Coding & Signal Processing for Holographic Data Storage

Vijayakumar Bhagavatula

2

AcknowledgementsAcknowledgementsAcknowledgements

Venkatesh VaddeMehmet KeskinozSheida NabaviLakshmi RamamoorthyKevin Curtis, Adrian Hill & Mark Ayres (InPhase)NIST (ATP program)

3

OutlineOutlineOutline

Holographic data storage (HDS) basicsEqualizationAdvanced detectionModulation codingError correction codingInterleavingOver-samplingSummary

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What is a Hologram?What is a Hologram?

5

Diff

ract

ed In

tens

ity

Sample Angle (Degrees)

Thickness determines width of Bragg peak

Δθ = λ/(2 sin(θο) L) first null

Λ = λ/ 2 sin(θο)

θ

L

Bragg SelectivityBragg Selectivity

Different data pages recorded with different angle referencesA data page reconstructed by addressing the volume hologram with a particular reference beam angle

6

Holographic Data StorageHolographic Data Storage

FeatureParallel access

(million vs. one bit data transfers)

Volumetric Storage(Overlap many datapages in one location)

Removable Media

BenefitFast data transfer rates

Ultrahigh storagedensities

Transportability

Recording Data

Reading Data

Modulator

Data tobe stored

Data Pages

StorageMedium

Reference Arm

Laser

LaserRecovered Data

RecoveredData Pages

Reference Arm

DetectorArray

Courtesy: InPhase Technologies

Initial MarketsProfessional Video ArchivalFixed Content: Banking, Insurance, Medical, Security & surveillance, Scientific dataData Archive

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Tapestry300r SpecsTapestry300r Specs

Courtesy: InPhase Technologies

8

Holographic Data Storage (HDS)Holographic Data Storage (HDS)Holographic Data Storage (HDS)

fL

fL

Detector (Camera)

Iij

IntensityData

FourierLens

θ

BinaryData

fL

fL

Spatial LightModulator

(SLM)

λ

LaserWavelength

FourierLens

ReferenceBeam

Medium

Aperture

dij

D

D

Δ

Fourier transform (FT) of data page recordedA typical 4-f architecture shown hereOther HDS architectures exist

9

Example Data PageExample Data Page

10

HDS Channels ChallengesHDS Channels Challenges

Two-dimensional inter-symbol interference due to page-oriented nature of holographic storage

Low signal-to-noise ratios when many pages are stored in the same volume (Diffraction efficiency decreases as the number of pages in a volume increases)

Complicated noise: electronic noise, optical noiseOutput photo-detector array detects intensity, not amplitudeInput SLM and Output CCD array mismatches can lead to

misalignment of pixelsNeed for balanced (i.e., equal gray level) input regionsNon-ideal SLM/CCD parameters (e.g., finite contrast, fill factors, etc.)

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HDS ChannelHDS ChannelHDS Channel

Equalization

ModulationdecodingDetection

Inputdata

Retrieveddata

Modulationencoding

Error correctionencoding

Errorcorrectiondecoding

Medium

W

R

I T

E

R

E A

D

12

HDS Channels: ApproachesHDS Channels: ApproachesHDS Channels: Approaches

EqualizationAdvanced detectionModulation codingError correction codingInterleavingOver-sampling

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HDS Channel ModelHDS Channel ModelHDS Channel Model

Impairments:Light source: non-flat input illuminationSpatial Light Modulator (SLM): finite contrast ratio, non-uniformity, non-full fill-factor, phase maskStorage medium: aperture, optical noiseCamera: non-full fill-factor, dark noise, electronic noise, quantization

fL

fL

Detector (Camera)

Iij

IntensityData

FourierLens

θ

BinaryData

fL

fL

Spatial LightModulator

(SLM)

λ

LaserWavelength

FourierLens

Beam

Medium

Aperture

dij

D

D

Δ

Reference

2( , ) ( , ) ( , ) ( , ) ( , )o eu x y a x y h x y n x y n x y K= ∗ + + +

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Light source10% decrease in intensity by the corners

SLMACR = 10Non-uniformity = 1%Number of phase mask levels = 16SLM fill-factor = 0.95 (area)

CameraCamera fill- factor = 0.40 (area)Dark noise = 10% (intensity)Number of detector quantization bits = 6

Simulator ParametersSimulator ParametersSimulator Parameters

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Histogram ComparisonHistogram ComparisonHistogram Comparison

RealReal SimulatedSimulated

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HDS Channels: ApproachesHDS Channels: ApproachesHDS Channels: Approaches

EqualizationAdvanced detectionSparse codingError correction codingInterleavingOver-sampling

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2-D Inter-symbol Interference

Channel

h(x,y)

Pixel SpreadFunction (PSF)

Impulse

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Coefficients wjm (3x3) are selected to minimize the mean squared error between the ideal and equalized outputs.Different set of coefficients for each 64×64 block in the page.

