Turbo Multiuser Detection for TD-CDMA

International Journal of Electronics and Communications

© Urban & Fischer Verlag http://www.urbanfischer.de/journals/aeue

Turbo Multiuser Detection for TD-CDMA Tobias Weber, Jochen Oster, Martin Weckerle and Paul Walter Baier Abstract In this paper the novel detection scheme multi-step joint detection for TD-CDMA mobile radio systems is presented. Multi-step joint detection uses the turbo principle for iteratively improving data detection. Extrinsic information obtained by FEC-decoding in a previous step is used for a joint reduction of interference and improvement of asymptotic efficiency of the linear multiuser detector. Multi-step joint detection helps to overcome the problems of small asymptotic efficiencies present in linear multiuser detectors like the zero forcing equalizer in TD-CDMA mobile radio systems in the case of high system loads. Simultaneously complexity is much lower than that of optimum nonlinear multiuser detectors based on the Viterbi algorithm. As an alternative to simulations the performance of multi-step joint detection can be theoretically determined under certain not too restrictive assumptions. Both approximate and simulative performance results are presented in the paper. It is shown that in typical mobile radio systems the required SNR at the receiver input can be reduced by approximately 10 dB compared to linear multiuser detection. Thus, multi-step joint detection helps to increase the permissible number of mobile stations per cell or to decrease the required cluster size in TD-CDMA mobile radio systems and thus improves spectrum efficiency and capacity. Keywords TD-CDMA, Multi-step joint detection, Turbo multiuser detection

1. Introduction TD-CDMA is an air interface, which utilizes a combination of the three basic multiple access schemes FDMA, TDMA and CDMA. For a comprehensive overview over TD-CDMA see [1, 2]. This air interface was selected by 3GPP as the standard of the Time Division Duplexing (TDD) mode of the 3G system IMT-2000. In the past the preferred detection scheme in both the uplink and the downlink of TD-CDMA was linear multiuser detection based on the zero forcing or the minimum mean square error equalizer [3, 4]. All active user signals of the considered cell and timeslot are jointly detected in a such a way that intracell interference is eliminated. Consequently, the only remaining disturbances are intercell interference and thermal noise. Extensive studies showed that linear multiuser detection offers a quite satisfying

Received November 23, 2000. Revised March 9, 2001. T. Weber, J. Oster, M. Weckerle, P.W. Baier, University of Kaiserslautern, Department of Electrical Engineering, Research Group for RF Communications, P. O. Box 3049, D-67653 Kaiserslautern, Germany. Correspondence to T. Weber. E-mail: [email protected] ¨ 56 (2002) No. 2, 120−130 Int. J. Electron. Commun. (AEU)

bit error behaviour [5] and leads to good system performance [6, 7]. Also the computational expense required to perform linear multiuser detection is affordable [8]. Nevertheless, especially in scenarios with high system loads, that is system loads close to the spreading factor, linear multiuser detection has the disadvantage of an undesirably small asymptotic efficiency [9], which can only be overcome by nonlinear detection schemes. The new multi-step joint detection is such a scheme and clearly outperforms linear multiuser detection. Nevertheless, it requires only moderate signal processing effort. Multi-step joint detection is an application of the general turbo principle [10]. One can view the convolutional FEC code and the CDMA spreading as a serially concatenated code. Turbo decoding of serially concatenated codes is based on alternately decoding the two codes and exchanging extrinsic information between the two decoders [10]. Whereas optimum decoding of the FEC code based on the Viterbi [11] or the BCJR [12] algorithms is state of the art, the complexity of optimum multiuser detection [9] based on the Viterbi or the BCJR algorithms is prohibitive. So one has to rely on suboptimum multiuser detectors which can make use of a priori information for implementing turbo multiuser detectors [13]. The crux of our approach consists in using the a priori information for a joint improvement of the asymptotic efficiency and reduction of the interference in the multiuser detector. Although quite simple, this detection principle offers quite promising performance results. Other multiuser detectors which can be used in turbo-schemes are described i.e. in [14–18]. In a way multi-step joint detection is related to serial and parallel interference cancellation techniques, which are, in contrast to multi-step joint detection, which uses multiuser detection techniques in each step, based on single user detection [19]. The paper is organized as follows. In Section 2 the well-known matrix-vector-model of signal transmission in TD-CDMA [4] is briefly recapitulated. Only the uplink transmission will be studied in the following, but the same techniques can also be applied to the downlink. The uplink is the more general case, as different user signals from the mobile stations reach the receiver in the base station over different mobile radio channels, whereas in the downlink all user signals propagate through the same mobile radio channel to a specific receiver in a mobile station. Section 3 contains a detailed description of the novel detection scheme multi-step joint detection. An in depth mathematical analysis of multi-step joint detection is presented in Section 4. In Section 5 the performance enhancements achievable by multi-step joint detection are 1434-8411/02/56/2-120 $15.00/0

