6.4 Turbo Multiuser Detection with Unknown Interferers

The turbo multiuser detection techniques developed so far assume that the spreading waveforms of all users are known to the receiver. Another important scenario, discussed in Chapter 3, is that the receiver has knowledge of the spreading waveforms of some but not all of the users in a system. Such a situation arises, for example, in a cellular system where the base station receiver knows the spreading waveforms of the in-cell users but not those of the out-of- cell users. In this section we discuss a turbo multiuser detection method that can be applied in the presence of unknown interference, which was first developed in [414].

6.4.1 Signal Model

Consider again the synchronous CDMA signal model (6.27). Here we assume that the spreading waveforms and the received amplitudes of the first ( < K ) users are known to the receiver, whereas the rest of the users are unknown to the receiver. Since some of the spreading waveforms are unknown, we cannot form the sufficient statistic (6.32). Instead, as done in Chapters 2 and 3, we sample the received continuous-time signal r(t) at the chip rate to convert it to discrete-time signal. The sample that corresponds to the j th chip of the i th symbol is given by

Equation 6.72

The resulting discrete-time signal corresponding to the i th symbol is then given by

Equation 6.73

Equation 6.74

with

where

Equation 6.75

is a Gaussian random variable; n [ i ] ~ N ( , s ² I _N ); s _k is the normalized discrete-time spreading waveform of the k th user , with c _n,k {+1, “1}; ; ; and .

Denote by the matrix consisting of the first columns of S . Denote the remaining = K “ columns of S by . These first signature sequences are unknown to the receiver. Let be the -vector containing the first bits of b [ i ], and let contain the remaining bits. Then we may write (6.74) as

Equation 6.76

Since we do not have knowledge of , we cannot hope to demodulate . We therefore write (6.76) as

Equation 6.77

where is regarded as an interference term that is to be estimated and removed by the multiuser detector before it computes the a posteriori log- likelihood ratios (LLRs) for the bits in .

6.4.2 Group -Blind SISO Multiuser Detector

The heart of the turbo group-blind receiver is the soft-input/soft-output (SISO) group-blind multiuser detector. The detector accepts, as inputs, the a priori LLRs for the code bits of the known users delivered by the SISO MAP channel decoders of these users, and produces, as outputs, updated LLRs for these code bits. This is accomplished by soft interference cancellation and MMSE filtering. Specifically, using the a priori LLRs and knowledge of the signature sequences and received amplitudes of the known users, the detector performs a soft-interference cancellation for each user, in which estimates of the multiuser interference from the other known users and an estimate for the interference caused by the unknown users are subtracted from the received signal. Residual interference is suppressed by passing the resulting signal through an instantaneous MMSE filter. The a posteriori LLR can then be computed from the MMSE filter output.

The detector first forms soft estimates of the user code bits as

Equation 6.78

where l ₂ ( b _k [ i ]) is the a priori LLR of the k th user's i th bit delivered by the MAP channel decoder. We denote hard estimates of the code bits as

Equation 6.79

and denote .

In the next step we form an estimate of interference of the unknown users, I [ i ], which we denote by . We begin by forming the preliminary estimate

Equation 6.80

where and d _k [ i ] is a random variable defined by

Equation 6.81

It will be seen that our ability to form a soft estimate for d _k [ i ] will allow us to perform the soft interference cancellation mentioned above. Clearly, d _k [ i ] can take on one of two values, 0 or 2 b _k [ i ]. The probability that d _k [ i ] is equal to zero is the probability that the hard estimate is correct and is given by

Equation 6.82

Recall that for b {+1, “1}, the probability that b _k [ i ] = b is related to the corresponding LLR by [cf. (6.42)]

