## 3 Code Matched Interleaver (CMI)
\[x_{i2}^{2}(\text{D})=(1+\text{D}^{ak_{2}})\text{D}^{\beta_{2}} \tag{13}\]  
where \(k_{2}\) = 1, 2, 3,... is a positive integer, \(\beta_{2}\) is the time delay, such that \(\beta_{2}\) = 1, 2, 3.... Let \(Q_{min}\) be the minimum parity check sequence generated by a weight-2 input sequence. Then the weight of \(j\)-th parity bit sequence is given as follows:  
\[\text{k}_{j}(Q_{min}-2)+2 \tag{14}\]  
where \(\text{k}_{j}\) = 1, 2, 3, 4..., \(j\) = 1, 2. The net weight of the turbo code generated by weight-2 input sequences is given as:  
\[d=2+\text{k}_{1}(Q_{min}-2)+2+\text{k}_{2}(Q_{min}-2)+2 \tag{15}\]  
\[d=6+\text{k}_{1}(Q_{min}-2)+\text{k}_{2}(Q_{min}-2) \tag{16}\]  
Suppose that the positions of \"1\"s in weight-2 information sequence \(x_{i1}\) are denoted by \(P_{1}\), \(P_{2}\). The information sequence \(x_{i1}\) after passing through interleaver scrambles the bits. The new position of \"1\"s becomes \(\pi(P_{1})\), \(\pi(P_{2})\) as shown in Figure 2.  
Let's take the most damaging case for weight-2 information sequence as  
\[|P_{1}-P_{2}|\text{mod }\alpha=0\text{ and }|\pi(P_{1})-\pi(P_{2})|\text{mod }\alpha=0 \tag{17}\]  
This case is the worst case because spacing between the consecutive two \"\(1\)\"s between the original and the interleaved sequences are both multiple of \(\alpha\). This will cause the interleaved information sequence still divisible by RSC-2 that will eventually result in low parity weight sequences. Thus, the net weight of turbo code word will be lowered and will decrease the overall performance of turbo code. Therefore, to prevent this worst-case scenario to happen, the interleaver must meet the following design mapping criteria:  
\[|P_{1}-P_{2}|\text{mod }\alpha=0\text{ and }|\pi(P_{1})-\pi(P_{2})|\text{mod }\alpha\
eq 0 \tag{18}\]  
However, not all the weight-2 input sequences are bad only those weight-2 input pattern that produce low weight code words that severely affects the BER performance should be removed in the interleaver design. Therefore a parameter, \(d_{max}^{\gamma}\) commonly known as threshold is usually defined where \(\gamma\) represents weight-2 or weight-4 sequences. If the weight of the code word lies outside the threshold value \(d_{max}^{\gamma}\), then that particular weight-2 sequence is not considered as bad pattern and interleaver must not remove that input pattern. The details of threshold \(d_{max}^{\gamma}\) are presented in [16]. The value of threshold \(d_{max}^{\gamma}\) varies for each turbo code. Applying threshold \(d_{max}^{\gamma}\) to Equation (13) we obtain the following inequality  
\[d=\omega+\omega(y_{p1})+\omega(y_{p2}) \tag{11}\]0 \[d=\omega+\omega(y_{p1})+\omega(y_{p2}) \tag{11}\]1  
Thus, the weight-2 information sequence is considered to be a bad sequence if it satisfies (17) and (20) simultaneously. The weight-3 information sequences, which result in low weight turbo code word is effectively removed by using S random constraint condition. The design steps of S-random interleaver are well formulated in [21]. We also consider a weight-4 sequence, which usually produces low weight parity sequence that degrades the performance of overall turbo code. A weight-4 sequence becomes weight-2 error pattern after interleaving which also produces bad weight code word. By same procedure, as weight-2 sequence, the new weight generated by weight-4 input sequence \(x_{i1}^{4}\) can be expressed as follows:  
\[d=\omega+\omega(y_{p1})+\omega(y_{p2}) \tag{11}\]2  
\[d=\omega+\omega(y_{p1})+\omega(y_{p2}) \tag{11}\]3 Weight-2 information sequence is mapped to weight-2 information sequence with similar pattern after permuted through interleaver. Suppose that the positions of \"1\"s are represented by \(P_{1}\), \(P_{2}\), \(P_{3}\) and \(P_{4}\) for weight-4 information sequence \(x_{i1}^{4}\) where, as \(P_{1}<P_{2}<P_{3}<P_{4}\). The information sequence \(x_{i1}^{4}\) is passed through an interleaver then positions of \"1\"s in the interleaved sequence is denoted by \(\pi(P_{1}),\pi(P_{2}),\pi(P_{2})\) and \(\pi(P_{4})\).