Logic DesignCS221

1st Term 2009-2010

combinational circuitscombinational circuits

Cairo University

Faculty of Computers and Information

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Administrivia project discussion lab of this week: simple game office hours:

tuesdays: 4:30-7 wednesdays: 12-4

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our story so far... digital circuit specification

inputs and outputs truth table of each output (or Boolean-algebraic representation) don't -care conditions (if any)

truth table -> K-map K-map -> simplified algebraic expression algebraic expression -> gate implementation

AND-OR, OR-AND, NAND, NOR, XOR, etc. combinational circuits (today's lecture)

no memory

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step 4: verification For small circuits:

assign variable names to intermediate signals derive truth table

For large circuits computer software

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step 4: verification

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example 2: binary adderDesign a circuit that adds two n-bit binary

numbers

0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 10

carry

Add two 5-bit numbers:

0 1 0 0 1

+ 1 1 0 1 0

--------------1100

1

0

1

1

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example 2: binary add/subtractwe can follow the same steps as before: step 1: specification

2n inputs n+1 outputs truth table

step 2: simplification step 3: implementation

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example 2: binary add/subtractwe will take another approach

bottom-up approach building circuit in blocks

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smallest block: half-adder circuit that adds two bits (half-adder) 2 inputs (x and y) and 2 outputs (Sum and

Carry)

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half-adder step 1: specification step 2: simplification

S = x'y + xy' C = xy

step 3: implementation

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full-adder: one step up 3 inputs (two bits and carry) 2 outputs: sum and carry

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full-adder: one step up 3 inputs (two bits and carry) 2 outputs: sum and carry implementation using half-adders

x

y

z

y

S0

C0

half-adder1

half-adder0

S1

C1

S

C

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11--0 1

0--0

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4-bit binary adder: one step up adds two 4-bit binary numbers (add an input

carry) K-map design: 9 inputs ... too much

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4-bit binary adder: one step up adds two 4-bit binary numbers (add an input

carry) using four full adders

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example: binary subtractor subtracts two n-bit binary numbers how many inputs? outputs?

7 0111 0111- 5 - 0101 +1010------ --------- + 1 --------- 10010

2's complement of 0101

7 0111 0111- 9 -1001 + 0110------ --------- + 1 --------- 01110

2's complement of 1001

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example: binary subtractorA – B = A + (1's complement of B) + 1

1

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overflow if the sum of two n-bit numbers occupies n+1

bits, an overflow has occurred computers must detect overflow...why? we will add a circuit to detect overflow overflow detection depends on whether the two

added numbers are signed or unsigned

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overflow detection unsigned: end carry => overflow signed: last bit is the sign

if the last two carries are equal => no overflowif the last two carries are not equal => overflow

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overflow detection C3 XOR C4 = 1 when? C3 XOR C4 = 0 when? V (overflow detection) = C3 XOR C4