hw2soln - ECE 152A Fall 2006 University of California Santa...

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ECE 152A – Fall 2006 10/17/2006 Homework #2 – Solution Page 1 of 62 University of California, Santa Barbara Department of Electrical and Computer Engineering ECE 152A – Digital Design Principles Homework #2 – Solution For problems 1 through 5, include (1) a truth table (for each output), (2) a minimized Boolean expression in either (or both) SOP or POS form and (3) either Karnaugh maps or Boolean algebra demonstrating how the function was minimized. From Previous Homework: Problem #1: Design a combinational circuit with three inputs and one output. The output is equal to logic 1 when the binary value of the input is less than 3. The output is logic 0 otherwise.
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ECE 152A – Fall 2006 10/17/2006 Homework #2 – Solution Page 2 of 62 Problem #2: A majority function is generated in a combinational circuit when the output is equal to 1 if the input variables have more 1’s than 0’s. The output is 0 otherwise. Design a 3-input majority function.
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ECE 152A – Fall 2006 10/17/2006 Homework #2 – Solution Page 3 of 62 Problem #3: Design a combinational circuit that detects an error in the representation of a decimal digit in BCD. The output of the circuit must be equal to logic 1 when the inputs contain any one of the six unused bit combinations in the BCD code.
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ECE 152A – Fall 2006 10/17/2006 Homework #2 – Solution Page 4 of 62 Problem #4: Design a combinational circuit that produces the binary sum of two, 2-bit numbers (A1 A0 and B1 B0). The three outputs of this circuit are Carry out (C0), Sum 1 and Sum 0 (S1 S0).
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ECE 152A – Fall 2006 10/17/2006 Homework #2 – Solution Page 5 of 62 Problem #4 (cont)
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ECE 152A – Fall 2006 10/17/2006 Homework #2 – Solution Page 6 of 62 Problem #5: Design a combinational circuit that multiplies 2, 2-bit numbers (A1 A0 and B1 B0). The four outputs are the four bits of the resulting product (P3 P2 P1 P0).
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ECE 152A – Fall 2006 10/17/2006 Homework #2 – Solution Page 7 of 62 Problem #5 (cont):
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ECE 152A – Fall 2006 10/17/2006 Homework #2 – Solution Page 8 of 62 From Midterm, Summer 2003 Problem #6. For the Boolean function given below: F(A,B,C,D) = A’B’C’ + A’BC’D’ + A’C’D + ACD (a) Simplify the function (in sum of products representation) using only Boolean algebra. Show each step in the simplification.
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ECE 152A – Fall 2006 10/17/2006 Homework #2 – Solution Page 9 of 62 (b) Determine the minimized product of sums representation for the function using only Boolean algebra. You may begin with either the original function or the simplified function derived in part (a). As above, show each step in the simplification.
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ECE 152A – Fall 2006 10/17/2006 Homework #2 – Solution Page 10 of 62 Problem #7. For the function below: F (A,B,C) = AB’ + B’C + A’BC’ (a) Express the function in sum of minterms (i.e., Σ ( minterm list in decimal form )) and product of Maxterms (i.e., Π ( Maxterm list in decimal form )) form.
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ECE 152A – Fall 2006 10/17/2006 Homework #2 – Solution Page 11 of 62 (b) Express the complement of the function (F’) in sum of minterms and product of Maxterms form. Again, the decimal representations of the minterm and Maxterm lists should be used.
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