Physis/Digital Circuit design

(See attached file for full problem description)

PHS 205 Digital Circuit Design
Homework Assignment #2
Due: June 3rd
Note:
• Please staple them if you turn in multiple pages
• Show your work in details
• Draw your design (truth table, circuit diagram) neatly

PART I. Truth Table; Boolean Algebra and Manipulation; Combinational Circuit Analysis

1. For the following circuits,
1) find an algebraic expression (Do not simplify the expression)
2) put it in SOP form.

(1)

(2)

2. Convert the given expression into a NAND-NAND expression. Assume all inputs are available both uncomplemented and complemented. DO NOT manipulate the functions to simplify the expression.
h(w,x,y,z) = z(x'y + w'x') + w(y'+ xz')

3. Reduce the following expression to a minimum SOP form. Show each step and the property used (4 terms and 9 literals in minimum).

G(A,B,C,D)= A'B'C'+ AB'D + BCD' + A'BD + CD + A'D

4. Express the following function to canonical (standard) SOP form.

h(a,b,c,d)= ab'c + bd + a'd'

5. Convert the following expression to SOP form.

(a'+ b + c')(b + c'+ d)(b'+ d')

6. Convert the following expression to POS form.

w'xy'+ wxy + xz

7. Prove algebraically that the following Boolean expression is true.

A' + AB + AC' + AB'C' = A' + B + C'

8. Find the complement of the following expression. Only single variables may be complemented in the answer.

h = (a + b)(b'+ c) + d'(a'b + c)

PART II. Karnaugh Maps; K-Map Manipulations

9. For each of the following, find all minimum SOP expressions. (If there is more than one solution, the number of solutions is given in parentheses)
1) h(a,b,c) = m(0,1,2,5,6,7) [2 Solutions]
2) f(a,b,c,d) = m(1,3,4,5,6,11,12,13,14,15) [2 Solutions]
3) F(W,X,Y,Z) = m(0,2,3,4,5,8,10,11,12,13,14,15)[4 Solutions]

10. For the given function, (i) list all prime implicants, indicating which are essential. (ii) Show the minimum SOP expression.
f(a,b,c,d) = m(0,3,4,5,8,11,12,13,14,15)

11. Map the given function and find the minimum SOP expression.
F(A,B,C,D) = AD + AB + A'CD'+ B'CD + A'BC'D'

12. Find all minimum SOP expressions.
1) f(w,x,y,z) = m(0,2,4,5,10,12,15)+d(8,14) [2 Solutions]
2) f(a,b,c,d) = m(5,7,9,11,13,14)+d(2,6,10,12,15) [4 Solutions]

13. For the following functions (a) list all essential prime implicants, (b) find all minimum SOP expressions, (c) find all minimum POS expressions. The number of solutions is given in parentheses.
f(a,b,c,d) = m(0,1,2,5,7,9) + d(6,8,11,13,14,15) [4 SOP and 2 POS solutions]