Extended Subspace Error Localization for Rate-Adaptive Distributed Source Coding

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1 Introduction Error Correction Extended Subspace Simulation Extended Subspace Error Localization for Rate-Adaptive Distributed Source Coding Mojtaba Vaezi and Fabrice Labeau McGill University International Symposium on Information Theory Istanbul, Turkey July 11, 2013

2 Introduction Error Correction Extended Subspace Simulation Outline 1 Introduction Definitions Motivation Outline 2 Error Correction in DFT Codes Coding-Theoretic Subspace-Based 3 Extended Subspace Algorithm Application 4 Simulation results

3 Introduction Error Correction Extended Subspace Simulation Definitions Motivation Outline Real BCH-DFT Codes Encoding G x R k X C k zero Y C n y R n DFT padding IDFT W k Σn k Wn H Figure: An (n, k) real BCH-DFT encoding scheme G consists of k columns from the IDFT matrix of order n The remaining n k columns of the IDFT matrix form H Σ n k inserts d = n k consecutive zeros in the transform domain = BCH code H (the extended parity check matrix) is defined such that [H T H T ] = W n

4 Introduction Error Correction Extended Subspace Simulation Definitions Motivation Outline Lossy DSC Practical code construction: Binary codes Problem: Distributed Source Coding (DSC) of continuous-valued sources - Common approach Wyner Ziv encoder Wyner Ziv decoder X Quantizer Slepian-Wolf Encoder Slepian-Wolf Decoder Reconstruction ˆX Y Y Binary codes (e.g., LDPC or Turbo codes) for Slepian-Wolf coding There are quantization loss and binning loss

5 Introduction Error Correction Extended Subspace Simulation Definitions Motivation Outline Lossy DSC Practical code construction: Real field codes - Alternative approach Encoder Decoder X Slepian-Wolf Encoder Q Q 1 Slepian-Wolf Decoder ˆX Y Similarities and differences There are still coding and quantization losses Coding is before quantization error correction in the real field (soft redundancy) Advantages 1 Correlation channel model is more realistic 2 Quantization error can be reduced by a factor of code rate (it vanishes if X and Y are completely correlated) 3 Better performance w.r.t. delay and complexity

6 Introduction Error Correction Extended Subspace Simulation Definitions Motivation Outline Lossy DSC Practical code construction x n Encoder s x ŝ x H Q Decoder n k n k Correlation Channel ˆx n y = x + e Figure: Wyner-Ziv coding using DFT codes: Syndrome approach. Suitable for delay-sensitive networks Highly vulnerable to variations in the correlation channel n

7 Introduction Error Correction Extended Subspace Simulation Definitions Motivation Outline Lossy DSC Practical code construction x n Encoder s x ŝ x H Q Decoder n k n k Correlation Channel ˆx n y = x + e Figure: Wyner-Ziv coding using DFT codes: Syndrome approach. Suitable for delay-sensitive networks Highly vulnerable to variations in the correlation channel Motivation for this works -To make rate-adaptive DSC based on DFT codes -To improve the decoding algorithm n

8 Introduction Error Correction Extended Subspace Simulation Definitions Motivation Outline Outline 1 Error Correction in DFT Codes Coding-Theoretic Approach Subspace-Based Approach 2 Extended Subspace Approach 3 Rate-Adaptive DSC 4 Simulation Results

9 Introduction Error Correction Extended Subspace Simulation Definitions Motivation Outline Outline 1 Error Correction in DFT Codes Coding-Theoretic Approach Subspace-Based Approach 2 Extended Subspace Approach 3 Rate-Adaptive DSC 4 Simulation Results Some related works -[Rath and Guillemot, 2004] -[Duartea and Baraniuk, 2013]

10 Introduction Error Correction Extended Subspace Simulation Coding-Theoretic Subspace-Based Error Correction in DFT Codes Decoding algorithms for a BCH-DFT code: 1 Detection (to determine the number of errors; ν t = n k 2 ) 2 Localization (to find the location of errors; i 1,..., i ν ) 3 Estimation (to calculate the magnitude of errors; e i1,..., e iν )

11 Introduction Error Correction Extended Subspace Simulation Coding-Theoretic Subspace-Based Error Correction in DFT Codes Decoding algorithms for a BCH-DFT code: 1 Detection (to determine the number of errors; ν t = n k 2 ) 2 Localization (to find the location of errors; i 1,..., i ν ) 3 Estimation (to calculate the magnitude of errors; e i1,..., e iν ) s = Hr = H(c + e) = He, s m = 1 ν e ip X α 1+m n p, m = 1,..., d = n k, p=1 and X p = e j2π n ip, p = 1,..., ν. s 1 s 2... s t s 2 s 3... s t+1 S t = s t s t+1... s 2t 1