1 1

( )( )1 1

ˆN N

kl jm k j l mj N m N

a w I K− −

− −=− + =− +

= +∑ ∑

Camera output from the (k, l) pixel

Coefficients of the (2N-1)×(2N-1) equalizer

Equalized output

MMSE EqualizerMMSE Equalizer

Bias

Error Correctionencoding

Channel EqualizationError

Correctiondecoding

DetectionInputdata

Retrieveddata

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Improvement from EqualizerImprovement from Equalizer

Before equalization After MMSE equalization

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BER ImprovementBER Improvement

About 15% improvement In BER

pagespages

BE

RB

ER

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Equalizers more useful at higher SNREqualizers more useful at higher SNREqualizers more useful at higher SNR

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HDS Channels: ApproachesHDS Channels: ApproachesHDS Channels: Approaches

EqualizationAdvanced detectionModulation codingError correction codingInterleavingOver-sampling

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DetectorDetector

Detection methods include Fixed thresholdAdaptive threshold1D Viterbi/DFE

Minimum probability of error detector based on log-likelihood ratios (LLRs)

Square root of camera output LLRs

Obtain mean of the block and use as

threshold to separate 1s and 0s

Process block-wise

for blocks of size n×n

Obtain means of 0s (m0) and 1s (m1) and average of standard deviation of

0s and 1s (σ)

Error Correctionencoding

Channel EqualizationError

Correctiondecoding

DetectionInputdata

Retrieveddata

Pr (y|x=1)___________

Pr (y|x=0)

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Optical Noise or Electronic Noise?Optical Noise or Electronic Noise?

0 200 400 600 800 1000 12000

5000

10000

15000Histogram of simulated page with dominant optical noise

Distribution of onesDistribution of zeros

0 200 400 600 800 1000 12000

500

1000

1500

2000

2500

3000

3500

4000

4500

5000Histogram of simulated page with dominant electronic noise

Distribution of onesDistribution of zeros

0 200 400 600 800 1000 12000

1000

2000

3000

4000

5000

6000Histogram of simulated page with equal optical and electronic noise

Distribution of onesDistribution of zeros

0 200 400 600 800 1000 12000

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 104 Histogram of real recovered page

Distribution of onesDistribution of zeros

All 4 pages have SNR of about 3 dB.

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BER using actual PDF 30% better than the BER using Gauss PDF.Using the actual PDF consistently improves BER for all pages.

0 10 20 30 40 50 60 70 80 90 100

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

Page no.

BE

R

BER

Actual PDF based LLRGaussian PDF based LLR

LLR Detector PerformanceLLR Detector Performance

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HDS Channels: ApproachesHDS Channels: ApproachesHDS Channels: Approaches

EqualizationAdvanced detectionModulation codingError correction codingInterleavingOver-sampling

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Balanced CodesBalanced Codes

Keep the average intensity over small regions same to achieve tolerance to illumination variations

2:4 BC with rate ½, 4C2 = 6 patterns

4:6 BC with rate 2/3, 6C3 = 20 patterns

6:8 BC with rate ¾, 8C4 = 70 patterns

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Sparse Modulation Codes Sparse Modulation Codes

Sparse modulation codes can increase the total storage capacity by 15% by using data pages that contain on average 25% ON pixels.B. M. King and M. A. Neifeld, “Sparse modulation coding for increased capacity in volume holographic storage,” Applied Optics, Vol. 39, No. 35, pp 6681-6688, 2000.

Error Correctionencoding

ChannelSparse

Modulationencoding

Inputdata

Retrieveddata

Error Correctiondecoding

Sparse Modulationdecoding

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Use of Sparse CodesUse of Sparse Codes

20 30 40 50 60 70 80 90 1000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

cSNR

Bit

erro

r rat

eBER of dense versus sparse pages

Dense data pages25% sparse (24,6,17)

Sparse codes can lead to improved BER performance

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HDS Channels: ApproachesHDS Channels: ApproachesHDS Channels: Approaches

EqualizationAdvanced detectionModulation codingError correction codingInterleavingOver-sampling

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LDPC CodesLDPC Codes

These are codes based on sparse parity check matrices.

Every column has same weight j (here j=2).

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

010001100001100010100010001100100100010010010001001001

H

N

M

Error Correctionencoding

Channel EqualizationError

Correctiondecoding

DetectionInputdata

Retrieveddata

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Hard/Soft DecisionsHard/Soft Decisions

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

-0.8-0.6-0.4-0.2

00.20.40.60.8

1

LLR

Exp

ecte

d V

alue

of a

k

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

-0.8-0.6-0.4-0.2

00.20.40.60.8

1

LLR

Exp

ecte

d V

alue

of a

k

Soft Decision

Hard Decision

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

-0.8-0.6-0.4-0.2

00.20.40.60.8

1

LLR

Exp

ecte

d V

alue

of a

k

-10 -8 -6 -4 -2 0 2 4 6 8 10-1

-0.8-0.6-0.4-0.2

00.20.40.60.8

1

LLR

Exp

ecte

d V

alue

of a

k

Soft Decision

Hard Decision

Dec

isio

n

Log-Likelihood Ratio (LLR) = log

Larger- magnitude LLRs represent more confident decisionsIterations can be used to improve decisions

