Abstract

Digital page-oriented volume holographic memory (POVHM) is a promising candidate for next-generation ultrahigh capacity optical data storage technology. As the capacity of the POVHMs increases, the bit error rate performance of the system is degraded due to increased interpixel interference (IPI) and noise. To improve the system performance under these adverse effects and to increase the capacity, joint iterative soft equalization–detection and error correction decoding might be attractive. To address that, by considering the nonlinearity inherent in the channel, an iterative soft equalization method that is optimized in the minimum mean-square error (MMSE) sense, called the iterative soft-MMSE (ISMMSE) equalization, is devised. The performance of the ISMMSE is evaluated by use of numerical experiments under different amounts of IPI and optical noise. Simulation results suggest that the ISMMSE is a good candidate for an ultrahigh capacity POVHM, which employs joint iterative equalization–detection and decoding.

© 2006 Optical Society of America

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    [CrossRef]
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2004 (1)

2003 (1)

2001 (1)

1999 (2)

1998 (1)

1997 (2)

1996 (3)

1995 (1)

1993 (1)

1974 (1)

L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory IT-20, 284-287 (1974).
[CrossRef]

Ashley, J.

Bahl, L. R.

L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory IT-20, 284-287 (1974).
[CrossRef]

Bashaw, M. C.

Bernal, M.-P.

Burr, G. W.

Chen, X.

Chou, W.

Chugg, K. M.

Cocke, J.

L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory IT-20, 284-287 (1974).
[CrossRef]

Coufal, H.

Edibi, A.

Fekri, F.

Grygier, R. K.

Gu, C.

Gunther, H.

Gurkan, K.

Ha, J.

Heanue, J. F.

Hesselink, L.

Hoffnagle, J. A.

Hong, J.

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, 1989).

Jefferson, C. M.

Jelinek, F.

L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory IT-20, 284-287 (1974).
[CrossRef]

Keskinoz, M.

M. Keskinoz and B. V. K. V. Kumar, "Discrete magnitude-squared channel modeling, equalization and detection for volume holographic storage channel," Appl. Opt. 43, 1368-1378 (2004).
[CrossRef] [PubMed]

M. Keskinoz and B. V. K. V. Kumar, "Application of linear minimum mean-squared-error equalization for volume holographic data storage," Appl. Opt. 38, 4387-4393 (1999).
[CrossRef]

M. Keskinoz and B. V. K. V. Kumar, "Linear minimum mean squared error (LMMSE) equalization for holographic data storage," in Proceedings of IEEE International Conference on Communications (IEEE, 1999), pp. 1957-1961.

M. Keskinoz, "Modeling, equalization and detection for two-dimensional quadratic storage channels," Ph.D. dissertation (Carnegie Mellon University, 2001).

King, B. M.

Kumar, B. V. K. V.

Macfarlane, R. M.

Mok, F. H.

Neifeld, M. A.

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes, 3rd ed. (McGraw-Hill, 1991).

Pisho-Nik, H.

Poor, H.

H. Poor, An Introduction to Signal Detection and Estimation, 2nd ed. (Springer-Verlag, 1994).

Psaltis, D.

Rahnavard, N.

Raviv, J.

L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory IT-20, 284-287 (1974).
[CrossRef]

Shelby, R. M.

Sincerbox, G. T.

Sornat, G.

Appl. Opt. (5)

IEEE Trans. Inf. Theory (1)

L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory IT-20, 284-287 (1974).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Lett. (5)

Other (5)

H. Poor, An Introduction to Signal Detection and Estimation, 2nd ed. (Springer-Verlag, 1994).

A. Papoulis, Probability, Random Variables and Stochastic Processes, 3rd ed. (McGraw-Hill, 1991).

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, 1989).

M. Keskinoz, "Modeling, equalization and detection for two-dimensional quadratic storage channels," Ph.D. dissertation (Carnegie Mellon University, 2001).

M. Keskinoz and B. V. K. V. Kumar, "Linear minimum mean squared error (LMMSE) equalization for holographic data storage," in Proceedings of IEEE International Conference on Communications (IEEE, 1999), pp. 1957-1961.

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Figures (7)

Fig. 1
Fig. 1

(Color online) Schematic of the four focal length POVHM systems.

Fig. 2
Fig. 2

POVHM channel model including ISI and noise sources where n ij o represents a circularly symmetric complex Gaussian noise whose components are modeled as statistically independent Gaussian random variables with zero mean and variance σ o 2 , and n ij e denotes the zero-mean additive white Gaussian noise with variance σ e 2 .

Fig. 3
Fig. 3

Schematic of the (ISMMSE).

Fig. 4
Fig. 4

(Color online) BER as a function of the SNR for the ISMMSE, IMSDFE, for the 3 × 3 LMMSE followed by the ATD and for the ATD only for the ONDC using a normalized aperture of 1.0, SLM and CCD areal-fill factors of 100 % , ACR of 5. IMSDFE and ISMMSE converge after one iteration.

