Coding & Signal Processing for Holographic Data Storage. Vijayakumar Bhagavatula

Transcription

1 Coding & Signal Processing for Holographic Data Storage Vijayakumar Bhagavatula

2 Acknowledgements Venkatesh Vadde Mehmet Keskinoz Sheida Nabavi Lakshmi Ramamoorthy Kevin Curtis, Adrian Hill & Mark Ayres (InPhase) NIST (ATP program) 2

3 Outline Holographic data storage (HDS) basics Equalization Advanced detection Modulation coding Error correction coding Interleaving Over-sampling Summary 3

4 What is a Hologram? 4

5 Bragg Selectivity L θ Λ = λ/ 2 sin(θ ο ) Diffracted Intensity Thickness determines width of Bragg peak Sample Angle (Degrees) Δθ = λ/(2 sin(θ ο ) L) first null Different data pages recorded with different angle references A data page reconstructed by addressing the volume hologram with a particular reference beam angle 5

6 Holographic Data Storage Feature Parallel access (million vs. one bit data transfers) Volumetric Storage (Overlap many data pages in one location) Removable Media Benefit Fast data transfer rates Ultrahigh storage densities Transportability Initial Markets Professional Video Archival Fixed Content: Banking, Insurance, Medical, Security & surveillance, Scientific data Data Archive Courtesy: InPhase Technologies 6 Recording Data Reference Arm Laser Reading Data Reference Arm Laser Storage Medium Modulator Recovered Data Data Pages Detector Array Data to be stored Recovered Data Pages

7 Tapestry3r Specs 7 Courtesy: InPhase Technologies

8 Holographic Data Storage (HDS) Fourier transform (FT) of data page recorded A typical 4-f architecture shown here Other HDS architectures exist Detector (Camera) Fourier Lens Intensity Data Spatial Light Modulator (SLM) Fourier Lens Medium D D θ I ij f L Aperture Reference Beam f L λ Laser Wavelength Δ d ij Binary Data f L f L 8

9 Example Data Page 9

10 HDS 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 noise Output photo-detector array detects intensity, not amplitude Input SLM and Output CCD array mismatches can lead to misalignment of pixels Need for balanced (i.e., equal gray level) input regions Non-ideal SLM/CCD parameters (e.g., finite contrast, fill factors, etc.) 1

11 HDS Channel Input data Retrieved data W R I T E Error correction encoding Medium Equalization Error correction decoding R E A D Modulation encoding Detection Modulation decoding 11

12 Equalization Advanced detection Modulation coding Error correction coding Interleaving Over-sampling HDS Channels: Approaches 12

13 HDS Channel Model Impairments: Light source: non-flat input illumination Spatial Light Modulator (SLM): finite contrast ratio, non-uniformity, non-full fillfactor, phase mask Storage medium: aperture, optical noise Camera: non-full fill-factor, dark noise, electronic noise, quantization Fourier Lens Spatial Light Modulator (SLM) λ Δ Laser Wavelength d ij Binary Data Medium f L D Fourier Lens D Aperture f L θ f Reference L Beam Detector (Camera) I ij f L Intensity Data uxy (, ) = axy (, ) hxy (, ) + n( xy, ) + n( xy, ) + K 13 o 2 e

14 Simulator Parameters Light source 1% decrease in intensity by the corners SLM ACR = 1 Non-uniformity = 1% Number of phase mask levels = 16 SLM fill-factor =.95 (area) Camera Camera fill- factor =.4 (area) Dark noise = 1% (intensity) Number of detector quantization bits = 6 14

15 Histogram Comparison Real Simulated 15

16 Equalization Advanced detection Sparse coding Error correction coding Interleaving Over-sampling HDS Channels: Approaches 16

17 2-D Inter-symbol Interference Impulse h(x,y) Channel Pixel Spread Function (PSF) 17

18 MMSE Equalizer Input data Error Correction encoding Channel Equalization Detection Error Correction decoding Retrieved data Equalized output N 1 N 1 aˆ = w I + K kl jm ( k j)( l m) j= N+ 1m= N+ 1 Camera output from the (k, l) pixel Bias Coefficients w jm (3x3) are selected to minimize the mean squared error between the ideal and equalized outputs. Different set of coefficients for each block in the page. 18 Coefficients of the (2N-1) (2N-1) equalizer

19 Improvement from Equalizer Before equalization After MMSE equalization 19

20 BER Improvement About 15% improvement In BER BER pages 2

21 Equalizers more useful at higher SNR 21

22 Equalization Advanced detection Modulation coding Error correction coding Interleaving Over-sampling HDS Channels: Approaches 22

23 Detector Input data Error Correction encoding Channel Equalization Detection Error Correction decoding Retrieved data Detection methods include Fixed threshold Adaptive threshold 1D Viterbi/DFE Minimum probability of error detector based on log-likelihood ratios (LLRs) Square root of camera output LLRs Process block-wise for blocks of size n n Obtain mean of the block and use as threshold to separate 1s and s Obtain means of s (m ) and 1s (m 1 ) and average of standard deviation of s and 1s (σ) Pr (y x=1) Pr (y x=) 23

