1 COMP541 Combinational Logic - 3 Montek Singh Jan 21, 2015.

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COMP541

Combinational Logic - 3

Montek Singh

Jan 21, 2015

Today’s Topics

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Synthesis: from truth table to logic implementation

Schematic drawing conventions

Non-Boolean values“Don’t Cares”, or X values“Floating values”, or Z values

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Mechanically Go From Truth Table to Function

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From Truth Table to Logic Equation Consider a truth table

Standard sum-of-products implementationOR of all product terms that are 1For each row where output is 1

– write the minterm» called “ON-set minterm”

– OR all of these minterms

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Standard Forms Not necessarily simplest F

But it is a systematic way to go from truth table to function

Definitions:“Literal”: a single variable, complemented or not Ā“Product terms”: AND of literals ĀBZ“Sum terms”: OR of product terms X + ĀThis is logical product and sum, not arithmetic

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Definition: Minterm Product term in which all variables appear

once (complemented or not)each minterm is 1 in exactly one row, 0 elsewhere

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Number of Minterms For n variables, there will be 2n minterms

Like binary numbers from 0 to 2n-1Often numbered same way (often in decimal)

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Maxterms Sum term in which all variables appear once

(complemented or not)each maxterm is 0 in exactly one row, 1 elsewhere

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Minterm related to Maxterm Minterm and maxterm with same subscripts

are complements

Example

33 MZYXYZXm

Mjm j

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Implementation: Sum of Minterms OR all of the minterms of truth table row with a

1“ON-set minterms”F = m0 + m2 + m5 + m7

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More General: Sum of Products Simplifying sum-of-minterms can yield a sum

of productsdifference is that each term need not be a minterm i.e., terms do not need to have all variables

Ex:

Implementation is still AND-ORbut products may contain fewer literals

simplifies to:

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Two-Level Implementation Sum of products has 2 levels of gates

ANDs followed by an ORequivalently: NANDs followed by a NAND

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More Levels of Gates? What’s best?

Hard to answerMore gate delays (more on this later)But maybe we only have 2-input gates

So multi-input ANDs and ORs have to be decomposed

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Complement of a Function Definition: 1s & 0s swapped in truth table Mechanical way to derive algebraic form

Take the dualRecall: Interchange AND and OR, and 1s & 0s

Complement each literal

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Complement of F Not surprisingly, just sum of the other

mintermssum of “OFF-set minterms”

Example:F = m0 + m2 + m5 + m7F’ = m1 + m3 + m4 + m6

simplifies to:

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Product of Maxterms Recall that maxterm is true except for its own

case So M1 is only false for 001

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Product of Maxterms Can express F as AND of all rows that should

evaluate to 0 i.e., product of OFF-set Maxterms!why?

a row in which F=0 (OFF-set)…… has a Maxterm that is 0which makes the product 0

or

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Complement of F Can express F’s

complement similarly:product of ON-set Maxterms!why?

a row in which F=1 (ON-set)…… has a Maxterm that is 0which makes F’ zero

or

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More General: Product of Sums Simplifying product-of-Maxterms can yield a

product of sumsdifference is that each term need not be a Maxterm i.e., terms do not need to have all variables

Ex:

Implementation is still OR-ANDbut each sum may contain fewer literals

simplifies to: HOW?? homework problem(hint: distributive property)

From Equations to Gates Simply parse the Boolean equation and

replace each operator with a gateAND, OR, NOT gatesparentheses indicate hierarchy

Example:

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Recap Working (so far) with AND, OR, and NOT Algebraic identities Algebraic simplification Minterms and maxterms Can now synthesize gate-level implementation

from truth table

Drawing Style Indicate inputs and outputs using

arrowsor: inputs at left/top, outputs at

right/bottom If possible, gates should flow from

left to rightor: top to bottom

Straight wires bestor: keep bends at a minimum (preferably

90 deg) Connections:

wires always connect at a “T” junctiona dot at a wire crossing indicates

connectionwire crossing without a dot means no

connection

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Circuit Schematic Rules (cont.)Wire connections

A dot where wires cross indicates a connectionWires crossing without a dot make no connectionWires always connect at a T junction

wires connectat a T junction

wires connectat a dot

wires crossingwithout a dot do

not connect

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Multiple Output Circuits: Example

A0

A1

PRIORITYCiIRCUIT

A2

A3

Y0

Y1

Y2

Y3

0

A1 A00 00 11 01 1

0

00

Y3 Y2 Y1 Y00000

0011

0100

A3 A20 00 00 00 0

0 0 0 1 0 00 10 11 01 10 0

0 10 10 11 0

0 11 01 01 10 00 1

1 01 01 11 1

1 01 11 11 1

0001

1110

0000

0000

1 0 0 01111

0000

0000

0000

1 0 0 01 0 0 0

Output assertedcorresponding tomost significantTRUE input

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Example: Priority Encoder Hardware

Example: Priority Encoder Hardware (contd.)

A1 A00 00 11 01 1

0000

Y3 Y2 Y1 Y00000

0011

0100

A3 A20 00 00 00 0

0 0 0 1 0 00 10 11 01 10 0

0 10 10 11 0

0 11 01 01 10 00 1

1 01 01 11 1

1 01 11 11 1

0001

1110

0000

0000

1 0 0 01111

0000

0000

0000

1 0 0 01 0 0 0

A3A2A1A0Y3

Y2

Y1

Y0

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Values that are not 0’s and 1’s

Don’t Cares (X)Floating values (Z)

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X values X is neither 1 nor 0

typically used to represent “unknown” or “illegal” values

Unknowne.g., an uninitialized value in a simulator

in hardware most flipflops will wake up to a 1 or a 0 value– but could be different each time it wakes up

Don’t Carean output specified as X means “don’t care”

i.e., left unspecified: whatever comes out is okay Illegal

e.g., contention at outputtwo gates fighting

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Actually: Several Meanings of X

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When used to specify an input valueMeans: “Don’t Care”: this particular input variable’s

value does not matter when determining the outputExample: Output F is 1 when the inputs A, B, C are

1X1Means F = AC // B is a Don’t Care

Unknown/uninitialized signal If a simulator cannot determine the value of a signal,

it will display it as XOther values that depend on this signal may also

become X Contention (illegal input value)

Sometimes a simulator will use X to denote the value of a node that is being pulled both to 0 and to 1

Example: Outputs of two gates are shorted; or a gate has p-transistor and n-transistor network simultaneously on!

Don’t Cares (X)

A1 A00 00 11 01 1

0000

Y3 Y2 Y1 Y00000

0011

0100

A3 A20 00 00 00 0

0 0 0 1 0 00 10 11 01 10 0

0 10 10 11 0

0 11 01 01 10 00 1

1 01 01 11 1

1 01 11 11 1

0001

1110

0000

0000

1 0 0 01111

0000

0000

0000

1 0 0 01 0 0 0

A1 A00 00 11 XX X

0000

Y3 Y2 Y1 Y00001

0010

0100

A3 A20 00 00 00 1

X X 1 0 0 01 X

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Example: Priority Encoder Hardware

More compact representation!

Z values Also neither 1 nor 0

but actually “floating” i.e., the output is neither connected to 0

(ground) nor to 1 (power supply) Could be undesirable:

actual voltage is highly susceptible to noisee.g., neighboring wires/gates could easily

influence value Could be by design:

useful in buses, memories, multiplexers, etc.usually one gate drives a wire to a 1 or 0all others “float” their outputs

example: tristate buffers/inverters cover in next lecture

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Next Mon next week:

Combinational building blocks

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