T. Weber et al.: Turbo Multiuser Detection for TD-CDMA 121

illustrated by both theoretical and simulative results. Finally, the paper is concluded in Section 6. Throughout the paper the problem of channel estimation, for which mature and well-established solutions [20, 21] exist, is not considered, that is, the channel impulse responses are assumed to be perfectly known at the receiver. The time discrete equivalent low-pass representation of signals is chosen [22]. Consequently, signals, spreading sequences and channel impulse responses are represented by complex vectors or matrices [4], which are printed in boldface. E{.} and I designate the expectation and the identity matrix, respectively, and [.]i, j is the element of the i-th row and j-th column of a matrix. ˆ. and ˆˆ. denote estimates.

The data vectors d (k) of eq. (1) result from uncoded binary data vectors
T (k) u(k) = u (k) . . . u , M 1 u km ∈ {−1, +1}, k = 1 . . . K ,

(3)

of length M by convolutional encoding with rate Rc and constraint length L c , interleaving and mapping on 4PSK symbols. Since the code is terminated, N=

M + Lc − 1 2Rc

(4)

holds [22]. The data symbols of eq. (1) are spread by a MS specific spreading sequence

2. Uplink transmission model


T (k) c(k) = c(k) , k =1...K , 1 . . . cQ

In the following the uplink data transmission in a time slotted CDMA mobile radio system, e.g. TD-CDMA, is considered. Only the transmission of an isolated data block will be investigated, i.e. no midamble for channel estimation [20, 21] is included in the model. The transmission of one data block corresponds to a single timeslot. Each frame contains several timeslots. For each mobile station (MS) data transmission takes place periodically once per frame. The transmission model is shown in Figure 1. The K MSs transmit the data vectors
T (k) , k =1...K , d (k) = d (k) 1 . . . dN

(1)

of length N. The data symbols are of the type 4PSK: d (k) n ∈ {+1 + j , +1 − j, −1 + j, −1 − j} , k = 1 . . . K, n = 1 . . . N .

Fig. 1. Uplink transmission model.

(2)

(5)

of length Q. The channel impulse response of the mobile radio channel from MS k to the base station (BS) can be represented by a MS specific vector
T (k) h(k) = h (k) . . . h , k =1...K , 1 W

(6)

of length W [23]. It is assumed that the time variant mobile radio channel does not change during the considered timeslot. However, it may change from frame to frame. The combined channel impulse response b(k) valid for MS k is defined as the convolution of the spreading sequence c(k) of eq. (5) and the channel impulse response h(k) of eq. (6) [4]:
T (k) b(k) = b(k) . . . b = c(k) ∗ h(k) . 1 Q+W−1

(7)

122 T. Weber et al.: Turbo Multiuser Detection for TD-CDMA With the combined channel impulse response b(k) of eq. (7) the NQ + W − 1 × N system matrix   A(k) ··· A(k) 1, 1 1, N   , A(k) =  (k)   A(k) NQ+W−1, 1 · · · A NQ+W−1, N

A(k) (n−1)Q+l, n

 (k)   bl for n = 1 . . . N, l = 1 . . . Q + W − 1, =   0 else, k =1...K ,

(8)

(k)

of MS k can be formed. The user signal r of length NQ + W − 1, which is the part of the received signal at the BS resulting from the transmission of the data vector d (k) by MS k, becomes r

(k)

(k) (k)

= A d , k =1...K .

K

r (k) + n .

(10)

k=1

For simplicity the noise vector n is assumed to represent white Gaussian noise in the following. The task of the data detector in the BS receiver consists in determining estimates uˆ (k) of the sent data vectors u(k) , k = 1 . . . K , of eq. (3). In order to perform this task, knowledge of the channel impulse responses h(k) , k = 1 . . . K , of eq. (6) and of the spreading sequences c(k) , k = 1 . . . K , of eq. (5) is required. This implies the knowledge of the system matrices A(k) , k = 1 . . . K , of eq. (8) at the data detector. As already mentioned, the problem of channel estimation is not treated in this paper. Therefore, the channel impulse responses h(k) , k = 1 . . . K , of eq. (6) are assumed to be perfectly known at the data detector. Table 1 presents typical parameters of a TD-CDMA mobile radio system. These parameters are used in the numerical calculations throughout this paper. Table 1. Parameters Parameter coding rate constraint length number of connections generators number of uncoded bits number of data symbols spreading factor number of MSs length of channel impulse response chip rate

3. Multi-step joint detection

(9)

r (k) of eq. (9) is a column vector of length NQ + W − 1. With the noise vector n of the length NQ + W − 1 representing intercell interference and thermal noise, the received signal at the BS becomes e=

Fig. 2. Turbo multiuser detector.

