K-map ( karnaugh map ) 8,4,-2,-1 to binary code conversion

Question

I'm taking computer science courses and some digital design knowledge is required, so I'm taking digital design 101.

Image above is representing the conversion process of 8,4,-2,-1 to binary using K-map (Karnaugh map).

I have no idea why 0001, 0011, 0010, 1100, 1101, 1110 are marked as 'X'.

For 0001, 0011, 0010, they could be expressed as 8,4,-2,-1 as 0111, 0110, 0101. And for 1100, 1101, 1110, 1110 can still be expressed as 1100 in 8,4,-2,-1 form as 1100. rests cannot be expressed in 8,4,-2,-1 since 1100 is the biggest amount of number in 8,4,-2,-1 binary form (I think).

Is there something I'm missing?

I understand the excess-3 to binary code conversion provided from my textbook example ( m10-m15 are marked as 'X' since excess-3 were used to express only 0-9.)

Try http://cs.stackexchange.com. This site is for actual practical programming, not theory. — Marc B, Jul 04 '16 at 20:53
@Minjae Checking with Wikipedia: the 9 is the largest value of BCD, that is why there is no coding for 10, 11 and 12. The solution corresponds with that definition. — Kit Ostrihon, Jul 10 '16 at 09:23

score 4 · Accepted Answer · answered Jul 10 '16 at 10:58

According to the definition of BCD, 1 decimal digit (NOT one number) is represented by 4 bits.

The 4 given inputs can therefore represent only values from interval from 0 to 9.

The corresponding and complete truth-table looks like this:

 decimal | 8  4 -2 -1 |  decimal  || BCD
 /index  | A  B  C  D |   result  || W X Y Z
----------------------------------||---------
     0   | 0  0  0  0 |     0     || 0 0 0 0 ~ 0
     1   | 0  0  0  1 |    -1     || X X X X
     2   | 0  0  1  0 |    -2     || X X X X
     3   | 0  0  1  1 |  -2-1=-3  || X X X X
     4   | 0  1  0  0 |     4     || 0 1 0 0 ~ 4
     5   | 0  1  0  1 |   4-1=3   || 0 0 1 1 ~ 3
     6   | 0  1  1  0 |   4-2=2   || 0 0 1 0 ~ 2
     7   | 0  1  1  1 |  4-2-1=1  || 0 0 0 1 ~ 1
     8   | 1  0  0  0 |     8     || 1 0 0 0 ~ 8
     9   | 1  0  0  1 |   8-1=7   || 0 1 1 1 ~ 7
    10   | 1  0  1  0 |   8-2=6   || 0 1 1 0 ~ 6
    11   | 1  0  1  1 |  8-2-1=5  || 0 1 0 1 ~ 5
    12   | 1  1  0  0 |   8+4=12  || X X X X
    13   | 1  1  0  1 | 8+4-1=11  || X X X X
    14   | 1  1  1  0 | 8+4-2=10  || X X X X
    15   | 1  1  1  1 | 8+4-2-1=9 || 1 0 0 1 ~ 9

The K-maps then match the truth-table by its indexes:

Using the K-maps, it can be indeed simplified to these boolean expressions:

W = A·B + A·¬C·¬D
X = ¬B·C + ¬B·D + B·¬C·¬D
Y = ¬C·D + C·¬D
Z = D

K-map ( karnaugh map ) 8,4,-2,-1 to binary code conversion

1 Answers1