SoP-form with three minterms. 3 1 f =...

Minterms

010101)3,2,1( xxxxxxmf ++== ∑William Sandqvist [email protected]

111101110000

3210

01 fxx

OR A minterm is a product of all variables and it describes the combination of ”1” and ”0” that toghether makes the term to adopt the value 1.

SoP-form with three minterms.

Minimization with Boolean algebra

010101)3,2,1( xxxxxxmf ++== ∑

William Sandqvist [email protected]

111101110000

3210

01 fxx

OR

Simplification with boolean algebra 10

1110

01101

0101

101

00101

)(

)1(

...)1(

)(

xxxxxx

xxxxx

xxxx

xxx

xxxxx

+==++=

=++=

=++=

=++=

=++=

As expected!

Maxterm


111101110000

3210

01 fxx

OR A maxterm är en sum-factor of all variables and it describes the combination of ”1” and ”0” that toghether makes the sum to adopt the value 0.

10)0( xxMf +== ∏This time we got the simple expression direct with the maxterm!

Venn-diagram OR

In a Venn diagram, one can see that a function can be expressed in a variety of ways, but it is not easy to see what is optimal. Another question is also how to draw Venn diagrams for more than three variables??


All minterms

OR, some of the minterms

Graphical minimization method


111101110000

3210

01 fxx

OR 0 1 1 1

x0 x1

0

1

0 1

m0 m1

m2 m3

x0 x1

0

1

0 1

1001

010132

)( xxxx

xxxxmm

=+=

=+=+

0110

010131

)( xxxx

xxxxmm

=+=

=+=+

01 xxf +=

Graphical minimization method


111101110000

3210

01 fxx

OR 0 1 1 1

x0 x1

0

1

0 1

01 xxf +=

Make ”groups” two ones that are "neighbors" (vertically or horizontally) the minterms could then be reduced to "what they have in common".

1x0x

Gate functions Graphical form


0 0 0 1

0 1 1 1

0 1 1 0

1 1 1 0

1 0 0 0

1 0 0 1

AND OR XOR

NAND NOR XNOR

x0 x1

0

1

0 1 x0 x1

0

1

0 1 x0 x1

0

1

0 1

x0 x1

0

1

0 1 x0 x1

0

1

0 1 x0 x1

0

1

0 1

01 xx +01 xx ⋅ 0101 xxxx +

{ } 0101 xxdMxx +⋅{ } 0101 xxdMxx ⋅+ 0101 xxxx +

3D boolean numberspace


The corners are coded with Gray-code. Between neighbor corners only one variable differs.

Ex. 3.4 Cube representation


012012012012012

012 )6,4,3,2,0(),,(

xxxxxxxxxxxxxxx

mxxxf

++++=

== ∑

This is the way to represent a function of three variables, on a 3D cube with Gray-coded corners.

Ex. 3.4 minimization with cube


A surface is represented by a variable, a side of a product term with two variables, and a corner a minterm with three variables. Cube methods can be generalized to "Hyper Cubes" with any number of variables.

012 xxx

0x

12xx

Hypercubes


A corner is a 0-dimension subspace, A side is a 1-dimension subspace, A surface is a 2-dimension subspace, A cube is a 3-dimension subspace …

There are minimization methods for hypercubes and they can apply for any number of variables! The methods based on hypercubes are suited to computer algorithms.


Graycode is a mirrored binarycode

One can easily construct a Gray code with an arbitrary number of bits needed for numbering the "corners" in "hypercubes" with!


























And so on …



How do you draw a 3D-cube?


You draw two 2D-cubes ( = squares), and then connects their corners.


How do you draw a 3D-cube? You draw two 2D-cubes ( = squares), and then connects their corners.

How do You draw a 4D-cube?


You draw two 3D-cubes (=cubes), and then connects their corners.


How do You draw a 4D-cube? You draw two 3D-cubes (=cubes), and then connects their corners.

4D-cube in Paris


4D-Donut


≡

4D-hypercube can be represented by a torus, a donut.

