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## Logic Gate Level

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### Logic Gate Level

Combinational Circuits, Part 1

Circuits

- Circuit: collection of devices physically connected by wires to form a network
- Net can be:
- combinational: input determines output
- sequential: output not entirely dependent on input
- We will focus first on combinational networks

Black box view of network

Inputs: a, b, c

Outputs: x, y

Describing combinational networks

- 3 common methods for describing combinational networks, ordered from most to least abstract:
- Truth tables
- Boolean expressions
- Logic diagrams

Truth tables

- Most like black box view:
- specifies what network does, not how
- lists output for every possible combination of input values
- Next slide illustrates truth table for a 3-input, 2-output combinational network
- Note that the table has 8 entries – in general, n inputs translates to 2n entries

Boolean algebra

- A Boolean algebraic expression describes both what a network does and how it does it
- Boolean values:
- 1 = true
- 0 = false
- Binary operations in order of precedence:
- = AND (may be omitted; ab same as a b)
- + = OR
- Unary operation: ’ = NOT

Fundamental properties of Boolean algebra

Note, in particular, how the distributive property of Boolean algebra differs from regular algebra – in regular algebra, multiplication (denoted with ) distibrutes over addition (+), but not vice versa, as the operations represented by these symbols (AND and OR) do in Boolean algebra

Duality

- Boolean algebra has symmetry that doesn’t exist in regular algebra: every property (and thus every expression) has a dual
- To construct the dual of an expression:
- exchange + and operations
- exchange 1 and 0
- Examples
- the dual of c(a + b) is c + ab
- the dual of x + 0 is x 1

Simplifying expressions

- Duality is a useful property in the simplification of expressions
- Associative properties also permit simplifications; when a single operator is involved in a compound expression, parentheses can be removed
- Example:
- (a + b) + c = a + b + c

Idempotent property

- Weird and fancy way to say that any value ORed with itself yields the same value:

x + x = x

- Proof:

x + x = (x + x) 1 by the identity property

(x + x) 1 = (x + x) (x + x’) by complement

(x + x) (x + x’) = x + (x x’) by distribution

x + (x x’) = x + 0 by complement

x + 0 = x by identity

- Same sequence of steps can be used to prove the dual – so any value ANDed with itself will also yield the same value

Notes on proofs

- A proof is a sequence of statements in which substitutions based on the fundamental properties of Boolean algebra are used to demonstrate the validity of a theorem
- Once a theorem is proven, we can immediately assert that its dual is true

More Boolean algebra properties

- Absorption:
- x + 1 = 1
- x 0 = 0
- x + xy = x
- x(x + y) = x
- DeMorgan’s Laws:
- (ab)’ = a’ + b’
- (a + b)’ = a’b’

More Boolean algebra properties

- To prove that one expression is the complement of another, show that:
- expr1 + expr2 = 1 and
- expr1 expr2 = 0
- DeMorgan’s laws:
- illustrate proving complements
- generalize to more than two variables – e.g. (abc)’ = a’ + b’ + c’
- Complements of complements:
- (x’)’ = x
- 1’ = 0
- 0’ = 1

Another useful operation

- Although any Boolean expression can be written as the combination of AND, OR and NOT operations, other operations are common
- The XOR (exclusive or) operation, denoted by the symbol has the following truth table for 2 variables (and generalizes, as the other operations do, to more than 2):

a b a b

1 1 0

1 0 1

0 1 1

0 0 0

- XOR takes precedence over OR, has lower precedence than AND

Logic diagrams

- Logic diagrams are drawings representing the most detailed version of a circuit
- Each Boolean operation (AND, OR, NOT) is represented by a gate symbol with multiple inputs and a single output
- The lines in the drawing represent the wires in the circuit

Logic diagrams

- It is possible to read a logic diagram as a kind of blueprint for a circuit, and to construct physical circuits from such blueprints
- It is actually far more common to use a slightly different set of gates in physical circuits, as these are easier to construct
- The next slide illustrates three of the most common gates

Common logic gates

NAND and NOR gates are logically equivalent to AND and OR gates followed by inverters. Since NAND is actually easier to build, ANDs are often constructed as inverted NAND gates.

Relationship between various representations of combinational nets

- As the illustration below suggests, there is a one-to-one correspondence between a boolean expression and a logic diagram
- Each of these might correspond to a single line in a truth table

Constructing logic diagram from Boolean expression

- For every draw a
- For every + draw a
- For every ’ draw a
- Show input(s) as line(s) coming into gate, output as line coming out
- Final output is output from final gate (not connected to input of another)

More complicated expression

- The logic diagram above depicts ((ab + bc’)a)’
- For a parenthesized expression, construct parenthesized portion first
- Note the connectors () used to indicate a continuation of the same input (for a and b)

Same expression, abbreviated

- Two inverters eliminated: first by including c’ as an input (instead of c) and second by making the third AND gate a NAND gate
- In this version, inputs a and b are shown multiple times – use same symbols, so we know they’re the same inputs

One more example

- To translate logic diagram to Boolean expression:
- label output of each gate (starting from left) with appropriate sub-expression
- output of last gate is full expression

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