## 3 Turbo Equalization Concept
### The Basic Architecture of the Iterative Receiver Using the Turbo Principle  
Figure 1 shows the structure of a transmitter in the presence of the convolutional encoder, the classic structure of the iterative receiver, and the model channel in the presence of intersymbol interference for BPSK modulation. Most turbo equalization architectures described in the literature use the same basic architecture, both for the SISO equalization module and for the SISO convolutional decoding module, using the BCJR algorithm [33]. In Figure 1, the input binary data sequence \(u_{i}\), assumed to be equiprobable, is encoded by a convolutional encoder, interleaved, mapped into BPSK symbols, and transmitted in a time-varying and frequency-selective medium. The signal from the channel output \(y_{i}\) was corrupted by a Gaussian noise, \(n_{i}\). At the reception, a turbo equalizer is used. It consists of three functions mentioned in [34]. Iterative processing is performed using these modules that exchange soft information and provide an estimate of the transmitted symbols \(L(\hat{u}_{i})\) for each iteration. Next, the expressions of the soft information for the MAP turbo-equalizer architecture will be presented. In this architecture, ISI cancellation is provided by a MAP detector. The principle of this architecture is that the multipath channel is similar to a convolutional code, which can be represented in the form of a trellis. Optimal detection of the symbols generated by the ISI channel can be achieved using the MAP algorithm. Figure 2 presents the block diagram of turbo equalization using the classic diagram. In general, the turbo equalizer corresponding to the transmission scenario of Figure 1 takes the form represented in Figure 2. It consists primarily of a MAP SISO equalizer, which  
Figure 1: Convolutional coded system, applying the turbo equalization, achieving equalization and channel decoding iteratively. Figure 2: Turbo equalizer using MAP criterion. takes as input both the vector \(y_{i}\) of observed data at the channel output and corrupted by Gaussian noise \(n_{i}\) and the a priori probabilistic information provided by the MAP SISO decoder, denoted here formally \(L_{e}^{D}(\hat{s})\)[5]. In the case of the first iteration, the term \(L_{e}^{D}(\hat{s})\) is equal to zero, meaning that no a priori information is provided by the channel decoder. The probabilistic information is propagated in the form of a Log-likelihood ratio (LLR), whose expression is written here for \(u_{i}\) with value in \(\{0,1\}\):  
\[L(u_{i}\,|\,y)=\ln\biggl{(}\frac{P(u_{i}=+1\,|\,y)}{P(u_{i}=-1\,|\,y)}\biggr{)}, \tag{1}\]  
where the \(P(u_{i})\) depicts the a priori probability of the input bits. The LLR is computed as the natural logarithm of the division between the probability of the transmitted bit \(u_{i}=+1\) and the one of the transmitted bit \(u_{i}=-1\), considering that \(y\) represents the received sequence. The notion of LLR is twice informatory because the sign of the quantity \(L(u_{i})\) gives the firm decision on \(u_{i}\) and its absolute value \(|L(u_{i})|\) evaluates the decoding credibility/likelihood of the channel [35]. The equalizer computes the a posteriori probability \(L^{E}(\hat{s})\). Therefore, the extrinsic information \(L_{e}^{E}(\hat{s})\) delivered by the equalizer is computed as follows:  
\[L_{e}^{E}(\hat{s})=L^{E}(\hat{s})-L_{e}^{D}(\hat{s}), \tag{2}\]  
The vector \(L_{e}^{E}(\hat{s})\) is then deinterleaved to give a new sequence \(L_{e}^{E}(\hat{s}^{\prime})\), which constitutes the a priori information for the MAP SISO decoder. The aim of subtracting the a priori LLRs \(L_{e}^{D}(\hat{s})\) from the a posteriori LLRs \(L^{E}(\hat{s})\) is to avoid the decoder obtaining information established on its own decisions, which were generated in the previous iteration in the turbo-equalization process. The MAP SISO decoder delivers the a posteriori information on the coded message estimates noted \(L^{D}(\hat{s}^{\prime})\).