4D-cubes in USA


3D-cube ⇒ 2D-map


00

a b c

a 0 1

b c 01 11 10

Gray-code → mirror

b = 0

b = 1

000 001

100 101

011 010

111 110

Karnaugh-map


Graphical method for minimization with "paper and pencil" minor Boolean functions? (For up to six variables)

Maurice Karnaugh

( The Map for Synthesis of Combinational Logic Circuits, AIEE, Nov. 1953 )


A function of four variables a b c d. Truth Table with 11 ”1” and 5 ”0”. The function can be expressed in SoP-form with 11 minterms or in PoS-form with 5 maxterms.


4D-cube ⇒ 2D-map The Karnaugh map is the truth table lined up in a different way.

The frames are ordered in such way that only one bit changes between two vertical frames or horisontal frames. This order is called Gray-code.


Two "neighbors" The frames "5" and "13" are "neighbors" in the Karnaugh map ( but they are distant from each other in the truthtable ). They correspond to two minterms with four variables, and the figure shows how, with Boolean algebra, they can be reduced to one term with three variables.

What the two frames have in common is that b = 1, c = 0 and d = 1; and the reduced term expresses just that.

Everywhere in the Karnaugh map where one can find two ones that are "neighbors" (vertically or horizontally) the minterms could be reduced to "what they have in common". This is called a grouping.

dcb


Four "neighbors" Frames "1" "3" "5" "7" is a group of four frames with "1" that are "neighbors" to each other. Here too, the four minterms could be reduced to a term that expresses what is common for the frames, namely that a = 0 and d = 1. Everywhere in Karnaugh map where one can find such groups of four ones such simplifications can be done, grouping of four.

da


Eight "neighbors"

All groups of 2, 4, 8, (... 2 N ie. powers of 2) frames containing ones can be reduced to a term, with what they have in common, grouping of n.

a


Karnaugh - toros

The Karnaugh map should be drawn on a torus (a donut). When we reach an edge, the graph continnues from the opposite side! Frame 0 is the "neighbor" with frame 2, but also the "neighbor" with frame 8 which is "neighbor" to frame 10.

db

4

12

5

13

7

15

The optimal groupings? One is looking for the bigest grouping as possible. In the example, there is a grouping with eight ones (frames 0, 1, 3, 2, 4, 5, 7, 6). Corners (0, 2, 8, 10) is a group of four ones. Two of the frames (0.10) has already been included in the first group, but it does not matter if a minterm is included multiple times. All ones in the logic function must either be in a grouping, or be included as a minterm. The "1" in frame 13 may form a group with "1" in frame 5, unfortunately there are no bigger grouping for this "1".


dcbdbadcbaf ++=),,,(


Grouping of "0"

The Karnaugh map is also useful for groupings of 0's. The groupings may include the same number of frames as the case of groupings of 1's. In this example, 0: s are grouped together in pairs with their "neighbors". Maxterms are simplified to what is in common for the frames.

))()((),,,( dbadcadbadcbaf ++++++=


Maps for other number of variables

Karnaugh maps with three and two variables are also useful.

The Karnaugh map can conveniently be used for functions of up to four variables, and with a little practice up to six variables.

6

Minimization Example


000 100

010 110

101

011 111

a

b

c 1 1

1 1 1

00 01

0

1

11 10 bc

a 001

)7,5,3,2,0(),,( ∑= mcbaf

0 1 3 2

4 5 7 6 0

2

5

3 7

Implicants


000 100

010 110

101

011 111

a

b

c 1 1

1 1 1

00 01

0

1

11 10 bc

a

Circle the adjacent min-terms

001

)7,5,3,2,0(),,( ∑= mcbaf

Little implicant terminology


• Implicant – a group of min-terms • Prime-implicant – a group of minterms that cant be made bigger. • Essential prime-implicant – must be included in order to cover the function. • Redundant prime-implicant – must not be included, the function can be covered by other implicants.

