# CSCE 212 Computer Architecture by maclaren1

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```									         CSCE 211 Digital Design

Lecture 4
Circuits Simplification
Topics
   Combinational Circuit Analysis
   Sums-of-Products Form
   Products-of-Sums Form
   Karnaugh Maps
   Lab 1 Handout (kits not in yet)

September 2, 2003
Overview
Last Time
   Basic Gates
   Boolean Algebra: Axioms, Theorems,
On last Time’s Slides(what we didn’t get to)
   Principle of Duality
   N-variable Theorems
New
   Combinational Circuit Analysis
   Algebraic analysis, Truth tables, Logic Diagrams
   Sums-of-Products and Products-of-Sums
   Circuit Simplification: Karnuagh Maps
–2–                                                            CSCE 211H Fall 2003
Principle of Duality
The Dual of an expression is Swapping 0 & 1, and
swapping AND & OR.
Given say
   X+X' = 1

Taking the dual yields
   X . X' = 0

Principle of Duality : If we completely parenthesize an
equation that is true and then take the dual of both
sides then the result is still true.
Why?
   Each axiom (A1-A5) has a dual (A1-A5

–3–                                                 CSCE 211H Fall 2003
Principle of Duality
Given say
   (X+Y).(X'+Z).(Y+Z) = (X+Y).(X'+Z)

Taking the dual yields
   (X.Y)+(X' .Z)+(Y.Z) = (X.Y)+(X'.Z)

Principle of Duality : If we completely parenthesize an
equation that is true and then take the dual of both
sides then the result is still true.

Why?
   Each axiom (A1-A5) has a dual (A1-A5

–4–                                                 CSCE 211H Fall 2003
N-variable Theorems
Generalized idempotency
T12    X+X+X…+X=X
T12’   X.X.X….X=X
DeMorgan’s theorems
T13    (X1 . X2 . X3 … . Xn)’ = X1’ + X2’ + X3’ … + Xn’
T13’ (X1 + X2 + X3 … + Xn)’ = X1’ . X2’ . X3’ … . Xn’

Prove using finite induction
Most important: DeMorgan’s theorems

–5–                                                   CSCE 211H Fall 2003
N-variable Dual Theorems
We will use F(X1, X2, … , Xn, +, ., ') to denote a
completely parenthesized boolean expression
The the dual can be denoted
FD(X1, X2, … , Xn, +, ., ') = F(X1, X2, … , Xn, ., +, ')
Demorgan’s theorem expressed using the dual operator
F(X1, X2, … , Xn)' = FD(X1 ', X2 ', … , Xn ')
Shannon provided the expansion theorems
   F(X1, X2, … , Xn ) = X1.F(1, X2, … , Xn ) + X1 '.F(0, X2, … , Xn )
   F(X1, X2, … , Xn ) = [X1+F(0, X2, … , Xn )] . [X1 '+F(1, X2, … , Xn )]

–6–                                                                 CSCE 211H Fall 2003
Combinational Circuit Analysis
A combinational circuit is one whose outputs are a
function of its inputs and only its inputs.
These circuits can be analyzed using:
1. Truth tables
2. Algebraic equations
3. Logic diagrams – timing considerations; graphical

–7–                                           CSCE 211H Fall 2003
Switching Algebra Terminology
Literal – a variable or the complement of a variable
Product term – a single literal or the AND of several
literals
Sum term – a single literal or the OR of several literals

Sums-of-products
Product-of-sums
Normal term – a product (sum) term in which no
variable appears twice
Minterm – a normal product term with n literals
Maxterm – a normal sum term with n literals
–8–                                               CSCE 211H Fall 2003
Boolean Algebra Proofs
Axioms
   Statements (boolean equations) that are assumed to be true that
form the basis of a mathematical system.
Theorems
   Statements that can be “proved” from the axioms and earlier
theorems.
Lemmas, Corollaries, Postulates
Proof by truth-table
   For a “possible theorem” with a small number of variables, we can
exhaustively consider all possible cases.
Algebraic Proofs
   Apply axioms and previously proven theorems to rewrite a
“possible theorem” until it is reduced to an equation known to be
true.
Induction
   Basis case: P(1)
–9–    Inductive hypothesis: Assuming P(n) show P(n+1)         CSCE 211H Fall 2003
Proof by truth-table
Prove Demorgan’s Law: (X+Y)' = X ' . Y '

