Coding Across Multicodes and Time in CDMA Systems Employing MMSE Multiuser Detector

When combining a multicode CDMA system with convolutional coding, two methods have been considered in the literature. In one method, coding is across time in each multicode channel while in the other the coding is across both multicodes and time. In this paper, a performance/complexity analysis of decoding metrics and trellis structures for the two schemes is carried out. It is shown that the latter scheme can exploit the multicode diversity inherent in convolutionally coded direct sequence code division multiple access (DS-CDMA) systems which employ minimum mean squared error (MMSE) multiuser detectors. In particular, when the MMSE detector provides su ﬃ ciently di ﬀ erent signal-to-interference ratios (SIRs) for the multicode channels, coding across multicodes and time can obtain signiﬁcant performance gain over coding across time, with nearly the same decoding complexity.


INTRODUCTION
In third generation code division multiple access (CDMA) systems, multimedia data transmission needs to support variable data rates.Multicode CDMA is one method to do this, the basic idea of which is to allocate more spreading codes to users with higher data rates [1].
There are two methods which have been suggested to combine a multicode CDMA system with convolutional coding.In one approach, a serial-to-parallel (S/P) conversion precedes the convolutional coding [2], while in the other approach the S/P conversion follows the convolutional coding [3].When S/P conversion precedes convolutional coding, the coding is only across time in each multicode channel, while when the order is reversed, the coding is across both multicodes and time.The latter approach has also been applied to multicarrier CDMA systems [4] and multiantenna CDMA systems [5].These systems assume that the temporal diversity associated with convolutional coding is fully exploited by interleaving.The case where the fading is slow and the de-lay constraints do not allow the adoption of an interleaver is considered in [5], where instead spatial diversity for the coded symbols is provided by the transmit antenna.None of these methods employ multiuser detection, which may be additionally required to get the desired performance.
To suppress interference among users in a multiuser environment, a minimum mean squared error (MMSE) multiuser detector is often employed [6] and can be combined with convolutional coding to further improve performance [7].In this paper, we consider a multicode CDMA system with convolutional coding and an MMSE detector.As in [5] (which does not employ multiuser detection), we focus on the situation where temporal diversity is not available, that is, the fading is too slow to employ an interleaver.Unless the crosscorrelations between all the users' spreading codes are equal, the signal-to-interference ratios (SIRs) for the users at the MMSE detector output are different.When only coding across time (CT) is employed and one of the multicodes of the desired user has significantly lower SIR at the MMSE detector output, it is shown that the average coded performance over all multicodes is dominated by the worst multicode SIR.However, when coding across multicodes and time (CMT) is used, each convolutionally coded symbol is transmitted over several multicodes with different SIRs, and it is shown that the average coded performance over all the multicodes is related to the average multicode SIR.Thus, the diversity inherent in multicodes at the MMSE multiuser detector output can be exploited with CMT.
We develop and analyze maximum-likelihood (ML) decoding for both CMT and CT in multicode CDMA systems.For CMT, the optimal decoding metric is rather complex.To reduce the complexity, three suboptimal decoding metrics are also considered, including one which has the same complexity as CT (we call this latter metric the "conventional" one in that it ignores the correlation due to multicoding and decodes the convolutional coding in a conventional way).In CMT, for metrics other than the conventional one, there exist both time-invariant and time-varying trellis representations.The time-invariant trellis is useful for analysis, while the time-varying trellises can significantly reduce the decoding complexity depending upon the number of multicodes and the code rate.
To find a true upper bound on the user bit error rate (BER) for the ML and suboptimal decoding, the Laplace transform method can be combined with the transfer function method [8].But using the residue method for the evaluation of the Laplace integral results in computationally complex symbolic operation to evaluate the transfer function.However, if the Gauss-Chebyshev quadrature rule is used for numerical integration, each pairwise error probability (PEP) can be expressed in a product form and the transfer function method can be applied.This is the approach taken in this paper.
The paper is organized as follows.In Section 2, the system model is presented.In Section 3, decoding metrics and trellises for CMT are proposed, and in Section 4 a theoretical upper bound on BER for each metric is derived.Simulation results are provided in Section 5 and some conclusions and future work are discussed in Section 6.

