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

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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




   

 



Last Updated : 05 Jan, 2022
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