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Quine-McCluskey Procedure and Minimal Sum-of-Product Solutions for Logic Functions - Prof., Assignments of Electrical and Electronics Engineering

University of Idaho (U of I)Electrical and Electronics Engineering
Prof. James F. Frenzel

Solutions to homework assignment #4 for ece 349 students, including the use of the quine-mccluskey procedure to find prime implicants and minimal sum-of-products solutions for given logic functions.

Typology: Assignments

Pre 2010

Uploaded on 08/19/2009

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koofers-user-d89 🇺🇸

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ECE 349 Homework Assignment #4 Solutions
Show your work! You will not receive full credit for the answer alone.
1. Use the Quine-McCluskey procedure to find all of the prime implicants for f(a, b, c, d) =
Pm(0,3,4,5,7,9,11,13).
The prime implicants are:
(0,4)a0c0d0,(4,5)a0bc0,(3,7)a0cd, (3,11)b0cd, (5,7)a0bd, (5,13)bc0d, (9,11)ab0d, and(9,13)ac0d.
2. Use the Quine-McCluskey procedure to find all of the prime implicants for f(a, b, c, d) =
Pm(0,1,3,5,6,7,8,10,14,15).
The prime implicants are:
(0,1)a0b0c0,(0,8)b0c0d0,(8,10)ab0d0,(10,14)acd0,(1,3,5,7)bc, and(6,7,14,15)a0d.
3. Using a prime implicant chart, find all minimum sum-of-products solutions for the
function of problem 1.
f=a0c0d0+a0cd +bc0d+ab0dand f=a0c0d0+a0bd +ac0d+b0cd
4. Using a prime implicant chart, find all minimum sum-of-products solutions for the
function of problem 2.
f=a0d+bc+b0c0d0+ab0d0and f=a0d+bc +b0c0d0+acd0and f=a0d+bc +a0b0c0+ab0d0
5. Use the branching method to find all of the minimal sum-of-product expressions for
f(a, b, c, d) = Pm(0,2,6,7,8,9,10,13,15).
f=b0d0+
a0cd0+bcd +ac0d
a0bc+
bcd+ac0d
abd+(ac0d
ab0c0
6. Use the branching method to find all of the minimal sum-of-product expressions for
f(w, x, y, z ) = Pm(7,12,14,15) + Pd(1,3,5,8,10,11,13).
f=
a0d+ab
bd+(ab
ad0
cd+(ab
ad0
7. Repeat problem 5 using Petrick’s method.
b0d0is essential and must be included in the solution. For the other terms, let A=a0cd0,
B=a0bc,C=bcd,D=abd,E=ac0d, and F=ab0c0.
(A+B)(B+C)(C+D)(D+E)(E+F) = (B+AC)(D+C E)(E+F)
= (B+AC)(DE +CE +D F ) = BDE +BCE +BDF +ACE +ACDF
The ACDF term is ignored because it has too many terms.
pf2

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ECE 349 Homework Assignment #4 Solutions

Show your work! You will not receive full credit for the answer alone.

  1. Use the Quine-McCluskey procedure to find all of the prime implicants for f (a, b, c, d) = ∑ m(0, 3 , 4 , 5 , 7 , 9 , 11 , 13). The prime implicants are: (0, 4)a′c′d′, (4, 5)a′bc′, (3, 7)a′cd, (3, 11)b′cd, (5, 7)a′bd, (5, 13)bc′d, (9, 11)ab′d, and(9, 13)ac′d.
  2. Use the Quine-McCluskey procedure to find all of the prime implicants for ∑ f (a, b, c, d) = m(0, 1 , 3 , 5 , 6 , 7 , 8 , 10 , 14 , 15). The prime implicants are: (0, 1)a′b′c′, (0, 8)b′c′d′, (8, 10)ab′d′, (10, 14)acd′, (1, 3 , 5 , 7)bc, and(6, 7 , 14 , 15)a′d.
  3. Using a prime implicant chart, find all minimum sum-of-products solutions for the function of problem 1. f = a′c′d′^ + a′cd + bc′d + ab′d and f = a′c′d′^ + a′bd + ac′d + b′cd
  4. Using a prime implicant chart, find all minimum sum-of-products solutions for the function of problem 2. f = a′d + bc + b′c′d′^ + ab′d′^ and f = a′d + bc + b′c′d′^ + acd′^ and f = a′d + bc + a′b′c′^ + ab′d′
  5. Use the branching method to find all of the minimal sum-of-product expressions for f (a, b, c, d) =

∑ m(0,^2 ,^6 ,^7 ,^8 ,^9 ,^10 ,^13 ,^ 15).

f = b′d′^ +

    

a′cd′+ bcd + ac′d

a′bc+

  

bcd+ ac′d

abd+

{ ac′d ab′c′

  1. Use the branching method to find all of the minimal sum-of-product expressions for f (w, x, y, z) =

∑ m(7,^12 ,^14 ,^ 15) +^

∑ d(1,^3 ,^5 ,^8 ,^10 ,^11 ,^ 13).

f =

      

a′d+ ab

bd+

{ ab ad′

cd+

{ ab ad′

  1. Repeat problem 5 using Petrick’s method. b′d′^ is essential and must be included in the solution. For the other terms, let A = a′cd′, B = a′bc, C = bcd, D = abd, E = ac′d, and F = ab′c′. (A + B)(B + C)(C + D)(D + E)(E + F ) = (B + AC)(D + CE)(E + F ) = (B + AC)(DE + CE + DF ) = BDE + BCE + BDF + ACE + ACDF The ACDF term is ignored because it has too many terms.
  1. Repeat problem 6 using Petrick’s method. There are no essential prime implicants. Let A = a′d, B = bd, C = cd, D = ab, E = ad′, and F = ac. (A + B + C)(B + C + D)(D + E)(D + E + F ) = (A + B + C)(B + C + D)(D + E) = (B + C + AD)(D + E) = BD + CD + AD + BE + CE
  2. Use Map-Entered-Variables to find a minimum sum-of-products for f (a, b, c, d) = a′b′c′d + a′d′^ + abc′^ + ab′d + ab′d′. I mapped c and got f = a′d′^ + ab′^ + ac′^ + b′c′.
  3. Use Map-Entered-Variables to find a minimum sum-of-products for g(a, b, c, d, e, f ) = b′d′e′^ + abd′e′^ + def + b′e′^ + bde′^ + b′c′d′e. I mapped a, c′, and f and got g = b′e′^ + de′^ + ae′^ + df + b′c′d′.
  4. (Optional 2 points) Use Map-Entered-Variables to find a minimum sum-of-products for f = a′c′d + b′cd + a′bcd′^ + a′bcd + ac′d. I mapped b and b′^ and got f = c′d + a′bc + b′d.