3/23/04 1 information rates for two-dimensional isi channels jiangxin chen and paul h. siegel center...
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3/23/04 1
Information Rates for Two-Dimensional ISI Channels
Jiangxin Chen and Paul H. Siegel
Center for Magnetic Recording Research
University of California, San Diego
DIMACS Workshop
March 22-24, 2004
3/23/04 2DIMACS Workshop
Outline
• Motivation: Two-dimensional recording
• Channel model
• Information rates
• Bounds on the Symmetric Information Rate (SIR)• Upper Bound
• Lower Bound
• Convergence
• Alternative upper bound
• Numerical results
• Conclusions
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Two-Dimensional Channel Model
• Constrained input array
• Linear intersymbol interference
• Additive, i.i.d. Gaussian noise
1
0
1
0
21
],[],[],[],[n
l
n
k
jinljkixlkhjiy
]j,i[x
],[ jih
)( 2,0~],[ Njin
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Two-Dimensional Processes
• Input process:
• Output process:
• Array
upper left corner:
lower right corner:
]j,i[XX
]j,i[YY
11 nj,mij,iY
]j,i[Y
]nj,mi[Y 11
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Entropy Rates
• Output entropy rate:
• Noise entropy rate:
• Conditional entropy rate:
n,m,n,m YH
mnlimYH 11
1
02
1eNlogNH
NHX|YHmn
limX|YH n,m,
n,m,
n,m
1111
1
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Mutual Information Rates
• Mutual information rate:
• Capacity:
• Symmetric information rate (SIR): Inputs are constrained to be independent, identically distributed, and equiprobable binary.
NHYHX|YHYHY;XI
],[ jixX
Y;XImaxC
XP
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Capacity and SIR
• The capacity and SIR are useful measures of the achievable storage densities on the two-dimensional channel.
• They serve as performance benchmarks for channel coding and detection methods.
• So, it would be nice to be able to compute them.
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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
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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
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Computing Sample Entropy Rate
• The forward recursion of the sum-product (BCJR)algorithm can be used to calculate the probability 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
nyP 1
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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
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SIR for Partial-Response Channels
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Capacity Bounds for Dicode
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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 .
(J. Chen and P.H. Siegel, ISIT 2004)
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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, we use conditional entropies to derive upper
and lower bounds on .
YH
YH
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Array Ordering
• Permuted lexicographic ordering:
• Choose vector , a permutation of .
• Map each array index to .
• Then precedes if
or and .
• Therefore, row-by-row ordering column-by-column ordering
21 k,kk 21,
21 t,t kk t,t21
21 t,t 21 s,s
kk ts11
kk ts11
kk ts22
:2,1k :1,2k
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Two-Dimensional “Past”
• Let be a non-negative
vector.
• Define to be the elements
preceding inside the region
(with permutation k )
4321 l,l,l,ll
j,il,k YPast
j,iY
42
31
lj,lilj,liY
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Examples of Past{Y[i,j]}
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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 |
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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
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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
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Two-Dimensional Boundary of Past{Y[i,j]}
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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
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Sketch of Proof
• Upper bound:
Note that
and that conditioning reduces entropy.
• Lower bound:
Markov property of , given “state
information” .
j,i,kj,il,k YPastYPast
j,iY
j,iYl,kStX
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Convergence Properties
• The upper bound on the entropy rate is
monotonically non-increasing as the size of the
array defined by increases.
• The lower bound on the entropy rate is
monotonically non-decreasing as the size of the
array defined by increases.
1Ul,kH
4321 l,l,l,ll
1Ll,kH
4321 l,l,l,ll
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Convergence Rate
• The upper bound and lower bound
converge to the true entropy rate at least
as fast as O(1/lmin) , where
1Ul,kH
k orderingcolumn -by-columnfor ,, min
k ordering row-by-rowfor , ,, min
321
431min lll
llll
1Ll,kH
YH
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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
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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
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Two-Dimensional “State”
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SIR Bounds for 2x2 Channel
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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.
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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 ,
,, |
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Choosing the Auxiliary Channel
• Assume is conditional probability distribution of the output from an auxiliary ISI channel
• A one-dimensional auxiliary channel permits a calculation based upon a larger number of columns in the output array.
• Conversion of the two-dimensional array into a one-dimensional sequence should “preserve” the statistical properties of the array.
• Pseudo-Peano-Hilbert space-filling curves can be used on a rectangular array to convert it to a sequence.
ji
lkjiyq yPast ,
,, |
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Pseudo-Peano-Hilbert Curve
j,iYPastj,iY l,,,,, 478712
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SIR Bounds for 2x2 Channel
Alternative upper bounds --------->
3/23/04 36DIMACS 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
3/23/04 37DIMACS Workshop
SIR Upper Bound for 3x3 Channel
3/23/04 38DIMACS Workshop
Concluding Remarks
• Upper and lower bounds on the SIR of two-dimensional finite-state ISI channels were presented.
• 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.
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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.
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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.