time-varying encoders for constrained systems - jonathan j. ashley, brian h. marcus jungwon lee...

Time-Varying Encoders for Constrained Systems- Jonathan J. Ashley, Brian H. Marcus

Jungwon [email protected]

EE392P Project: Time-Varying Encoders for Constrained Systems

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Outline

Introduction Time-Varying Finite-State Encoders Approximate EigenvectorsTime-Varying State Splitting Sliding-Block Decoders Conclusion


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Introduction Data-to-codeword assignment Can significantly affect the complexity

and performance of the code p small: reasonably good assignments

by an ad hoc approach p large: hard to find a good

assignment Break down the coding problem into

smaller subproblems Periodically time-varying constraints


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Introduction Example 1: 16:19 -> 8:9/8:10 low-complexity & low error rate

16:19 block-decodable code

8:9/8:10 block-decodable code

8:9/8:10 (0,1)- SBD code

1 isolated channel error

2 bytes 1 byte 2 bytes

2-bit channel error

2 or 4 bytes 1 or 2 bytes 2 or 3 bytesx x

9

10

x

9

10

x

9

10

xx

9

10


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Introduction Example 2: PRML (4,4)

Set S of all the binary sequences such that maximum run of 0’s is 4, and maximum run of 0’s in both the odd and even interleaves is

4. G is a labeled graph whose states are the ordered pairs

i: length of the run of 0’s in the preceding interleave j: length of the run of 0’s in the current interleave Six of 25 possible states are not reachable. So, G has only

19 states. Two phase rate 8:8/8:9 (0,1)-SBD code Neither rate 16:17 block code nor block-decodable code PRML(4,4) is minimal to support a code at rate 16:17 –

capacities of PRML(3,4) and PRML(4,3) are less than 16/17.

}4 , 0: ) , {( j i j i


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Introduction Generality of time-varying encoders States in the different phases can have different

lengths of input tags and output labels. States in the same phase have the same length. Not restricted to codes constructed by state-

splitting Time-varying approach can be used for look-

ahead or combined look-ahead/state-splitting encoders.

Time-varying state splitting as an application of variable-length state splitting


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Time-Varying Encoders k-partite labeled graphs Vertices divided into k subsets called phases:

States: k copies of vertex set of G Edges:

G

0V 1kV1V

1,,0 kVV

qGG

V

:qG

:G u v

(u,i) in Vi (v,i+1) in Vi+1110 iq

eee

1iqe1e0e


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Time-Varying Encoders presented by set of sequences in S, each expressed as a

sequence of nonoverlapping qi-blocks

Given and ,a -encoder is a labeled graph s.t.a) each sequence of output labels obtained by

traversing the graph belongs to S;b) each state in phase i has exactly ni outgoing

edges;c) is lossless.

qSS qGG

S ),,( 10 knnn nS , G

G


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Time-Varying Encoders A tagged encoder: ni-ary input labels for every state in phase i Rate encoder into S: tagged -encoder, wheregiven Proposition 1: Let . The following are equivalent.1) There is a rate encoder into S.2) There is a rate encoder into S.3)

),,(),,,( 1010 kk qqqppp

qp :

nS ,

nS ,qpp SSn k ),2,,2( 10

1

0

1

0

,k

ii

k

ii qqpp

qp :qp :

)(/ SCqp


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Time-Varying Encoders Proof of proposition 1

1) => 2): restrict to the states of phase 0 2) => 1): assume k=2

2) <=> 3): results of the ordinary state-splitting algorithm

PRML (4,4): capacity = 0.9614 > 16/17 = 0.9412 There is a rate 8:8/8:9 encoder.

1010 / yyyxxx

),,( 10 uxx

:G u v

:G ),,( 1 vxa0/ ya

),,( vba1/ yb

bpap block -each and block -each for 10


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Approximate Eigenvectors k-partite matrix: adjacency matrix of a k-partite graph Adjacency matrix of :

PRML (4,4): rate 8:8/8:9 encoder

qGG

000

000

0

00

000

1

2

1

0

k

k

qG

qG

qG

qG

G

A

A

A

A

AA

0

09

8

G

G

A

AA


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Approximate Eigenvectors : diagonal matrix indexed by the states of with diagonal entries where u is a state in phase i -approximate eigenvector: a nonnegative integer vector x such that Proposition 2: Let be a k-partite labeled graph, , and . Then, there is an -approximate eigenvector if and only if .

nDD

),( nA0, xDxxA

),,( 10 knnn

kkqG

SCSqCpk

i

p AAAq

i )()()(2222 )()(1

0

)(/1 An k

G GAA

),( nA

)(/ SCqp

G

iuu nD ,

1

0

k

i inn


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Approximate Eigenvectors Proof of proposition 2 =>

-approximate eigenvector is an ordinary -approximate eigenvector.

<=

There is an -approximate eigenvector x.

ny is -approximate

eigenvector.