Value Rc = 1/2 Lc = 5 Na = 7 g1 = [10011] g2 = [11101] M=6 N = 10 Q = 16 1 ≤ K ≤ 16 W = 57 f c = 3.84 Mchip/s

Figure 2 shows the general structure of the turbo multiuser detector. It mainly consists of a multiuser detector and a FEC decoder, which exchange extrinsic information. In contrast to conventional multiuser detectors, which only use the received signal e, the multiuser detector applied (k) here uses some a priori information dˆˆ delivered by the FEC decoder from a previous iteration to obtain esti(k) mates dˆ of the data vectors d (k) of eq. (1). The optimum scheme would use nonlinear maximum a posteriori symbol estimators, implemented e.g. with the BCJR algorithm [12], for both the multiuser detector and the FEC decoder [13]. Due to the prohibitively high complexity of the optimum multiuser detector, suboptimum multiuser detectors have to be used instead. In general, a priori information can be used for the following three purposes in a multiuser detector: • reduction of the multiple access interference, • reduction of the intersymbol interference and • reducing the uncertainty on the data symbol to be detected. Considering the detection of a specific data symbol d (k) n , • in the first case a priori information on the data symbols of the other mobile stations, • in the second case a priori information on the other data symbols of the same mobile station and • in the third case a priori information on the specific data symbol itself is used. As multiple access interference is the dominant detrimental effect in mobile radio systems, we consider only the reduction of multiple access interference by turbo techniques. Furthermore it will be shown that we do not need a soft output FEC decoder because at the low bit error rates under consideration the soft bits [10], which are the expectation of the bits, are almost equal to the hard quantized bits. As the differences of the bit error rates of sequence estimation and symbol estimation are negligible for the settings in Table 1, a conventional Viterbi decoder can be used for FEC decoding. This has the advantage that we do not need an esti-

T. Weber et al.: Turbo Multiuser Detection for TD-CDMA 123

mate of the SNR at the output of the FEC decoder, as it would be required for obtaining soft outputs. Considering multiple access interference reduction, the best we can do if the data vectors d (k) of the interferers are perfectly known is to reconstruct the received interference signals r (k) and subtract them from the received signal e to get an interference reduced received signal. If the data vectors d (k) of the interferers are not perfectly known, we have to use their expectations E{d (k) |·} conditioned on the output of the previous iteration, reconstruct the conditioned expectations of the received interference signals E{r (k) |·} and subtract them from the received signal e. This minimizes the variance, i.e. maximizes the SNR of the remaining interference reduced signal. As already mentioned, the difference between expectations and quantized values is small at the bit error rates under consideration here and, therefore, we will use quantized values for signal reconstruction and elimination. For explaining the basic ideas of multi-step joint detection the K MSs are partitioned into two groups. The first group g = 1 comprises the MSs k = 1 . . . K g whereas the second group g = 2 comprises the MSs k = K g + 1 . . . K , e.g. if K g = 8 the first group will contain the MSs k = 1 . . . 8 and the second group will contain the MSs k = 9 . . . 16. Now, with d (k) of (1), the data vectors
T  T T (1) d (1) . . . d (K g ) G = d

(11)

and d (2) G




= d

(K g +1) T

...d

(K) T

(12)

and the system matrices  (1)  (K g ) A(1) G = A ... A

(13)

 (K g +1)  A(2) . . . A(K) G = A

(14)

and

are introduced. In eqs. (13) and (14) the matrices in parentheses are formed according to eq. (8). Utilizing eqs. (11) to (14), the partial received signal originating in the MSs of group g can be written as (g)

(g) (g)

(15)

If zero forcing equalization is used [24], the estimate (g) (g) dˆ G,FZ of the data vector d G of eqs. (11) and (12), respectively, of group g is obtained as
−1 (g) (g)∗T (g) (g)∗T dˆ G,ZF = AG AG AG e , g = 1, 2 . (g)

and
 T T (K g +1) T u(2) . . . u(K) , G = u

(16)