12 Introduction Error Correction Extended Subspace Simulation Coding-Theoretic Subspace-Based Error Correction in DFT Codes Coding-Theoretic Approach 1 Detection (ν =?) ν = µ iff S µ is nonsingular but S µ+1 is singular 2 Localization (X i =?) Define error-locator polynomial as ν Λ(x) = (1 xx i ) = Λ 0 + Λ 1 x Λ ν x ν i=1 Find Λ 1,..., Λ ν by solving S ν [Λ ν... Λ 1 ] T = [s ν+1... s 2ν ] T Evaluate Λ(ω i ), i = 1... N, ω = e j 2π N, to find the roots 3 Estimation (Y i =?) Determine error magnitudes by solving the set of following linear equations s 1 s 2. s 2t = X 1... X ν X Xν X1 2t... Xν 2t Y 1 Y 2. Y 2ν

13 Introduction Error Correction Extended Subspace Simulation Coding-Theoretic Subspace-Based Error Correction in DFT Codes Subspace-Based Approach Form the error-locator matrix of order m as X 1 X 2... X ν V m = X m 1 1 X m Xν m 1 Define the syndrome matrix (for m = d/2 ) by S m = V m DVd m+1 T s 1 s 2... s d m+1 s 2 s 3... s d m+2 = s m s m+1... s d. where D is a diagonal matrix of size ν with nonzero diagonal elements d p = 1 n e ip X α p, p = 1,..., ν.

14 Introduction Error Correction Extended Subspace Simulation Coding-Theoretic Subspace-Based Error Correction in DFT Codes Subspace-Based Approach Eigen-decompose the covariance matrix R m = S m Sm H [ ] e 0 R m = [U e U n ] [U 0 e U n ] H, n - e and n contain the ν largest and m ν smallest eigenvalues -U e and U n contain the eigenvectors corresponding to e and n The columns in U e span the channel-error subspace spanned by V m thus, U H e U n = 0 V H m U n = 0 Let v = [1, x, x 2,..., x m 1 ] T where x is a complex variable, then F (x) m ν j=1 v H u n,j = m ν j=1 m 1 f ji x i. F (x) is sum of m ν polynomials {f ji } m ν j=1 of order m 1. i=0

15 Introduction Error Correction Extended Subspace Simulation Coding-Theoretic Subspace-Based Error Correction in DFT Codes Subspace-Based Approach U n U e Figure: Subspace method: graphical representation

16 Introduction Error Correction Extended Subspace Simulation Coding-Theoretic Subspace-Based Error Correction in DFT Codes Subspace-Based Approach U n Ũ n Ũ e U e Figure: Subspace method: graphical representation

17 Introduction Error Correction Extended Subspace Simulation Coding-Theoretic Subspace-Based Error Correction in DFT Codes Subspace-Based Approach U n Ũ Û n Ũ e U e Figure: Subspace method: graphical representation V m

18 Introduction Error Correction Extended Subspace Simulation Coding-Theoretic Subspace-Based Error Correction in DFT Codes Subspace-Based Approach U n Ũ Û n Ũ e U e Figure: Subspace method: graphical representation V m Subspace vs. coding-theoretic method There are m ν = d 2 ν polynomials rather than just one, and they have higher degrees of freedom Subspace method performs better than the coding-theoretic approach

19 Introduction Error Correction Extended Subspace Simulation Algorithm Application Extended Subspace Decoding Motivation Main idea: Increasing the dimension of the estimated noise subspace the number of polynomials with linearly independent coefficients and/or their degree grow.

20 Introduction Error Correction Extended Subspace Simulation Algorithm Application Extended Subspace Decoding Motivation Main idea: Increasing the dimension of the estimated noise subspace the number of polynomials with linearly independent coefficients and/or their degree grow.

21 Introduction Error Correction Extended Subspace Simulation Algorithm Application Extended Subspace Decoding Motivation Main idea: Increasing the dimension of the estimated noise subspace the number of polynomials with linearly independent coefficients and/or their degree grow. Construction: The extended syndrome matrix S m is defined for d > d, and similar to S m it is decomposable as S m = V m DV T d m+1. To form S m we need d syndrome samples while we only have d samples. s m = 1 n ν p=1 e ip X α 1+m p, m = 1,..., d,

22 Introduction Error Correction Extended Subspace Simulation Algorithm Application Extended Subspace Decoding Extended Syndrome s m = { sm, 1 m d, s m d, d < m d, where s = He, is the extended syndrome of error. Recal: H consists of those k columns of the IDFT matrix of order n not used in H (used in G).

23 Introduction Error Correction Extended Subspace Simulation Algorithm Application Extended Subspace Decoding Extended Syndrome s m = { sm, 1 m d, s m d, d < m d, where s = He, is the extended syndrome of error. Recal: H consists of those k columns of the IDFT matrix of order n not used in H (used in G). Q: How can we compute s?