Implicants


000 100

010 110

101

011 111

a

b

c

1 1

1 1 1

00 01

0

1

11 10 bc

a

Redundant implicants - not both are necessary (one of them must be included) to cover the function.

001

)7,5,3,2,0(),,( ∑= mcbafacbacaf

acbccaf

++=

++=

Two-level minimization


Minimal Sum-Product Implementation


a b c

f

1 1

1 1 1

00 01

0

1

11 10 bc

a

)7,5,3,2,0(),,( ∑= mcbaf

acbccaf ++=

Programmable Logic Array (PLA)


Programmable AND and OR - matrix. The programming consists of "burning fuses" to the connections you do not want. (now an obsolete technology)



The Gate-matrix has so many inputs that one use to drav the circuit in a simplified way.


OR plane

AND plane


PLA. Both AND- and OR-matrix are programable

( a flexibility that costs )

1x 2x 3x

1f 2f

1P

2P

3P

4P

Simplified drawing

Programmable Array Logic (PAL)


1x 2x 3x

1P

2P

3P

4P

1f

2f

PAL. Only the AND-matrix is programable.

(an economic compromise)

PAL, what functions hides behind the crosses?


1x 2x 3x

1P

2P

3P

4P

1f

2f

?

?

Sub-cube with four varibles


1

1 1 1 1

1 1

00 01 11 10

00

01

11

10

x1x0

x3x2

We always group a full sub-cube when possibly!!!

3xf =

1


1 1

00 01 11 10

00

01

11

10

x1x0

x3x2

0xf =

1 1 1

1 1 1

Sub-cube with four varibles

We always group a full sub-cube when possibly!!!

XOR can be helpful


1 1 1 1 1 1

1 1

00 01 11 10

00

01

11

10

x1x0

x3x2 If two groups of four minterms can not form a group of eight, the XOR / XNOR function may be helpful.

020202 xxxxxxf ⊕=+=

This is under the assumption that there exists an efficient implementation of the XOR function.

4D the order of the Minterms ...


0 5

3 2 7

00 01

00

01

11

10

11 10

1 4 6

MSB LSB

12 9

15 14 11

13 8 10

)14,7,3,0(),,,( 0123 ∑= mxxxxf

x1x0

x3x2

5D five variables

1

00 01 11 10

00

01

11

10

x1x0

x3x2

1

00 01 11 10

00

01

11

10

x1x0

x3x2

x4 0 1


Same in both diagrams, independent of x4. 0123 xxxx

04 =x 14 =x

Mirroring with 5 variables


1

110 111 101 100

1

000 001 011 010

00

01

11

10

x4x1x0

x3x2

0123 xxxx

Mirror!

And with an other order …


110 111 101 100

1

000 001 011 010

00

01

11

10

x2x1x0

x4x3

1

0123 xxxx

Karnaugh-map with 6 variables


x4

0 1

1

00 01 11 10

00

01

11

10

1

00 01 11 10

00

01

11

10

1

00 01 11 10

00

01

11

10

1

00 01 11 10

00

01

11

10

x3x2

x1x0

x3x2

x1x0

x3x2

x1x0

x3x2

x1x0

0

x5

1

0123 xxxx

Independent of x5 and x4.

04 =x 14 =x

05 =x

15 =x

Mirroring with 6 variables


1

110 111 101 100

1

000 001 011 010 000

001

011

010

110

111

101

100

x4x1x0

x5x3x2

1

1

0123 xxxx

Note!


1

110 111 101 100

1

000 001 011 010 000

001

011

010

110

111

101

100

x2x1x0

x5x4x3

1

1

0123 xxxx

And with an other order …

Groupings of 0’s


Group the zeroes if they are fewer than the ones!

