# CRCSI: A Generic Block Interleaver for the Next Generation Terrestrial Broadcast Systems
## 3 Designing the Block Interleaver
### CRCSI Design
#### 3.2.2 Cyclic Shift  
For mobile reception scenarios, since the Doppler effect caused by the mobility of user terminals is inevitable, it is necessary to resist continuous error caused by the time-selective fading. Time interleaving is realized by dispersing the neighboring data cells belonging to different OFDM symbols. The cyclic shift scheme is a simple and effective method to implement time interleaving. As mentioned in TBI, a row-by-row cyclic shift to the left achieves diagonal-wise reading. However, \(p=1\) and \(s(i)=i\bmod N\) does not sufficiently break up the relationship between two adjacent elements. The traditional method involves continuously reading different OFDM symbols and subcarriers, which cannot maximize the error correction capability of the codewords. In order to better characterize the excellent performance of the block interleaver in resisting cluster errors, we adopt the two-dimensional interleaver form in the design phase [13]. It is worth pointing out that this two-dimensional representation is not only applicable to two-dimensional arrays, but also applicable to one-dimensional data. The sphere-packing model gives a theoretical resolution for the error correction upper bound in a two-dimensional array with arbitrary shapes. It is indicated that the \(t\)-interleaved array is less likely to be corrupted by the interference, where \(t\) represents the radius of the sphere associated with the array [20], that is, the interleaving ability. The definition is as follows:  
\[\mathcal{S}_{2,t}=\left\{ \begin{array}{ll}|x_{1}|+|x_{2}|\leq(t-1)/2,&\text{ if $t$ is odd}\\ |x_{1}|+|x_{2}-1/2|\leq(t-1)/2,&\text{ if $t$ is even}\end{array} \right.\]  
Let \(S_{2,t}\) be a sufficiently large \(t\)-interleaved two-dimensional array. We calculate the elements in \(\mathcal{S}_{2,t}\), then the degree of interleaving is bounded by  
\[|\mathcal{S}_{2,t}|\geq\lceil\frac{t^{2}}{2}\rceil, \tag{17}\]  
where \(t\geq 1\) and \(t\) is less than or equal to the minimum value of the row and column in an array. When the size of the data block is given, the elements in the \(t\)-interleaved sphere are bounded by (17). Conversely, when the number of elements is \(N\), the interleaving radius \(t\) has the upper bound:  
\[t\leq\sqrt{2N}. \tag{18}\]  
Correspondingly, the smallest number of elements with the interleaving ability [21]\(t\) is  
\[N^{*}=\frac{t^{2}}{2}. \tag{19}\]  
Finally, the upper bound \(t\) of the cyclic shift operation and the lower bound \(N^{*}\) of the interleaving degree for a block interleaver have been developed. As defined in \(\pi_{1}\), the incremental interval \(p\) needs to be a prime number so as to iterate over all the columns. We now set the cyclic shift value \(p_{t}\) as  
\[p_{t}=2\lceil\frac{t}{2}\rceil-1. \tag{20}\]  
Then, the permuting order \(C^{t}\) in a sphere model becomes \(C^{t}_{i}=p_{t}i\bmod N^{*}\). Based on above  
Thus, the preshift vector \(\mathbf{s}^{t}\) is rewritten as  
\[s_{i+1}^{t}=\left\{ \begin{array}{ll}s_{i}^{t}+p_{t},&s_{i}^{t}+p_{t}\leq N\\ \left[(s_{i}^{t}+p_{t})\bmod N^{*}\right]\bmod p_{t},&s_{i}^{t}+p_{t}>N\end{array} \right. \tag{23}\]  
where \(i=0,1,\cdots,M-1\) and \(s_{0}^{t}=0\). For the \(5\times 12\) block, a new preshift vector \(s^{t}\) is determined according to Equation (23). It is observed in Figure 3b, \(p_{t}=3,Q=12\) and \(\mathbf{s}^{t}=(0,3,5,8,1)\) makes spare columns exploited. Following the above analysis, the proposed CRCSI interleaver design algorithm can be summarized as follows. For a given size of the data block, first rearrange the row data in the data block according to the designed cross-row index method. Then, the cyclic shift operation is performed row-by-row to maximize the time diversity. Finally, the data are read out sequentially according to the reading order. For ease of understanding, a schematic diagram is also illustrated in Figure 4.  
After designing the mapping rule, the reading order \(k\) shall be determined.