),( nA ),( nAk

kk AAn )()( )(/1 An k

),( nAk

10

11

120

122

1210

1211

000

/

)/(

)/(

0 assuming

kkk

kk

kk

nxAy

nnxAAy

nnnxAAAy

xxy

).()(

),,,,( 110

nyDnyA

yyyy

n

k

),( nA


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Approximate Eigenvectors Modified Franaszek Algorithm

PRML (4,4):

}

/* isecomponentw },min{ and apply * /

};,min{y

;

{ )( while

;0

;

1

xxAD

yx

yx

x

y

)2,3,3,2,4,4,2,5,5,3,3,4,5,5,3,4,5,5,5(

)1,2,2,1,3,3,1,3,3,2,2,3,3,4,2,3,3,4,4(

},{);5,,5,5(

1

0

10

x

x

xxx


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Approximate Eigenvectors Merging state v into another state u u and v are in the same phase. u and v have the same approximate

eigenvector entry. Any output label sequence generated from

u can also be generated from v.

PRML (4,4): has eight states. -approximate eigenvector is

, where .

H

))2,2(,( 88HA

)3,4,2,4,5,3,4,5(),2,1,3,3,4,2,3,4( 10 xx},{ 10 xxx


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Time-Varying State Splitting

x-consistent partition:

, where i is the phase of state u

x-consistent splitting: state splitting based on such a partition

0,

)(,,2,1for ,

)()(

1

)(

)()(

))(()2()1(

)(

ru

uN

ru

ru

rui

Eee

uNuuuu

xxx

uNrxnx

EEEE

ru


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Encoder construction given and Find an -approximate eigenvector using

the modified Franaszek algorithm. Do a sequence of x-consistent splittings. End with a k-partite lossless presentation

s.t.the entries of the induced -approximate eigenvector are at most 1.

After deleting states with vector entry 0, each state of phase i has out-degree at least .

The resulting graph is an -encoder.

qGG ),( nA

G

K

),( nAK

ipin 2

),( nS

)2,,2( 10 kppn


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More merging u and v are in the same phase. Any output label sequence generated from u can

also be generated from v. The weight m of u is less than the weight n of v.=> m descendants of u can be merged with m of the n

descendants of v.

PRML (4,4): After state-splitting, 22 states in phase 0 and 30

states in phase 1 After merging, 8 states in phase 0 and 11 states in

phase 1


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Sliding-Block Decoders Sliding-block decodable: begin in the same phase

(m,a)-definite: begin in the same phase

x-m’/y-m

e-m’x0’/y0

e0’ xa’/ya

ea’

x-m/y-m

e-m

x0/y0

e0 xa/ya

ea

x0 = x0 ’

x-m’/y-m

e-m’x0’/y0

e0’ xa’/ya

ea’

x-m/y-m

e-m

x0/y0

e0 xa/ya

ea

e0 = e0 ’


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Sliding-Block Decoders Finite-type constrained system G: finite memory : (m,0)-definite State splitting of an (m,a)-definite graph

always yields an (m,a+1)-definite graph.

Proposition 3:Let S be a finite-type constrained system. Let

and .

Then, there is a sliding-block decodable rateencoder into S if and only if .

),,(),,,( 1010 kk qqqppp

1

0

1

0

,k

ii

k

ii qqpp

qp :

)(/ SCqp

qG


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Sliding-Block Decoders Proposition 4:

Then, (0,a)-sliding-block decodable. PRML (4,4): The compatibility conditions with one block of anticipation and no memory are satisfied.

x0/y0

e0(u,i)

x1/y1

e1 xa/ya

ea

x0’/y0

e0’(u’,i)

x1’/y1

e1’ xa’/ya

ea’

x0 = x0 ’


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Sliding-Block Decoders The memory and/or anticipation of a sliding-block decoder can vary from phase to phase. Case of only two phases.

Let x={x0, x1}. If , then only one round of splitting will be needed because states of phase 0 need not to be split; for any edge starting in phase1, its terminal state is in

phase 0 and so its weight is 1; it follows that there is an x-consistent splitting that reduces all weights of phase 1 to 1.

The anticipation of the resulting decoder will be 1 in phase 0 and 0 in phase 1 in the finite-type case.

10 x


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Sliding-Block Decoders Proposition 5:Let x={x0, x1, …, xk-1}. If , then at most k-

1 rounds of splitting are required. In the finite-type case, in phase j, the sliding-

block decoder will have anticipation at most k-j-1.

Sliding block decoder window size In this case, the sliding-block decoder will

stop looking ahead periodically. If the sliding-block decoder has no memory,

the code is actually block-decodable on blocks of length .

10 x

1

0

k

i iqq


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Conclusion The paper showed how to adapt the state-splitting algorithm to the time-varying setting. The framework of time-varying encoders is useful to design high-rate codes with reduced decoder error propagation and reduced complexity. PRML (4,4) constraint is an example of a constrained system whose design benefits a lot from the time-varying approach.


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References Main paper:

J. J. Ashley, and B. H. Marcus, “Time-Varying Encoders for Constrained Systems: An approach to Limiting Error Propagation”, IEEE Trans. on Inform. Theory, vol. 46, May 2000.

Other papers: R. L. Adler, J. Friedman, B. Kitchens, and B. H. Marcus, “State

splitting for variable-length graphs,” IEEE Trans. Inform. Theory, vol. IT-32, pp. 108-113, Jan. 1986.

B. H. Marcus, P. H. Siegel, and J. K. Wolf, “Finite-state modulation codes for data storage,” IEEE J. Select. Areas Commun., vol. 10, pp.5-37, Jan. 1992.

C. D. Heegard, B. H. Marcus, and P. H. Siegel, “Variable-length state splitting with applications to average runlength-constrained (ARC) codes”, IEEE Trans. Inform. Theory, vol. 37, pp.759-777, May 1991.

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