## 3 4-SSB System Design
### The Concept of Application of 4-SSB into OFDM M-QAM 4-SSB Uncoded System Using the Shadow Equalizer
#### 4.3.3 The MF-SIC in 4-SSB Scheme  
The proposal of MF-SIC is 4-SSB in shown in Figure 4.  
The two symbols enter the MMSE to decrease error estimation so that the candidates of symbol detection can be reflected. Without the loss of generality, we represent the process through the pseudocode algorithm of MF-SIC-SAC [13] as in Algorithm 1. The system starts by detecting each data stream \(\hat{d}_{1},\hat{d}_{2},\dots,\ \hat{d}_{N_{T}}\) which can be obtained in each stage. The constellation is located within the shadow. The algorithm of soft decision \(\widetilde{d}_{n}[i]\ =\ w_{n}^{H}\ d[i]\) is applied to check the decision of symbol is reliable or non-reliable as follows [13]:  
1. Reliable Decision: if the algorithm of SAD considers that the symbol \(\hat{d}_{n}[i]\) is reliable, the function of hard slice decision is used for both data stream for the hard decision as well as the cancellation function for the next symbol. The conventional in SIC algorithm will match hard slice decision as follows: \[\hat{d}_{n}[i]\ =\ Q\Big{(}\widetilde{d}_{n}[i]\Big{)}\] (30)
2. The symbol is decided as non-reliable if the previous exhaustive decision is non-reliable. Then several candidate's vectors are generated, and the aforementioned conventional SIC will decide the candidate's symbol as minimal as possible for saving energy saving. Algorithm 1 selects which candidate symbol is the nearest one to the original symbol.  
Figure 4: The shadow equalizer in the 4-SSB MIMO System.  
Particularly, the selection algorithm is used to choose the best candidate symbol in 4-SSB. The procedure has two steps. First, multiple feedback (MF) is generated. As shown in Figure 4, the MF algorithm is applied after SAC makes the non-reliable decision.  
To optimize the symbol candidate in the first iteration, we select the nearest \(M\) constellation points by the soft decision of MMSE filter \(\vec{d}_{n}[i]\). This filter is defined as the function of multiple feedbacks as the following expression:  
\[\mathcal{L}=\ M\mathrm{F}\Big{(}\vec{d}_{n}[i]\Big{)} \tag{31}\]  
The constellation of the M-QAM symbol candidates can be represented by the subset of \(\mathcal{L}=[\ c_{1},\ c_{2},\ \ldots,\ c_{M}]^{T}\subseteq\mathrm{A}\). The nearest M constellation refers to the signal to noise ratio (SNR) as a higher value of SNR implies a smaller value of \(M\).  
In Algorithm 1, we also introduce a new factor called \(\hat{z}_{n}[i]\) which is represented by \(M\ x\ 1\) vectors. The output of MF generates a nearest constellation of symbol candidate represented by:  
\[\hat{z}_{n}[i]=\ [\ c_{1},\ c_{2},\ \ldots,\ c_{M}]^{T} \tag{32}\]  
This stage is significant to make the decision feedback optimal. The previous step of MF-SIC is to prevent the system from making error propagation.  
The second step is for the candidate symbol. In this step, the objective is to select optimal candidates from the symbol \(\hat{z}_{n}[i]\). Let \(b_{m}\) denote a new vector symbol of the optimal candidates with the range of \(1\leq m\leq M\).  
As each \(b_{m}\) represents \(m\)th candidate in each n to \(N_{T}\) stage of processing, the candidate vector symbol is expressed by the following transform matrix:  
\[b_{m}[i]=\ \Big{[}\hat{d}_{n}[i],\ldots,\hat{d}_{n-1}[i],\ b_{n,m}[i],\ldots,b_{ j,m}[i],\ldots,b_{N_{T},m}[i]\Big{]}^{T}. \tag{33}\]This vector value is filtered by the MMSE filter described as follows:  
\[b_{j,m}=\ Q\Big{(}w_{j}^{H}\hat{d}_{j,m}[i]\Big{)}, \tag{34}\]  
where \(\hat{d}_{j,m}\) denotes the cancelled receiver feedback vector in the \(m\)th stage. The reason of applying MMSE vector is to decrease of the complexity of the system:  
\[b_{m}=\ \Big{[}b_{1,m},\ldots,b_{N_{T},m}\Big{]}^{T}. \tag{35}\]  
The algorithm then selects the optimal feedback corresponding to the following constraint:  
\[m_{opt}=\arg\min_{1\leq m\leq M}\|d_{i,SSB,n}-Hb_{m}[i]\|^{2}, \tag{36}\]  
\[\hat{d}_{n}=\ (\hat{\varepsilon}_{n})_{m_{opt}}. \tag{37}\]