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**Chapter 3 Combinational Logic Design**

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**Combinational Circuits**

(logic gates) n binary inputs m binary outputs n input bits 2n possible binary input combinations For each possible input combination, there is one possible output value truth table Boolean functions (with n input variables) Examples: adders, subtractors, comparators, decoders, encoders, and multiplexers.

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**Possible Design Steps Find out the number of inputs and outputs**

Derive the truth table that defines the required relationship between inputs and outputs Obtain a simplified Boolean function for each output Draw the logic diagram Verify the correctness of the design

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**Example: Design Process**

BCD-to-2421 Converter Verbal specification: Given a BCD digit (i.e. {0, 1, …, 9}), the circuit computes 2421 code equivalent of the decimal number Step 1: how many inputs and how many outputs? four inputs and four outputs Step 2: Obtain the truth table 0000 0000 1001 1111 etc.

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BCD-to-2421 Converter Truth Table Inputs Outputs A B C D x y z t 1

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BCD-to-2421 Converter Step 3: Obtain simplified Boolean expression for each output Output x: A B C D x 1 The rest X 10 11 01 00 CD AB 1 x x x x x x x = BD + BC + A

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**Boolean Expressions for Outputs**

Output y: CD AB 00 01 11 10 1 X A B C D y z 1 The rest X Output z: CD AB 00 01 11 10 1 X

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**Boolean Expressions for Outputs**

Output t: CD AB 00 01 11 10 1 X A B C D T 1 The rest X t = D Step 4: Draw the logic diagram x = BC + BD + A y = A + BD’ + BC z = A + B’C + BC’D

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**Example: Logic Diagram**

B x = BC + BD + A C D y = A + BD’ + BC z = A + B’C + BC’D t = D

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**Example: Verification**

Step 5: Check the functional correctness of the logic circuit Apply all possible input combinations And check if the circuit generates the correct outputs for each input combinations For large circuits with many input combinations, this may not be feasible. Statistical techniques may be used to verify the correctness of large circuits with many input combinations

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**Design Example Specification BCD to Excess-3 code converter**

Transforms BCD code for the decimal digits to Excess-3 code for the decimal digits BCD code words for digits 0 through 9: 4-bit patterns 0000 to 1001, respectively Excess-3 code words for digits 0 through 9: 4-bit patterns consisting of 3 (binary 0011) added to each BCD code word

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**Design Example (continued)**

Formulation (Find truth table) Conversion of 4-bit codes can be most easily formulated by a truth table Variables - BCD: A,B,C,D Variables - Excess-3 W,X,Y,Z Don’t Cares - BCD to 1111

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**Design Example (continued)**

W = A + BC + BD X = C + D + B Y = CD + Z = B C D A 1 3 2 4 5 7 6 12 13 15 14 8 9 11 10 X w z y x B B C D C D D

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**Design Example (continued)**

Technology Mapping A B C D W X Y Z

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