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Finite State Mac ine (FSM! Design

• FSMs are different from counters in the sense that they have external!"s# and state transitions are de$endent on these !"s and the current

state%• &xam$le ' "ro(lem StatementThere is a (it)serial !" line% Desi*n an FSM that out$uts a +,-if an even . of 1-s have (een received on the !" line and theout$uts a +1- other/ise%

Note ' f a synchronous se0uential circuit is (ein* desi*ned# the countin*of the . of 1s occur every cloc cycle%

FSMx !$ y

345 345

x

. of 1s

even6,7

odd617

even627

odd687 odd687

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<

FFs

&xternal !"s &xternal !"s

m1 m2

nn

3om(%4o*ic

345

FFsn

345

n

ut$ut4o*ic

m2

Next State3om(%4o*ic

m1

&xternal!"s

&xternal ut$uts

Mealy Machine Model Moore Machine Model

even ↓odd

Time t ' &ven !" ∆ = $ro$a*ation delay of lo*ic of Mealy M!3

t t>∆ t>T 345 t>T 345 >∆2

&venx=1

!"=,

!"=1

6Mealy7

dd !"=1

6Moore7

∆2 = $ro$a*ation delay of !"

lo*ic unit of Moore M!3

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?

State Transition Ta(le

6&ven)"arity 3hec er7

&ven State' , @ dd State' 1@ State aria(le A

A x A > y 1 y 2 D A TA, , , , , , ,

, 1 1 , 1 1 1

1 , 1 1 1 1 ,

1 1 , 1 , , 1

"resentState

n$ut NextStateMoore

!"Mealy

!"D)FF&xcit%

T)FF&xcit%

n$ut varia(lesto com(% lo*ic

B

FF

N%S% C !"

4o*ic

345

x y2

A D A

r

FFs

y1

N%S%4o*ic

!"4o*

ic

DAAB

x

DA= A⊕x @ TA= xy1 = A for Moorey2 = A⊕x for Mealy

ut$ut functions

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State=,&ven

State=1dd

1!11!,

,!1

9eset ,State=,&ven

11

,

9eset:,;

State=1dd

:1;

x

FF

N%S%4o*ic

345

B

B DD)

Mealy MooreAssume sin*le (it state information stored in a D)FF

345

x

D

B6s

tate7y26Mealy !"7y1 Moore !"

State Transition is occurrin*

State Transition is occurrin*

S%T% is com$lete%

S%T% is com$lete%

oddoddevenevenoddeven

,!,

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Moore M!3 m$lementation

D B

B9 345

y2x=1

A,

a7 D)FF

T B

B9

A y 2x

345

(7 T)FF

Moore !" is synchroni ed /ith cloc %

Mealy M!3 m$lementation

D B

B9 345

y1

x=1A

, 1 T B

B9

x

345

y1

a7 D)FF (7 T)FF

Mealy !" is not synchroni ed /ith cloc %

9eset

9eset

9eset

9eset

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Di erence 3et1een Mealy an$ Moore Mac ine

Mealy Moore617 !"s de$end on the $resent !"s de$end only on the

state and $resent !"s $resent state627 The !" chan*e asyn Since the !"s chan*e)chronously /ith the /hen the state chan*es#

ena(lin* cloc ed*e and the state chan*e is synchronous /ith the ena(lin* cloc ed*e# !"s chan*e synchronously

/ith this cloc ed*e687 A counter is not a Mealy A counter is a Moore

machine machine6

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Another exam$le' A sim$le vendin* machineHere is ho/ the control is su$$osed to /or % The vendin* machine delivers

a $ac a*e of *um after it has received 1? cents in coins% The machine has a sin*lecoin slot that acce$ts nic els and dimes# one coin at a time% A mechanical sensor

indicates to the control /hether a dime or a nic el has (een inserted into the coin slot%The controller-s out$ut causes a sin*le $ac a*e of *um to (e released do/n a chuteto the customer%

ne further s$ecification' Ie /ill desi*n our machine so it does not *ivechan*e% A customer /ho $ays /ith t/o dimes is out ? centsJ

endin*Machine

FSM

345

9eset

3oinSensor Kum9eleaseMechanism

$en

endin* Machine (loc dia*ram

,3 ≥1?31,3?3

States'