The estimates dˆ G,ZF obtained by eq. (16) are not free from intracell MAI, because not all MSs k = 1 . . . K are in-

(18)

(g) respectively, can be introduced. From the estimates dˆ G,ZF (g) of eq. (16) estimates uˆ G,ZF of the uncoded data vectors u(1) can be obtained by demapping, deinterleaving and decoding. (g) (g) Alternatively an estimate dˆ G,MF of the data vector d G of eqs. (11) and (12), respectively, of group g could be obtained by matched filtering [25]: (g) (g) dˆ G,MF = AG e , g = 1, 2 . ∗T

(19)

The ratio (g) η j = 


(g)∗T (g) AG AG

1 −1 

·



(g)∗T (g) AG AG

j, j

g = 1, 2 ,  j=

T

r G = AG d G , g = 1, 2 .

cluded in the zero forcing equalization; the non included user signals r (k) of eq. (9) are approximately considered as white noise.
T T (1) (K g ) T u(1) = u . . . u (17) G

1 . . . Kg N

,

 j, j

if g = 1, (20)

1 . . . (K − K g )N if g = 2 ,

(g) (g) of the SNRs of the j-th elements of dˆ G,ZF and dˆ G,MF of eqs. (16) and (19), respectively, is termed asymptotic ef(g) ficiency [9]. The asymptotic efficiencies η j of eq. (20) can be looked at as the price to be paid for intracell interference reduction by zero forcing equalization. The (g) asymptotic efficiencies η j only depend on the correlation properties of the combined channel impulse responses b(k) of eq. (7) of the MSs k included in group g. Typically (g) the asymptotic efficiencies η j decrease dramatically if the numbers K g and K − K g , respectively, of jointly detected MS signals come close to the spreading factor Q, see Figure 3. On the other hand the power of the intracell interference signals not included in the joint detection process increases, if the number K g and K − K g , respectively, of jointly detected MS signals is reduced. Therefore, the following dilemma exists if zero forcing equalization is used: It is not possible to simultaneously (g)

• maximize the asymptotic efficiency η j of eq. (20) and • minimize the crosscoupling power of the interfering signals not included in the joint detection process. Said dilemma can be resolved by an at least approximate elimination of the signals not included in the joint detection process of each group [26]. This is the basic idea

124 T. Weber et al.: Turbo Multiuser Detection for TD-CDMA (g)

ered (i) is a received signal of group g with reduced interference. 2. Performing groupwise multiuser detection with the zero forcing equalizer:
−1 (g) (g)∗T (g) (g)∗T (g) dˆ G (i) = AG AG AG ered (i) , g = 1, 2 .

(22)

(g)

3. Calculating an estimate uˆ G (i) of the uncoded data (g) vectors uG (i) of eqs. (17) and (18), respectively, by deinterleaving, demapping and FEC decoding of (g) dˆ G (i) of eq. (22). (g) 4. Generating improved estimates ˆdˆ (i) of the data (g)

Fig. 3. Average asymptotic efficiency η(1) versus the number K g of jointly detected MSs (1) simulated; orthogonal spreading sequences; COST207 bad urban channel model (2) simulated; white Gaussian combined channel impulse responses b(k) (3) approximation by eq. (34).

of multi-step joint detection, see Figure 4. In the following i is the step index of the actual iteration. The signal processing of each iteration i and group g = 1, 2 consists of the following five steps: 1. Approximate elimination of the user signals r (k) of eq. (9) of group 3 − g by subtracting the reconstructed (3−g) received signal rˆ G (i − 1) of the MSs of the other group (3 − g) from the received signal e of eq. (10): (g)

(3−g)

ered (i) = e − rˆ G

(i − 1), g = 1, 2 .

(21)

G

vectors d G (i) of eqs. (11) and (12), respectively, by (g) re-encoding, re-mapping and re-interleaving of uˆ G (i). (g) 5. Reconstructing the received signal r G of eq. (15) originating in the MSs of group g: (g) (g) (g) ˆ rˆ G (i) = AG · dˆ G (i), g = 1, 2 .

(23)

In the following iteration these hopefully improved (g) reconstructions rˆ G (i), g = 1, 2 , of eq. (23) will be used for the elimination in the first step, see eq. (21). (g)

For the first iteration i = 1 the signals rˆ G (0), g = 1, 2, may be initialized by 0, because no reconstructions of the received signals from previous iterations are available. The signal processing effort required for linear multiuser detection of all MSs with the zero forcing equalizer and for multi-step joint detection are comparable. In both cases the numbers of operations required for the matrix inversions in the zero forcing equalizers are dominant and are proportional to the third power of the number

Fig. 4. Multi-step joint detection scheme; ZF: Zero Forcing; Rec.: Reconstruction.

