Abstract

As storage density increases, the performance of volume holographic storage channels is degraded, because intersymbol interference and noise also increase. Equalization and detection methods must be employed to mitigate the effects of intersignal interference and noise. However, the output detector array in a holographic storage system detects the intensity of the incident light’s wave front, leading to loss of sign information. This sign loss precludes the applicability of conventional equalization and detection schemes. We first address channel modeling under quadratic nonlinearity and develop an efficient model named the discrete magnitude-squared channel model. We next introduce an advanced equalization method called the iterative magnitude-squared decision feedback equalization (IMSDFE), which takes the channel nonlinearity into account. The performance of IMSDFE is quantified for optical-noise-dominated channels as well as for electronic-noise-dominated channels. Results indicate that IMSDFE is a good candidate for a high-density, high-intersignal-interference volume holographic storage channel.

© 2004 Optical Society of America

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References

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1999 (3)

1998 (2)

1997 (2)

1996 (3)

1995 (1)

1993 (1)

Ashley, J.

Bashaw, M. C.

Bernal, M. P.

Bernal, M.-P.

Burr, G. W.

Chen, X.

Chugg, K. M.

Coufal, H.

Grygier, R. K.

Gu, C.

Gunther, H.

Gurkan, K.

Heanue, J. F.

Hesselink, L.

Hoffnagle, J. A.

Hong, J.

Jain, A. K.

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

Jefferson, C. M.

Keskinoz, M.

M. Keskinoz, 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, B. V. K. V. Kumar, “Linear minimum mean squared error (LMMSE) equalization for holographic data storage,” in Proceedings of IEEE International Conference on Communications (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 1957–1961.

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

King, B. M.

Kumar, B. V. K. V.

M. Keskinoz, 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]

V. Vadde, B. V. K. V. Kumar, “Channel modeling and estimation for intrapage equalization in pixel-matched volume holographic data storage,” Appl. Opt. 38, 4374–4386 (1999).
[CrossRef]

M. Keskinoz, B. V. K. V. Kumar, “Linear minimum mean squared error (LMMSE) equalization for holographic data storage,” in Proceedings of IEEE International Conference on Communications (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 1957–1961.

V. Vadde, B. V. K. V. Kumar, “Channel estimation and intra-page equalization for digital volume holographic storage,” in Optical Data Storage 1997 Topical Meeting, H. Birecki, J. Z. Kwiecien, eds., Proc. SPIE3109, 250–255 (1997).
[CrossRef]

Macfarlane, R. M.

Marcus, B.

Mok, F. H.

Neifeld, M. A.

Papoulis, A.

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

Proakis, J. G.

J. G. Proakis, Digital Communications, 3rd ed. (McGraw-Hill, New York, 1995).

Psaltis, D.

Quintanilla, M.

Shelby, R. M.

Sincerbox, G. T.

Sornat, G.

Vadde, V.

V. Vadde, B. V. K. V. Kumar, “Channel modeling and estimation for intrapage equalization in pixel-matched volume holographic data storage,” Appl. Opt. 38, 4374–4386 (1999).
[CrossRef]

V. Vadde, B. V. K. V. Kumar, “Channel estimation and intra-page equalization for digital volume holographic storage,” in Optical Data Storage 1997 Topical Meeting, H. Birecki, J. Z. Kwiecien, eds., Proc. SPIE3109, 250–255 (1997).
[CrossRef]

Appl. Opt. (5)

J. Opt. Soc. Am. A (2)

Opt. Lett. (5)

Other (6)

J. G. Proakis, Digital Communications, 3rd ed. (McGraw-Hill, New York, 1995).

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

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

M. Keskinoz, B. V. K. V. Kumar, “Linear minimum mean squared error (LMMSE) equalization for holographic data storage,” in Proceedings of IEEE International Conference on Communications (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 1957–1961.

V. Vadde, B. V. K. V. Kumar, “Channel estimation and intra-page equalization for digital volume holographic storage,” in Optical Data Storage 1997 Topical Meeting, H. Birecki, J. Z. Kwiecien, eds., Proc. SPIE3109, 250–255 (1997).
[CrossRef]

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

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Figures (18)

Fig. 1
Fig. 1

Schematic of the four-focal-length holographic data storage system.

