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ECE201Notes(5)

# ECE201Notes(5) - ECE 201 Digital Logic Chapter 5 1 of 30...

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ECE 201 Digital Logic Chapter 5 1 of 30 MSI Components and PLD’s 5.1 Binary Adders and Subtractors x y C S 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0 S = x’y + xy’ C = xy S C x y y’ x x’ y S C x y’ y x x’ y S = x’y + xy’ C = xy S = (x + y)(x’ + y’) C = xy S C x’ y y’ x S C x y’ y x’ S = (C + x’y’)’ C = xy S = (x+y)(x’+ y’) C = (x’+y’)’ S C x y S = x y C = xy Which implementation above is best?

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ECE 201 Digital Logic Chapter 5 2 of 30 Full Binary Adder x y c in C out S 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1 S = x’y’c in + x’y c in ’ + x y’c in + x y c in 00 01 01 11 10 00 x c out c in y 0 1 0 1 1 1 0 0 C out = xy + xc in + yc in S C out x’ x x y y y y’ y C in C in C in C in C in C in y’ x’ x x C out C in x y S S = C in (x y) = C in ’(xy’ + x’y) + C in (xy’ + x’y)’ = C in ’(xy’+x’y) + C in (xy+x’y’) = xy’ C in ’ + x’y C in ’ + xy C in + x’y’ C in C out = xy + C in (x y) = xy + C in (xy’+x’y) = xy + C in (xy’+x’)(xy’+y) = xy + C in (y’ + x’)(x + y) = xy + C in (xy)’(x + y) = [xy + (xy)’][xy + C in (x + y)] = xy + C in x + C in y
ECE 201 Digital Logic Chapter 5 3 of 30 Binary Subtractor x y B in B out D 0 0 0 0 0 0 0 1 1 1 0 1 0 1 1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 D = x’y’b in + x’y b in + xy’b in + xy b in 00 01 01 11 10 00 x b out b in y 0 1 0 1 1 1 0 0 b out = x’y + x’ b in + y b in Does a Computer need an adder and subtractor? 4-Bit Adder/Subtractor

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ECE 201 Digital Logic Chapter 5 4 of 30 Look-Ahead Carry Generator A i B i C i C i+1 P i G i S i P i = A i B i and G i = A i B i (Note, P’s and G’s do NOT depend on C’s) so S i = P i C i and C i+1 = G i + P i C i giving C 2 = G 1 + P 1 C 1 , C 3 = G 2 + P 2 C 2 = G 2 + P 2 (G 1 + P 1 C 1 ) = G 2 + P 2 G 1 + P 2 P 1 C 1 , and C 4 = G 3 + P 3 C 3 = G 3 + P 3 G 2 + P 3 P 2 G 1 + P 3 P 2 P 1 C 1 and so on... When are the outputs C 1 , C 2 , C 3 , C 4 , etc... ready (valid)? How many gates would there be for a 32-bit generator?
ECE 201 Digital Logic Chapter 5 5 of 30 Four-Bit Adder with Look-Ahead Carry Generator How long does each circuit above take to calculate a sum? 5.2 Decimal Adders When a digital system is primarily involved with decimal numbers, a decimal adder may be practical. This addition can be done by first performing a binary addition, and then correcting for the case where the sum is greater than 9. Binary 4-bit Adder Binary 4-bit Adder Add Six 0 0 y 3 y 3 x 3 x 3 s 3 s 3 c out c out c in c out c in c in A 3 B 3 y 2 y 2 x 2 x 2 s 2 s 2 A 2 B 2 y 1 y 1 x 1 x 1 s 1 s 1 A 1 B 1 y 0 y 0 x 0 x 0 s 0 s 0 A 0 B 0

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ECE 201 Digital Logic Chapter 5 6 of 30 Binary Multiplier Consider multiplying two 2-bit binary numbers: B 1 B 0 A 1 A 0 ________________________________ A 0 B 1 A 0 B 0 A 1 B 1 A 1 B 0 __________________________________________________ M 3 M 2 M 1 M 0 A 0 A 1 B 1 M 1 M 3 B 1 B 0 M 0 M 2 B 0 HA HA 2-Bit by 2-Bit Binary Multiplier
ECE 201 Digital Logic Chapter 5 7 of 30 5.3 Comparators Consider doing conditional testing on two binary numbers: A = A 3 A 2 A 1 A 0 and B = B 3 B 2 B 1 B 0 The two numbers are equal if A i = B i for all i. We can use the equality function to test each case: x i = A i B i + A i ’B i Where equality holds if all x i ’s hold, or: F 1 = (A == B) = x 3 x 2 x 1 x 0

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• Spring '08
• ried
• Binary-coded decimal, Programmable logic device, Programmable logic array, Programmable Array Logic, D1 D2 D3

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ECE201Notes(5) - ECE 201 Digital Logic Chapter 5 1 of 30...

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