,
, is a noise related vector. It is well known that the optimal solution maximizing requires an exhausted bit search. This combi-natorial problem is shown to be NP hard and the required computation complexity is on the order of . Although the ML criterion and the minimum BER criterion are different, their solutions are close especially for high SNR ratios. When the asynchronous transmission is considered, it has been shown that the complexity of the optimal receiver, implemented by a matched filter bank followed by the Viterbi algorithm, remains . The ML receiver re-quires the information of the signal amplitudes, signature waveforms, and signal delays for all users. When the criterion of minimum BER is utilized, the optimum detection, implemented with the backward-forward dynamic programming, still requires the complexity of . In this case, the variance of background noise is also necessary. These requirements along with the high computational complexity makes the optimal receiver infeasible for real-world imple-mentation.
§ 2.2 Linear Suboptimal Receivers
The optimal MUD has been regarded as powerful yet complicated. The suboptimal MUD was developed to reduce the complexity while still provide performance gain. In this section, we describe the suboptimal linear multiuser receivers. The linear receiver performs a linear trans-formation on the received signal vector. The first linear multiuser receiver is called the decor-relating detector or simply the decorrelator, whose name stems from the fact that the detector simply inverts the correlation matrix in (2.6). Let
(2.7)
Then, the receiver output is given by
(2.8)
As shown, the interference from the MAI is eliminated completely. However, noise becomes colored and its level may be enhanced. When the noise level dominates the MAI, the perfor-mance of the decorrelator is degraded. The decorrelator is also the joint ML solution for the simultaneous estimate of channel gains and transmission bits. The solution can be found by the minimization of a least-squares criterion.
(2.9)
In contrast to the optimal MUD, the decorrelator does not require user signal amplitudes. In addition, it was shown that the near-far resistance is equal to the that of the optimal receiver.
The fluctuation of the interference powers do not have any influence on the performance of the decorrelator.
Another commonly cited suboptimal linear receiver is the LMMSE receiver whose transfor-mation matrix is defined by
(2.10)
After some matrix manipulation, we can obtain the transformation matrix for the LMMSE mul-tiuser receiver as
(2.11)
Comparing the decorrelator with LMMSE receivers, we can observe that the LMMSE receiver becomes the decorrelator as approaches zero. On the other hand, the LMMSE receiver will degenerate into the matched filter when noise approaches infinity. This means that LMMSE multiuser receiver performs a compromise between noise enhancement and interference cancel-lation. When the LMMSE receiver is used, the signal amplitudes as well as the noise variance have to be known, in addition to the signal spreading codes and received signal delays.
−10 −5 0 5 10 15 10−4
10−3 10−2 10−1 100
a2/a 1 ( dB)
BER
Conv. receiver Decorrelator LMMSE Optimal ML
Figure 2.1: BER performance comparison of multiuser receivers for the first user (
dB, and).
It can be observed that the matrix inversion is required in linear receivers. In order to reduce the computational burden, adaptive implementation was proposed. Let
,
. Rewrite (2.9) as
²
(2.12)
where and
are the chip-sampled sequences of and
, respectively, and is the processing gain. The estimate of gives the channel gains and bits, which are those in (2.8). The adaptive implementation of the decorrelator does not involve the complicated matrix inversion operation. Similarly, the LMMSE receiver can have an adaptive implementation. Let
. It can be easily shown that the LMMSE receiver output is
(2.13)
where is an matrix. Thus, we can use the MMSE criterion to
0 1 2 3 4 5 6 7 8 9 10 10−2
10−1 100
Eb/N 0
BER
Conv. receiver Decorrelator LMMSE Optimal ML
Figure 2.2: BER performance comparison of multiuser receivers ( , and).
derive . Thus,
(2.14)
Note that some transmission bits are required for training. The MUD performance measure includes the BER, the asymptotic multiuser efficiency, and the near-far resistance [7]. We have carried out some simulations to evaluate the performance of the receivers described above.
Figure 2.1 shows the result for BER vs. interference power. Here, the user number is two, the code correlation is, and
dB (
). Note that the-axis of the figure is the power ratio of the two users. The conventional receiver suffers from the interference from the second user, and its performance degrades rapidly when the normalized interference power increases up to 5 dB. The ML receiver has the best near-far resistance among the four detectors.
The decorrelator exhibits a constant near-far resistance in all interference power ratios. The LMMSE receiver is degenerated to the conventional receiver when the interference is weak while to the decorrelator when the interference is strong; it performs very similarly to the ML
2 3 4 5 6 7 8 9 10 10−4
10−3 10−2 10−1 100
User number
BER
Conv. receiver Decorrelator LMMSE Optimal ML
Figure 2.3: BER performance comparison of multiuser receivers (random codes, , and
dB.)
receiver in weak interference. Figure 2.3 shows the BER vs. for ten equicorrelated users ( ). The single-user receiver suffers from MAI and perform poorly in most cases. The linear receivers perform similarly to the ML receiver when the number of users is small.