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ECE 462 Logic Design
Homework 2
This homework (Questions 1-5) have an assignment of 60 points. If you do the extra credit question at the end, you will get an equivalent of 15 additional homework points towards your total. Partially correct answers will also be graded. You are highly encouraged to attempt Question 6.
**Prime implicants, Consensus theorem
a)** Is it possible to have a 4-variable function f(w,x,y,z) that has more prime- implicants than non-prime implicants? If yes, justify with an example. If no, prove that this can never happen. [4 points] b) We studied that the consensus theorem states the following. xy + x'z + yz = xy + x'z If xy and x’z are two prime implicants of a given function, where x is a variable and y and z are two product terms, is their consensus yz (if yz ≠ 0) also an implicant? Is it a prime implicant? If yes, show proof. If no, give a counterexample. [5 points]
Quine-McCluskey method
2) A two-level circuit C with 4 inputs and 4 outputs is to be designed to convert a decimal digit from standard BCD (8 4 2 1) code to excess-3 code. The circuit may be composed of all NANDS or all NORs. Use the tabular (Quine- McCluskey) method to obtain the lowest cost two-level logic circuit for C. Give an SOP and POS expression and a logic circuit for C. [8 points]
3) Use the greedy algorithm based on heuristics taught in class for solving the covering problem in the following a b c d e f g h i 3 l A x x x 2 l B x x x x 2 l C x x x 4 l D x x x x x 3 l E x x x x 2 l F x x x x x x
NOTE: 3 l means there are 3 literals
Can you determine how far your solution is from the exact solution? Optimize for minimum cost of literals [8 points]
Technology Mapping
3) In the problem, you will technology map the given circuit, minimizing for area. The library is as follows
Function Cost INV 1 NAND2 2 NAND3 3 NOR2 2 AOI21 3
In both a) and b) parts of this question, illustrate the way you decomposed the library cells into pattern trees. Show the best cost solution trace through the subject tree and show your final cover for the circuit.
a) Use the dynamic programming method we used in class with no modifications. [6 points] b) Modify the basic method by inserting a pair of series inverters into every wire that does not already have an inverter at one end (including root and leaves of the subject tree). You will need to modify your library slightly by
6) EXTRA CREDIT QUESTION (Prime implicants, minterms)
An essential prime implicant is a prime implicant that covers some 1-cell or minterm that is not covered by any other prime implicant of the function. A prime implicant is one that ceases to be an implicant if any one of its literals is removed. An implicant is a minterm, which, when true, implies that the function evaluates to true.
A Boolean function z(x 1 , x 2 , …, xn) is called positive or positive unate if there is some SOP expression for z in which, for every i, 0 ≤ i ≤ n, the positive literal xi appears but its complement xi’ does not. Eg: z(x 1 , x 2 , x 3 , x 4 ) = x 1 x 2 + x 2 x 3 x 4 is positive unate. Similarly z is called negative unate if there is some SOP expression in which, for every i, only the complemented literal xi’ appears. Eg: z(x 1 , x 2 , x 3 ) = x 1 ’x 2 ’^ + x 2 ’x 3 ’^ is negative unate. (Unate forms of a Boolean function are often used for decomposing it.)
Prove that every prime implicant of a positive or negative unate function is essential, which implies that the function has a unique minimal SOP form. Hint: Start with an example and see if the above is true. Then, generalize your result to all such examples.
