11.1 Introduction to Combinational Logic Synthesis (1)

7/28/2019 11.1 Introduction to Combinational Logic Synthesis (1)

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INTRODUCTION TO COMBINATIONAL

LOGIC SYNTHESIS

N.ABISHA


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INTRODUCTION

COMBINATIONAL

Logic circuits that do not possess an internal state

Built from elementary logic gates ,such as NAND

and NOR

Cannot have directed cycles-memory elements


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BEHAVIOR

Consists of m inputs and n outputs can be described by a

Boolean function f: , where B={0,1} .

Combinational logic synthesis problem be stated as the

problem of generating some circuitry that will implement the

behavior given by a Boolean function while minimizing the

cost function.

Practically, function is incompletely specifiedbecause

values of some outputs do not matter for some specific input

patterns.

They require third output value called the dont care denoted

by -.

nm

BB


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contd

Thus, the Boolean function is redefined as , where

Y={0,1,-}.

Inputs are partitioned to three sets:

On-set output should be 1

Off-set - output should be 0

Dc-set output is dont care

A Boolean function whose dc-set is empty is called ful ly

specified.

nm

YB:f


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Minterm

The term literal denotes a Boolean variable or its complement .

Any point in can be identified by a product of the m distinct

literals . Such a product is called a minterm.

So, - minterm for

mB

321.. xxx

3B

fig.(1)


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Boolean Function

Fully specified Boolean function is described by the points in

its on-set, it can be specified by sum ofminterms.

Consider, the on-set of the Boolean function given in figure,

the sum of minterms corresponding to this function is :

321321321321321321x.x.xx.x.xx.x.xx.x.xx.x.xx.x.xf

fig.(3)


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contd

The main problem of using the sum-of-products canonical

form in practice is the size of the representation.

For a Boolean function with m variables has points which

means that a SOP will consists of O( ) minterms.

The specification of a Boolean function by its SOP canonical

form is very similar to a specification by means of a truth

table.

The truth table enumerates all points in Boolean space and

states for each point whether it belongs to the on-set, off-set or

dc-set.

m

2

m2


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contd

By definition , any truth table for a function ofm Boolean

variables will have exactly entries.m2

Fig.(4) The truth table for function of fig.(3)


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d


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contd In this example, it turns out that two irredundant prime covers

are possible, corresponding to the expressions:

They are illustrated in the fig.5(a) and (b).

3132212

2132311

...

...

xxxxxxf

and

xxxxxxf

Fig.(5 )two possible coverings by prime implicants(a , b)


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contd

Although the last two expressions are both minimal, the

existence of two different expressions shows that minimization

alone does not lead to a canonical representation.

The complete set of prime implicants off a function , on the

other hand, is a canonical form, calledBlakes canonical

form.


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Parity function The representation general cubes instead of minterms does not

always lead to a reduction of the representation size. A well known fully specified function with this property is the

par ity function.

Its on-set consists of those minterms in which the number of

non complemented Boolean variables is odd.

Fig.(6)


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Example

library ieee;

use ieee.std_logic_1164.all;

entity example is

port (x1,x2,x3,x4,x5:in std_logic;

y1,y2:out std_logic);

end example;

architecture behavioral of example isbegin

react:process(x11,x2,x3,x4,x5)

begin

if x1='1' and x2='0'

then

y1


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contd

The on-sets are given by:

and the dc-sets are given by:


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Boolean algebra such as:


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Thank you

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11.1 Introduction to Combinational Logic Synthesis (1)