1/11/05 1 capacity of noiseless and noisy two-dimensional channels paul h. siegel electrical and...
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1/11/05 1
Capacity of Noiseless and Noisy Two-Dimensional Channels
Paul H. Siegel
Electrical and Computer Engineering
Center for Magnetic Recording Research
University of California, San Diego
1/11/05 2LANL Workshop
Outline
• Shannon Capacity
• Discrete-Noiseless Channels• One-dimensional
• Two-dimensional
• Finite-State Noisy Channel• One-dimensional
• Two-dimensional
• Summary
1/11/05 5LANL Workshop
The Formula on the “Paper”
Capacity of a discrete channel with noise [Shannon, 1948]
For noiseless channel, Hy(x)=0, so:
Gaylord, MI: C = W log (P+N)/NBell Labs: no formula on paper
(“H = – p log p – q log q” on plaque)
(x))H – (H(x)Max C y
H(x)Max C
1/11/05 6LANL Workshop
Discrete Noiseless Channels(Constrained Systems)
• A constrained system S is the set of sequences generated by walks on a labeled, directed graph G.
Telegraph channel constraints [Shannon, 1948]
DOT
DASH
DOT
DASH
LETTER SPACE
WORD SPACE
1/11/05 7LANL Workshop
Magnetic Recording Constraints
Forbidden words F={101, 010}
01
0 0
1 1Biphase
Even
10
0 0
1 1
1
0
1
0
Forbidden word F={11}
Runlength constraints (“finite-type”: determined by finite
list F of forbidden words)
Spectral null constraints (“almost-finite-type”)
1/11/05 8LANL Workshop
(d,k) runlength-limited constraints
• For , a (d,k) runlength-limited
sequence is a binary string such that:
• F={11} forbidden list corresponds to
1 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 1 0
kd0
),1(),( kd
ksd s1' econsecutivbetween '0#
1/11/05 9LANL Workshop
Practical Constrained Codes
Finite-state encoder Sliding-block decoder (from binary data into S) (inverse mapping from S to data)
m data bits n code bits
Encoder Logic
Rate m:n
(states)
Decoder Logic
n bits
m bitsWe want: high rate R=m/n low complexity
1/11/05 10LANL Workshop
Codes and Capacity
• How high can the code rate be?• Shannon defined the capacity of the constrained system S:
where N(S,n) is the number of sequences in S of length n.
Theorem [Shannon,1948] : If there exists a decodable code at rate R= m/n from binary data to S, then R C.
Theorem [Shannon,1948] : For any rate R=m/n < C there exists a block code from binary data to S with rate km:kn, for some integer k 1.
S,n Nn
Cn
log1
lim
1/11/05 11LANL Workshop
Computing Capacity:Adjacency Matrices
• Let be the adjacency matrix of the graph G representing S.
• The entries in correspond to paths in G of length n.
GA
nGA
0
0
1
01
11GA
1/11/05 12LANL Workshop
Computing Capacity (cont.)
• Shannon showed that, for suitable representing graphs G ,
where , i.e.,the spectral radius of the matrix .
• Assigning “transition probabilities” to the edges of G, the constrained system S becomes a Markov source x, with entropy H(x). Shannon proved that
and expressed the maximizing probabilities in terms of the spectral radius and corresponding eigenvector of .
GA ρlogC
GG lue of Aan eigenva is: λ λ maxAρ GA
x HmaxC
GA
1/11/05 13LANL Workshop
Maxentropic Measure
• Let denote the largest real eigenvalue of , with corresponding eigenvector
• Then the maxentropic (capacity-achieving) transition probabilities are given by
• The stationary state distribution is expressed in terms of corresponding left and right eigenvectors.
MBBB ,,1
ij
i
jij
A
B
BP
GA
1/11/05 14LANL Workshop
Computing Capacity (cont.)
• Example:
• More generally, , where is the
largest real root of the polynomial
and
),1(),( kd
6942.02
51log
C
kdkd logC ,, kd ,
kxxxxf dkkkd for ,1)( 1
,
. 1for ,12 ,1, dCC ddd
1/11/05 15LANL Workshop
Constrained Coding Theorems
• Stronger coding theorems were motivated by the problem of constrained code design for magnetic recording.