T. Weber et al.: Turbo Multiuser Detection for TD-CDMA 125 (g)

of jointly detected MSs [27]. Consequently, a single zero forcing equalizer for a single group and iteration according to the concept of multi-step joint detection needs less operations than the zero forcing equalizer of linear multiuser detection. On the other hand two groups have to be processed in parallel in multi-step joint detection. In subsequent iterations the inverted matrices from the first iteration can be reused.

For the SNR γin (i) of the received signal of reduced (g) interference ered (i) of eq. (21) at the inputs of the zero forcing equalizers follows

4. Approximate analysis of multi-step joint detection

With the average asymptotic efficiency

In performance assessments of communication systems one is not so much interested in specific signals which can be observed in a certain situation but rather in average performance criteria like e.g. bit error rates or SNRs. Therefore, the following performance analysis has the goal to calculate the interdependencies of such performance criteria of the signals in a multi-step joint detection system. Only low bit error rates, e.g. below 10−3 , have to be considered, as this is a typical requirement in a communication system. In the following it is assumed that the signals of all MSs k are received with the same average power C=

|A(k) d (k) |2 , k =1...K . 2(NQ + W − 1)

With the covariance matrix  1  Rnn = E n · n∗T = σ 2 · I 2

(24)

(25)

of the received white noise n the SNR at the receiver input becomes γ=

C . σ2

(26)

With the spreading factor Q and the code rate Rc the E b /N0 is determined by E b /N0 =

1 Qγ , 2Rc

(27)

if 4PSK modulation is used. The quality of the recon(g) structed signals rˆ G (i), g = 1, 2 , of eq. (21) can be described by the reconstruction error   (g) (g) E |ˆr G (i) − r G |2   ε(g) (i) = , g = 1, 2 . (28) (g) E |r G |2 For (g)

rˆ G (0) = 0, g = 1, 2 ,

(29)

    C/ σ 2 + (K − K g )ε(2) (i − 1)C if g = 1, γin(g) (i) =  C/ σ 2 + K ε(1) (i − 1)C if g = 2 . g (31)

  (g) η(g) = E η j , g = 1, 2 ,

(32) (g)

and the spreading factor Q the average SNR γout (i), g = (g) 1, 2 , of the estimates dˆ G (i), g = 1, 2 , of eq. (22) at the outputs of the zero forcing equalizers is determined by (g)

γout (i) =

  (g) E |d G |2

 (g)  (g) E |dˆ G (i) − dG |2

= Qγin(g) (i)η(g) , g = 1, 2 .

(33)

If the combined channel impulse responses b(k) , k = 1 . . . K , of eq. (7) are considered as white Gaussian noise, the average asymptotic efficiency can be well approximated by [28, 29]  



 Q − K g + 1 / (Q + 1) if g = 1 , η(g) ≈   Q − K + K + 1 / (Q + 1) if g = 2 , g

(34)

see curve (3) in Figure 3. A similar approximation formula is presented in [30]. Also in mobile radio scenarios with sufficiently large delay spreads, e.g. COST207 bad urban scenarios [31], eq. (34) yields a good approximation of the average asymptotic efficiency η(g) , g = 1, 2 , of eq. (20). Demodulation of 4PSK symbols consists in separating real and imaginary parts of the complex data (g) vectors dˆ G (i), g = 1, 2 , of eq. (22), which are used as soft inputs for Viterbi decoders. At low bit error rates deinterleaving converts burst errors into isolated bit errors. In the following the free distance of the convolutional code is denoted by df . The bit error rate of the uncoded (g) data vectors uˆ G at the output of the Viterbi decoders for (g) sufficiently large SNRs γout (i) of eq. (33) at the input of the Viterbi decoder, i.e. low bit error rates, is approximately [22, 32] (g) Pb (i)

  1 (g) ≈ exp − γout (i)df , 2

(35)

the reconstruction error of eq. (28) in the first iteration is ε(g) (0) = 1, g = 1, 2 .

(30)

see Figure 5. If only low bit error rates are considered, we only have to take into account the effect of a single bit

126 T. Weber et al.: Turbo Multiuser Detection for TD-CDMA

(g)

Fig. 7. Approximation of the reconstruction error ε(g) (i), g = 1, 2.

