The excess-3 code can be converted back to BCD in the same manner.

Let A, B, C and D be the bits representing the binary numbers, where D is the LSB and A is the MSB, and Let w,x, and z be the bits representing the gray code of the binary numbers, where z is the LSB and w is the MSB.

The truth table for the conversion is given below. The X’s mark don’t care conditions.

The truth table of Excess 3 to BCD Code

$A = wx + wyz$

$B = \overline{x}\overline{y} + \overline{x}\overline{z} +xyz$

$C = y+y\overline{z}$

$D = \overline{z}$

The corresponding digital circuit -

Here $E_3, E_2, E_1, and E_0$ corresponds to $w, x, y, and z$ and $B_3, B_2, B_1, and B_0$ corresponds to $A, B, C, and D$