SYSTEM MODELS
The transmitter block diagram of a multicode CDMA system with CT and CMT for desired user k is shown in Figure 1, and the corresponding receiver block diagrams are shown in Figure 2. The mth information bit for user k is d k [m].Let 1/T k denote the information bit rate, m k the number of multicodes, and R k the code rate, for user k.The ith coded bit of multicode j for user k is c k, j [i] ∈ {0, 1}, and the cor- The code symbol rate for each multicode channel of user k is 1/m k T k R k .For the purpose of designing and implementing the MMSE multiuser detector, it is simplest to require that these code symbol rates are the same for all users, that is, 1/m k T k R k = 1/T for some T. A special case of this occurs when the user information bit rates are all integral multiples of the lowest bit rate 1/T b = 1/ max k T k , and the multi- codes are assigned such that m k = T b /T k and all users employ the same rate 1/n convolutional code (CC).For ease of presentation we will assume this case in the sequel.(A more general but similarly treated situation is the case when m k = T b /T k for some users k ∈ A and all other users are assigned m j = T b /T j multicodes for j / ∈ A. Each user j / ∈ A employs the same rate 1/n CC.The code rate R k , k ∈ A, becomes T b /T k m k n and the integer m k ≥ 1 must satisfy m k > T b /T k n to make T b /T k m k n a realizable CC rate which should be less than 1.)Note that the processing gains of CT and CMT are equal when the chip rates of the spreading waveforms of the two systems are equal.However, the encoder and decoder of CMT requires m k times faster processing speed at both the encoder and decoder than CT.
In CDMA systems, the transmitted signal for user k is where A k, j is the received signal amplitude of the jth multicode of the kth user, and a k, j (t) is the normalized spreading waveform of the jth multicode of the kth user with support in [0, T].The spreading waveform a k, j (t) is where where n(t) is (real) white Gaussian noise with power spectral density σ 2 , and K is the number of active users.The matched filter output of multicode j of user k in the ith code symbol interval is The vector given by [6] where R is the crosscorrelation matrix T .The covariance matrix of the Gaussian noise vector Since we do not employ an interleaver in our system, {b i } is a correlated sequence due to the convolutional coding.Furthermore, for CMT (but not CT) the components of b i associated with the same user but different multicodes are correlated for the same reason.However, the correlation between code symbols becomes smaller as the constraint length gets larger.We will assume that b k, j [i] and b k, j [i ] are independent for i = i or j = j .Our simulations show this to be a good approximation with rate 1/2 or 1/3 codes and constraint length greater than or equal to 5.This approximation is employed in the design of the MMSE detector below, and to design and analyze the decoding metrics in Sections 3 and 4.
The MMSE detector output for multicode j of desired user k in the ith symbol interval is r k, j [i] = ω T k, j y i , where [6] and e k, j is the vector whose elements are all zeros except for the ( k−1 p=1 m p + j)th element which is 1.Thus the MMSE detector for all K p=1 m p multicodes can be expressed as

