## 3 DI-TIE Search Algorithm for ISF Quantization  
DI-TIE is presented as a two-stage search algorithm, i.e., a dynamic mechanism as the first stage and an intersection mechanism as the second. For the sake of discussion, a TIE algorithm is firstly stated as follows. As presented in [18], a codeword captured from the preceding frame is treated as a reference codeword \(\mathbf{c}_{r}\) in the current frame. Subsequently, the Euclidean distance between an input vector \(\mathbf{x}\) and \(\mathbf{c}_{r}\), symbolized as \(d(\mathbf{c}_{r},\mathbf{x})\), is evaluated. A set, composed of all the codewords \(\mathbf{c}_{i}\) satisfying the condition \(d(\mathbf{c}_{r},\mathbf{c}_{i})<2d(\mathbf{c}_{r},\mathbf{x})\), is referred to as a candidate search group (CSG), denoted by \(CSG(\mathbf{c}_{r})\) and formulated as  
\[\text{TIE}:\;CSG(\mathbf{c}_{r})=\left\{\mathbf{c}_{i}|d(\mathbf{c}_{r}, \mathbf{c}_{i})<2d(\mathbf{c}_{r},\mathbf{x})\right\},\;1\leq i\leq CNum, \tag{8}\]  
where \(CNum\) represents the total number of codewords. \(CSG(\mathbf{c}_{r})\) is the search space over which a current codebook search is performed. In addition, a lookup table listing all the codewords sorted by the Euclidean distance is prebuilt for the TIE search algorithm.  
The dynamic mechanism refers to a mechanism exclusively developed to downsize a search space dynamically. With \(N(\mathbf{c}_{r})\) representing the number of candidate codewords contained in \(CSG(\mathbf{c}_{r})\), the aim is to find a way to reduce the value of \(N(\mathbf{c}_{r})\). For this sake, each candidate codeword in \(CSG(\mathbf{c}_{r})\), \(\mathbf{c}_{k}\), \(1\leq k\leq N(\mathbf{c}_{r})\), is treated in turn as a reference codeword. For instance, \(\mathbf{c}_{k}\) at \(k=1\), represented as \(\mathbf{c}_{k}|_{k=1}\), is initially appointed a reference codeword, according to which \(CSG(\mathbf{c}_{k})\) and \(N(\mathbf{c}_{k})\) are found directly in the lookup table, and merely one time evaluation of \(d(\mathbf{c}_{k},\mathbf{x})\) is required each time \(CSG(\mathbf{c}_{k})\) is determined. Thus, if the updated \(CSG(\mathbf{c}_{k})\) satisfies the condition \(N(\mathbf{c}_{k})<N(\mathbf{c}_{r})-k\), the search space can be downsized, and then move forward to stage 2. Otherwise, the above-stated mechanism is repeatedly performed on the rest of \(\mathbf{c}_{k}\) until the end.  
In stage 2, intersection of the sets \(CSG(\mathbf{c}_{r})\) and \(CSG(\mathbf{c}_{k})\) is performed, in an attempt to locate the optimal codeword that lies in each CSG, and in an effort to downsize the search space for search performance improvement. \(CSG(\mathbf{c}_{r})\) is updated as  
\[CSG(\mathbf{c}_{r})=CSG(\mathbf{c}_{k})\cap CSG(\mathbf{c}_{r}). \tag{9}\]  
The \(N(\mathbf{c}_{r})\) and the reference codeword are updated until a search task is done. The DI-TIE search algorithm is illustrated as a flowchart in Figure 1, and is described below.  
\begin{table}
\begin{tabular}{c l l} \hline \hline  & \multicolumn{2}{c}{**Structure of S-MSVQ**} \\ \hline \multirow{3}{*}{Stage 1} & CB1: & CB2: \\  & \(\mathbf{r_{1}}\) (1–9 order of \(\mathbf{r}\)) & \(\mathbf{r_{2}}\) (10–16 order of \(\mathbf{r}\)) \\  & (8 bits) & & (8 bits) \\ \hline \multirow{3}{*}{Stage 2} & CB11: & CB12: & CB21: \\  & \(t^{(2)}\),1-5 & \(t^{(2)}\),1-9 & \(t^{(2)}\),2-1-7 \\  & (7 bits) & (7 bits) & (6 bits) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Structure of S-MSVQ in AMR-WB in the 6.60 kbit/s coding mode.