## 1 Introduction  
The importance of coding techniques has achieved much attention in modern technology. In the 90s, the Turbo coding technique was proposed to improve the performance of communication systems. Then the era of many modern iterative decoding techniques starts which includes the iterative, turbo, convolutional and repeat accumulate (RA) [1; 2; 3; 4]. Such error correcting techniques were utilized in many applications specially in mutli-user detection, forward error control, data estimation and control signals. The reader is referred to [5; 6; 7; 8; 9] for more reading and application dependent techniques. Multi-input and user detection is closely related to the detection of data over the multiple transmission in a network, many users are sending the information over a channel in one time and access it simultaneously. In such situation, there is another issue arised known as multiple access interference (MAI). The major objective behind using the MUD is to utilized the inherent structure of MAI via joint detection of many transmitter [10; 11; 12; 13; 14; 15; 16]. To discuss the coded code division multiple access (C-CDMA) system, it was explained in reference [17] that the Viterbi decoder is the optimal maximum-likelihood decoders. These are operational on the joint trellis of multi-access links. But the bad things for such systems is the greater complexity which is the major hindrance of its implementation. The prohibitive computational complexity of optimal coded MUD necessitates using suboptimal detectors in practical implementations. Even when functionality at the receiver is split into the roles of MUD and single user decoding, using an optimal multiuser detector is still computationally costly. For this reason, using suboptimal detectors is common in iterative systems, which incurs a performance loss in terms of possible system load, output bit error rate (BER) and speed to convergence [18; 19]. The tables of symbols and acronyms are given in Tables 1 and 2. In iterative systems with optimal components, the sum-product algorithm tells us one should pass extrinsic information between system components. However, in systems with sub-optimal building blocks, the most suitable soft information to exchange within the system remains an open problem; however, other schemes, such as passing a-posteriori probability (APP) information, have been demonstrated and can yield better results [20; 21]. This work considers a coded multiuser receiver that utilizes a (suboptimal) parallel interference cancellation (PIC) component for its MUD stage. A recursive log-likelihood ratio (LLR) filter is derived, considering the LLR streams of the previous and current iterative cycles. The filter produces an output that maximizes the output LLR sequences based on knowledge of the statistics (mean and variance) of the two input LLR streams and the noise correlation between the two LLR streams [22; 23; 24; 25]. In this paper, we focus on the design of a novel filter and compare its performance in different scenarios. The proposed work is an extension of the maximum ratio combining (MRC) filter. In contrast, in the previous work, it has been acknowledged that: _combining over iterations introduces correlations, which lead to sub-optimal performance_. Here we ponder the question: 'Knowing that correlations have been introduced, can we utilize this knowledge and perhaps derive a filter that provides better results despite these correlations?' This idea is investigated in the proposed research work. The key contributions were established upon which the research element of this work is built:  
* Extrinsic feedback is used in a coded CDMA system.