DECODING METRICS AND TRELLISES
For both CT and CMT, the decoder input of desired user k is given by However, for CT m k decoders are required and the input of each decoder with index j i can be expressed as where I k is m k × m k identity matrix and represents elementwise product of two matrices.The covariance matrix of nk i is σ 2 Rk where ( Rk ) j,l = ω T k, j Rω k,l , j = 1, . . ., m k , l = 1, . . ., m k .In CT, the kth user employs m k encoders and the trellis stage index t j for jth encoder is related to the input bit index m by t j (m) = (m − 1)/m k + 1 and m(t j ) = (t j − 1)m k + j, j = 1, . . ., m k .Assuming i k i and īk i are well approximated as Gaussian [6], the branch metric corresponding to the t j th transition reduces to In CMT, m k code symbols are aligned at the same symbol interval and their MMSE detector outputs are correlated with each other.The covariance matrix Φk of īk i is given by Φk = Rk Āk ( Āk ) T ( Rk ) T and that of nk i is σ 2 Rk .Assuming īk i is wellapproximated as Gaussian, the branch metric corresponding to the tth transition is given by where i t and īt depend on the choice of decoding trellis discussed in the sequel.
To reduce the complexity, suboptimal metrics are considered.If the Viterbi decoder does not consider i k i , the corresponding suboptimal metric is given by If the decoder ignores the correlation of īk i and ni , the corresponding suboptimal metric is given by Finally, the suboptimal metric corresponds to both of the above simplifications and is just the conventional decoding metric for vector observations in additive white Gaussian multiple access interference and (independent) AWGN.
We first consider the time-invariant trellis for CMT.In the time-invariant trellis, one transition corresponds to m = LCM(n, m k )/n transitions in the code trellis and any combination of m k and n can be treated.For the time-invariant trellis we have i t = ĩ(t − 1) + 1 and īt = ĩt, where ĩ = LCM(n, m k )/m k in the branch metrics above.The case where n is divisible by m k corresponds to a special case of this timeinvariant trellis with m = 1, ĩ = n/m k .Here a transition occurs at each information bit input and the state transitions are equivalent to those of the CC.In this case we say that m k and n are matched.If we define the decoding complexity as the number of visited branches per decoded bit, the decoding complexity of the CC is 2 ν+1 , where ν is the memory of the CC.The decoding complexity of the time-invariant trellis is that of the CC multiplied by 2 m−1 / m, when we employ metrics other than that in (15) (if the conventional metric (15) is used, the decoding complexity is equal to that of the CC regardless of m).
Mismatched cases of m k and n can be dealt with using the time-invariant-trellis approach.But with a time-invariant trellis we have to search over 2 m branches at each state for each transition, and for some m k and n this complexity can be prohibitive.However, if we design the trellis so that a transition occurs for information input bits corresponding to an appropriate number of observation vectors of the decoder depending on m k and n, it turns out that the decoding complexity can be significantly reduced.But because of the mismatch of m k and n, the trellis becomes time varying.
To describe time-varying trellises for CMT, we first define the maximum number of information bit inputs m to generate all of the m k code symbols which cause multicode interference with each other at each CC state transition.When m k and n are matched, m = 1 which reduces to the case of CC.When m k < n and n is not divisible by m k , m = 2 because multicode interference can affect code symbols corresponding to two transitions in the code trellis.When In the following time-varying trellis discussion, we are interested in mismatch of m k and n, that is, when m ≥ 2, and we consider both fixed and variable state augmentation approaches corresponding to whether the CMT trellis obtained by augmenting the input bits and/or state in the CC trellis varies from one transition to another (in addition to the number and/or structure of the CMT branch metrics).
We first consider fixed state augmentation time-varying trellises.
(i) When m k < n and n is not divisible by m k , the CC state is augmented by 1 bit, but the number of information input bits per transition is kept equal to one.For this fixed state augmentation time-varying trellis, īt = max i (im k ≤ tn) and i t = ī(t − 1) + 1 in the branch metrics in ( 12), ( 13), (14), and (15).The computational complexity of the fixed state augmentation time-varying trellis decoding is that of the CC multiplied by 2 Time varying trellises were considered above based on the feedforward convolutional encoder.However, we can also develop time varying trellises with a recursive systematic convolutional code (RSCC).With RSCC, the information bit vector input can be dealt with because we know the primitive state transition in the code trellis.We can also treat the augmented state in RSCC.
The complexity relative to the CC complexity of timeinvariant and two time-varying trellises are shown in Figure 3 for each n and m k ."TI" denotes time-invariant trellis, "TV, fixed" time-varying trellis with fixed state augmentation and "TV, variable" time varying trellis with variable state augmentation.

Numerical approach
To find a tight numerical upper bound on the BER, the Laplace transform method based on Gauss-Chebyshev quadrature rule is employed.We will focus on the analysis of CMT in this section, since the analysis of CT is simpler and conventional.
The optimal metric in (12) considers interference among multicode symbols i k i , and correlation of īk i and n k i for the same multicode symbols.If i k i and/or the correlation of īk i and n k i are disregarded in the Viterbi decoder, it corresponds to employing one of the suboptimal metrics in ( 13), (14), and (15).The Viterbi decoder using the metric in ( 13) is i k imismatched, and that using the metric in ( 14) is ( īk The evaluation of the PEP for a mismatched Viterbi decoder was analyzed in [9] but a true upper bound for BER using the transfer function method was not treated there.This problem was considered recently in [10] and the following derivation is based on the results in [10].For the analysis, we employ the time-invariant trellis.We evaluate the PEP, P(b k,c → b k,i ) of choosing the incorrect code symbol sequence b k,i when b k,c was transmitted.Let M denote the length of the error event.
For the analysis of the Viterbi decoder employing the metrics in ( 12) and ( 13), and for the metrics in ( 14) and (15), let where for the i k i -matched Viterbi decoder employing metrics in ( 12) and ( 14), d k,c t, j = Rk A k b k,c (t−1) ĩ+ j and d k,i t, j = Rk A k b k,i (t−1) ĩ+ j , and for the i k i -mismatched Viterbi decoder employing metrics in ( 13) and ( 15 12) and ( 13) is and for the metrics in ( 14) and ( 15) is The PEP is P(b k,c → b k,i ) = P (U k ≤ 0) and is given by for positive real constant c.By using the Gauss-Chebyshev rule, it can be obtained as [11] P where τ p = tan((2p − 1)π/(2P)) and E P is the error term.By using ( 18), (19), and (21), we obtain the PEP in a product form and can find the union bound on BER by applying the transfer function method.Since uniformity does not hold for CMT, the transfer function bound is computed based on matrix branch labels instead of scalar branch labels.Let N s be the number of states of the timeinvariant trellis, which is 2 ν , equal to that of CC.Let e be the error code vector c k,c (i, j) ⊕ c k,i (i, j), where c k,c and c k,i are correct and incorrect output code bits corresponding to the transition from state j to state i, respectively.Let c k,l,c (i, j) = [(c k,c (i, j)) (l−1)mk+1 , . . ., (c k,c (i, j)) lmk ] T , c k,l,i (i, j) = [(c k,i (i, j)) (l−1)mk+1 , . . ., (c k,i (i, j)) lmk ] T and e k,l = [(e) (l−1)mk+1 , . ., (e) lmk ] T .For branch metrics ( 12) and ( 13) let the N s × N s matrix G λ (e) be defined by where b k,l,c (i, j whose elements are all ones.Similarly, for metrics ( 14) and ( 15), let G λ (e) be where where G λ (e) N 1 = N s −1, and e(i, j) is the error code vector corresponding to the transition from error state j to error state i.The first error event probability (FEEP) is bounded by where The FEEP can also be evaluated by calculating residues of T λ /λ.This requires the inverse of the symbolic matrix in (24) and as the state number of the trellis grows, it is infeasible to compute due to numerical problems.The Gauss-Chebyshev quadrature rule allows us to avoid numerical problems associated with the symbolic operation.When evaluating BER, the number of bits for each transition should be considered in the transfer function.In this case, we employ the first-order difference method in [12], avoiding the symbolic operation.