Theorem[Adler-Coppersmith-Hassner, 1983] Let S be a finite-type constrained system. If m/n C, then there exists a rate m:n sliding-block decodable, finite-state encoder. (Proof is constructive: state-splitting algorithm.)
Theorem[Karabed-Marcus, 1988]Ditto if S is almost-finite-type.(Proof not so constructive…)
1/11/05 16LANL Workshop
Two-Dimensional Constrained Systems
• Band-recording and page-oriented recording technologies require 2-dimensional constraints, for example:
• Two-Dimensional Optical Storage (TwoDOS) - Philips
• Holographic Storage - InPhaseTechnologies
• Patterned Magnetic Media – Hitachi, Toshiba, …
• Thermo-Mechanical Probe Array – IBM
1/11/05 18LANL Workshop
Constraints on the Integer Lattice Z2
• : constraint in x - y directions:
),1(),( kd
Hard-Square ModelHard-Square Model
11 11
11
11
11
11
1111
11
11
11
11
1111
11
11 Independent SetsIndependent Sets
11,
1
1F
,1sqS
1/11/05 19LANL Workshop
(d,k) Constraints on the Integer Lattice Z2
• For 2-dimensional (d,k) constraints , the capacity is given by:
• The only nontrivial (d,k) pairs for which is known precisely are those with zero capacity, namely [Kato-Zeger, 1999] :
mn
N limC
kdnm
nm
kd,,
,
,
kdC ,
2 , 0, dkC kd
kdsqS ,
01, ddC , d>0, d>0
1/11/05 20LANL Workshop
(d,k) Constraints on Z2 – Capacity Bounds
• Transfer matrix methods provide numerical bounds on [Calkin-Wilf, 1998] , [Nagy-Zeger, 2000]
• Variable-rate “bit-stuffing” encoders for yield best known lower bounds on for d >1 [Halevy, et al., 2004]:
,dC
6858789116180 7558789116170 ,1 .. C
,dsqS
,1C
d Lower bound d Lower bound
2 0.4267 4 0.2858
3 0.3402 5 0.2464
)1(121
)(max lim /},min{12210,
,dnmdpnm
d oppdp
phC
1/11/05 21LANL Workshop
2-D Bit-Stuffing RLL Encoder
• Source encoder converts binary data to i.i.d bit stream (biased bits) with , rate penalty .
• Bit-stuffing encoder inserts redundant bits which can be identified uniquely by decoder.
• Encoder rate R(p) is a lower bound of the capacity. (For d=1, we can determine R(p) precisely.)
Data Source
Bit-stuffingDecoder
SourceEncoder
B it-stuffingEncoder
DiscreteNoiselessChannel
SourceDecoder
Data S ink
pp, 10Pr 1Pr )( ph
),( d
1/11/05 22LANL Workshop
2-D Bit-Stuffing (1,∞) RLL Encoder
• Biased sequence: 1 1 1 0 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0
11 00
00
00
00
0000 11
11
11
11
11
00
00
00
11
00
00
00
00
00
00
00
00
00
00
00
0000
00
00
Optimal bias Pr(1) = p = 0.3556
R(p)=0.583056 (within 1% of capacity)
Optimal bias Pr(1) = p = 0.3556
R(p)=0.583056 (within 1% of capacity)
1/11/05 23LANL Workshop
Enhanced Bit-Stuffing Encoder
• Use 2 source encoders, with parameters p0 , p1 .