(g)

Fig. 5. Bit error rate Pb (i) versus SNR γout (i); g = 1, 2.

is a linear process, and due to re-interleaving, only isolated bit errors will occur at low bit error rates. Therefore,   (g) E |AG ∆(g) (i)|2   ε(g) (i) = (g) (g) E |AG d G |2 (g)

= 4Pbu , g = 1, 2 , Fig. 6. Re-encoding in the case of a single bit error; Na = 7 connections at the shift register.

error on the re-coded data, see Figure 6. With increasing time the bit error passes through the shift register of the convolutional re-encoder, and every time it is included in a modulo-2 sum it changes the corresponding output bit. Therefore, the number of bit errors at the output of the reencoder is equal to the number Na of connections at the shift register. With the rate Rc of the convolutional code the uncoded bit error rate (g)

(g)

Pbu (i) = Na Rc Pb (i)

(36)

from the re-encoded bits follows. The effect of bit errors ˆ (g) on the re-modulated signals dˆ G (i) can be described by an error signal ∆(g) (i), g = 1, 2 , with average energy   (g) 1  (g) 2  E |∆ (i)| = 4E |d (g) |2 Pbu , g = 1, 2 . 2

(37)

(39)

follows for the reconstruction error of eq. (28). As can be seen from Figure 7 this approximation is quite good for (g) low bit error rates Pb , g = 1, 2 . The offset between the exact and the approximated curves at low bit error rates (g) Pb (i), g = 1, 2, results from the finite length of the data (g) vectors. At high bit error rates Pb (i), g = 1, 2 , the exact curve asymptotically approaches the value 3 dB, which (g) means that the reconstructed signal rˆ G (i) of eq. (23) and (g) the true signal r G (i) of eq. (15) are almost uncorrelated signals with the same power. If soft bits are used for signal reconstruction and elimination, the curve asymptotically approaches the value 0 dB, because nothing can be eliminated at high bit error rates. Starting from a given SNR γ of eq. (26) at the receiver input and an initial value for ε(g) (0) of eq. (28), e.g. ε(g) (0) = 1, g = 1, 2 , the approximate SNRs and bit error rates obtained by multi-step joint detection in subsequent iterations i can be iteratively calculated by using eqs. (31), (33), (34), (35), (36) and (39). Alternatively, the limit reached after infinitely many iterations can be calcu(g) lated by determining the stationary values γin , g = 1, 2 , of the SNRs at the inputs of the zero forcing equalizers, which are the solution of the nonlinear system of equations

The reconstruction γin(1) =

(g)

(g) (g) ˆ rˆ G (i) = AG · dˆ G (i), g = 1, 2 ,

(38)

γ 1 + 4Na Rc γ e

(Q−K g +1)Q (2) − 2(Q+1) γin df

(40) (K − K g )

T. Weber et al.: Turbo Multiuser Detection for TD-CDMA 127

and γin(2) =

γ −

1 + 4Na Rc γ e

(Q−K+K g +1)Q (1) γin df 2(Q+1)

(41) Kg

obtained from eqs. (31), (33), (34), (35), (36) and (39). (g) From the SNRs γin , g = 1, 2 , obtained in this way the bit (g) error rates Pb can be directly calculated by resorting to eq. (35). If we were able to perfectly reconstruct the signals (g) r G (i), the reconstruction error will be ε(g) (i) = 0 .

(42)

This allows us to derive a lower bound for the performance of multi-step joint detection. Substituting eq. (42) in eq. (31) yields the SNRs γin(g) (i) =

C = γ, g = 1, 2 , σ2

(43)

at the inputs of the zero forcing equalizers. From eqs. (33) and (34) we obtain     Q − K g + 1 / (Q + 1) Qγ if g = 1 , (g) γout (i) =   Q − K + K + 1 / (Q + 1) Qγ if g = 2 g (44) for the SNRs at the outputs of the zero forcing equalizers, i.e. the lower bound for the performance of multistep joint detection corresponds to the performance of independent parallel zero forcing equalizers for the number of users in each group with no additional interference. Substituting eq. (44) in eq. (35) yields the bit error rates 
 Q−K g +1   if g = 1 ,  exp − 2(Q+1) Qdf γ (g) Pb (i) = (45)
    exp − Q−K+K g +1 Qdf γ if g = 2 . 2(Q+1)