Prob. of bit 1Prob. of bit 0

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Code specificationsCode specificationsRate = ½, regular LDPC code with column weight = 3, quasi cyclic H matrix. SumSum--product algorithm (SPA) for LDPC decoderproduct algorithm (SPA) for LDPC decoder

SPA determines the maximum a posterior (MAP) estimate of the true value xkfrom the PDF of the true value xk given the observations y1, y2, …, yN and the parity-check matrix H

l1 l2 l3

m1 m2

l4 l5

25 mlq →12 lmr →

( )

∑

∑

∏

∈→

∈→→

∈→

−→

→

+=

+=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

=

)(

\)(

\)(

1

)()()(

)()()(

2/)(tanhtanh2)(

)()(

''

''

lMmlmll

mlMmlmlml

lmLlmllm

lml

rLpLqL

rLpLqL

qLrL

pLqL

LDPC DecoderLDPC DecoderError

Correctionencoding

Channel EqualizationError

Correctiondecoding

DetectionInputdata

Retrieveddata

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BER/BLER

1. 0E- 11

1. 0E- 10

1. 0E- 09

1. 0E- 08

1. 0E- 07

1. 0E- 06

1. 0E- 05

1. 0E- 04

1. 0E- 03

1. 0E- 02

1. 0E- 01

1. 0E+00

0 0. 5 1 1. 5 2 2. 5 3

Eb/ N0 ( dB)

BLER( 30 i t er at i on,Scal ed mi n- sum, FPGA)

BER( 30 i t er at i on,Scal ed mi n- sum, FPGA)

Rate ½PEG QC-LDPC Code with Block Length of 32768 bits

BER/BLERBER/BLER

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7 bit quantized LLRs used for decoder (sum-product algorithm) input.Iterations terminated either by checksum criteria or a maximum of 20 iterations.

Decoder Performance Decoder Performance

000000# errors

1.7×10-61.7×10-61.7×10-61.7×10-61.7×10-61.7×10-6Decoded BER

0.08320.07840.07140.07860.08810.0810Raw BER

Page 100Page 80Page 60Page 40Page 20Page 1

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Combining Sparse and LDPC CodesCombining Sparse and LDPC Codes

LDPC codeword size of 32768, rate 1/2 was used.For a target BER of 10-3, there is an SNR gain of 1.75 dB when we used sparse codes.

1 1.5 2 2.5 3 3.5 4 4.5 510-6

10-5

10-4

10-3

10-2

10-1

hSNR

Bit

erro

r rat

e

LDPC encoded dataLDPC and sparse code encoded data

37

HDS Channels: ApproachesHDS Channels: ApproachesHDS Channels: Approaches

EqualizationAdvanced detectionModulation codingError correction codingInterleavingOver-sampling

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Optical quality is usually best at the center and the worst at the edgesSame magnification error causes more displacement as we move away from the centerInterleaving is important to handle the non-stationarityEach codeword spread from the center to the outer edges in the same manner.

Non-stationarityNon-stationarity

Mrr

(M-1)r

x

y

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BER with Magn. ErrorBER with Magn. Error

0 5 10 15 2010-5

10-4

10-3

10-2

10-1

Iteration number

BER

Mag factor =101%, no interleavingMag factor =101%, with interleavingMag factor =110%, no interleavingMag factor =110%, with interleavingMag factor =120%, no interleavingMag factor =120%, with interleavingMag factor =150%, no interleavingMag factor =150%, with interleaving

With interleaving the data pages with magnification error up to 50% was decoded with no errors.

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HDS Channels: ApproachesHDS Channels: ApproachesHDS Channels: Approaches

EqualizationAdvanced detectionModulation codingError correction codingInterleavingOver-sampling

41

M. Ayres et al, Applied Optics, 2006Using more detector pixels than SLM pixels (e.g., 3x3 SLM pixels mapped to 4x4 detector)Use reserved blocks in data page to estimate the relative shift between the SLM block and the detector blockUse the estimated shifts to determine an appropriate “equalizing filter” to map the detector outputs to corresponding input bits.

Over-sampling ApproachOver-sampling Approach

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SNR vs. MisalignmentSNR vs. Misalignment

• 4/3 oversampling

• SNR actually peaks at a slight offset, presumably because the 4 x 4 sample grid is very asymmetric when one is aligned to the SLM pixel.

• Surprisingly uniform (< 1 dB) over misalignment range.

Courtesy: Mark Ayres, InPhase Technologies

43

SummarySummary

Equalizers are more useful at higher SNRsAWGN-based detectors suboptimalNeed powerful ECC to handle low SNRSparse coding can help; but comes with rate loss & soft decoding needed to make it compatible with LDPC decodingInterleaving important to deal with the HDS channel non-stationarityOver-sampling approach a promising technology