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