00
00
00 00
00
11
Optimal bias
Pr(1) = p0 = 0.328167
Optimal bias
Pr(1) = p0 = 0.328167
Optimal bias
Pr(1) = p1 = 0.433068
Optimal bias
Pr(1) = p1 = 0.433068
R(p0 , p1)=0.587277 (within 0.1% of capacity)R(p0 , p1)=0.587277 (within 0.1% of capacity)
1/11/05 24LANL Workshop
Non-Isolated Bit (n.i.b.) Constraint on Z2
• The non-isolated bit constraint is defined by the forbidden set:
• Analysis of the coding ratio of a bit-stuffing encoder yields:
0.91276 ≤ Csqnib ≤ 0.93965
1
101
1
,
0
010
0
F
nibsqS
1/11/05 25LANL Workshop
Constraints on the Hexagonal Lattice A2
• : constraints:
),1(),( kd
Hard-Hexagon ModelHard-Hexagon Model
,1hexS
1
1,11,
1
1F
111
1
111
1
1
1/11/05 26LANL Workshop
Hard Hexagon Capacity
• Capacity of hard hexagon model is known precisely! [Baxter,1980]*
212
4
22
3
22
2
21254511
1221
12211
12211
1134
aaaa
cccc
cccc
c
,1hexC
31
3131
21
31
)1()1(8
3
4
1
3311979
2501
11363
124
bbac
b
a
and where,log 4321,1
hhexC
So, So, 480767622.0,1 hexC
1/11/05 27LANL Workshop
Hard Hexagon Capacity
• Alternatively, the hard hexagon entropy constant satisfies a degree-24 polynomial with (big!) integer coefficients.
• Baxter does offer this disclaimer regarding his derivation, however:
*“It is not mathematically rigorous, in that certain analyticity properties of κ are assumed, and the results of Chapter 13 (which depend on assuming that various large-lattice limits can be interchanged) are used. However, I believe that these assumptions, and therefore (14.1.18)-(14.1.24), are in fact correct.”
*“It is not mathematically rigorous, in that certain analyticity properties of κ are assumed, and the results of Chapter 13 (which depend on assuming that various large-lattice limits can be interchanged) are used. However, I believe that these assumptions, and therefore (14.1.18)-(14.1.24), are in fact correct.”
1/11/05 28LANL Workshop
(d,k) Constraints on A2 – Capacity Bounds
• Zero capacity region partially known [Kukorelly-Zeger, 2001].
• Variable-to-fixed length “bit-stuffing” encoders for
yield best known lower bounds on for d>1
,dhexS
,dhexC
d Lower bound d Lower bound
2 0.3387 4 0.2196
3 0.2630 5 0.1901
[Halevy, et al., 2004]:[Halevy, et al., 2004]:
)1(31
)(max lim /},min{210,
,dnm
pnm
dhex o
pdp
phC
1/11/05 29LANL Workshop
Practical 2-D Constrained Codes
• There is no comprehensive algorithmic theory for constructing encoders and decoders for 2-D constrained systems.
• Very efficient bit-stuffing encoders have been defined and analyzed for several 2-D constraints, but they are not suitable for practical applications [Roth et al., 2001] , [Halevy et al., 2004] , [Nagy-Zeger, 2004].
• Optimal block codes with m x n rectangular code arrays have been designed for small values of m and n, and some finite-state encoders have been designed, but there is no generally applicable method [Demirkan-Wolf, 2004] .
• There is no comprehensive algorithmic theory for constructing encoders and decoders for 2-D constrained systems.
• Very efficient bit-stuffing encoders have been defined and analyzed for several 2-D constraints, but they are not suitable for practical applications [Roth et al., 2001] , [Halevy et al., 2004] , [Nagy-Zeger, 2004].
• Optimal block codes with m x n rectangular code arrays have been designed for small values of m and n, and some finite-state encoders have been designed, but there is no generally applicable method [Demirkan-Wolf, 2004] .
1/11/05 30LANL Workshop
Concluding Remarks
• The lack of convenient graph-based representations of 2-D constraints prevents the straightforward extension of 1-D techniques for analysis and code design.
• There are strong connections to statistical physics that may open up new approaches to understanding 2-D constrained systems (and, perhaps, vice-versa).
1/11/05 31LANL Workshop
Noisy Finite-State ISI Channels (1-Dim.)
