Combinational Logic Combinational Logic: the Outputs Depend on the Present Values of Inputs. in Other Words, They Are Logic Combinations of the Inputs. Sequential Logic: the Outputs Depend Not Only on the Present but
2012
Combinational Logic
Combinational Logic: The outputs depend on the present values of inputs. In other words, they are logic combinations of the inputs. Sequential Logic: The outputs depend not only on the present but also on the past inputs. Problem Statement
Truth Table
min. number of gates min. number of inputs to gates
Simplification
min. propagation time min. number of interconnections
Implementation
type of gates
Adders
Half-adder performs addition of two bits. Full-adder performs addition of three bits.
Half-adder Inputs x 0 0 1 1 y 0 1 0 1 Outputs C 0 0 0 1 S 0 1 1 0 x y S C
S xy xy x y C x y
Feza Kerestecioğlu, K.H.Ü.
1
EE 205 Lecture Notes
2012
Adders
Half-adder x y
x y
S C
S xy xy C x y
S
C
x y
S
x y
S C
C
S xy xy (1, 2) (0, 3) ( x y )( x y)
S (0, 3) xy xy xy C
Adders
Full-adder Inputs x 0 0 0 0 1 1 1 1 y 0 0 1 1 0 0 1 1 z 0 1 0 1 0 1 0 1 Outputs C 0 0 0 1 0 1 1 1 S 0 1 1 0 1 0 0 1 x y z S z x y C x yz 00 x 0 1 1 y 01 11 10 1 1 1 x yz 00 x 0 1 y 01 11 10 1 1 1 1
z S xyz xyz xyz xyz
z C xy yz xz
Feza Kerestecioğlu, K.H.Ü.
2
EE 205 Lecture Notes
2012
Adders
Full-adder can also be implemented with half-adders: 1. Add x and y by a half-adder 2. Add z to the output of Step 1. 3. OR the carry outputs of Step 1 and 2.
x y
S1
S
HA
C1
HA
C2
C
z
S ( x y) z ( xy xy) z ( xy xy) z ( xy xy) z ( xy xy) z ( x y)( x y) z xyz xyz ( xy xy) z xyz xyz xyz xyz
C xy ( x y) z xy ( xy xy) z xy xyz xyz xy xyz xyz xyz xyz xy ( x x) yz ( y y) xz xy yz xz
Subtractors
Half-subtractor performs subtraction of one bit from another. Full-subtractor performs subtraction of sum of two bits from a third one.
Half-subtractor Inputs x 0 0 1 1 y 0 1 0 1 Outputs B 0 1 0 0 D 0 1 1 0 x y D B
D xy xy x y B x y
Feza Kerestecioğlu, K.H.Ü.
3
EE 205 Lecture Notes
2012
Subtractors
Full-subtractor xyz Inputs Outputs x 0 0 0 0 1 1 1 1 y 0 0 1 1 0 0 1 1 z 0 1 0 1 0 1 0 1 B 0 1 1 1 0 0 0 1 D 0 1 1 0 1 0 0 1 z x y D1 D z D xyz xyz xyz xyz z B xy yz xz x y yz 00 x 0 1 1 01 11 10 1 1 1 x yz 00 x 0 1 y 01 11 10 1 1 1 1
HS
B1
HS
B2
B
Code Conversion
Example: BCD to Excess-3 conversion CD 00 AB BCD Excess-3 A B C D w x y z 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 1 1 1 1 0 1 0 1 0 0 0 1 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 X X X X 10 00 01 0 0 CD 00 AB 00 0 01 1
01 11 10 0 1 X 1 0 1 X X 0 1 X X
01 11 10 1 0 X 1 1 0 X X 1 0 X X
11 X 1
11 X 10 0
w A BD BC
CD 00 AB 00 1 01 1 01 11 10 0 0 X 0 1 1 X X 0 0 X X
x BD BC BCD
CD 00 AB 00 1 01 1 01 11 10 0 0 X 0 0 0 X X 1 1 X X
11 X 10 1
11 X 10 1
y CD CD
z D
Feza Kerestecioğlu, K.H.Ü.
4
EE 205 Lecture Notes
2012
Code Conversion
Example: BCD to Excess-3 conversion
w A BD BC A B(C D) x BD BC BCD B(C D) B(C D) B (C D) y CD CD C D z D A w x
B C
y D z
Code Conversion
Example: Gray code to binary conversion Gray 0 0 0 0 1 1 1 1 0 0 1 1 1 1 0 0 0 1 1 0 0 1 1 0 Binary y 0 0 1 1 0 0 1 z 0 1 0 1 0 1 0 A B C A 0 0 0 0 1 1 1 1
xA
BC 00 0 1 1 1 01 11 10 1 1 A BC 00 0 1 1 01 11 10 1 1 1
A B C x
y AB AB
A B
y ABC ABC ABC ABC ( A B)C ( A B)C ( A B) C
x y z
1 1
Feza Kerestecioğlu, K.H.Ü.
5
EE 205 Lecture Notes
2012
EX-OR, Equivalence and Error Handling
EX-OR function: x y xy xy (1, 2)
x0 x
x 1 x
x1 x2 xn1 xn xi
i 1 n
xx 0
x x 1
x y x y ( x y)...
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