University of California, San Diego

M.S. Exam: Logic Design (CSE140)

Spring 2015

NAME : ____________________________________________

 

 

Question 1. [20 pts]

(a) [10 pts] Using Boolean algebra theorems, simplify 𝐹 to a minimum sum-of-products representation. Write all steps to your answer. You do not need to write the names of the Boolean algebra theorems used.

𝐹  (𝐴,𝐵,𝐶,𝐷) = 𝐴 + 𝐵! + 𝐶! ⋅ 𝐴 + 𝐵! + 𝐷! ⋅ (𝐵! + 𝐶! + 𝐷!)

(b) [10 pts] Find the minterm expansion of 𝑓.

𝑓 𝑎, 𝑏, 𝑐,𝑑, 𝑒 = 𝑎!𝑏!𝑒 + 𝑎𝑑 + 𝑏𝑐𝑑′

𝑓 = 𝑚                                                                                                                                                                                                      

 

 

Question 2. [30 pts]

(a) [10 pts] Fill in the Karnaugh map to represent the following function F. (d = don’t-care)

𝐹 𝐴,𝐵,𝐶,𝐷 =   𝑚 0, 2, 5, 8, 10, 12, 13, 14, 15 +   𝑑 4    

(b) [5 pts] List all of the essential prime implicants of F (there are three essential prime implicants).

(c) [5 pts] List all of the prime implicants of F (there are five prime implicants).

(d) [5 pts] List all possible minimal sum of product representations for F.

AB CD 00 01 11 10

00

01

11

10

 

 

(e) [5 pts] Implement F (A, B, C, D) using exactly one 4:1 MUX and a minimum number of 2:1 MUXes.

 

 

Question 3. [20 pts] (a) [10 pts] Fill in the following truth table for the J-K flip-flop. Here, Q+ is synonymous with Q(t+1).

J K Q Q+

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1 (b) [15 pts] Complete the following timing diagram for the J-K flip-flop (Initial state of Q = 0).

Clock

J

K

Q

 

 

Question 4. [30 pts] The sequential circuit shown below has input (X), output (Z) and three-bit state (Q1, Q2, Q3).

(a) [10 pts] Write down the state equations.

Q1(t+1) =

Q2(t+1) =

Q3(t+1) =

Z =

(b) [10 pts] Fill in the state transition table.

Q1 Q2 Q3 X = 0 X = 1

Q1+ Q2+ Q3+ Z Q1+ Q2+ Q3+ Z S0 0 0 0 S1 0 0 1 S2 0 1 0 S3 0 1 1 S4 1 0 0 S5 1 0 1 S6 1 1 0 S7 1 1 1

 

 

(c) [10 pts] Draw the state machine diagram below. Is this a Moore or a Mealy machine?