• Binary input process
• Linear intersymbol interference
• Additive, i.i.d. Gaussian noise
1
0
][][][][n
k
inkixkhiy
][ix
][ih
)( 2,0~][ Nin
1/11/05 32LANL Workshop
Example: Partial-Response Channels
• Impulse response:
• Example: Dicode channel
1
0
)1)(1(][)(
NiN
i
DDDihDh
)1()( DDh
-1
10 0
00 11
1/11/05 33LANL Workshop
Entropy Rates
• Output entropy rate:
• Noise entropy rate:
• Conditional entropy rate:
nn YH
nYH 1
1lim
02
1eNlogNH
NHXYHn
XYH nn
n
11 |1
lim|
1/11/05 34LANL Workshop
Mutual Information Rates
• Mutual information rate:
• Capacity:
• Symmetric information rate (SIR): Inputs are constrained to be independent, identically distributed, and equiprobable binary digits.
NHYHX|YHYHY;XI
][ixX
Y;XImaxC
XP
1/11/05 35LANL Workshop
Finding the Output Entropy Rate
• For one-dimensional ISI channel model:
and
where
n
nYH
nYH 1
1lim
nnn yYpEYH 111 log
nYYYY n ,2,11
1/11/05 36LANL Workshop
Sample Entropy Rate
• If we simulate the channel N times, using inputs with specified (Markovian) statistics and generating output realizations
then
converges to with probability 1 as .N nYH 1
Nknyyyy kkkk ,,2,1,][,,]2[,]1[ )()()()(
N
k
kypN 1
)(log1
1/11/05 37LANL Workshop
Computing Sample Entropy Rate
• The forward recursion of the sum-product (BCJR)algorithm can be used to calculate the probability p(y1
n) of a sample realization of the channel output.
• In fact, we can write
where the quantity is precisely the normalization constant in the (normalized) forwardrecursion.
n
i
ii
n y|yplogn
yplogn 1
111
11
11i
i y|yp
1/11/05 38LANL Workshop
Computing Entropy Rates
• Shannon-McMillan-Breimann theorem implies
as , where is a single long sample realization of the channel output process.
ny1
YHyplogn .s.a
n 11
n
1/11/05 40LANL Workshop
Computing the Capacity
• For Markov input process of specified order r , this technique can be used to find the mutual informationrate. (Apply it to the combined source-channel.)
• For a fixed order r , [Kavicic, 2001] proposed a Generalized Blahut-Arimoto algorithm to optimize the parameters of the Markov input source.
• The stationary points of the algorithm have been shown to correspond to critical points of the information rate curve [Vontobel,2002] .
1/11/05 42LANL Workshop
Markovian Sufficiency
Remark: It can be shown that optimized Markovian processes whose states are determined by their previous r symbols can asymptotically achieve the capacity of finite-state intersymbol interference channels with AWGN as the order r of the input process approaches .
(This generalizes to 2 dimensional channels.)
[Chen-Siegel, 2004]
1/11/05 43LANL Workshop
Capacity and SIR in Two Dimensions
• In two dimensions, we could estimate by calculating the sample entropy rate of a very large simulated output array.
• However, there is no counterpart of the BCJR algorithm in two dimensions to simplify the calculation.
• Instead, conditional entropies can be used to derive upper and lower bounds on .
YH
YH
1/11/05 45LANL Workshop
Conditional Entropies
• For a stationary two-dimensional random field Y on the integer lattice, the entropy rate satisfies:
(The proof uses the entropy chain rule. See [5-6])
• This extends to random fields on the hexagonal lattice, via the natural mapping to the integer lattice.
j,i,kj,i YPastYHYH |
1/11/05 46LANL Workshop
Upper Bound on H(Y)
• For a stationary two-dimensional random field Y,
where
1Ul,k
kHminYH
j,il,kj,i YPastYHYH |Ul,k 1
1/11/05 47LANL Workshop
Two-Dimensional Boundary of Past{Y[i,j]}
• Define to be the boundary
of .
• The exact expression for
is messy, but the geometrical concept is
simple.
j,il,k YPast
j,il,kStrip Y
j,il,kStrip Y
1/11/05 49LANL Workshop
Lower Bound on H(Y)
• For a stationary two-dimensional hidden Markov field Y,
where
and is the “state information” for
the strip .
