EECE 320 Homework 3, Fall 2016

Prof. M. Mansour American University of Beirut Department of Electrical and Computer Engineering EECE 320 – Digital Systems Design Homework 3 Problem 1 [10 pts]: A 4-bit “comparator” circuit receives two 4-bit unsigned numbers P=P3P2P1P0 and Q=Q3Q2Q1Q0. Design a minimal sum-of-products circuit that produces a 1 output if and only if P < Q. Modify your design to work with 2’s complete numbers. Problem 2 [10 pts]: Prove whether or not the following expression is a minimal sum. Do it without using K-maps. F = S’.T.U.V.W + S’.T.U’W.Y + S’.T.V.W.X’.Y Problem 3 [14 pts]: Consider the following functions F1 = A’B’C’D’ + A’C’D + BC’D’ + ABD + AB’C’ and F2 = !W,X,Y,Z(4,5,9,13,15) + d(0,1,7,11,12). Determine the following for each of F1 and F2: (a) Canonical SOP, (b) Canonical POS, (c) Complete SOP, (d) Complete POS, (e) Minimal SOP, (f) Minimal POS (g) Essential prime-implicants. Problem 4 [20 pts]: Find the minimal SOP and minimal POS for each of the following 5-variable functions: (a) F1 = !V,W,X,Y,Z(5,7,13,15,16,20,25,27,29,31), (b) F2 = !V,W,X,Y,Z(0,7,8,9,12,13,15,16,22,23,30,31), (c) F3 = "V,W,X,Y,Z(4,5,10,12,13,16,17,21,25,26,27,29), (d) F4 = !V,W,X,Y,Z(4,6,7,9,11,12,13,14,15,20,22,25,27,28,30) + d(1,5,29,31). Problem 5 [10 pts]: Find the minimal-output sum-of-products expressions for F = !X,Y,Z(0,1,2), G = !X,Y,Z(1,4,6), and H = !X,Y,Z(0,1,2,4,6). Draw the corresponding minimal logic circuit. Problem 6 [20 pts]: For each of the following logic expressions, find all of the static hazards in the corresponding two- level AND-OR and OR-AND circuit, and then design a hazard-free circuit that realizes the same logic function: (a) F1 = WY + W’Z’ +XY’Z, (b) F2 = W’X’ + Y’Z + W’XYZ + WXYZ’, (c) F3 = (W+Y’+Z’)(W’+X’+Z’)(X’+Y+Z), and (d) F4 = (W+Y+Z’)(W+X’+Y+Z)(X’+Y’)(X+Z). Problem 7 [20 pts]: Find the minimal SOP for the following 6-variable function: F = !U,V,W,X,Y,Z(2,4,5,6,12,13,14,15,16,17,18,19,20,21,28,29,30,31,34,38,50,51,60,61,62,63) Problem 8 [10 pts]: Design a 4-bit adder logic circuit. First design a logic circuit that adds two bits A0 and B0 and a carry in Cin0, and generates a sum bit S0 and an output carry bit Cout0. Write the minimized Boolean equations of this block. Then use this block to design a circuit that adds A3A2A1A0 and B3B2B1B0. Show all your connections. Problem 9 [10 pts]: Repeat the previous problem, but now for a circuit that subtracts two numbers A3A2A1A0 and B3B2B1B0. Hint: Use the relationship between addition and subtraction in 2’s complement. Problem 10 [10 pts]: Design a circuit that increments a 4-bit number A3A2A1A0 by 1. Write logic expressions for all outputs, and simplify them as much as you can. Start by designing a 1-bit incrementor circuit using its truth table.