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

Prerequisite – Number System and base conversions

Excess-3 binary code is an 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 its corresponding Excess-3 code by simply adding 3 to it. Since we have only 10 digits(0 to 9) in decimal, we don’t care about the rest and marked them with a cross( X ).
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 is don’t care condition.

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.

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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 is don’t care condition.

K-Map for D-

K-Map for C-

K-Map for B-

K-Map for A-
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Corresponding minimized boolean expressions for Excess-3 code bits –
$A = wx+wyz\\ B = x^\prime y^\prime + x^\prime z^\prime +xyz\\ C = y^\prime z+ yz^\prime \\ D = z^\prime$
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

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Last Updated : 05 Jan, 2022