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1,

L The fi*ure (elo/ sho/ the Moore and Mealy machine state transition dia*rams%

, cent:,;

? cent:,;

1, cent:,;

1? cent:1;

Moore machine

1? cent

1, cent

? cent

, cent

Mealy machine

Moore and Mealy machine state dia*rams for the vendin* machine FSM

9eset ! , N 6 N 6 + D9eset

9eset9eset ! ,

N ! ,

D ! ,

N ! ,

N>D!1 N>D

N

D

N

D

D!1

N D

N ,! D

N D N ,! D

9eset 7!, + D 9eset 7!,

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B 1 B 2 D N B 1> B 2> $en $en

, , , , , , , , , 1 , 1 , , 1 , 1 , , , 1 1 x x x x , 1 , , , 1 , , , 1 1 , , , 1 , 1 1 , 1 1 1 x x x x 1 , , , 1 , , , , 1 1 1 , 1 1 , 1 1 , 1 1 1 x x x x 1 1 , , 1 1 1 1 , 1 1 1 1 1 1 , 1 1 1 1 1 1 x x x x

"resent State n$uts Next State Moore ut$ut Mealy ut$ut

&ncoded vendin* machine state transition ta(le%

B > = D

B B > D, , ,, 1 11 , ,1 1 1

LState transition ta(le for Moore and Mealy M!3%6Next state also *ives D)FF excitation7%

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m$lementation usin* D)FFs

,, ,1 11 1,

,,

,1

11

1,

B 1B ,DN ,, ,1 11 1,

,,

,1

11

1,

B 1B ,DN ,, ,1 11 1,

,,

,1

11

1,

B 1B ,DN

, , 1 1

, 1 1 1

x x x x

1 1 1 1

, 1 1 ,

1 , 1 1

x x x x

, 1 1 1

, , 1 ,

, , 1 ,

x x x x

, , 1 ,

5)ma$ for D 1 5)ma$ for D , 5)ma$ for $en 6Moore7

D1 = B 1 > D > B , N DQ N Q N QQ N D ⋅+⋅+⋅+⋅= 11,,,

"&N = B1

B,

"&N = B1 B , > D B, > D B1 > N B1MooreMealy

,, ,1 11 1,

,,

,1

11

1,

B 1B ,DN

, , 1 ,

, , 1 1

x x x x

, 1 1 1

5)ma$ for $en 6Mealy7

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m$lementation usin* )5 FFS

B 1 B 2 D N B 1> B 2> 1 5 1 , 5 ,

, , , , , , , x , x , 1 , 1 , x 1 x 1 , 1 , 1 x , x 1 1 x x x x x x , 1 , , , 1 , x x , , 1 1 , 1 x x 1 1 , 1 1 1 x x ,

1 1 x x x x x x 1 , , , 1 , x , , x , 1 1 1 x , 1 x 1 , 1 1 x , 1 x 1 1 x x x x x x 1 1 , , 1 1 x , x ,

, 1 1 1 x , x , 1 , 1 1 x , x , 1 1 x x x x x x

9ema$$ed next)state functions for the vendin* machine exam$le%

B B>

5 , , , x, 1 1 x1 , x 11 1 x ,

)5 &xcitation

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1?

