Error-Correcting Codes and Frames with Erasures

1. Error-Correcting Codes and Frames with Erasures Amanda S., Amy, Izzie, Katie SPWM July 30th, 2011

2. What it is • An error-correcting code is an algorithm for expressing a sequence of numbers • Any errors which are introduced can be detected and corrected (within certain limitations) based on the remaining numbers • study of these codes known asCoding Theory

3. Coding Theory • Transmits codes for reliable transmission of information across noisy channels • Implores: • Finite fields • Group theory • Polynomial algebra • A branch of information theory

4. Error-Correcting and Compression • Interested in: • Detecting errors • Correcting errors • Examples where this is useful • CD’s • Computer memory malfunction glitch

5. More Specifically • Start with signal • Some corruption occurs • Impossible to know that it is not the original signal

6. Doubling the Bit • Instead we double every bit • After corruption, bits are changed • Problem occurs with not knowing if 01 is supposed to be 00 or 11

7. Tripling the Bit • Next we try tripling • After corruption, bits are changed • We can now detect and correct the error • Unfortunately, memory needed has been tripled

8. Using Less Memory • Original message: • Replace every two bit string with five bits • Apply to original message to get

9. New String • Memory increases by a factor of 2.5 rather than 3 • 2 code words are represented by a strand of 5 • Can only correct single-flip errors

10. Change in Ideas • Previously been discussing flipped bits, but now we will look at lost coefficients • Applies to Equal-Norm Tight Frames • Continuing to use the idea of perfectly reconstructing a signal despite corruption

11. Carrying Over to Equal-Norm Tight Frames • Vectors can be written as elements in a frame and this representation may or may not be unique • Frames are used in signal processing because: • Resilience to additive noise • Resilience to quantization • Numerical stability of reconstruction • Freedom to capture signal characteristics

12. The Purpose of Frames • Information overflow at different nodes in the network • Majority of loss due to unpredictable transport time • If data is lost, retransmission requires more time and is not feasible • Potential for large delay is unacceptable • Because of independence between data, it is impossible to reconstruct what is lost

13. Equal-Norm Parseval Tight Frames (ENPTF) • The ENPTF’s are the frames that will be explored • Minimizes mean-squared error if and only if it is tight • To examine robust data transmission • Robust – resistance to the allowed number of erasures in a frame that is still frame • Erasure – missing coefficient in a frame

14. Mercedes-Benz Frame • Want this vector in the form: • Say we want to send the vector . Then, the coefficients are computed as follows:

15. Loss of Coefficient • Once message is sent, the third coefficient is lost. We want to recover this using the first two coefficients: • We define a new analysis operator to be: • We find the synthesis operator: • We compute the frame operator:

16. We then found • Then, using , we are able to reconstruct f to be: • This is the f that we had started with, so we were able to reconstruct our signal with the loss of a coefficient.

17. Another Example • Another frame in is the Harmonic Tight Frame (HTF) • Note this frame can be formed by

18. Robust to Erasures • In an n-dimensionalHilbert Space, we want to find a frame that is robust to m-n erasures • m is the number of vectors in the ENTPF • We look specifically at being robust to one erasure.

19. Definition • A frame is said to be robust to k erasures if is still a frame, for any index set of erasures, and .

20. Proposition • Let be a set of vectors in . The following are equivalent: • is a frame robust to one erasure. • There are scalars , for so that

21. Proof • : Choose maximal for which there are nonzero ’s, and We claim that . We proceed by contradiction. If , choose . Since is robust to one erasure, there are scalars , not all zero, so that is erased, it can be recovered from the rest as or

22. Case 1 • Assume that for all . Then, . Recall our definition of We can write: Therefore, and has nonzero coefficients on every , plus a nonzero coefficient on contradicting the maximality of . Thus, our assumption that for all is false.

23. Case 2 • At least one for some . By definition, for all , we can choose an so that Now, and has nonzero coordinates on , for all , as well as for a coordinate on , again contradicting the maximality of . Thus, our assumption that at least one is false, so for all .

24. Proof Cont’d • : Assume , for all and Then for each we have: That is, any vector lost can be recovered using the rest and so is robust to the erasure , for an arbitrary . ∎

25. Works Cited • Casazza, Peter G. and JelenaKovacevic, “Equal-Norm Tight Frames with Erasures.” Adc. Comput. Math. 18, 287-430. (2003). • Daubechies, I. and S. Hughes. “Error-Correcting and Compression – Part 1: “How come a scratched CD can still play flawlessly?”.” course notes, Math Alive, http://ww.math.princepton.edu/math_alive/2/Notes1.pdf. • Weisstein, Eric W. "Coding Theory." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CodingTheory.html • Weisstein, Eric W. "Error-Correcting Code." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Error-CorrectingCode.html