Fig. 5
Fig. 5

(Color online) BER as a function of the SNR for the ISMMSE, IMSDFE, for the 3 × 3 LMMSE followed by the ATD and for the ATD only for the ONDC using a normalized aperture of 0.9, SLM and CCD areal-fill factors of 100 % , ACR of 5. ISMMSE and IMSDFE converge after two iterations.

Fig. 6
Fig. 6

(Color online) BER as a function of the SNR for the ISMMSE, IMSDFE, for the 3 × 3 LMMSE followed by the ATD and for the ATD only for the ONDC using a normalized aperture of 0.85, SLM and CCD areal fill factors of 100 % , ACR of 5. ISMMSE and IMSDFE converge after two iterations.

Fig. 7
Fig. 7

(Color online) BER as a function of the SNR for the ISMMSE, IMSDFE, for the 3 × 3 LMMSE followed by the ATD and for the ATD only for the ONDC using a normalized aperture of 0.8, SLM and CCD areal-fill factors of 100 % , ACR of 5. ISMMSE and IMSDFE converge after two iterations.

Equations (39)

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G k m = - β Δ / 2 β Δ / 2 [ x + ( k γ Δ / 2 ) x + ( k + γ Δ / 2 ) h A ( x ) d x ] × [ x + ( m γ Δ / 2 ) x + ( m + γ Δ / 2 ) h A ( x ) d x ] d x .
G k m = r = 1 R λ r ( v r ) k ( v r ) m ,
G k m g k g m * = λ max ( v ) k ( v ) m ,
I i j = | d i j h i j + n ij o | 2 + n ij e ,
SNR = μ 4 σ o 4 + 4 μ σ o 2 + σ e 2 ,
μ 1 P 2 ,
SNR = { μ 2 σ o ( σ o 2 + μ ) μ 2 σ o for   μ σ o 2   and       the   ONDC, μ σ e for   the   ENDC .
SNR = { μ 2 σ o 1 P for  μ σ o 2   and   the   ONDC , μ σ e 1 P 2 for   the   ENDC .
d ^ i = E { d i } + k = N 1 N 2 w k ( I i k E { I i k } ) = E { d i } + w T ( I i E { I i } ) ,
w = C o v ( I i , I i ) 1 C o v ( d i , I i ) ,
d ^ i = E { d i } + Cov ( d i , I i ) T C o v ( I i , I i ) 1 ( I i E { I i } ) .
C d I ( k , p ) = Re [ h p k 4 ( α 1 + α 0 ) ( α 1 α 0 ) 2 ( k h k * ) ] ,
C II ( k , p ) = 1 8 ( α 1 2 α 0 2 ) 2 Re ( ( r h r * ) 2 r h r h p k + r + | r h r | 2 [ r h r h p k + r * ] ) + ( α 1 α 0 ) 4 16 ×   ( | r h r h p k + r | 2 + | r h r * h p k + r | 2 ) ( α 1 α 0 ) 4 8 [ r | h r | 2 | h p k + r | 2 ] + σ o 2 [ ( α 1 + α 0 ) 2 | r h r | 2 + ( α 1 α 0 ) 2 ×   ( r | h r | 2 ) ] δ p k + 4 σ o 4 δ p k + σ e 2 δ p k ,
δ r = { 0 , r 0 1 , r = 0
E { I i } = E { | x i + n i o | 2 + n i e } = E { | x i + n i o | 2 } + E { n i e } 0 = E { | x i | 2 } + E { x i } E { n i     o * } 0 + E { x i * } E { n i o } 0 + E { | n i o | 2 } = k = p = h k h p * E { d i k d i p } + 2 σ o 2 .
E { I i } = | k = h k E { d i k } | 2 + k = | h k | 2   Var { d i k } + 2 σ o 2 .
L ( d i ) = ln [ P ( d i = α 1 ) / P ( d i = α 0 ) ] .
P ( d i = α 1 ) = e L ( d i ) 1 + e L ( d i ) ,
P ( d i = α 0 ) = 1 1 + e L ( d i ) .
E { d i } = ( α 1 + α 0 ) 2 + ( α 1 α 0 ) 2 tanh [ L ( d i ) 2 ] ,
Var { d i } = ( α 1 2 + α 0 2 ) 2 + ( α 1 2 - α 0 2 ) 2 tanh [ L ( d i ) 2 ] E 2 { d i } .
L E ( d i ) = ln P ( d ^ i | d i = α 1 ) P ( d ^ i | d i = α 0 ) .
E { I i k L ( d i ) = 0 } = | p = h p E { d i k p } + h k × ( α 1 + α 0 2 E { d i } ) | 2 + p = | h p | 2   Var { d i k p } + | h k | 2 × [ ( α 1 α 0 ) 2 4 Var { d i } ] + 2 σ o     2 for   k = N 1 , , N 2 .