Equation 6.83

On substituting b = sign {tanh [ ½ l ₂ ( b _k )[ i ]} in (6.83), we find that

Equation 6.84

Therefore, d _k [ i ] is a random variable that can be described as

Equation 6.85

We now perform an eigendecomposition on G G ^T / M where 1]]. We denote by U _u the matrix of eigenvectors corresponding to the largest eigenvalues. The span of the columns of U _u represents an estimate of the subspace of the unknown users (i.e., the interference subspace). Ideally , that is, when d [ i ] = in (6.80), U _u contains the signal subspace spanned by the unknown interference . To refine our estimate of I [ i ], we project g [ i ] onto U _u . The result is

Equation 6.86

Denote and . Since, ideally, we have

Equation 6.87

Now we subtract the interference estimate from the received signal and form a new signal

Equation 6.88

where

Equation 6.89

with

Equation 6.90

For each known user we perform a soft interference cancellation on z [ i ] to obtain

Equation 6.91

where

with a soft estimate for d _k [ i ], given via (6.85) by

Equation 6.92

Substituting (6.88) into (6.91), we obtain

Equation 6.93

An instantaneous linear MMSE filter is then applied to r _k [ i ] to obtain

Equation 6.94

The filter is chosen to minimize the mean-square error between the code bit b _k [ i ] and the filter output z _k [ i ]:

Equation 6.95

where the expectation is with respect to the ambient noise and the interfering users. The solution to (6.95) is given by

Equation 6.96

It is easy to show that

Equation 6.97

where . The covariance matrix D [ i ] has the dimensions 2 x 2 and may be partitioned into four diagonal x blocks in the following manner:

Equation 6.98

The diagonal elements of D ₁₁ [ i ] are given by

Equation 6.99

Using (6.85), the diagonal elements of D ₂₂ [ i ] are given by

Equation 6.100

where

Equation 6.101

The diagonal elements of D ₁₂ [ i ] and D ₂₁ [ i ] are identical and are given by

Equation 6.102

It is also easy to see that

Equation 6.103

where e _k is a -vector whose elements are all zero except for the k th element, which is 1. Substituting (6.97) and (6.103) into (6.96), we may write the instantaneous MMSE filter for user k as

Equation 6.104

As before, we make the assumption that the MMSE filter output is Gaussian; we may write

Equation 6.105

where m _k [ i ] is the equivalent amplitude of the k th user's signal at the filter output, and h _k [ i ] ~ N (0, ) is a Gaussian noise sample. Using (6.97) and (6.104), the parameter m _k [ i ] is computed as

Equation 6.106

Equation 6.107

where (6.107) follows from (6.97), (6.104), and (6.106).

Finally, exactly the same as (6.64), the extrinsic information, l ₁ ( b _k [ i ]), delivered by the SISO multiuser detector is given by

Equation 6.108

This group-blind SISO multiuser detection algorithm is summarized as follows.

Algorithm 6.3: [Group-blind SISO multiuser detector ”synchronous CDMA]

• Given { l ₂ ( b _k [ i ])}, form soft and hard estimates of the code bits:

  Equation 6.109

  

  
  

  Equation 6.110

  

  
  

  Denote

  

• Let

  Equation 6.111

  

  
  

  Equation 6.112

  

  
  

  Perform an eigendecomposition on G G ^T / M ,

  Equation 6.113

  

  
  

  Set U _u equal to the first columns of U .

• For i = 0,1, ..., M “ 1:

  “ Refine the estimate of the unknown interference by projection:

  Equation 6.114

  

  
  

  “ Compute according to

  Equation 6.115

  

  
  

  where a _k [ i ] is defined in (6.101). Define

  

  “ Subtract [ i ] from r [ i ] and perform soft interference cancellation:

  Equation 6.116

  

  
  

  where .

  “ Calculate D [ i ], according to (6.99) “(6.102).

  “ Calculate and apply the MMSE filters:

  Equation 6.117

  

  
  

  Equation 6.118

  

  
  

  where , , and where and .

  “ Compute m _k [ i ] according to (6.106).

  “ Compute the a posteriori LLRs for code bit b _k [ i ] according to (6.108).