• For each MS k an individual channel impulse response h(k) , k = 1 . . . K , of eq. (6) is randomly generated for each burst. A channel model with a delay-powerspectrum corresponding to the COST207 bad urban channel model [31] is adopted. With the chip frequency f c this model leads to channel impulse responses h(k) , k = 1 . . . K , of length W less than 57. • The transmit powers are adaptively controlled in such a way that the received power of each of the K MSs averaged over one burst attains the same value C, see eq. (24), i.e. a fast power control is assumed. • Additive white Gaussian noise n is assumed. First, linear multiuser detection with the zero forcing equalizer, which can be considered a special case of multistep joint detection with K g = K , i.e. no MS in the second group, is considered. In this special case no performance enhancement by multiple iterations can be achieved, because no signals can be eliminated. From eqs. (31), (33) and (34) we obtain (1) γout =

Q − K +1 Qγ Q +1

(46)

for the SNR at the output of the zero forcing equalizer. Substituting eq. (46) in eq. (35) yields the bit error rate   Q − K +1 (1) (47) Pb = exp − Qdf γ . 2(Q + 1) Figure 8 shows the bit error rate Pb(1) for linear multiuser detection versus E b /N0 of eq. (27). The dashed curves are approximately calculated by eq. (47) and the solid curves are simulated. The parameter is the number K of MSs. Obviously, the approximation of eq. (47) is quite good. Furthermore the dramatic degradation of detection quality occuring when the number K of MSs comes close to the spreading factor Q can be observed. For reasons of comparison the curve for the AWGN channel is also included in Figure 8.

5. Numerical results The numerical calculations are performed for the parameter values of Table 1. For the simulations the following conditions concerning the scenario are chosen: • Ideal chip impulse bandpass filtering with a sincimpulse resulting in a bandwidth of 3.84 MHz is assumed. • Orthogonal spreading sequences c(k) are generated by scrambling the 16 Walsh functions of length 16 with a common binary sequence of length 16, which is randomly generated on a burst-by-burst basis. Each of the up to K = 16 MSs is assigned one of these spreading sequences c(k) .

Fig. 8. Bit error rate Pb(1) for linear multiuser detection versus E b /N0 ; parameter: K g = K .

128 T. Weber et al.: Turbo Multiuser Detection for TD-CDMA lated bit error rates Pb(1) after the first, second and third iteration for both multi-step joint detection with soft values and multi-step joint detection with hard quantized values for interference reconstruction and elimination. It can be seen that the main difference is a slightly slower convergence if hard quantized values are used. Finally, the influence of the group partitioning described by the parameter K g on the bit error rate Pb(1) (i) is depicted in Figure 11. This figure shows the approximated bit error rate Pb(1) (i) of the MSs of the first group g = 1 versus the group size K g for a constant SNR γ = −3.7 dB

(50)

of eq. (26) at the receiver input after the first, second and third iteration and after infinitely many iterations. The total number K of MSs is again equal to 16. The minimum Fig. 9. Bit error rate Pb(1) of the first group for multi-step joint detection versus E b /N0 ; parameter: K = 16, K g = 8.

Multi-step joint detection introduces an additional parameter K g , describing the partitioning of MSs into two groups. First, two groups of equal size will be considered, i.e. K = 2K g .

(48)

Figure 9 shows approximately calculated and simulated bit error rates Pb(1) of the first group obtained by multi-step joint detection versus E b /N0 of eq. (27) for K = 16

(49)

MSs in total. Bit error rates Pb(1) after the first, second and third iteration are depicted. Here again the approximations are quite close to the simulated curves. For reasons of comparison the simulated curve valid for the AWGN channel, the approximately calculated limit for infinitely many iterations and the curve for linear multiuser detection with the zero forcing equalizer are also included in Figure 9. Of course it is not possible to calculate the limit for infinitely many iterations directly. Instead the value obtained after many, e.g. 100 iterations is used. Already after only three iterations the bit error rate Pb(1) (3) is quite close to the limit Pb(1) (∞), which can be reached with infinitely many iterations. Therefore, only few iterations are required. Furthermore the E b /N0 of eq. (27) required for a typical bit error rate Pb(1) of 10−3 is about 10 dB smaller than the one required by linear multiuser detection with the zero forcing equalizer, which is, in comparison to the bound given by the AWGN channel, a remarkable performance improvement. Of course the bound given by the AWGN channel cannot be reached by multi-step joint detection, as in the optimum case of perfect interference reconstruction the performance of multi-step joint detection will be equivalent to the performance of the zero forcing equalizer for eight MSs. As already mentioned the effect of hard quantization on the performance is rather small. Figure 10 shows simu-

Fig. 10. Bit error rate Pb(1) of the first group for multi-step joint detection versus E b /N0 ; parameter: K = 16, K g = 8.

