# Technical Document Extraction ## Network Diagram (Left Side) ### Nodes - **Top Layer Nodes**: - `X_{2,1}` (central node) - **Intermediate Layer Nodes**: - `X_{1,1}`, `X_{1,2}`, `X_{1,3}`, `X_{1,4}`, `X_{1,5}` - **Bottom Layer Nodes**: - `X_{0,1}`, `X_{0,2}` ### Connections - **Top-to-Intermediate Connections**: - `x̃_{1,1,1}`, `x̃_{1,1,2}`, `x̃_{1,1,3}`, `x̃_{1,1,4}`, `x̃_{1,1,5}` - `x̃_{1,2,1}`, `x̃_{1,2,2}`, `x̃_{1,2,3}`, `x̃_{1,2,4}`, `x̃_{1,2,5}` - `x̃_{1,3,1}`, `x̃_{1,3,2}`, `x̃_{1,3,3}`, `x̃_{1,3,4}`, `x̃_{1,3,5}` - `x̃_{1,4,1}`, `x̃_{1,4,2}`, `x̃_{1,4,3}`, `x̃_{1,4,4}`, `x̃_{1,4,5}` - `x̃_{1,5,1}`, `x̃_{1,5,2}` - **Intermediate-to-Bottom Connections**: - `x̃_{0,1,1}`, `x̃_{0,1,2}`, `x̃_{0,1,3}`, `x̃_{0,1,4}`, `x̃_{0,1,5}` - `x̃_{0,2,1}`, `x̃_{0,2,2}`, `x̃_{0,2,3}`, `x̃_{0,2,4}`, `x̃_{0,2,5}` ### Function Blocks - **Top Layer Functions**: - `φ_{1,1,1}`, `φ_{1,1,2}`, `φ_{1,1,3}`, `φ_{1,1,4}`, `φ_{1,1,5}` - `φ_{1,2,1}`, `φ_{1,2,2}`, `φ_{1,2,3}`, `φ_{1,2,4}`, `φ_{1,2,5}` - `φ_{1,3,1}`, `φ_{1,3,2}`, `φ_{1,3,3}`, `φ_{1,3,4}`, `φ_{1,3,5}` - `φ_{1,4,1}`, `φ_{1,4,2}`, `φ_{1,4,3}`, `φ_{1,4,4}`, `φ_{1,4,5}` - `φ_{1,5,1}`, `φ_{1,5,2}` - **Bottom Layer Functions**: - `φ_{0,1,1}`, `φ_{0,1,2}`, `φ_{0,1,3}`, `φ_{0,1,4}`, `φ_{0,1,5}` - `φ_{0,2,1}`, `φ_{0,2,2}`, `φ_{0,2,3}`, `φ_{0,2,4}`, `φ_{0,2,5}` ## Graph and Equations (Right Side) ### Main Graph - **Function**: `φ(x)` - **Equation**: - `φ(x) = ∑_{i=0}^7 c_i B_i(x)` - **Time Axis**: - Range: `t_{-3}` to `t_8` - Key Points: `t_{-3}`, `t_{-2}`, `t_{-1}`, `t_0`, `t_1`, `t_2`, `t_3`, `t_4`, `t_5`, `t_6`, `t_7`, `t_8` - **Grid Extension**: - **Extended Function**: `φ(x) = ∑_{i=0}^{12} c'_i B'_i(x)` - **Extended Time Axis**: - Range: `t'_{-3}` to `t'_{13}` - Key Points: `t'_{-3}`, `t'_{-2}`, `t'_{-1}`, `t'_0`, `t'_1`, `t'_2`, `t'_3`, `t'_4`, `t'_5`, `t'_6`, `t'_7`, `t'_8`, `t'_9`, `t'_{10}`, `t'_{11}`, `t'_{12}`, `t'_{13}` - **Basis Functions**: - `B_0(x)`, `B_1(x)`, `B_2(x)`, `B_3(x)`, `B_4(x)`, `B_5(x)`, `B_6(x)`, `B_7(x)` - Extended Basis Functions: `B'_0(x)`, `B'_1(x)`, ..., `B'_{12}(x)` - **Coefficients**: - Original: `c_0`, `c_1`, ..., `c_7` - Extended: `c'_0`, `c'_1`, ..., `c'_{12}` ### Legend - **Colors**: - `B_0(x)`: Red - `B_1(x)`: Green - `B_2(x)`: Blue - `B_3(x)`: Purple - `B_4(x)`: Cyan - `B_5(x)`: Orange - `B_6(x)`: Pink - `B_7(x)`: Dark Blue - Extended Basis Functions: Not explicitly labeled but implied by continuation. ### Additional Notes - **Grid Extension Indicator**: Dashed arrow pointing from original to extended time axis. - **G Values**: - `G_1 = 5` (original grid size) - `G_2 = 10` (extended grid size) - **Key**: `k = 3` (possibly a parameter for the network or grid configuration).