X      Y    (X+Y) (X+Y)'   X'   Y'   X' . Y'
0     0
0     1
1     0
1     1

Note the table considers all possible cases and in each
case the value in the column for (X+Y)' is equal to the
value in the column for X' . Y'
So, (X+Y) ' = X' . Y'
– 10 –                                                  CSCE 211H Fall 2003
Algebraic Simplification
Simplify F = A.B.C’.D + D.C.A + B.C.D + A’.B’.C.D

– 11 –                                               CSCE 211H Fall 2003
Proof by Induction
Thereom 13 (X1 . X2 . X3 … . Xn)’ = X1’ + X2’ + X3’ … + Xn’
Proof: Basis Step n = 2, (X1 . X2)’ = X1’ + X2’ was
proven using a truth-table.
Now suppose as inductive hypothesis that
(X1 . X2 . X3 … . Xn)’ = X1’ + X2’ + X3’ … + Xn’
Then consider
     (X1 . X2 . … . Xn . Xn+1)’
   = ((X1 . X2 . X3 … . Xn ) . Xn+1)’ by associativity
   = (X1 . X2 . X3 … . Xn )’ + Xn+1’  by the basis step
   = (X1’ + X2’ + X3’ … + Xn’) + Xn+1’ by the inductive hypothesis

– 12 –                                                           CSCE 211H Fall 2003
Logic Diagrams (Xilinx)

– 13 –                     CSCE 211H Fall 2003
Timing Analysis
We will do some extensive timing analysis in the labs
but for right now we will assume the delay for and an
AND-gate and an OR-gate is “d”

When we fabricate circuits there are a couple special
circumstances:
1. Inverters (Not gates) cost nothing
2. Circuits are usually fabricated from “NANDs”

– 14 –                                          CSCE 211H Fall 2003
Circuit Simplification
Why would we want to simplify circuits?
   To minimize time delays
   To minimize costs
   To minimize area

– 15 –                                     CSCE 211H Fall 2003
Sums-of-Products

What is the delay of sums-of-products circuit?
– 16 –                                            CSCE 211H Fall 2003
Products-of-Sums

– 17 –              CSCE 211H Fall 2003
Circuit Simplification
Minterms – a product term in which every variable occurs
once either complemented or uncomplemented

X             Y           F      minterm
0             0           1        X’ . Y’
0             1           0        X’ . Y
1             0           1        X . Y’
1             1           1        X.Y

Sum of minterms form:
F(X,Y) = X’ . Y’ + X . Y’ + X . Y
F(X,Y) = SUM(0, 2, 3)           (use capital sigma)
– 18 –                                                             CSCE 211H Fall 2003
Karnaugh Maps
Tabular technique for simplifying circuits
two variable maps          three variable map
X                       XY
0   1            00    01   11   10
Y                    Z
0        00 10       0        000 010 110 100
1        01 11       1        001 011 111 101

X
XY
Y           0   1             00   01   11   10
Z
0       0   2
0     0    2     6    4
1       1   3       1     1    3     7    5

– 19 –                                                 CSCE 211H Fall 2003
Karnaugh Map Simplification
F(X,Y,Z) =

XY   00   01   11   10
Z
0    1    1    1    1
1              1    1    Z

Sum of minterms form
F(X,Y,Z)=
Minimize ? Fewer gates, fewer inputs
F(X,Y,Z)=
F(X,Y,Z)=
– 20 –                                         CSCE 211H Fall 2003
Karnaugh Map Terminology
F(X,Y,Z) =