24 Introduction Error Correction Extended Subspace Simulation Algorithm Application Extended Subspace Decoding Extended Syndrome s m = { sm, 1 m d, s m d, d < m d, where s = He, is the extended syndrome of error. Recal: H consists of those k columns of the IDFT matrix of order n not used in H (used in G). Q: How can we compute s?

25 Introduction Error Correction Extended Subspace Simulation Algorithm Application Extended Subspace Decoding Extended Syndrome s m = { sm, 1 m d, s m d, d < m d, where s = He, is the extended syndrome of error. Recal: H consists of those k columns of the IDFT matrix of order n not used in H (used in G). Q: How can we compute s? Let us try Hr = Hc + He He So to have Hr = s, either Hc must vanish or we should remove it.

26 Introduction Error Correction Extended Subspace Simulation Algorithm Application Extended Subspace Decoding Extended Syndrome Hc = 0 could happen in the special case of rate 1 2 all error indices are even In general, we need to find a way to remove Hc codes when We exploit the gain from the extended subspace decoding by transmitting extra samples Rate-adaptive DFT codes

27 Introduction Error Correction Extended Subspace Simulation Algorithm Application Extended Subspace Decoding Extended Syndrome Hc = 0 could happen in the special case of rate 1 2 all error indices are even In general, we need to find a way to remove Hc codes when We exploit the gain from the extended subspace decoding by transmitting extra samples Rate-adaptive DFT codes

28 Introduction Error Correction Extended Subspace Simulation Algorithm Application Extended Subspace Decoding Extended Syndrome Hc = 0 could happen in the special case of rate 1 2 all error indices are even In general, we need to find a way to remove Hc codes when We exploit the gain from the extended subspace decoding by transmitting extra samples Rate-adaptive DFT codes Applications 1 Rate-adaptive DSC (syndrome & parity approaches) 2 Rate-adaptive channel coding 3 Rate-adaptive distributed joint source-channel coding

29 Introduction Error Correction Extended Subspace Simulation Algorithm Application Rate-Adaptive DSC Syndrome Approach x n Encoder s x ŝ x H Q Decoder n k n k Correlation Channel n y ˆx n Figure: The Wyner-Ziv coding using DFT codes: Syndrome approach.

30 Introduction Error Correction Extended Subspace Simulation Algorithm Application Rate-Adaptive DSC Syndrome Approach x n Encoder s x ŝ x H Q Decoder n k n k Correlation Channel n y ˆx n Figure: The Wyner-Ziv coding using DFT codes: Syndrome approach. Rate Adaptation: 1 Decoder: Request for extra syndrome samples 2 Encoder: Transmit s x = Hx sample by sample 3 Decoder: Compute s y = Hy = s x + s e and s e = s y s x 4 Decoder: Use the extended subspace decoding

31 Introduction Error Correction Extended Subspace Simulation Algorithm Application Rate-Adaptive DSC Parity Approach x k Encoder p ˆp G sys Q Decoder n k n k Correlation Channel k y ˆx k Figure: The Wyner-Ziv coding using DFT codes: Parity approach.

32 Introduction Error Correction Extended Subspace Simulation Algorithm Application Rate-Adaptive DSC Parity Approach x k Encoder p ˆp G sys Q Decoder n k n k Correlation Channel k y ˆx k Figure: The Wyner-Ziv coding using DFT codes: Parity approach. Rate Adaptation: 1 Parity Puncturing: Performance is poor 2 Syndrome Augmentation: Very similar to the syndrome-based DSC

33 Introduction Error Correction Extended Subspace Simulation Numerical results Syndrome-based DSC Relative frequency of correct localization of errors Subspace 0.3 Extended Subspace (1 sample) Extended Subspace (2 samples) 0.2 Extended Subspace (3 samples) Extended Subspace (4 samples) CEQNR (db) Figure: Rate-adaptation using a (10,5) DFT code and extended subspace method. The the code rate is increased from 0.5 to 0.9 by a step of 0.1.

34 Introduction Error Correction Extended Subspace Simulation Numerical results Syndrome-based DSC 1 Relative frequency of correct localization of errors Subspace Extended Subspace 3 errors 4 errors CEQNR (db) Figure: Rate-adaptation for a (17,9) DFT code based on 4 additional syndrome samples.

35 Introduction Error Correction Extended Subspace Simulation Conclusions Summary: The extended subspace algorithm of DFT codes Improves decoding at the expense of increasing the code-rate Is suitable for rate-adaptive DSC Makes possible to correct more than t = n k 2 errors The gain comes from diminishing the effect of quantization error by averaging several error localization polynomials. Generalization: To further improve the subspace decoding without increasing the code rate.

36 Introduction Error Correction Extended Subspace Simulation Thank you!

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