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
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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