review
DESCRIPTION
Review. D: n bit binary number D = (d n-1 ∙ ∙ ∙ d 1 d 0 ) 2 If D is an unsigned binary number D = ( 2 n-1 d n-1 + ∙ ∙ ∙ 2 1 d 1 + 2 0 d 0 ) 10 If D is a sign-magnitude binary number D = + ( 2 n-2 d n-2 + ∙ ∙ ∙ 2 1 d 1 + 2 0 d 0 ) 10 if d n-1 =0 - PowerPoint PPT PresentationTRANSCRIPT
KU College of EngineeringElec 204: Digital Systems Design 1
Review
• D: n bit binary number
D = (dn-1 ∙ ∙ ∙ d1 d0)2
• If D is an unsigned binary number
D = (2n-1 dn-1+∙ ∙ ∙ 21 d1 + 20 d0)10
• If D is a sign-magnitude binary number
D = + (2n-2 dn-2+∙ ∙ ∙ 21 d1 + 20 d0 )10 if dn-1=0
= – (2n-2 dn-2+∙ ∙ ∙ 21 d1 + 20 d0 )10 if dn-1=1
(–D) = (d’n-1 dn-2 ∙ ∙ ∙ d1 d0)2
• If D is in two`s complement system
D = (-2n-1 dn-1+ 2n-2 dn-2 + ∙ ∙ ∙ 21 d1 + 20 d0)10
(–D) = 2n – D = (2n-1) – D + 1 = (d’n-1 d’
n-2 ∙ ∙ ∙ d’1 d’
0)2 + 1
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• Two’s complement multiplication– Shift and two’s complement addition except for the last
step. Remember MSB represent (-2n-1)-5 1011
x –3 1101 0000 initial partial product, which is zero. 1011 11011 partial product 0000 111011 partial product 1011 11100111 0101 shifted-and-negated1 00001111
Review
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BCD: Binary-Coded Decimal
• 0-9 encoded with their 4-bit unsigned binary representation (0000 – 1001). The codewords (1010 – 1111) are not used.
• 8-bit byte represent values from 0 to 99.
• BCD Addition:
Carry 1 1
448 0100 0100 1000
+ 489 0100 1000 1001
937 Sum 1001 1101 10001
Add 6 + 0110 + 0110
BCD sum 1 0011 1 0111
BCD result 1001 0011 0111
Review
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2. Combinational Logic Circuits
• Boolean Algebra– switching algebra
– deals with Boolean values --- 0, 1
• Positive-logic convention– analog voltages LOW, HIGH 0, 1
• Negative logic --- seldom used• Signal values denoted by variables
(X, Y, FRED, etc.)
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Boolean operators
• Complement: X (opposite of X)• AND: X Y• OR: X + Y
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• Literal: a variable or its complement– X, X, FRED, CS_L
• Expression: literals combined by AND, OR, parentheses, complementation– X+Y
– P Q R
– A + B C
– ((FRED Z) + CS_L A B C + Q5) RESET
• Equation: Variable = expression– P = ((FRED Z) + CS_L A B C + Q5)
RESET
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Basic Logic Gates
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Theorems
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More Theorems
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Duality
• Swap 0 & 1, AND & OR– Result: Theorems still true
• Why?– Each axiom (T1-T5) has a dual (T1-T5
• Counterexample:X + X Y = X (T9)X X + Y = X (dual)
X + (X Y) = X (T9)X (X + Y) = X (dual)(X X) + (X Y) = X (T8)
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N-variable Theorems
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DeMorgan Symbol Equivalence
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Similar for OR
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Complement of a function
• F1 = XYZ’ + X’Y’Z
• F1’ = (XYZ’ + X’Y’Z)’ = (XYZ’)’ (X’Y’Z)’
= (X’+Y’+Z) (X+Y+Z’)
• Complement = take dual +complement each literal
• Dual of F1 = (X+Y+Z’) (X’+Y’+Z)
• F1’ = (X’+Y’+Z) (X+Y+Z’)
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• Standard Forms:– Product and sum terms
• Minterm: A product term in which all variables appear exactly once, either complemented or not (2n minterms)– For a two variable function, minterms are
• X’Y’, X’Y, XY’, XY
• m0 , m1 , m2 , m3
• Maxterms: A sum term that contains all variables in complemented or uncomplemented form
• X+Y, X+Y’, X’+Y, X’+Y’
• M0 , M1 , M2 , M3
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• Alternative representations– F(X,Y,Z) = X’Y’Z’ + X’YZ’ +XY’Z + XYZ
= m0 + m2 + m5 + m7
=
– F’(X,Y,Z) = X’Y’Z + X’YZ + XY’Z’ + XYZ’
= m1 + m3 + m4 + m6
=
– F(X,Y,Z) = (m1 + m3 + m4 + m6)’ = m1’ m3’ m4’ m6’
= M1 M3 M4 M6
=
)7,5,2,0(m
)6,4,3,1(m
)6,4,3,1(M
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• Maxterms are seldom used, we’ll use minterms rather.
• Properties of minterms:– There are 2n minterms. 1-1 with binary numbers 0-(2n-1)
– Every Boolean function can be expressed as sum of minterms.
– Absent minterms belong to complement function
– A function that include all minterms is equal to logic 1.