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CSEE 3827: Fundamentals of Computer Systems
Lecture 2 January 26, 2009 Martha Kim mak2191@columbia.edu
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Agenda
- TA office hours
- Boolean algebra
- Logic gates
- Circuit fabrication
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CSEE 3827: Fundamentals of Computer Systems Lecture 2 January 26, - - PowerPoint PPT Presentation
CSEE 3827: Fundamentals of Computer Systems Lecture 2 January 26, - - PowerPoint PPT Presentation
CSEE 3827: Fundamentals of Computer Systems Lecture 2 January 26, 2009 Martha Kim mak2191@columbia.edu 1 1 Agenda TA office hours Boolean algebra Logic gates Circuit fabrication 2 2 TA Office Hours TA Room, first floor
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TA Office Hours
Roopa Kakarlapudi Tuesdays 5-6:30PM Harsh Parekh Mondays 11-12:20PM; Tuesdays 3:30-5PM Nishant Shah Wednesdays 10-11:30AM
TA Room, first floor of Mudd (see: http://ta.cs.columbia.edu/tamap.shtml)
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Boolean Logic
- Binary digits (or bits) have two values: {1,0}
- All logical functions can be implemented in terms of three logical operations:
x x
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1 x y x y 1 1 1 1
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.
x y
x + y
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1
1
1
1 1
1
NOT AND OR
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Boolean Logic 2
- Precedence rules just like decimal system
- Implied precedence: NOT > AND > OR
- Use parentheses as necessary
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AB + C = (AB) + C (A + B)C = ((A) + B)C
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Boolean Logic: Example
D X A
L=DX + A
1 1 1 1 1 1 1 1 1 1 1 1
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Boolean Logic: Example
D X A X DX
L=DX + A
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1 1
1 1 1
1
1
1 1 1
1 1
1 1 1
1 1 1 1 1
1
(M&K Table 2-2)
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Boolean Logic: Example 2
X Y XY + XY
1 1 1 1
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Boolean Logic: Example 2
X Y X Y XY XY XY + XY
1 1 1 1
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1
1
1
1 1
1 1
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Boolean Algebra: Identities and Theorems
OR AND NOT X+0 = X X1 = X (identity) X+1 = 1 X0 = 0 (null) X+X = X XX = X
(idempotent)
X+X = 1 XX = 0 (complementarity) X = X (involution) X+Y = Y+X XY = YX (commutativity) X+(Y+Z) = (X+Y)+Z X(YZ) = (XY)Z (associativity) X(Y+Z) = XY + XZ X+YZ = (X+Y)(X+Z) (distributive) X+Y = X Y XY = X + Y
(DeMorgan’s theorem)
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Boolean Algebra: Example
F = XYZ + XYZ + XZ
Simplify this equation using algebraic manipulation.
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Boolean Algebra: Example
F = XYZ + XYZ + XZ
XY(Z + Z) + XZ (by reverse distribution) XY1 + XZ (by complementarity) XY + XZ (by identity)
Simplify this equation using algebraic manipulation.
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Boolean Algebra: Example 2
F = AB + AB F =
Find the complement of F.
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Boolean Algebra: Example 2
F = AB + AB F = AB + AB
(AB) (AB) (by DeMorgan’s) (A + B) (A + B) (by DeMorgan’s) (A + B) (A + B) (by involution)
Find the complement of F.
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Boolean Algebra: Why? These circuits consume area, power, and time
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Logic gate area
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Information signaled through voltage level
1.3 v 0.0 v 0.0 v (AND)
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Idealized timing diagram of AND gate
(AND)
A B Q
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Actual signal timing has delays
- transition time: time required for
- utput to change (RC delay:
- hms x farads = time
- propagation time: time from
input change to output change
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Returning to boolean algebra...
F = XYZ + XYZ + XZ
XY(Z + Z) + XZ (by reverse distribution) XY1 + XZ (by complementarity) XY + XZ (by identity)
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Returning to boolean algebra...
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Universal gates: NAND, NOR
x y z = xy
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1
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1 1
XY
x y z = x+y
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1 1 1 1
X+Y
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Universal how?
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Boolean algebra practice 1
1 = AB + BC + AB + BC
B (A + A) + B (C+C) (by distribution) B + B (by complementarity) 1 (by complementarity)
Prove that this boolean equation is true using algebraic manipulation.
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Boolean algebra practice 2
X + Y = XY + XY + XY
XY + XY + XY + XY (by idempotence) X (Y + Y) + Y (X + X) (by distribution) X 1 + Y 1 (by null) X + Y (by identity)
Prove that this boolean equation is true using algebraic manipulation.
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Boolean algebra practice 3
F = (VW + X)Y + Z F = (VW + X)Y + Z
((VW + X)Y)Z (by DeMorgan’s) ((VW + X) + Y)Z
(by DeMorgan’s & involution)
(VW X + Y)Z (by DeMorgan’s) ((V + W)X + Y)Z (by DeMorgan’s) ((V + W)X + Y)Z (by null)
Find the complement of F.
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Integrated circuit fabrication
raw silicon crystallization
- f molten silicon
silicon ingots wafer
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Integrated circuit fabrication 2
wafer processed wafer
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Integrated circuit fabrication 3
processed wafer dicing packaging
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Integrated circuit fabrication 4
packaged die test
$$$
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A more detailed tutorial on integrated circuit fabrication:
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http://www.necel.com/fab/en/flow.html
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Next class: more boolean algebra, duals
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