Department of Electrical and Computer Engineering University of Wisconsin - Madison. ECE/CS 352 Digital System Fundamentals.

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1 Last (family) name: First (given) name: Student I.D. #: Circle section: Lipasti Kim Department of Electrical and Computer Engineering University of Wisconsin - Madison ECE/CS 352 Digital System Fundamentals Quiz #2 Solution Thursday, March 6, 2003, 7:5 8:30 PM Instructions:. Closed book examination. 2. No calculator, hand-held computer or portable computer allowed. 3. Five points penalty if fail to enter name, ID#, or instructor selection. 4. No one shall leave room during last 5 minutes of the examination. 5. Upon announcement of the end of the exam, stop writing on the exam paper immediately. Pass the exam to aisles to be picked up by a T. The instructor will announce when to leave the room. 6. Failure to follow instructions may result in forfeiture of your exam and will be handled according to UWS 4 cademic misconduct procedures. ECE/CS 352 Spring 2003 Quiz #2 March 6, 2003

2 Problem Points Score Total 00. (20 points) oolean Circuit nalysis and Design (a) (0 points) Find the Product of Maxterms (POM) representation for the following logic circuit. F(,,C,D) =? M( 0,,2,3,,2,3,4,5) F(,,C,D) C D F = ( + )( + )( + (CD) ) = ( + )( + )( + C + D ) Expanding maxterms gives 00xx, xx, x0 => 0,,2,3,,2,3,4,5 ECE/CS 352 Spring 2003 Quiz # 2 March 6,

3 (b) (0 points) Convert the following logic schematic diagram into NND-only realization. You may use two -input NND gates and inverters. ssume the complements of all input oolean variables are NOT available. a b c d b F(a,b,c,d,e) e b a F(a,b,c,d,e) c e d b ECE/CS 352 Spring 2003 Quiz # 2 March 6,

4 2. (5 points) Minimized 2-level implementation Implement the following function as a minimized 2-level NOR/OR circuit. Show your work below. ssume both true and complemented inputs are available. F(a,b,c,d) =? M (,2,6,7,8,9,2,3,4,5) NOR/OR means circle s in K-map, take complement at end. F = a c d + a b c + b cd + ab c Taking complement: F = ((a+c+d)(a+b +c)(b+c +d )(a +b+c )) ECE/CS 352 Spring 2003 Quiz # 2 March 6,

5 3. (20 points) Complement representation and arithmetic (a) (0 points) Find the radix complement and diminished radix complement of the following octal number (assume radix r=8, number of digits n = 4): (332) 8 Radix complement: (4457) 8 Diminished radix complement: (4456) 8 (b) (0 points) dd the following 6-digit signed binary numbers, using s complement representation on the left, then using two s complement on the right. You must show all your work, including all arithmetic operations and carries using the space provided. Indicate whether overflow occurred, and how you determined this condition. (5 points) s Complement (5 points) 2 s Complement Overflow: yes Overflow: no ECE/CS 352 Spring 2003 Quiz # 2 March 6,

6 4. (20 points) Multiplexer-based design Implement the following 4-input oolean function using a single 8: mux. Find the solution with the fewest complemented inputs to the mux. Show your work below, and label the inputs to the mux below with your solution. F(,, C, D) =? M(, 3, 3, 4, 5). 0 D7 C D6 D5 D4 D3 F D2 0 D D0 S2 S S0 F D0 0 0 D 0 0 D2 0 D3 0 0 D4 0 D5 0 D6 D7 D S0 S S ECE/CS 352 Spring 2003 Quiz # 2 March 6,

7 5. (25 points) Design of an optimized self-adder. (a) (20 points) You are to optimize an existing 4-bit carry-lookahead adder for the special case of self-addition. In other words, as shown below, the adder only accepts a single input 0-3 and adds it to itself, producing a sum S 0-3 and a carryout c out. C out 4-bit Carry Lookahead C 4 g 3 p 3 C 3 g 2 p 2 C 2 g p C g 0 p 0 S g p g p g p g p F C in S F C in S F C in H S S Fill in the unoptimized (bit position zero is done for you) and optimized columns below. Show your work below and on the next page. it Term Unoptimized Optimized 0 g p S S 2 S S C g 0 + p g p 0 S C 0 C 2 g + p C g p S C 2 C 3 g 2 + p 2 C 2 2 g p S C 3 2 C 4 g 3 + p 3 C 3 3 ECE/CS 352 Spring 2003 Quiz # 2 March 6,

8 (b) (5 points) Draw your optimized self-adder implementation below using ND, OR, XOR gates and inverters only. C out 0 S 3 3 S 2 2 S S 0 0 ECE/CS 352 Spring 2003 Quiz # 2 March 6,

9 asic Identities of oolean lgebra. X + 0 = X 2. X = X 3. X + = 4. X 0 = 0 5. X + X = X 6. X X = X 7. X + X = 8. X X = 0 9. X =X 0. X + Y = Y + X. X Y = Y X Commutative Law 2. X + (Y + Z) = (X + Y) + Z 3. X (YZ) = (XY) Z ssociative Law 4. X(Y + Z) = XY + XZ 5. X + YZ = (X + Y)(X + Z) Distributive Law 6. X + Y = X Y 7. X Y = X + Y DeMorgan s Law Useful oolean Identities 8. X + XY = X 9. X + XY = X + Y 20. XY + XY = X 2. XY + XZ + YZ = XY + XZ 22. ( X + Y)( X + Y) = X Y + X Y 23. ( X + Y)( X + Y) = X Exclusive-Or Identities 24. X 0 = X 25. X = X 26. X X = X X = 28. X Y = X Y 29. X Y = X Y 30. X Y = Y X 3. (X Y ) Y = X (Y Z) = X Y ZX ECE/CS 352 Spring 2003 Quiz # 2 March 6,