Homework 6 3-to-8 binary decode

Homework 6
Max marks: 50 marks

Problem 1 Show how the function
f(w1, w2, w3) = Pm(0, 2, 3, 4, 5, 7) can be
implemented using a 3-to-8 binary decoder and
an OR gate. [1, Prob 4.1] (10 marks)
Problem 2 Consider the function f = ¯w2w¯3 +
w1w2. Derive a circuit for f that uses only one
2-to-1 multiplexer and no other gates (assuming
inputs are available in both uncomplemented and
complemented form). [1, Prob 4.4](10 marks)
Problem 3 Figure 1 shows the notation for a
BCD to 7-segment display and Table 1 shows
the corresponding truth table. The inputs corresponding to the missing rows in the truth table
should be considered as don’t care.
1. implement segment “a” using an 8:1 mux
and no other logic gate, (10 marks)
2. implement segment “a” using a 4:1 mux and
one other gate, (10 marks)
3. implement segment “f ” with 4:1 mux and
no other logic gate. Assume inputs
are available in both uncomplemented and
complemented form. (Hint: There are
(
4C2 = 6) possible pairs of control inputs: (w3, w2), (w2, w1), (w1, w0), (w0, w3),
(w0, w2), (w1, w3). There are 6 don’t care
conditions. With two control inputs of the
multiplexer and one input, you can represent an expression with up to 4-SOP-terms
of size three-literals or less. You might the
arrive at the answer sooner, if you try to
write the minimal SOP expression first and
find the two inputs that occur most often in
all the terms. Those two inputs are most
likely to be the chosen pair of control inputs.) (10 marks)
Figure 1: Seven segment display and BCD-to7-segment display converter. When a = 1 the
corresponding segment of the display lights up.
To display the number 8, you will turn on all the
seven segments, while to display 1, you will turn
on b = 1, c = 1 and turn off = 0 the rest. The
full truth-table for the seven-segment display is
shown in Table 1.
Row w3 w2 w1 w0 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
Table 1: Truth table for BCD to seven-segment
display as shown in Figure 1. The missing combinations of inputs should be considered as dont
care.
References
[1] S. Brown and Z. Vranesic. Fundamentals of
Digital Logic with Verilog Design: Third Edition. McGraw-Hill Higher Education, 2013.
1