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# BCD to Excess-3 conversion

To understand the process of converting BCD to Excess-3, it is required to have knowledge of Number System and Number Base Conversion.

The Excess-3 binary code is an example of a self-complementary BCD code. A self-complementary binary code is a code which is always complimented in itself. By replacing the bit 0 to 1 and 1 to 0 of a number, we find the 1â€™s complement of the number. The sum of the 1â€™st complement and the binary number of a decimal is equal to the binary number of decimal 9.

The process of converting BCD to Excess-3 is quite simple from other conversions. The Excess-3 code can be calculated by adding 3, i.e., 0011 to each four-digit BCD code. Below is the truth table for the conversion of BCD to Excess-3 code. In the below table, the variables A, B, C, and D represent the bits of the binary numbers. The variable â€˜Dâ€™ represents the LSB, and the variable â€˜Aâ€™ represents the MSB. In the same way, the variables w, x, y, and z represent the bits of the Excess-3 code. The variable â€˜zâ€™ represents the LSB, and the variable â€˜wâ€™ represents the MSB. The â€˜donâ€™t care conditionsâ€™ is expressed by the variable â€˜Xâ€™.

Decimal NumberBCD CodeExcess-3 Code
ABCDWxyz
000000011
100010100
200100101
300110110
401000111
501011000
601101001
701111010
810001011
910011100
101010XXXX
111011XXXX
121100XXXX
131101XXXX
141110XXXX
151111XXXX

Now, we will use the K-map method to design the logical circuit for the conversion of BCD to Excess-3 code as:

So,

w=A+BC+BD
x=Bâ€™ C+Bâ€™ D+BCâ€™ Dâ€™
y=CD+Câ€™Dâ€™
z=Dâ€™

Example: (100001011001)BCD

To find the Excess-3 code of the given Excess-3 code, first, we will make the group of 4 bits from right to left. Then, we will add 0011 in each group of 4 bits in order to get the excess-3 code.

## Excess-3 to BCD conversion

The process of converting Excess-3 to BCD is opposite to the process of converting BCD to Excess-3. The BCD code can be calculated by subtracting 3, i.e., 0011 from each four-digit Excess-3 code. Below is the truth table for the conversion of Excess-3 code to BCD. In the below table, the variables w, x, y, and z represent the bits of the Excess-3 code. The variable â€˜zâ€™ represents the LSB, and the variable â€˜wâ€™ represents the MSB. In the same way, the variables A, B, C, and D represent the bits of the binary numbers. The variable â€˜Dâ€™ represents the LSB, and the variable â€˜Aâ€™ represents the MSB. The â€˜donâ€™t care conditionsâ€™ is defined by the variable â€˜Xâ€™.

Decimal NumberExcess-3 CodeBCD Code
ABCDWxyz
00000XXXX
10001XXXX
20010XXXX
300110000
401000001
501010010
601100011
701110100
810000101
910010110
1010100111
1110111000
1211001001
131101XXXX
141110XXXX
151111XXXX

Now, we will use the K-map method to design the logical circuit for the conversion of Excess-3 code to BCD as:

So,

w=AB+ACD
B=xâ€™ yâ€™+xâ€™ zâ€™+xyz
C=yâ€™ z+yzâ€™
D=zâ€™

Example: (101110001100)Excess-3

To find the BCD code of the given BCD number, first, we make the group of 4 bits from right to left. Then, we subtract 0011 in each group of 4 bits in order to get the BCD code.