,, ,1 11 1,

,,

,1

11

1,

B 1B ,DN

,, ,1 11 1,

,,

,1

11

1,

B 1B ,DN ,, ,1 11 1,

,,

,1

11

1,

B 1B ,DN

,, ,1 11 1,

,,

,1

11

1,

B 1B ,DN

, , x x

, 1 x x

x x x x1 1 x x

x x , ,

x x , ,

x x x x

x x , ,

, x x ,

1 x x 1

x x x x

, x x 1

x , , x

x 1 , x

x x x x

x , , x

5)ma$ for 1 5)ma$ for 5 1

5)ma$ for , 5)ma$ for 5 ,5)ma$s for )5 fli$)flo$ im$lementation of vendin* machine%

1 = D > B , N 5 1 = , DQ N Q J ⋅+⋅= 1,, N Q K ⋅= 1,

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1

B

B9

B 1

345

5

B

B9

B,

345

5

"&N N

B ,

D

N

D

B 1

N

,Q1Q

,Q1Q

)5 fli$)flo$ im$lementation for the vendin* machine exam$le 6Moore7%

Similarly# a Mealy im$lementation@ only the "&N function chan*es%

9eset

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1E

Oasic Ste$s in the FSM Desi*n "rocedure

1% Understand the $ro(lem and the different information classes6minimal num(er7 re0uired to solve it%

2% 3onvert these information classes into distinct states# and

determine the state transition dia*ram of the FSM%

8% &ncode states in (inary# and o(tain state transition ta(le and FFexcitation for desired FF ty$e%

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Note' 6A7' is 8y>, = 8y . of 1-s received%

• No/ the transitions (et/een the8 classes of information is clear'6A7→ 6O17→ 6O27→ 6A7

1 received 1 received 1 received

• Hence these classes of information can (e considered states ofthe re0uired as states of the re0uired FSM'

These 8 states can (e re$resented (y 8y> # i = ,#1#2

i=,

i=1

i=2 i=2:,;

i=1:,;

i=,:1;

9eset 9eset,!1

,!,

,!,

1!,

1!1

,,

1,

,1 1

1

1

,

,

,

Moore MachineMealy Machine

n$ut

ut$ut

1!,

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2,

FSM Iord "ro(lem 2'

• Desi*n a system that out$uts a +1- /henever it receives'6a7 A multi$le of 8 . of 1-s AND 6(7 A non) ero even . of ,-s

&%*%# 66,#27 # 68#27 # 68#i# i = ,#1#2:8 classes;

) For . of ,-s' 2 >Q# Q = ,#1For Q = ,# /e need to distin*uish (et/een ero 6 = ,7and non) ero 6 R ,7 . of ,-s

) Thus /e have 8 classes'2 >,# = , 6 , 72 >,# R , 6 non) ero even 72 >1 6 odd 7

. of ,-s

. of 1-s

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The relevant . of 1-s can (e re$resented (y i = ,# 1# 2 6 . of 1-s = 8y>i 7

L The relevant . of ,-s can (e re$resented (y Q= , , # , R, # 1

6 . of ,-s = 2 >Q 7 /here the su(scri$t of the , indicates /hether=, or R,%

L Since at any $oint time# a certain . of 1-s and . of ,-s /ill have

(een received# the state of the system /ill (e *iven (ya com(ination of relevant . of 1-s and . of ,-s%

L There are com(inations'

,# 1# 2# P , , # , R, # 1 = {6,#, , 7# 6,#,R, 7# 6,#17# 61#,, 7# 61#,R, 7# 61#17# 62#,, 7# 62#,R, 7# 62#17} . of 1-s . of ,-s

3artesian"roduct

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6,#, , 7

61#,, 76,#, R, 7

62#,, 7

62#17

61#17 6,#17

62#,R, 7

61#,R, 7

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6,#, , 7

61#,, 7 6,#, R, 7

62#,, 7

62#17

61#17 6,#17

62#,R, 7

61#,R, 7

9eset Note' , R, ≡ 2 >Q# Q = ,

R ,

1!,

1!,

,!, 1!,

,!,

,!,

,!,

,!,

,!,

,!1

1!,

1!,

,!,

,!,

1!,

1!,

1!1

1!,