1 1 1 1 1 0 1 1 1 1 1 1 1 1 1

00 01 11 10

00

01

11

10

x1x0

x3x2

1

{ } )( 0123

0123

xxxxdM

xxxxff

+++==

===

”0” as minterm – totaly wrong!

invert, then its correct! 0123 xxxxf =

Don’t care


• Often you can simplify the specification of the logical function because we know that certain combinations can never occur • For these combinations, we use a value of "do not care" • There are different symbols for "do not care" in use ’d’, ’D’, ’-’, ’Φ’, ’x’

Incompletely specified functions


x 1 x 0 x 3 x 2

0

00 01 11 10

0 0 1

1 1 0 1

d d d d

0 0 0 1

00

01

11

10

x 2 x 1

x 1 x 0

(a) SOP implementation

Two implementations of the function f ( x3,…, x0) = Σ m(2, 4, 5, 6, 10) + D(12, 13, 14, 15).

0

00 01 11 10

0 0 1

1 1 0 1

d d d d

0 0 0 1

00

01

11

10 x 2 x 1 + ( )

x 1 x 0 + ( )

(b) POS implementation

x 1 x 0 x 3 x 2

Another notation (-) …


x 1 x 0 x 3 x 2

0

00 01 11 10

0 0 1

1 1 0 1

- - - -

0 0 0 1

00

01

11

10

x 2 x 1

x 1 x 0

(a) SOP implementation

0

00 01 11 10

0 0 1

1 1 0 1

- - - -

0 0 0 1

00

01

11

10 x 2 x 1 + ( )

x 1 x 0 + ( )

(b) POS implementation

x 1 x 0 x 3 x 2

Two implementations of the function f ( x3,…, x0) = Σ m(2, 4, 5, 6, 10) + D(12, 13, 14, 15).

Funktions with multiple outputs


Different outputs can share prime-implicants!!!

0301 xxff +=

Multi-level synthesis


Do You need mere levels than two?


One can realize any combinational circuits with two levels(AND-OR, OR-AND)

– The assumption then is that all inputs are also available in inverted form? (as in PAL, PLA)

Why multi-level logic?


• The number of inputs in a circuit could be limited

• High fan-in leads to long delays

• It may be more cost effective to use an implementation with more levels

Two strategies for multi-level logic


1 Factoring 2 Functional decomposition

Factoring


f = x1x 2x3x 4 x5x6 + x1x2x 3x 4 x 5x6

We have only AND gates with a maximum of 4 inputs?

Factor the expression

)( 532532641 xxxxxxxxxf +⋅=

641 xxx

Factoring


)( 532532641 xxxxxxxxxf +⋅=

x 6

x 4

x 1

x 5

x 2

x 3

x 2

x 3

x 5

Functional decomposition


• You can often reduce the complexity of a logic function by reusing functions multiple times • To implement this it means to reuse a "circuit" at several points in the construction



f = x 1x2x3 + x1x 2x3 + x1x2x4 + x 1x 2x4

Functions can be seen as composed of sub-functions.



f = x 1x2x3 + x1x 2x3 + x1x2x4 + x 1x 2x4

= (x 1x2 + x1x 2)x3 + (x1x2 + x 1x 2)x4XOR XNOR

f = gx3 + g x4

= g = g

Can You think up this? …

Factor x3 respective x4

( XOR och XNOR )


0 1 1 0

1 0 0 1

XOR XNOR x2 x1

0

1

0 1 x2

x1

0

1

0 1

2121 xxxx + 2121 xxxx +

XNORXOR =

With an efficient implementation of the XOR gate (with few transistors) you can see solutions in Karnaugh map even when it is not possible to make any groups!



1 x 2

x 3 x 4

f

g

h

x

x 1

x 2

x 3

x 4

f g Implementation

2121 xxxxg +=

f = gx3 + g x4

Algorithms for minimization


• Karnaugh minimization gives a good insight into how to minimize logic functions

• But to minimize complex functions using computer there are better algorithms

• Chapter 4.9 and 4.10 in Brown / Vranesic provides an introduction to the minimization algorithms (for the interested student)

Summary


• Karnaugh map is a good tool for minimizing logic functions with few variables

• There are algorithms for both two-level and multi-level minimization

Date post:	15-Nov-2019
Category:	Documents
Upload:	others
View:	2 times
Download:	0 times

SoP-form with three minterms. 3 1 f =...

Documents