ESC201 : Introduction to

Electronics

HW5: Solution

Solution Credit: Prof. B. Mazhari

Q.1 Simplify the following 4-variable functions into sum-of-products form using K-map.

a.  (1,5, 6, 7,14) b.  (0, 4, 6,8)

c.  (0,1, 4, 6,8,9,14) d.  (1, 4, 7,11,13,14)
b)
a) x3x4
x3x4 x1x2 00 01 11 10
x1x2 00 01 11 10
00 1 0 0 0
00 0 1 0 0

01 1 0 0 1
01 0 1 1 1
11 0 0 0 0
11 0 0 0 1

10 0 0 0 0 10 1 0 0 0

F = x1 x3 x4 + x1 x2 x3 + x2 x3 x4 F = x2 x3 x4 + x1 x2 x4
c) d)
x3x4 x3x4
x1x2 00 01 11 10 x1x2 00 01 11 10

00 1 1 0 0 00 0 1 0 0

01 1 0 0 1 F = x2 x3 + x1 x3 x4 + x2 x3 x4 01 1 0 1 0

11 0 0 0 1 11 0 1 0 1

10 1 1 0 0 10 0 0 1 0

Cannot be minimized
Q.3 Simplify the following expressions into sum-of-products form using the don’t care
conditions (d) into account.

F ( A, B, C , D) =  (4,5, 7,12,13,14) F ( A, B, C , D) =  (1, 2,12,13,14)

a. b.
d ( A, B, C , D) =  (1,9,11,15) d ( A, B, C , D) =  (8,9,10,11)

a)
CD b)
CD
AB 00 01 11 10
AB 00 01 11 10
00 0 x 0 0
1 1
00
01 1 1 1 0
01
11 1 1 x 1
11 1 1 1
10 0 x x 0
10 x x x x

F = AB + BC + BD
F = AC + AD + BCD + BC D
Q.4 Construct a 16 x 1 multiplexer with two 8 to 1 and one 2 to 1 multiplexers. Use
block diagrams
I0 0
I1 1
I2 2
I3 3
I4 4
I5 5
I6 6
I7 7
0 y
1
S0 S1 S2

S3
I8 0
I9 1
I10 2
I11 3
I12 4
I13 5
I14 6
I15 7

S0 S1 S2
Q.5 Implement the following Boolean function using one 4 to 1 multiplexer and
external gates. (Hint: Connect inputs A and B to the control or selection lines of the
mux and then use basic gates to apply appropriate combinations of C & D to the input
lines of the Mux.) .
F ( A, B, C , D) =  (1,3,4,11,12,13,14,15)

AB CD F
00 00 0
00 01 1 F=D
00 10 0
00 11 1
01 00 1
00
D
01 01 0 F=C D 01
F
01 10 0 10
C
01 11 0 11
1
10 00 0
10 01 0 F=C D
A B
10 10 0
10 11 1
11 00 1
11 01 1 F=1
11 10 1
11 11 1
Q.6 Show how one can multiply a 4-bit number A with a 3-bit number B using basic gates
and 4-bit adder.
multiplicand
B3 B2 B1 B0

A2 A1 A0 multiplier

A0B3 A0B2 A0B1 A0B0

A1B3 A1B2 A1B1 A1B0 Partial products

A2B3 A2B2 A2B1 A2B0

C6 C5 C4 C3 C2 C1 C0

A0 B0
B3 B2 B1

A1 B0
B3 B2 B1

4-bit Adder
Cout S3 S2 S1 S0
A2 B0
B3 B2 B1

4-bit Adder
Cout S3 S2 S1 S0 C1 C0
C6 C5 C4 C3 C2
Q.7 Design a circuit that would implement the following function:
IF (A = B) then Y = C + D ELSE Y = C-D. A ,B,C, D are four bit numbers

FA FA FA FA

C3 C2 C1 C0

A=B
4-BIT COMPARATOR
D3 D2 D1 D0

Adder/Subtractor

A0..A3 B0..B3
Q.8 A PN flip-flop has four operations, reset to 0, hold, complement and set to 1 when
inputs PN are 00,01,10,11 respectively. Tabulate the characteristic table, excitation table
and show how the PN FF can be converted to a D FF.

P N Q(t+1) State Q(t) Q(t+1) P N

0 0 0 Reset 0 0 0 X
0 1 Q(t) Hold 0 1 1 X

1 0 Q(t) Toggle 1 0 X 0

1 1 1 Set 1 1 X 1

characteristic table excitation table

General circuit for converting a FF with inputs X1, X2 into a different FF with inputs
Y1, Y2.
Y1 x1 Q
Y2
CC
Q
x2

D Q Q(t+1) P N
P Q
D 0 0 0 0 X
?
Q 0 1 0 X 0
N
1 0 1 1 X
1 1 1 X 1

→ P=N=D
D P Q

N
Q.10 Design a sequential circuit with two D flip-flops A and B and one input x such that
when x = 0, the state of the circuit remains the same. When x = 1, the circuit goes
through the state transitions from 00 to 01, to 11, to 10 and back to 00, and repeats.

Present State Input Next State

A B x A B DA DB Bx Bx
A 00 01 11 10 A 00 01 11 10
0 0 0 0 0 0 0
0 0 1 0 1 0 1 0 1 0 0 1 1 1
0 1 0 0 1 0 1
0 1 1 1 1 1 1
1 1 1 1 1 1

1 0 0 1 0 1 0
1 0 1 0 0 0 0 DA = A.x + B.x DB = A.x + B.x
1 1 0 1 1 1 1
1 1 1 1 0 1 0

x D Q A
A
A
D Q B