SIR-based approach
In Section 4.1, we presented a method to compute a tight upper bound for both CT and CMT numerically and the results are in good agreement with the simulation results (see Section 4).However, it does not explain qualitatively under what conditions CMT performs better than CT.Here, we present another method which can give some insight.
According to [6], the residual interference at the MMSE detector output is well approximated as Gaussian distributed and the probability of error of multicode j of user k at the MMSE detector output is given by P k, j e = Q( SIR k, j ), where We use the decoding metrics in (11) for CT.In CMT, we assume that E[( that is, all elements of (n k i + īk i + i k i ) are uncorrelated each other.Hence for CMT, we use the decoding metric and we see that decoding can be done in the code trellis with this metric.We assume that a uniform interleaver with interleaver size N i [13] is employed at the encoder output.The effect of the uniform interleaver is to make the SIR difference on each multicode channel disappear and hence uniformity holds for CMT.Thus, we can evaluate the union bound on BER by considering only the all-zero path as a reference path.
Consider an error event of length M and Hamming distance d from the all-zero path in the time-invariant trellis.If N i = M mn, the uniform interleaver maps a given codeword with d "1" bits into Ni d distinct permutations with equal probability 1/ Ni d .For each permutation index p, the set of indices of "1" bits in the trellis diagram is I p .Then the SIR resulting from the comparison of two paths of Hamming distance d in CMT is given by where l is the coded bit index in the trellis, since N i is an integer multiple of m k .Hence the PEP of the path of distance d in CMT is obtained as In CT, the PEP of the path of distance d is given by when it is averaged over multicodes.
) by Jensen's inequality.The proposition follows.
From the proposition, we can assert that CMT works better than CT.Furthermore, as SIR k, j , j = 1, . . ., m k , become more different, the advantage of CMT over CT gets larger.The PEP of CMT in this section provides optimistic performance since it is assumed that the elements of random vector (n k i + īk i + i k i ) are uncorrelated.However, the analysis gives some intuition where the multicode diversity comes from.
Next we comment on when the MMSE detector (8) actually provides different SIRs in (29).Note that R is a symmetric matrix with all diagonal terms equal to 1.As an extreme case, if all A k, j are equal and all off-diagonal terms of R are equal, all off-diagonal terms of M also become equal and equality of all SIR k, j follows.Generally, the variation of crosscorrelations of two spreading codes increases when they are not synchronized.In ISI channels, when we consider the effective signature waveform of each user, the variation of crosscorrelations can be significant because the spreading codes are not synchronized in the effective signature waveform's crosscorrelation.

