The 13th Design Contest in Okinawa・ Design Specification- 5
5. BCH decoder circuit design example
As explained in the previous section, two key functions are syndrome computation and Table access.
In figure 4, two linear feedback shift register circuits are shown in order to compute two syndrome S1(x) and S2(x). In the figure, flip flops of linear feedback shift register is named t**, and final computation results will appear in registers s**. Figure 5 shows the operational timing diagram of the circuit. In first 11 cycles, the switch is connected to A port, then feedback operation happens. Since first 11 cycle corresponds to 14th to 4th order of the xn and the power is larger than order of syndromes, feedback operation is required. In last 4 cycles, the switch is connected to B port and simple exor operation and shift operation are performed.
START signal is asserted to indicate the beginning and t** flip flop outputs are forced to ‘0’ by AND gates for initializing. As you can see from figure 5, at the cycle of 14, syndrome values appear in s** flip flops. Access the table 5 by the 8 syndrome bits and you can get the error status.
Figure 4 : Syndrome computation circuit example
Figure 5 : Timing diagram of Syndrome computation
In order to further explain the syndrome computation, EXCEL worksheet is used. In figure 6, syndrome are computed from input YELLOW cell of (1,0,0,0,0,0,0,1,1,1,0,1,0,0,0). In this case, the input does not include errors. Then the computed syndromes in RED cells are all 0s.
Figure 7 is another example. The YELLOW inputs are (1,0,0,1,0,0,0,1,1,1,0,1,1,0). In this example, there two errors at r2 (bit position = 2) and u3 (bit position = 11). The syndrome results are (s10, s11, s12, s13)=(0101), and (s20, s21, s22, s23)=(0110). Use the table 5, then you can get two positions of 2, 11. By flipping those two position bits, error correction can be performed.
Figure 6 : Syndrome computation example (no error case)
Figure 7 : Syndrome computation example (2 bits error case)
This excel work sheet is linked as follows. Please make use of the sheet.