(1)

Fig. 11. Bit error rate Pb of the first group for multi-step joint detection versus the size K g of the first group; parameters: K = 16, γ = −3.7 dB.

T. Weber et al.: Turbo Multiuser Detection for TD-CDMA 129

bit error rate Pb(1) is obtained for K g < K/2 .

(51)

Therefore, it is impossible to simultaneously minimize the (g) bit error rates Pb (i), g = 1, 2, of both groups. Nevertheless, still good results are obtained if both groups have equal size. The bit error rate Pb(1) (i) of the first step increases with increasing number K g of MSs included in the zero forcing equalization of group g = 1. This results from the fact that the SNR γ of eq. (26) at the input of the receiver is so low that the negative effect of decreased asymptotic efficiency η(1) overrules the positive effect of reduced interference in the case of a larger number of MSs included in the detection process. This means that at very low SNRs γ of eq. (26) at the receiver input linear multiuser detection with the zero forcing equalizer performs worse than single user detection.

6. Conclusions The novel detection scheme multi-step joint detection was introduced. This scheme follows the turbo principle and is based on partitioning the MSs into two groups, with the signals of each group being jointly detected. By crosswise reconstruction and elimination of user signals, the detection results can be improved iteratively. The analysis of multi-step joint detection is based on new techniques for approximately calculating the performance. These techniques can also be applied to conventional TD-CDMA mobile radio systems with multiuser detection. In multistep joint detection only few iterations suffice for obtaining almost optimum performance. Especially in mobile radio systems with high system loads the detector performance can be significantly improved compared to linear multiuser detection albeit the signal processing efforts are still comparable. In summary it may be said that multistep joint detection is a promising nonlinear detection scheme for TD-CDMA mobile radio systems.

Acknowledgement The authors are indebted to Deutsche Forschungsgemeinschaft (DFG) and to Siemens AG for sponsoring parts of this work. They are also grateful for fruitful discussions with their colleagues Anja Klein, Martin Haardt and Gerhard Ritter from Siemens AG. Special thanks are extended to the staffs of Central Computer Facility (RHRK) and of the Center for Microelectronics of the University of Kaiserslautern (ZMK) for sustained technical support.

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Tobias Weber received the Dipl.-Ing. degree in electrical engineering in 1996 from the University of Kaiserslautern, Germany. Since then he has been a member of the staff of the Research Group for RF Communications at the University of Kaiserslautern. From 1996 to 1999 he was active in the development of a hardware demonstrator for a 3rd generation mobile radio system, where his work focused on future signal processing concepts. In 1999 he received the Ph.D. degree from the University of Kaiserslautern. His main research interest are interference reduction techniques for future mobile radio systems.

Jochen Oster received the Dipl.-Ing. degree in electrical engineering and information technology from the University of Kaiserslautern, Germany, in 1997. Since then he has been a Research Assistant with the Research Group for Radio Frequency Communications of the University of Kaiserslautern, Germany. Currently he is working on his Ph.D. thesis. His main research interests are in the arae of digital mobile radio systems, interference reduction and multiuser detection techniques.

Martin Weckerle received the Dipl.-Ing. degree in electrical engineering in 1997 from the University of Kaiserslautern, Germany. He is currently working toward the Ph.D. degree as a research assistant at the Research Group for RF Communications of the University of Kaiserslautern. His work is on adaptive array processing, especially for CDMA mobile radio systems with multiuser detection.

Paul W. Baier was born in Backnang, Germany, in 1938 and graduated from the Technical University Munich, Germany, where he after graduation acted as a senior lecturer until 1970. In 1970 he joined the Central Telecommunications Laboratories of Siemens AG, Munich, where he became head of the spread spectrum development group and was engaged in various topics of communication engineering. Since 1973 he has been a Professor for Electrical Communications and Director of the Institute for RF Communications and Fundamentals of Electronic Engineering at the University of Kaiserslautern, Germany. His main research interests are spread spectrum techniques, impulse compression radars, imaging radars, mobile radio systems and adaptive antennas. The basics of the TD-CDMA component of the UMTS Terrestrial Radio Access System (UTRA) adopted by ETSI in January 1998 and also forming part of the IMT-2000 standard agreed upon by 3 GPP were developed by him in cooperation with Siemens. He (co)authored two books and over 100 papers and supervised over 40 doctoral theses. He is a member of the U.R.S.I. Member Committee Germany and a Fellow of the IEEE and he was awarded the Innovation Prize 1999 of the Mannesmann Mobile Radio Foundation and the VDE ring of honour 2000.