Chapter 6
Page 120
The circuit for the four status flags is
Page 123
Instead of selecting one output line like the decoder in Figure 6-8, a BCD to seven-segment decoder selects several of the seven output lines for each 4-bit BCD input.
We start with a truth table that shows which segments are activated to display the numeral, 0 – 9, corresponding to the four xi BCD bits.
| Digit | x3 | x2 | x1 | x0 | a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
| 1 | 0 |
0 |
0 |
1 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
| 2 | 0 |
0 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
0 |
1 |
| 3 | 0 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
0 |
1 |
| 4 | 0 |
1 |
0 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
1 |
| 5 | 0 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
| 6 | 0 |
1 |
1 |
0 |
1 |
0 |
1 |
1 |
1 |
1 |
1 |
| 7 | 0 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
0 |
0 |
0 |
| 8 | 1 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
| 9 | 1 |
0 |
0 |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
1 |
The functional relationship between the four inputs and each of the seven outputs can be expressed as a sum of minterms.
a(x) = ∑(0,2,3,5,6,7,8,9)
b(x) = ∑(0,1,2,3,4,7,8,9)
c(x) = ∑(0,1,3,4,5,6,7,8,9)
d(x) = ∑(0,2,3,5,6,8)
e(x) = ∑(0,2,6,8)
f(x) = ∑(0,4,5,6,8,9)
g(x) = ∑(2,3,4,5,6,8,9)
We’ll use Karnaugh maps to simplify the functions for each of the seven segments.
a(x) = x1 ∨ x3 ∨ (¬x0 ∧ ¬x2) ∨ (x0 ∧ x2)
b(x) = ¬x2 ∨ (¬x0 ∧ ¬x1) ∨ (x0 ∧ x1)
c(x) = x0 ∨ ¬x1 ∨ x2
d(x) = x3 ∨ (¬x0 ∧ ¬x2) ∨ (x1 ∧ ¬x2) ∨ (¬x0 ∧ x1) ∨ (x0 ∧ ¬x1 ∧ x2)
e(x) = (¬x0 ∧ ¬x2) ∨ (¬x0 ∧ x1)
f(x) = x3 ∨ (¬x0 ∧ ¬x1) ∨ (¬x0 ∧ x2) ∨ (¬x1 ∧ x2)
g(x) = x3 ∨ (¬x0 ∧ x1) ∨ (x1 ∧ ¬x2) ∨ (¬x1 ∧ x2)
This leads to a circuit for our decoder.
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We start with a truth table that shows the relationship of x compared to y. (Remember that GT and LT refer to signed values.)
| x1 | x0 | y1 | y0 | EQ | GT | LT |
|---|---|---|---|---|---|---|
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
1 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
0 |
1 |
1 |
0 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
0 |
0 |
1 |
1 |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
0 |
0 |
1 |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
1 |
0 |
1 |
1 |
0 |
0 |
1 |
1 |
1 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
0 |
1 |
0 |
0 |
1 |
1 |
1 |
1 |
0 |
0 |
1 |
0 |
1 |
1 |
1 |
1 |
1 |
0 |
0 |
We can specify the connections in the PLD for our comparator directly from the truth table.
