U.S. patent application number 12/331268 was filed with the patent office on 2009-12-10 for method and apparatus for fast encoding of data symbols according to half-weight codes.
This patent application is currently assigned to Digital Fountain, Inc.. Invention is credited to Michael Luby, M. Amin Shokrollahi.
Application Number | 20090307565 12/331268 |
Document ID | / |
Family ID | 35908165 |
Filed Date | 2009-12-10 |
United States Patent
Application |
20090307565 |
Kind Code |
A1 |
Luby; Michael ; et
al. |
December 10, 2009 |
METHOD AND APPARATUS FOR FAST ENCODING OF DATA SYMBOLS ACCORDING TO
HALF-WEIGHT CODES
Abstract
Efficient methods for encoding and decoding Half-Weight codes
are disclosed and similar high density codes are disclosed. The
efficient methods require at most 3(k-1)+h/2+1 XORs of symbols to
calculate h Half-Weight symbols from k source symbols, where h is
of the order of log(k).
Inventors: |
Luby; Michael; (Berkeley,
CA) ; Shokrollahi; M. Amin; (Preverenges,
CH) |
Correspondence
Address: |
QUALCOMM INCORPORATED
5775 MOREHOUSE DR.
SAN DIEGO
CA
92121
US
|
Assignee: |
Digital Fountain, Inc.
Fremont
CA
|
Family ID: |
35908165 |
Appl. No.: |
12/331268 |
Filed: |
December 9, 2008 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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11202933 |
Aug 11, 2005 |
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12331268 |
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60600932 |
Aug 11, 2004 |
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Current U.S.
Class: |
714/780 ;
714/E11.032 |
Current CPC
Class: |
H03M 13/19 20130101;
H03M 13/3761 20130101; H03M 13/51 20130101; H03M 13/03 20130101;
H03M 13/29 20130101; H03M 13/3776 20130101; H03M 13/1102 20130101;
H03M 13/373 20130101 |
Class at
Publication: |
714/780 ;
714/E11.032 |
International
Class: |
H03M 13/05 20060101
H03M013/05; G06F 11/10 20060101 G06F011/10 |
Claims
1. In a decoder having an input to receive signals representative
of a plurality of received symbols representing a transmitted
signal encoding a plurality of source symbols, the encoding
including coding according to a Half-Weight code, wherein a
sequence of the source symbols and h Half-Weight symbols are
included in the source symbols and the decoder is in a state
wherein some of the sequence of source symbols and Half-Weight
symbols are missing symbols and some are received symbols, a method
of removing dependencies of the Half-Weight symbols on received
source symbols in a process of decoding, comprising: decoding
source symbols from received symbols using a dynamic decoder; and
performing on the order of 3(k-1)+log(k)/2+1 XOR symbol operations
to remove the dependencies of the Half-Weight symbols on the
received source symbols, where h is of the order of log(k) and k is
the number of input symbols to be recovered.
2. The method of claim 1, wherein the number of XOR symbol
operations is not greater than a predetermined constant times
k+h.
3. The method of claim 1, further comprising a solving process as
part of the process of decoding wherein the solving process
comprises solving for the missing source symbols based on the
received Half-Symbols, after the dependencies of the received
Half-Symbols on the received source symbols have been removed in
the performing step.
4. The method of claim 1, wherein dependencies from more than one
set of Half-Weight symbols are removed in parallel.
5. The method of claim 1, wherein the dynamic decoder is a chain
reaction decoder capable of recovering source symbols from received
symbols when source symbols are generated by a fountain encoder and
the received symbols are allocated among a set of distinct encoded
symbols greater than a number of symbols that would be expected
given expected transmission conditions and number of input symbols
to recover.
Description
CROSS-REFERENCES TO RELATED APPLICATIONS
[0001] This application is a divisional patent application of U.S.
Non-provisional patent application Ser. No. 11,202,933 filed Aug.
11, 2005 which claims priority from co-pending U.S. Provisional
Patent Application No. 60/600,932 filed Aug. 11, 2004 for Luby et
al. "File Download System Using Fountain Codes" (hereinafter
"Luby-Prov") which is hereby incorporated by reference, as if set
forth in full in this document, for all purposes.
[0002] Related applications with common assignment include U.S.
Pat. No. 6,307,487 to Luby entitled "Information Additive Code
Generator and Decoder for Communication Systems" (hereinafter "Luby
I"), U.S. patent application Ser. No. 10/032,156 filed Dec. 21,
2001 for Shokrollahi et al., entitled "Multi-Stage Code Generator
and Decoder for Communication Systems" (hereinafter "Shokrollahi
I") and U.S. patent application Ser. No. 11/125,818 filed May 9,
2005 for Luby et al., entitled "File Download System" (hereinafter
"Luby05"), each of which is hereby incorporated by reference, as if
set forth in full in this document, for all purposes.
