Digital Logic | Code Converters – BCD(8421) to/from Excess-3

Prerequisite – Number System and base conversions

Excess-3 binary code is a unweighted self-complementary BCD code.
Self-Complementary property means that the 1’s complement of an excess-3 number is the excess-3 code of the 9’s complement of the corresponding decimal number. This property is useful since a decimal number can be nines’ complemented (for subtraction) as easily as a binary number can be ones’ complemented; just by inverting all bits.
For example, the excess-3 code for 3(0011) is 0110 and to find the excess-3 code of the complement of 3, we just need to find the 1’s complement of 0110 -> 1001, which is also the excess-3 code for the 9’s complement of 3 -> (9-3) = 6.

Converting BCD(8421) to Excess-3 –

As is clear by the name, a BCD digit can be converted to it’s corresponding Excess-3 code by simply adding 3 to it.
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,\:y,\: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.
 \begin{tabular}{||c|c|c|c||c|c|c|c||} \hline  \multicolumn{4}{||c||}{BCD(8421)} & \multicolumn{4}{|c||}{Excess-3}\\ \hline  A & B & C & D & w & x & y & z \\ \hline \hline  0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\  \hline  0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\  \hline  0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 \\  \hline  0 & 0 & 1 & 1 & 0 & 1 & 1 & 0 \\  \hline \hline  0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 \\  \hline  0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 \\  \hline  0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 \\  \hline  0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 \\  \hline \hline  1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 \\  \hline  1 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\  \hline  1 & 0 & 1 & 0 & X & X & X & X \\  \hline  1 & 0 & 1 & 1 & X & X & X & X \\  \hline \hline  1 & 1 & 0 & 0 & X & X & X & X \\  \hline  1 & 1 & 0 & 1 & X & X & X & X \\  \hline  1 & 1 & 1 & 0 & X & X & X & X \\ \hline  1 & 1 & 1 & 1 & X & X & X & X \\ \hline \hline \end{tabular}
To find the corresponding digital circuit, we will use the K-Map technique for each of the Excess-3 code bits as output with all of the bits of the BCD number as input.

Corresponding minimized Boolean expressions for Excess-3 code bits –
 w = A+BC+BD\\ x = B^\prime C + B^\prime D +BC^\prime D^\prime\\ y = CD + C^\prime D^\prime \\ z = D^\prime
The corresponding digital circuit-

Converting Excess-3 to BCD(8421) –

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,\:y,\: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.
 \begin{tabular}{||c|c|c|c||c|c|c|c||} \hline  \multicolumn{4}{||c||}{Excess-3} & \multicolumn{4}{|c||}{BCD}\\ \hline  w & x & y & z & A & B & C & D \\ \hline \hline  0 & 0 & 0 & 0 & X & X & X & X \\  \hline  0 & 0 & 0 & 1 & X & X & X & X \\  \hline  0 & 0 & 1 & 0 & X & X & X & X \\  \hline  0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\  \hline \hline  0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\  \hline  0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 \\  \hline  0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 \\  \hline  0 & 1 & 1 & 1 & 0 & 1 & 0 & 0 \\  \hline \hline  1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \\  \hline  1 & 0 & 0 & 1 & 0 & 1 & 1 & 0 \\  \hline  1 & 0 & 1 & 0 & 0 & 1 & 1 & 1 \\  \hline  1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\  \hline \hline  1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \\  \hline  1 & 1 & 0 & 1 & X & X & X & X \\  \hline  1 & 1 & 1 & 0 & X & X & X & X \\ \hline  1 & 1 & 1 & 1 & X & X & X & X \\ \hline \hline \end{tabular}
K-Map for D-

K-Map for C-

K-Map for B-

K-Map for A-

Corresponding minimized boolean expressions for Excess-3 code bits –
 A = wx+wyz\\ B = x^\prime y^\prime + x^\prime z^\prime +xyz\\ C = y^\primez+ yz^\prime \\ D = z^\prime
The corresponding digital circuit –
Here E_3,\:E_2,\:E_1,\:and\:E_0 correspond to w,\:x,\:y,\:and\:z and B_3,\:B_2,\:B_1,\:and\:B_0 correspond to A,\:B,\:C,\:and\:D.

Excess-3 to BCD
Image Reference- sanfoundry.com

References-

Digital Design, 5th edition by Morris Mano and Michael Ciletti
Excess-3 – Wikipedia

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