Fig. 2
Fig. 2

Physical model of a VHSC.

Fig. 3
Fig. 3

Model of a VHSC as a bank of discrete magnitude-squared channels. Output intensity I ij that is due to binary input data d ij can be obtained by summing the outputs of a bank of DMSCs (i.e., a discrete linear shift-invariant channel followed by magnitude-squared operation).

Fig. 4
Fig. 4

NMSE as a function of normalized aperture width w for several CCD areal fill factors. SLM fill factor, 100%; amplitude contrast ratios (a) 2, (b) 5, and (c) 100.

Fig. 5
Fig. 5

VHSC model including ISI and noise sources, where n ij o represents 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 zero mean additive white Gaussian noise with variance σ e 2 .

Fig. 6
Fig. 6

CCD intensity array I ij convolved with a LMMSE equalizing kernel w ij to produce the equalized data ij . LMMSE equalizer w ij is designed by minimizing the mean of the squared error, e ij , between the ideal binary input data d ij and the equalized data ij .

Fig. 7
Fig. 7

Equalized signal ij converted to detected binary sequence ij by passage through an ATD.

Fig. 8
Fig. 8

Schematic of IMSDFE.

Fig. 9
Fig. 9

BER as a function of SNR for IMSDFE, for 3 × 3 LMMSE followed by ATD and for ATD only for the ONDC. Normalized aperture, 1.0; SLM and CCD areal fill factors, 100%; ACR, 5. IMSDFE converges after one iteration. Here and in the following figures, ite means “iterations.”

Fig. 10
Fig. 10

BER as a function of SNR for IMSDFE for 3 × 3 LMMSE followed by ATD and for ATD only for the ENDC. Normalized aperture, 1.0; SLM and CCD areal fill factors, 100%; ACR, 5. IMSDFE converges after one iteration.

Fig. 11
Fig. 11

BER as a function of SNR for IMSDFE for 3 × 3 LMMSE followed by ATD and for ATD only for the ONDC. Normalized aperture, 0.9; SLM and CCD areal fill factors, 100%; ACR, 5. IMSDFE converges after two iterations.

Fig. 12
Fig. 12

BER as a function of SNR for IMSDFE for 3 × 3 LMMSE followed by ATD and for ATD only for the ENDC. Normalized aperture, 0.9; SLM and CCD areal fill factors, 100%; amplitude contrast ratio, 5. IMSDFE converges after two iterations.

Fig. 13
Fig. 13

BER as a function of SNR for IMSDFE for 3 × 3 LMMSE followed by ATD for the ONDC. Normalized aperture, 0.9; CCD areal fill factors, 100%; ACR, 5.

Fig. 14
Fig. 14

BER as a function of SNR for IMSDFE for 3 × 3 LMMSE followed by ATD for the ENDC. Normalized aperture, 0.9; CCD areal fill factors, 100%; ACR, 5.

Fig. 15
Fig. 15

BER versus SNR for IMSDFE for 3 × 3 LMMSE followed by ATD for the ONDC. Normalized aperture, 0.9; SLM and CCD areal fill factors, 100%. IMSDFE provides SNR gains of 0.25 and 5 dB over LMMSE for ACRs of 5 and 2, respectively, to achieve a target BER of 10-3.

Fig. 16
Fig. 16

BER versus SNR for IMSDFE for 3 × 3 LMMSE followed by ATD for the ENDC. Normalized aperture, of 0.9; SLM and CCD areal fill factors, 100%. For the ENDC the IMSDFE appears to yield 0.5-dB SNR gain over LMMSE for an ACR of 2 whereas it does not improve the performance of LMMSE when an ACR of 5 is used to achieve a target BER of 10-3.

Fig. 17
Fig. 17

BER versus SNR for IMSDFE employing a 3 × 3 LMMSE equalizer (3 × 3 IMSDFE), IMSDFE employing a 5 × 5 LMMSE equalizer (5 × 5 IMSDFE), 3 × 3 LMMSE followed by ATD, and 5 × 5 LMMSE followed by ATD for the ONDC. Normalized aperture, 0.9; SLM and CCD areal fill factors, 100%; ACR, 5.