6.4.3 Sliding Window Group-Blind Detector for Asynchronous CDMA

It is not difficult to extend the results of Section 6.4.2 to asynchronous CDMA. The received signal due to user k (1 k K ) is given by

Equation 6.119

where t _k is the delay of the k th user's signal, { c _j,k } is a signature sequence of 1's assigned to the k th user, and y ( t ) is a normalized chip waveform of duration T _c = T/N . The total received signal, given by

Equation 6.120

is match-filtered to the chip waveform and sampled at the chip rate. The n th matched-filter output during the i th symbol interval is

Equation 6.121

Substituting (6.119) into (6.121), we obtain

Equation 6.122

where . Then

Equation 6.123

Denote

Equation 6.124

and for j = 0, 1, ..., I _k “1,

Equation 6.125

Then

Equation 6.126

By stacking successive received sample vectors, we define

Equation 6.127

Equation 6.128

where . Then we can write the received signal in matrix form as

Equation 6.129

Define the set of matrices such that is the matrix composed of columns jK + 1 through jK + of the matrix H . We define the matrix . The size of is N I x (2 I - 1). We denote by the matrix that contains the remaining (2 I - 1) columns of H . We define [ i ] and [ i ] by performing a similar separation of the elements of b [ i ]. Then we may write (6.129) as

Equation 6.130

This equation is the asynchronous analog to (6.76). We can obtain estimates of with straightforward modifications to Algorithm 6.3.

Simulation Examples

We next present simulation results to demonstrate the performance of the proposed turbo group-blind multiuser receiver for asynchronous CDMA. The processing gain of the system is seven and the total number of users is seven. The number of known users is either two or five, as noted on the figures. The spreading sequences are randomly generated and the same sequences are used for all simulations. All users employ the same rate- ½, constraint-length-3 convolutional code (with generators g ₁ = [110] and g ₂ = [111]). Each user uses a different random interleaver, and the same interleavers are used in all simulations. The block size of information bits for each user is 128. The maximum delay in symbol intervals is 1. All users use the same transmitted power and the chip pulse waveform is a raised cosine with roll-off factor 0.5.

Figure 6.11 illustrates the average bit-error-rate performance of the known users for the group-blind turbo receiver and the conventional turbo receiver discussed in Section 6.3 for the first four iterations. The number of known users is five. For the sake of comparison, we have included plots for the conventional turbo receiver when all of the users are known. The three sets of plots in this figure are denoted in the legend by "GBMUD," "TMUD," and "ALL KNOWN," respectively. Note that the curves for the first iteration are identical for GBMUD and TMUD. Hence we have suppressed the plot of the first iteration for TMUD, to improve clarity. Notice that iteration does not significantly improve the performance of the conventional turbo receiver, whereas the group-blind receiver provides significant gains through iteration at moderate and high signal-to-noise ratios. We can also see that the use of more than three iterations does not provide significant benefits.

Figure 6.11. Performance of a group-blind iterative multiuser receiver with five known users. Curves denoted GBMUD are produced using a turbo group-blind multiuser receiver and those denoted TMUD are produced using a standard turbo multiuser receiver. Also included are plots for TMUD when all users are known.

In Fig. 6.12, the number of known users has been changed to two. As we would expect, there is performance degradation for both conventional and group-blind turbo receivers. In fact, the conventional receiver gains nothing through iteration for this scenario because there are now five users whose interference is simply ignored. It is also apparent that the group-blind turbo receiver will not be able to mitigate all of the interference of unknown users, even for a large number of iterations. This is due, in part, to the use of an imperfect interference subspace estimate in the SISO group-blind multiuser detector.

Figure 6.12. Performance of a group-blind iterative multiuser receiver with two known users. Curves denoted GBMUD are produced using a turbo group-blind multiuser receiver and those denoted TMUD are produced using a standard turbo multiuser receiver. Also included are plots for TMUD when all users are known.