SIMULATION RESULTS
Computer simulations have been done for a BPSK system.Signature sequences with processing gain N = 15 were generated randomly and fixed. 1 The code rates of CCs were 1/n = 1/2, 1/3, and 1/4, and the constraint length L is set to 5. The generator matrix is 23 35 , 25 33 37 , and 25 27 33 37 for rates 1/2, 1/3, and 1/4 codes, respectively, and the decoding delay is 6L.The cases where the desired user k has two, three, or four times faster information bit rates than others and all other users have the same data rates are considered.Thus, m k = 2, 3, or 4, and m j = 1 for j = k.The total number of users K is determined by 15 − (m k − 1) and the total number of CDMA channels C = K p=1 m p is fixed at 15.The CDMA channel index c k, j for jth multicode of kth user is given by c k, j = k−1 p=1 m p + j.The case of matched m k and n was simulated with m k = 2, n = 2, and the results are shown in Figures 4 and 5. From the figures and Table 1, we see that when uncoded BERs of multicode channels at the MMSE detector output are similar, the performance improvement of CMT compared with CT is negligible; however, when the performances of multicode channels at MMSE detector output are very different, the performance gain of CMT over CT becomes significant.Also, the simulated BER of the MMSE detector is in good agreement with analytical BER.Simulation results of mismatched m k and n are shown in Figures 6 and 7

CONCLUSION
In this paper, we investigated two coding schemes combined with MMSE multiuser detection in multicode CDMA systems.Four metrics for CMT were considered along with different decoding trellises.Time invariant-trellises are employed for the analysis, and to reduce decoding complexity, time-varying trellises are used.
CMT generally provides better performance than CT with all four metrics.As the variation of SIRs among the multicodes increases at the MMSE detector output, the gain of CMT over CT becomes more significant.The Gauss-Chebyshev quadrature rule yielded a tight upper bound on BER using transfer function method.An SIR-based method provides insight into where the multicode diversity comes from.
Saul B. Gelfand received the Ph.D. degree in electrical engineering and Computer Science from MIT in 1987.Previously he was with Scientific Systems, Inc., Cambridge, MA, and Bolt Beranek and Newman Inc., Cambridge, MA.In these positions he worked on multitarget detection and tracking problems in sonar signal processing.Since 1987 he has been with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, Ind, where he is a Professor.His major contributions lie in the formulation and analysis of recursive algorithms for detection, estimation, filtering, and optimization of stochastic systems, with primary application to digital communications and biomedical signal processing.Professor Gelfand has previously served as Editor for Equalization for the IEEE Transactions on Communications, and Chair of the Communications Theory Symposium at the 2000 IEEE International Conference on Communication.

Figure 1 :
Figure 1: Transmitter of desired user k.(a) Coding across time, (b) coding across multicodes and time.

Figure 2 :
Figure 2: Receiver of desired user k.(a) Coding across time, (b) coding across multicodes and time.
is the spreading sequence, N is the processing gain, and ψ(t) is 1 .If n is divisible by m k then the CMT trellis is just the CC trellis.(ii) When m k > n, for some transitions more than one input bit is required and there is no simple fixed state augmentation time-varying trellis.It is possible to modify the variable state augmentation time-varying trellis, below to enforce a fixed state augmentation, but such an approach has no advantage either intuitively or computationally so we do not elaborate on it here.When m k < n and n is not divisible by m k , at the tth transition, if (t − 1)n is not divisible by m k , the CC state is augmented by 1 bit and otherwise it is not augmented.The number of information input bits at the tth transition is one if (t − 1)n is not divisible by m k and is two otherwise.For this variable state augmentation time-varying trellis, īt = (x t + 1)n/m k and i t = (x t − 1)n/m k + 1 when (t − 1) is divisible by ( m − 1) and otherwise īt = y t n/m k and (14)m − 1)/ m, which is always less than the factor of 2 for the fixed state augmentation time-varying trellis decoding.When m = 2, the variable state augmentation time-varying trellis reduces to a time-invariant trellis with the same decoding complexity as CC.If n k > n and m k is not divisible by n, at the tth transition, if (t − 1)m k is not divisible by n, the CC state is augmented by 1 bit and otherwise it is not augmented.The number of information input bits at the tth transition is given by tm k /n − (t − 1)m k /n .For this variable state augmentation time-varying trellis, īt = i t = t in the branch metrics in (12), (13),(14), and (15).The computational complexity of the variable state augmentation time varying trellis decoding is upper bounded by that of the CC multiplied by 2 m−1 / m • ĩ.If m k is divisible by n then the CMT trellis is just m k /n transitions in the CC trellis.
= t or j = j .The Laplace transform of the pdf of U k = with equality if and only if all SIR k, j , j = 1, . . ., m k , are equal.

Table 1 :
Comparison of CT and CMT when n = 2 and m k = 2 (10 dB SNR).