1Ll,k
kHmaxYH
j,iYl,kStX,j,il,kj,i YPastYHYH |Ll,k 1
j,iYl,kStX
j,iYl,kStrip
1/11/05 50LANL Workshop
Computing the SIR Bounds
• Estimate the two-dimensional conditional entropies
over a small array.
• Calculate to get
for many realizations of output array.
• For column-by-column ordering, treat each row
as a variable and calculate the joint probability
row-by-row using the BCJR forward
recursion.
BAP
BAH
BPBAP ,,
iY
mYYYP ,,, 21
1/11/05 51LANL Workshop
2x2 Impulse Response
• “Worst-case” scenario - large ISI:
• Conditional entropies computed from 100,000 realizations.
• Upper bound:
• Lower bound:
(corresponds to element in middle of last column)
5.05.0
5.05.0],[1 jih
1 ,log
2
1min 0
10,3,7,7,1,2 eNH U
01
0,3,7,7,1,2 log2
1eNH L
1/11/05 53LANL Workshop
Computing the SIR Bounds
• The number of states for each variable increases exponentially with the number of columns in the
array.
• This requires that the two-dimensional impulse response have a small support region.
• It is desirable to find other approaches to computing bounds that reduce the complexity, perhaps at the cost of weakening the resulting bounds.
1/11/05 54LANL Workshop
Alternative Upper Bound
• Modified BCJR approach limited to small impulse response support region.
• Introduce “auxiliary ISI channel” and bound
where
and is an arbitrary conditional
probability distribution.
2Ul,kHYH
ydjilk
jiyqjilk
jiypH yPastyPastUlk
,
,,log,
,,, |2
,
ji
lkjiyq yPast ,
,, |
1/11/05 55LANL Workshop
3x3 Impulse Response
• Two-DOS transfer function
• Auxiliary one-dimensional ISI channel with memory
length 4.
• Useful upper bound up to Eb/N0 = 3 dB.
011
121
110
101],[2 jih
1/11/05 57LANL Workshop
Concluding Remarks
• Recent progress has been made in computing information rates and capacity of 1-dim. noisy finite-state ISI channels.
• As in the noiseless case, the extension of these results to 2-dim. channels is not evident.
• Upper and lower bounds on the SIR of two-dimensional finite-state ISI channels have been developed.
• Monte Carlo methods were used to compute the bounds for channels with small impulse response support region.
• Bounds can be extended to multi-dimensional ISI channels.
• Further work is required to develop computable, tighter bounds for general multi-dimensional ISI channels.
1/11/05 58LANL Workshop
References
1. D. Arnold and H.-A. Loeliger, “On the information rate of binary-input channels with memory,” IEEE International Conference on Communications, Helsinki, Finland, June 2001, vol. 9, pp.2692-2695.
2. H.D. Pfister, J.B. Soriaga, and P.H. Siegel, “On the achievable information rate of finite state ISI channels,” Proc. Globecom 2001, San Antonio, TX, November2001, vol. 5, pp. 2992-2996.
3. V. Sharma and S.K. Singh, “Entropy and channel capacity in the regenerative setup with applications to Markov channels,” Proc. IEEE International Symposium on Information Theory, Washington, DC, June 2001, p. 283.
4. A. Kavcic, “On the capacity of Markov sources over noisy channels,” Proc. Globecom 2001, San Antonio, TX, November2001, vol. 5, pp. 2997-3001.
5. D. Arnold, H.-A. Loeliger, and P.O. Vontobel, “Computation of information rates from finite-state source/channel models,” Proc.40th Annual Allerton Conf. Commun., Control, and Computing, Monticello, IL, October 2002, pp. 457-466.
1/11/05 59LANL Workshop
References
6. Y. Katznelson and B. Weiss, “Commuting measure-preserving transformations,” Israel J. Math., vol. 12, pp. 161-173, 1972.
7. D. Anastassiou and D.J. Sakrison, “Some results regarding the entropy rates of random fields,” IEEE Trans. Inform. Theory, vol. 28, vol. 2, pp. 340-343, March 1982.