L E ( d i ) = ln ( 1 2 π σ e ( 1 / 2 σ 2 ) [ d ̂ i μ ( i ) ] 2 1 2 π σ e ( 1 / 2 σ - 2 ) [ d ̂ i μ - ( i ) ] 2 ) = ln ( σ σ ) + 1 2 σ 2 [ d ^ i μ ( i ) ] 2 1 2 σ     2 [ d ^ i μ ( i ) ] 2 .
μ ( i ) = E { d ^ i L ( d i ) = } = E { d ^ i d i = α 1 } = k = N 1 N 2 w k [ E { I i k | d i = α 1 } E { I i k | L ( d i ) = 0 } ] ,
μ ( i ) = E { d ^ i L ( d i ) = } = E { d ^ i d i = α 0 } = k = N 1 N 2 w k [ E { I i k | d i = α 0 } E { I i k | L ( d i ) = 0 } ] ,
E { I i k d i = α 1 } = | p = h p E { d i k p } + h k ( α 1 E { d i } ) | 2 + p = | h p | 2  Var { d i k p } ( | h k | 2  Var { d i } ) + 2 σ o 2 for   k = N 1 , , N 2 ,
E { I i k d i = α 0 } = | p = h p E { d i k p } + h k ( α 0 E { d i } ) | 2 + p = | h p | 2   Var { d i k p } ( | h k | 2   Var { d i } ) + 2 σ o 2 for   k = N 1 , , N 2 .
σ 2 = E { d ^ i 2 d i = α 1 } E 2 { d ^ i d i = α 1 } = k = - N 1 N 2 p = - N 1 N 2 w k w p cov ( I i k , I i p d i = α 1 ) ,
C ( k , p ) = 1 8 Re { [ ( α 1 + α 0 ) r h r * + h - k * ( α 1 α 0 ) ] ×  [ ( α 1 + α 0 ) r h r + h p ( α 1 α 0 ) ] ×  [ ( α 1 α 0 ) 2 r h r * h p k + r h - k * h * ×  ( α 1 α 0 ) 2 ] } + 1 8 Re { [ ( α 1 + α 0 ) r h r * + h - k * ( α 1 α 0 ) ] [ ( α 1 + α 0 ) r h r * + h - p * ( α 1 α 0 ) ] [ ( α 1 α 0 ) 2 r h r h p k + r h - k h p ( α 1 α 0 ) 2 ] } + ( α 1 α 0 ) 4 16
× [ | r h r h p k + r h k h p | 2 + | r h r h p k + r * h k h - p * | 2 2 ( r | h r | 2 | h p k + r | 2 | h k | 2 | h p | 2 ) ] + σ o     2 [ | ( α 1 + α 0 ) r h r + h k ( α 1 α 0 ) | 2 + ( α 1 α 0 ) 2 ( r | h r | 2 | h k | 2 ) ] δ p k + 4 σ o 4 δ p k + σ e 2 δ p k .
σ 2 = E { d ^ i     2 d i = α 0 } E 2 { d ^ i d i = α 0 } = k = N 1 N 2 p = N 1 N 2 w k w p cov ( I i k , I i p d i = α 0 ) ,
C - ( k , p ) = 1 8 Re { [ ( α 1 + α 0 ) r h r * h - k * ( α 1 α 0 ) ] ×   [ ( α 1 + α 0 ) r h r - h p ( α 1 α 0 ) ] ×   [ ( α 1 - α 0 ) 2 r h r * h p k + r h - k * h - p ×   ( α 1 α 0 ) 2 ] } + 1 8 Re { [ ( α 1 + α 0 ) ×   r h r * - h - k * ( α 1 α 0 ) ] [ ( α 1 + α 0 ) ×   r h r * - h - p * ( α 1 α 0 ) ] [ ( α 1 - α 0 ) 2 ×   r h r h p k + r h - k h p ( α 1 α 0 ) 2 ] }
+ ( α 1 α 0 ) 4 16 [ | r h r h p k + r h k h p | 2 + | r h r h p k + r * h k h - p * | 2 2 ( r | h r | 2 | h p k + r | 2 | h k | 2 | h p | 2 ) ] + σ o     2 [ | ( α 1 + α 0 ) r h r - h k ( α 1 - α 0 ) | 2 + ( α 1 - α 0 ) 2 ( r | h r | 2 h k 2 ) ] δ p k + 4 σ o 4 δ p k + σ e 2 δ p k .
H 0 :   ( I ^ ij n ) 0 = | h 00 α 0 + ( k , l ) = ( L , L ) ( k , l ) ( 0 , 0 ) ( L , L ) h kl d ˜ i k , j l n 1 | 2 for   n = 1 , 2 , , H 1 :   ( I ^ ij n ) 1 = | h 00 α 1 + ( k , l ) = ( L , L ) ( k , l ) ( 0 , 0 ) ( L , L ) h kl d ˜ i k , j l n 1 | 2
d ˜ i , j n = { 0 , if   | I i j ( I ^ ij n ) 0 | < | I i j ( I ^ ij n ) 1 | , 1 , otherwise.
c ij n = k = L , l = L k 0 , l 0 k = L , l = L h k l d ˜ i k , j l n - 1 .
c i k , j l 1 = p , q = p k , q 1 h p q E { d i k p , j l q } ,
c i k , j l 2 = p , q = p k , q 1 | h p q | 2   Var { d i k p , j l q } .

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