (4.37) s C C C C C C u v u v u v describes the correlation between two BPSK modulated signals ukand vkiftheyhave a mutual delay κ and one of the information symbols a or a  changes its sign during u v correlation.192 CODE DIVISION MULTIPLE ACCESS Orthogonal Spreading Codes With respect to MUI, orthogonal spreading codes would be optimum because they suppress interfering signals perfectly. An example of orthogonal codes are Hadamard codes or Walsh sequences (Harmuth 1964, 1971; Walsh 1923) that were already introduced as forward error correction (FEC) codes in Section 3.2.4. For a synchronous transmission over frequency- nonselective channels, signals can be perfectly separated because 1 u= v w w φ κ= 0= (4.38) C C u v 0else holds. Walsh sequences exist for those lengths N for which N , N /12 or N /20 are powers s s s s of 2. The number of Walsh sequences of a given length N equals exactly N . s s Since autocorrelation and crosscorrelation properties cannot be perfect at the same time, we can conclude that the autocorrelation properties of Walsh sequences are rather bad. Moreover, even the crosscorrelation can take quite large values for κ= 0. Hence, for asynchronous transmissions or frequency-selective mobile radio channels, the orthogonality is destroyed and severe interference makes a reliable transmission nearly impossible. w A solution to this problem is to combine Walsh sequences c k with an outer scrambling u s code c k. This second code does not perform an additional spreading because the duration w s of its chips is the same as for c k. In cellular networks, c k is usually identical for all u users within the same cell (Dahlman et al. 1998; Gilhousen et al. 1991) and allows for cell identification. Therefore, the new code w s c k= c k· c k (4.39) u u maintains orthogonality for synchronous signal components and suppresses asynchronous parts, like a random code. Maximum Length Sequences (m-sequences) To have an efficient implementation, spreading codes are often generated by feedback shift registers. Figure 4.15 shows an example with a register length of m= 9. The register can be initialized arbitrarily with ±1 and generates a periodic sequence. If the polynomial m g(D)= g + g D+··· g D describing the feedback structure of the register is prime, 0 1 m m the period of the sequence is maximized to 2 − 1 and the register passes through all m 2 − 1 states except the all-one-state within one period. Therefore, these sequences are termed m-sequences or maximum length sequences. One important property of m-sequences is that they have a near-optimum autocorrelation function 1for κ= 0 m−seq φ κ= (4.40) C C u u −1/N else. s Hence, they are called quasi-orthogonal since they tightly approach an impulse-like shape. The power of asynchronous replicas of the desired signal can be suppressed by −2 a factor N . With respect to the crosscorrelation, m-sequences perform much worse. s Moreover, given a certain spreading factor N , there exist only a few m-sequences. This s dramatically limits the applicability in CDMA systems because only few users can be

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