Error-Correcting Codes and Frames with Erasures
Amanda S., Amy, Izzie, Katie
SPWM July 30th, 2011
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 as Coding Theory
Coding Theory Transmits codes for reliable transmission
of information across noisy channels Implores:
Finite fields Group theory Polynomial algebra
A branch of information theory
Error-Correcting and Compression Interested in:
Detecting errors Correcting errors
Examples where this is useful CD’s Computer memory malfunction glitch
More Specifically Start with signal
Some corruption occurs
Impossible to know that it is not the original signal
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
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
Using Less Memory Original message:
Replace every two bit string with five bits
Apply to original message to get
00
→ 00001
01
→ 01010
10
→ 10100
11
→ 11111
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
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
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
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
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
Mercedes-Benz Frame
Want this vector in the form:
Say we want to send the vector . Then, the coefficients are computed as follows:
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:
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.
Another Example Another frame in is the Harmonic Tight
Frame (HTF)
Note this frame can be formed by
Robust to Erasures
In an n-dimensional Hilbert 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.
Definition
A frame is said to be robust to k
erasures if is still a frame, for any index
set of erasures, and .
Proposition Let be a set of vectors in . The following
are equivalent:1. is a frame robust to one erasure.2. There are scalars , for so that
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
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.
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 .
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 . ∎
Works Cited Casazza, Peter G. and Jelena Kovacevic, “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
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