1. CSE 370 – Winter 2002 - Number syst.; Logic functions- 1
Overview
Last lecture:
Abstraction and hierarchy in digital design
Combinational vs. sequential systems
High-level examples of combinational and sequential designs
The whole course in 2 examples!
In section
Number systems
Binary; 2’s complement
Today
Number systems (very quick review)
Combinational logic
Logic functions and truth tables
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2. The basics binary numbers
Base conversion (binary, octal, decimal, hexadecimal)
Positional number system
1012=510 (1 * * * 20)
1478=10310 (1 * * * 80) Octal (base 8)
2C116=70510 (2 * * * 160) hexadecimal (base 16)
Do it also for fractions
Conversion between binary/octal/hex
Binary:
Octal: | 011 | 110 | 001=23618
Hex: | 1111 | 0001=4F116
Addition and subtraction are trivial, but worth practicing
See Katz, appendix A
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3. The basics base conversion
Conversion from decimal to binary/octal/hex: divide by the base
The digits are generated from least significant (rightmost) to most significant (leftmost)
Why does it work (recall positional system)
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4. CSE 370 – Winter 2002 - Number syst.; Logic functions- 4
Number systems
How do we write negative binary numbers?
We want same number of positive and negative representations (or as close to same as possible)
How do we represent zero?
How do we recognize easily if a number is positive or negative?
Historically: Three approaches
Sign and magnitude
Ones complement
Twos complement
Twos complement makes addition and subtraction easy
Used almost universally in present-day systems
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5. Approach 1: Sign and magnitude
The most-significant bit (msb) is the sign digit
0  positive
1  negative
The remaining bits are the number’s magnitude
Problem 1: Two representations for zero
0 = 0000 and also –0 = 1000
Problem 2: Arithmetic is cumbersome
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6. Approach 2: Ones complement
Negative number: Bitwise complement of positive number
0011  310
1100  –310
Solves the arithmetic problem
Remaining problem: Two representations for zero
0 = 0000 and also –0 = 1111
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7. Approach 3: Twos complement
Negative number: Bitwise complement plus one
0011  310
1101  –310
Number wheel
Note: one more negative number
0000
0001
0011
1111
1110
1100
1011
1010
1000
0111
0110
0100
0010
0101
1001
1101
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
– 8
– 7
– 6
– 5
– 4
– 3
– 2
– 1
Only one zero!
msb is the sign digit
0  positive
1  negative
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8. Twos complement (con’t)
Complementing a complement restores the original number
Arithmetic is easy
We ignore the carry
Same as a full rotation around the wheel
Subtraction = negation and addition
Easy to implement in hardware
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9. CSE 370 – Winter 2002 - Number syst.; Logic functions- 9
Overflow
Conditions: Sign bit changes
Summing two positive numbers gives a negative result
Summing two negative numbers gives a positive result
0000
0001
0011
1111
1110
1100
1011
1010
1000
0111
0110
0100
0010
0101
1001
1101
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
– 8
– 7
– 6
– 5
– 4
– 3
– 2
– 1
0000
0001
0011
1111
1110
1100
1011
1010
1000
0111
0110
0100
0010
0101
1001
1101
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
– 8
– 7
– 6
– 5
– 4
– 3
– 2
– 1
6 + 4  –6
–7 – 3  +6
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10. Combinational logic topics
Logic functions, truth tables, and switches
NOT, AND, OR, NAND, NOR, XOR, . . .
minimal set
Axioms and theorems of Boolean algebra
proofs by re-writing
proofs by perfect induction
Gate logic
networks of Boolean functions
time behavior
Canonical forms
two-level
incompletely specified functions
Simplification
Boolean cubes and Karnaugh maps
two-level simplification
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11. Logic functions and truth tables
X Y
Buffer (identity) X
NOT X X’
AND X.Y XY
OR X+Y X Y
X Y X Y X
X Y Z Z Y
X Y Z X Z Y
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12. Logic functions and truth tables (ct’d)
X Y Z
NAND X.Y XY
NOR X+Y
XOR X  Y X Z Y
X Y Z X Z Y
X Y Z X Z Y
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13. Possible logic functions of two variables
There are 16 possible functions of 2 input variables:
in general, there are 2**(2**n) functions of n inputs X F Y
X Y possible functions (F0–F15) 1 X Y
not Y
not X
X xor Y
X = Y
X and Y
X nand Y
X or Y
X nor Y
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14. Cost of different logic functions
Different functions are easier or harder to implement
each has a cost associated with the number of switches (transistors) needed
0 (F0) and 1 (F15): require 0 switches, directly connect output to low/high
X (F3) and Y (F5): require 0 switches, output is one of inputs
X' (F12) and Y' (F10): require 2 switches for "inverter" or not-gate
X nor Y (F4) and X nand Y (F14): require 4 switches (cf. Katz page 677)
X or Y (F7) and X and Y (F1): require 6 switches
X = Y (F9) and X  Y (F6): require 16 switches
thus, because NOT, NOR, and NAND are the cheapest they are the functions we implement the most in practice
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15. Minimal set of functions
All logic functions can be implemented from NOT, AND, and OR
Can we do it with only 2 of the 3?
Can we implement all logic functions from NOT, NOR, and NAND?
For example, implementing X and Y is the same as implementing not (X nand Y)
In fact, we can do it with only NOR or only NAND
NOT is just a NAND or a NOR with both inputs tied together
X’ = X nand X X’ = X nor X
NAND and NOR are "duals", that is, it’s easy to implement one using the other
X nand Y  not ( (not X) nor (not Y) ) X nor Y  not ( (not X) nand (not Y) )
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16. From (logic) expressions to gates
More than one way to map expressions to gates
e.g., Z = A' • B' • (C + D) = (A' • (B' • (C + D))) T2 T1
use of 3-input gate A Z A B T1 B Z C C T2 D D
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17. What is the optimal gate realization?
We can use the theorems and axioms of Boolean algebra to optimize our designs
Design goals vary depending on the technology. For example:
Reduce the number of inputs? (variables, e.g., A, X and their complements, e.g., A are also called literals)
Reduce number of gates?
Reduce number of levels?
More options (e.g., limited fan-in and fan-out)
How do we explore the trade-offs?
CAD tools
Logic minimization: Reduce number of gates and complexity
Logic optimization: Minimization vs. speed and delay
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18. The Mathematics: Boolean algebra
A Boolean algebra is an algebraic structure that consists of
a set of elements B; B = {0, 1}
binary operations { + , • }; + is logical OR, • is logical AND
and a unary operation { ' }; ' is logical NOT
such that the following axioms (laws) hold:
1. the set B contains at least two elements, a, b, such that a  b 2. closure: a + b is in B a • b is in B 3. commutativity: a + b = b + a a • b = b • a 4. associativity: a + (b + c) = (a + b) + c a • (b • c) = (a • b) • c 5. identity: a + 0 = a a • 1 = a 6. distributivity: a + (b • c) = (a + b) • (a + c) a • (b + c) = (a • b) + (a • c) 7. complementarity: a + a' = 1 a • a' = 0
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19. Logic functions and Boolean algebra
Any logic function that can be expressed as a truth table can be written as an expression in Boolean algebra using the operators: ', +, and •
X Y X • Y
1 1 1
X Y X' X' • Y
X Y X' Y' X • Y X' • Y' ( X • Y ) + ( X' • Y' )
( X • Y ) + ( X' • Y' )  X = Y
Boolean expression that is true when the variables X and Y have the same value
and false, otherwise
X, Y are Boolean algebra variables
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