XY   00   01   11   10
Z
0         1    1
1              1    1    Z

Implicant set - rectangular group of size 2i of adjacent
containing ones
Each implicant set of size 2i of corresponds to a
product term in which i variables are true and the
rest false
Implicant Sets:
– 21 –                                            CSCE 211H Fall 2003
Karnaugh Map Terminology
F(X,Y,Z) =

XY   00   01   11   10
Z
0

1                        Z

Prime implicant – an implicant set that is as large as
possible
Implies – We say P implies F if everytime P(X1, X2, … Xn)
is true then F (X1, X2, … Xn) is true also.
If P(X1, X2, … Xn) is a prime implicant then P implies F

– 22 –                                            CSCE 211H Fall 2003
Karnaugh Map Terminology
F(X,Y,Z) =

XY   00   01   11   10
Z
0         1    1    1
1              1    1    Z

Prime implicants –

If P(X1, X2, … Xn) is a prime implicant then P implies F
and if we delete any variable from P this does not
imply F.

– 23 –                                            CSCE 211H Fall 2003
Karnaugh Map Simplification
F(X,Y,Z) =

XY   00   01   10   11
Z
0

1                        Z

F(X,Y,Z) =

– 24 –                                         CSCE 211H Fall 2003
Karnaugh Map Simplification
F(X,Y,Z) =

XY   00   01   10   11
Z
0

1                        Z

F(X,Y,Z) =

– 25 –                                         CSCE 211H Fall 2003
4 Variable Map Simplification
F(W,X,Y,Z) =
X

WX   00   01       11       10
YZ
00 0000 0100 1100 1000

01 0001 0101 1101 1001
Z
11 0011 0111 1111       1011
Y
10 0010 0110 1110 1010

W

– 26 –                                            CSCE 211H Fall 2003
4 Variable Map Simplification
F(W,X,Y,Z) =
X

WX   00   01       11       10
YZ
00     0    4        12       8

01   1    5        13       9
Z
11   3    7        15       11
Y
10   2    6        14       10

W

– 27 –                                            CSCE 211H Fall 2003
Karnaugh Map Simplification
F(W,X,Y,Z) =
X

WX   00   01       11       10
YZ
00

01
Z
11
Y
10

W

– 28 –                                            CSCE 211H Fall 2003
Karnaugh Map Simplification
F(W,X,Y,Z) =
X

WX   00   01       11       10
YZ
00

01
Z
11
Y
10

W

– 29 –                                            CSCE 211H Fall 2003
Karnaugh Map Simplification
F(W,X,Y,Z) =
X

WX   00   01       11       10
YZ
00

01
Z
11
Y
10

W

– 30 –                                            CSCE 211H Fall 2003
Larger Maps
Five variable maps - Figure X4.72 page 307

Six variable maps - Figure X4.74 page 308

But who cares, use Quine-McKluskey section 4.5

– 31 –                                        CSCE 211H Fall 2003
Don’t Care Conditions
F(W,X,Y,Z) =
X

WX   00   01       11       10
YZ
00

01
Z
11
Y
10

W

– 32 –                                            CSCE 211H Fall 2003
Don’t Care Conditions
F(W,X,Y,Z) =
X

WX   00   01       11       10
YZ
00

01
Z
11
Y
10

W

– 33 –                                            CSCE 211H Fall 2003
Don’t Care Conditions
F(W,X,Y,Z) =
X

WX   00   01       11       10
YZ
00

01
Z
11
Y
10

W

– 34 –                                            CSCE 211H Fall 2003
Don’t Care Conditions
F(W,X,Y,Z) =
X

WX   00   01       11       10
YZ
00

01
Z
11
Y
10

W

– 35 –                                            CSCE 211H Fall 2003
Summary
Homework
1. 4.6b
2. 4.31
3. 4.39b
4. 4.40
5. 4.55
6. 4.13d
7. 4.19c,e

– 36 –        CSCE 211H Fall 2003

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