24 Optical Noise or Electronic Noise? Histogram of simulated page with dominant electronic noise Distribution of ones Distribution of zeros Histogram of simulated page with equal optical and electronic noise 6 Distribution of ones Distribution of zeros Histogram of simulated page with dominant optical noise Distribution of ones Distribution of zeros 4.5 x 14 Histogram of real recovered page Distribution of ones Distribution of zeros All 4 pages have SNR of about 3 db. 24

25 LLR Detector Performance BER Actual PDF based LLR Gaussian PDF based LLR BER Page no. BER using actual PDF 3% better than the BER using Gauss PDF. Using the actual PDF consistently improves BER for all pages. 25

26 Equalization Advanced detection Modulation coding Error correction coding Interleaving Over-sampling HDS Channels: Approaches 26

27 Balanced 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 = 2 patterns 6:8 BC with rate ¾, 8C4 = 7 patterns 27

28 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 , 2. Input data Sparse Modulation encoding Error Correction encoding Channel Error Correction decoding Sparse Modulation decoding Retrieved data 28

29 Use of Sparse Codes.7.6 BER of dense versus sparse pages Dense data pages 25% sparse (24,6,17).5 Bit error rate csnr Sparse codes can lead to improved BER performance 29

30 Equalization Advanced detection Modulation coding Error correction coding Interleaving Over-sampling HDS Channels: Approaches 3

31 31 LDPC Codes LDPC Codes These are codes based on sparse parity check matrices. Every column has same weight j (here j=2). = H N M Error Correction encoding Channel Equalization Error Correction decoding Detection Input data Retrieved data

32 Hard/Soft Decisions Expected Value of a k Decision LLR Log-Likelihood Ratio (LLR) = log 32 Soft Decision Hard Decision Prob. of bit 1 Prob. of bit Larger- magnitude LLRs represent more confident decisions Iterations can be used to improve decisions

33 LDPC Decoder Input data Error Correction encoding Channel Equalization Code specifications Rate = ½, regular LDPC code with column weight = 3, quasi cyclic H matrix. Sum-product algorithm (SPA) for LDPC decoder Detection Error Correction decoding Retrieved data L( q L( r L( q L( q l m m l l m l ) = L( p ) ) = 2 tanh ) = L( p ) = L( p l l l ) + 1 ) + m M ( l) ' l L( m)\ l ' m M ( l)\ m L( r tanh L( r m l ( L( q ' ) / 2) m l ) ' ) l m m 1 m 2 l 1 l 2 l 3 rm 2 l 1 ql 5 m 2 l 4 l 5 SPA determines the maximum a posterior (MAP) estimate of the true value x k from the PDF of the true value x k given the observations y 1, y 2,, y N and the parity-check matrix H 33

34 BER/BLER 1. E+ B E R / B L E R 1. E E E E E E E E E- 9 BLER( 3 i t er at i on, Scal ed mi n-sum, FPGA) Rate ½ PEG QC- LDPC Code with Block Length of bits 1. E E- 11 BER( 3 i t er at i on, Scal ed mi n-sum, FPGA) Eb/ N ( db) 34

35 Decoder Performance 7 bit quantized LLRs used for decoder (sum-product algorithm) input. Iterations terminated either by checksum criteria or a maximum of 2 iterations. Page 1 Page 2 Page 4 Page 6 Page 8 Page 1 Raw BER Decoded BER # errors 35

36 Combining Sparse and LDPC Codes 1-1 LDPC encoded data LDPC and sparse code encoded data 1-2 Bit error rate hsnr LDPC codeword size of 32768, rate 1/2 was used. For a target BER of 1-3, there is an SNR gain of 1.75 db when we used sparse codes. 36

37 Equalization Advanced detection Modulation coding Error correction coding Interleaving Over-sampling HDS Channels: Approaches 37

38 Non-stationarity Optical quality is usually best at the center and the worst at the edges Same magnification error causes more displacement as we move away from the center Interleaving is important to handle the non-stationarity Each codeword spread from the center to the outer edges in the same manner. y (M-1)r r Mr x 38

39 BER with Magn. Error 1-1 BER Iteration number Mag factor =11%, no interleaving Mag factor =11%, with interleaving Mag factor =11%, no interleaving Mag factor =11%, with interleaving Mag factor =12%, no interleaving Mag factor =12%, with interleaving Mag factor =15%, no interleaving Mag factor =15%, with interleaving With interleaving the data pages with magnification error up to 5% was decoded with no errors. 39

40 Equalization Advanced detection Modulation coding Error correction coding Interleaving Over-sampling HDS Channels: Approaches 4

41 Over-sampling Approach M. Ayres et al, Applied Optics, 26 Using 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 block Use the estimated shifts to determine an appropriate equalizing filter to map the detector outputs to corresponding input bits. 41

42 SNR 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 42

43 Summary Equalizers are more useful at higher SNRs AWGN-based detectors suboptimal Need powerful ECC to handle low SNR Sparse coding can help; but comes with rate loss & soft decoding needed to make it compatible with LDPC decoding Interleaving important to deal with the HDS channel non-stationarity Over-sampling approach a promising technology 43