Fig. 18
Fig. 18

BER versus SNR for IMSDFE employing a 3 × 3 LMMSE equalizer (3 × 3 IMSDFE), IMSDFE employing a 5 × 5 LMMSE equalizer (5 × 5 IMSDFE), 3 × 3 LMMSE followed by ATD, and 5 × 5 LMMSE followed by ATD for the ENDC. Normalized aperture, 0.9, SLM and CCD areal fill factors, 100%; ACR, 5.

Equations (29)

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h A x ,   y = h A x h A y ,
H A f x = 1 | f x | D 2 λ f L 0 otherwise .
p x ,   y = Π x α Δ Π y α Δ ,
s x ,   y = k l   d kl p x - k Δ ,   y - l Δ ,
r x ,   y = s x ,   y h A x ,   y   = k , l   d kl p x - k Δ ,   y - l Δ h A x ,   y   = k , l   d kl h x - k Δ ,   y - l Δ ,
I ij = i - β / 2 Δ i + β / 2 Δ j - β / 2 Δ j + β / 2 Δ   | r x ,   y | 2 d y d x ,
I ij = i - β / 2 Δ i + β / 2 Δ j - β / 2 Δ j + β / 2 Δ k l   d kl h x - k Δ ,   y - l Δ 2 d y d x .
I ij = - β Δ / 2 β Δ / 2 - β Δ / 2 β Δ / 2 k l   d kl h x + i - k Δ ,   y + j - l Δ 2 d y d x .
I ij = - β Δ / 2 β Δ / 2 - β Δ / 2 β Δ / 2 k = - L L l = - L L   d i - k , j - l h x + k Δ ,   y + l Δ 2 d y d x .
I ij = -   β Δ / 2 β Δ / 2 - β Δ / 2 β Δ / 2 k = - L L l = - L L m = - L L n = - L L × d i - k , j - l d i - m , j - n h x + k Δ ,   y + l Δ h * x + m Δ ,   y + n Δ d y d x ,
I ij = k = - L L l = - L L m = - L L n = - L L   d i - k , j - l d i - m , j - n G km G ln ,
G km = - β Δ / 2 β Δ / 2   h x + k Δ h * x + m Δ d x ,
h x = p x h A x = x - α / 2 Δ x + α / 2 Δ   h A x d x ,
G km =   - β Δ / 2 β Δ / 2 x + k - α / 2 Δ x + k + α / 2 Δ   h A x d x × x + m - α / 2 Δ x + m + α / 2 Δ   h A * x d x d x .
G km =   r = 1 R   λ r v r k v * r m ,
I ij =   r = 1 R r = 1 R k = - L L l = - L L m = - L L n = - L L × d i - k , j - l d i - m , j - n λ r λ r v r k v * r m v r l v * r n   = r = 1 R r = 1 R   | d ij λ r λ r v r i v r j | 2 .
G km g k g m * = λ max v k v * m
I ˆ ij = | d ij h ij | 2 ,
w = D / D N ,
I ij = | d ij     h ij + n ij o | 2 + n ij e ,
I ij = | d ij h ij | 2 I ij s + 2   Re d ij h ij n ij o * + | n ij o | 2 + n ij e n ij .
SNR   E I ij s σ n =   μ σ n ,
SNR =   μ 4 σ o 4 + 4 μ σ o 2 + σ e 2 1 / 2 .
SNR = μ 2 σ o σ o 2 + μ μ 2 σ o μ σ o 2   and   ONDC μ / σ e ENDC .
μ     1 / P 2 ,
SNR = μ / 2 σ o     1 / P μ σ o 2   and   ONDC μ / σ e     1 / P 2 ENDC .
R dI m = w m R II m ,
H 0 : I ˆ ij n 0 = h 00 1 ε + k = - L L l = - L L k ,   l 0 ,   0   h kl d ˜ i - k , j - l n - 1 2 , H 1 : I ˆ ij n 1 = h 00 + k = - L L l = - L L k ,   l 0 ,   0   h kl d ˜ i - k , j - l n - 1 2 , n = 1 ,   2 , ,
d ˜ i , j n = 0 if   | I ij - I ˆ ij n 0 | < | I ij - I ˆ ij n 1 | 1 otherwise .

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