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Error-Correcting Codes and Frames with Erasures Error-Correcting Codes and Frames with Erasures

Error-Correcting Codes and Frames with Erasures - PowerPoint Presentation

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Error-Correcting Codes and Frames with Erasures - PPT Presentation

Amanda S Amy Izzie Katie SPWM July 30 th 2011 What it is An errorcorrecting 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 ID: 685430

robust frame error frames frame robust frames error erasures tight correcting signal coefficient lost erasure errors memory coefficients math

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Slide1

Error-Correcting Codes and Frames with Erasures

Amanda S., Amy,

Izzie

, Katie

SPWM July 30

th

, 2011Slide2

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

C

oding

T

heorySlide3

Coding Theory

Transmits codes for reliable transmission of information across noisy channels

Implores:

Finite fields

Group theory

Polynomial algebra

A branch of information theorySlide4

Error-Correcting and Compression

Interested in:

Detecting errors

Correcting errors

Examples where this is useful

CD’s

Computer memory malfunction glitchSlide5

More Specifically

Start with signal

Some corruption occurs

Impossible to know that it is not the original signal

 Slide6

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

 Slide7

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

 Slide8

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

→ 11111Slide9

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

 Slide10

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 corruptionSlide11

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 characteristicsSlide12

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 lostSlide13

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 frameSlide14

Mercedes-Benz Frame

Want this vector in the form:

Say we want to send the vector

. Then, the coefficients are computed as follows:

 Slide15

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:

 Slide16

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.

 Slide17

Another Example

Another frame in

is the Harmonic Tight Frame (HTF)

Note this frame can be formed by

 Slide18

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 ENTPFWe look specifically at being robust to one erasure.Slide19

Definition

A frame

is said to be robust to

k

erasures if

is still a frame, for

any index set of

erasures,

and

.

 Slide20

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

 Slide21

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

o

r

 Slide22

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.

 Slide23

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

.

 Slide24

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

.

 Slide25

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.htmlWeisstein, Eric W. "Error-Correcting Code." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Error-CorrectingCode.html