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COLLEGE

COQUITLAMEstablished 1982

Multifunction Digital Systems

Arithmetic Circuits (continued)

Multi-function Systems

February 09, 0!" #eoffrey $ien !

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%n&alid 'ut uts

• $he sum or difference of t o n-bit integers may roduce aresult not ithin the range of &alues re resented by n bits*

– Must be able to detect hether a result is &alid

•

+nsigned addition – +nsigned addition of t o n-bit o erands may roduce an -bit sum* %f

the carry-out of unsigned addition is 0 , the sum is &alid*

•

Signed addition &alid if t o o erands ha&e different signs – '&erflo addition of t o ositi&e o erands roduces negati&e sum

– +nderflo addition of t o negati&e o erands roduces ositi&e sum

•

February 09, 0!" #eoffrey $ien

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'ther Arithmetic Functions

• %ncrementing – Adding a fi.ed &alue to an arithmetic &ariable – Most often the fi.ed &alue is !, called counting (up) – e*g* , – %m lemented in a functional bloc/ called incrementer

• Decrementing – Subtracting a fi.ed &alue from an arithmetic &ariable – Fi.ed &alue is often !, called counting (down) – e*g* , – Functional bloc/ is called decrementer

•

February 09, 0!" #eoffrey $ien

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'ther Arithmetic Functions

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1ero fill – filling an m-bit o erand ith 0s to become an n-bit o erand ith – Filling is usually a lied to the MS2 (most significant bit) end of the

o erand, but can also be done on the 3S2 (least significant bit) end

• 4.am le !!!! 0!0! filled to !" bits – MS2 end 0000 0000 !!!! 0!0! – 3S2 end !!!! 0!0! 0000 0000

•

February 09, 0!" #eoffrey $ien 5

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4.tension

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4.tension of an m-bit signed o erand to n bits () hile reser&ing the o erand s numerical &alue

– de ends on the encoding used

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4.am le sign magnitude, 6 bits to !" bits – 00!0 !!0! becomes 0000 0000 00!0 !!0! – !0!0 !!0! becomes !000 0000 00!0 !!0!

• ! s com lement, 6 bits to !" bits – 00!0 !!0! becomes 0000 0000 00!0 !!0! – !0!0 !!0! becomes !!!! !!!! !0!0 !!0!

• s com lement, 6 bits to !" bits – 00!0 !!0! becomes 0000 0000 00!0 !!0! – !0!0 !!0! becomes !!!! !!!! !0!0 !!0!

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2us 8orts

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As the number of in uts out uts of an entity increases, ado t amore com act ay of re resenting entities and functionals ecifications*

– :hen a grou of bits are logically related (e*g* encoding integers ormulti le control signals), re lace the indi&idual signals ith a singlein ut or out ut ort called a bus ort

– An arro indicates in ut or out ut, and a slash ith a number indicatesthe si;e of the binary se

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2us 8orts

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Functional s ecifications associated ith entities that include bus orts can be e. ressed algebraically in terms of the bus ort labels

– $his is more com act than defining the beha&iour as a set of 2ooleane. ressions (i*e* a 2oolean function for each bit of out ut)

• e*g* a 5-bit full adder

February 09, 0!" #eoffrey $ien =

Functional s ecification

> denotes binary concatenation

c0 is filled to 5 bits

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Multi-Function Digital Systems

•

Many larger digital systems can erform one of se&eraldifferent ossible functions and send result to a single out ut

ort – Function select in uts are s ecial control in uts hich s ecify hich

function is to be erformed• A function select table enumerates the combinations of control

in uts and s ecifies an algebraic descri tion of the beha&iourfor the selected control in ut assignment

• Design a roach – design solution for each indi&idual function – incor orate necessary s itching logic (e*g* multi le.ers) to direct

selected function s out ut to the de&ice out ut ort


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•

Multi-function systems are characterised by these features – one or more data in uts – t o or more distinct beha&iours – one data out ut – control in uts to choose hich function to a ly

• e*g* a circuit that o tionally com lements a 5-bit in ut2eha&ioural descri tion


function

0

!

function

0

!

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•

$he system on the re&ious slide has 7 bits of in ut – the function table ill ha&e ro s – determine the solution to a sim ler &ersion of the roblems, i*e* ith

only ! bit of in ut* – $his roblem-sol&ing a roach is called bit-slice design – !-bit com lementer function table

– $his is sim ly an ?'@ gate* @e licate for the other bit ositions toobtain the 5-bit solution

•

February 09, 0!" #eoffrey $ien !0

4.am le !-bit com lementer

0 0 0

0 ! !

! 0 !

! ! 0

0 0 0

0 ! !

! 0 !

! ! 0

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•

2eha&ioural descri tion

– the functional s ecification consists of a function select table

describing and , and a 2oolean e. ression defining• is defined the same ay regardless if or

•

February 09, 0!" #eoffrey $ien !!

4.am le Adder Subtractor

function

0

!

function

0

!

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•

A 5-bit full adder can be used to im lement the addition – substiting FA5 s orts, e obtain

• $o im lement subtraction as s com lement

–

FA5 can be used to im lement subtraction if is first com lemented,and is set to !

• use the 5-bit com lementer from earlier e.am le• control in uts on com lementer and on FA5 must be defined as functions

of on the adder subtractor

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0 0 0

! ! !

0 0 0

! ! !

ill be connected directly to and 0

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• A se arate circuit can im lement the &alidity out ut as a sumof roducts

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4.ercise

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Design a multi le.er ith 5-bit data in uts, and use it todesign a system ith the follo ing beha&ioural descri tion

•

February 09, 0!" #eoffrey $ien !5

function

0

!

function

0

!

A

2

s e

l

D

5

5

5

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08 Multifunction 1161111