BACKGROUND OF THE INVENTION
[0003] The present invention relates to systems and methods for
encoding and/or decoding data, and more particularly to systems and
methods for encoding and/or decoding using multi-stage fountain
codes, herein referred to collectively as "MS Codes."
[0004] Fountain codes have been described previously such as in
Luby I, Shokrollahi I and Luby05. As described therein, sometimes
referred to "chain reaction codes" in view of some of the decoding
techniques, fountain codes provide a form of forward
error-correction that enables data reconstruction from a received
data set of a given size, without regard to the particular data
packets received and provide for sequences of encoded that does not
need to repeat over normal use, such that a set of data can be
encoded with a fountain code to generate as much output sequence
data as needed. Unlike many coding techniques that generate a fixed
amount of output for a given input and thus have a predetermined
"code rate", fountain codes can be "rateless" codes. Fountain codes
are often considered information additive codes in that randomly
selected output symbols are likely to contribute (add) information
to other randomly selected output symbols, whereas random
selections of among fixed rate code symbols are more likely to be
duplicative of selections already received.
[0005] As a result, communication systems employing fountain codes
are able to communicate information much more efficiently compared
to traditional FEC codes transmitted via data carousel or
acknowledgement-based protocols, as described in Luby I, or
Shokrollahi I or Luby05.
[0006] Transmission of data between a sender and a recipient over a
communications channel has been the subject of much literature.
Preferably, but not exclusively, a recipient desires to receive an
exact copy of data transmitted over a channel by a sender with some
level of certainty. Where the channel does not have perfect
fidelity (which covers most of all physically realizable systems),
one concern is how to deal with data lost or garbled in
transmission. Lost data (erasures) are often easier to deal with
than corrupted data (errors) because the recipient cannot always
tell when corrupted data is data received in error. Many
error-correcting codes have been developed to correct for erasures
and/or for errors. Typically, the particular code used is chosen
based on some information about the infidelities of the channel
through which the data is being transmitted and the nature of the
data being transmitted. For example, where the channel is known to
have long periods of infidelity, a burst error code might be best
suited for that application. Where only short, infrequent errors
are expected, a simple parity code might be best.
[0007] Data transmission between multiple senders and/or multiple
receivers over a communications channel has also been the subject
of much literature. Typically, data transmission from multiple
senders requires coordination among the multiple senders to allow
the senders to minimize duplication of efforts. In a typical
multiple sender system sending data to a receiver, if the senders
do not coordinate which data they will transmit and when, but
instead just send segments of the file, it is likely that a
receiver will receive many useless duplicate segments. Similarly,
where different receivers join a transmission from a sender at
different points in time, a concern is how to ensure that all data
the receivers receive from the sender is useful. For example,
suppose the sender is wishes to transmit a file, and is
continuously transmitting data about the same file. If the sender
just sends segments of the original file and some segments are
lost, it is likely that a receiver will receive many useless
duplicate segments before receiving one copy of each segment in the
file. Similarly, if a segment is received in error multiple times,
then the amount of information conveyed to the receiver is much
less than the cumulative information of the received garbled data.
Often this leads to undesirable inefficiencies of the transmission
system, or may require transmitter and/or receiver
coordination.
[0008] Often data to be transmitted over a communications channel
is partitioned into equal size input symbols. The "size" of an
input symbol can be measured in bits, whether or not the input
symbol is actually broken into a bit stream, where an input symbol
has a size of M bits when the input symbol is selected from an
alphabet of 2.sup.M symbols.
[0009] A coding system may produce output symbols from the input
symbols. Output symbols are elements from an output symbol
alphabet. The output symbol alphabet may or may not have the same
characteristics as the alphabet for the input symbols. Once the
output symbols are created, they are transmitted to the
receivers.
[0010] The task of transmission may include post-processing of the
output symbols so as to produce symbols suitable for the particular
type of transmission. For example, where transmission constitutes
sending the data from a wireless provider to a wireless receiver,
several output symbols may be lumped together to form a frame, and
each frame may be converted into a wave signal in which the
amplitude or the phase is related to the frame. The operation of
converting a frame into a wave is often called modulation, and the
modulation is further referred to as phase or amplitude modulation
depending on whether the information of the wave signal is stored
in its phase or in its amplitude. Nowadays this type of modulated
transmission is used in many applications, such as satellite
transmission, cable modems, Digital Subscriber Lines (DSL), and
many others.
[0011] A transmission is called reliable if it allows the intended
recipient to recover an exact copy of the original data even in the
face of errors and/or erasures during the transmission. Recovery of
erased information has been the subject of much literature and very
efficient coding methods have been devised in this case. Fountain
codes, as described in Luby I or Shokrollahi I or Luby05, are
efficient coding methods for recovery of erasures in a wide variety
of settings.
[0012] One solution that has been proposed to increase reliability
of transmission is to use Forward Error-Correction (FEC) codes,
such as Reed-Solomon codes, Tornado codes, or, more generally, LDPC
("low density parity codes"). With such codes, one sends output
symbols generated from the content instead of just sending the
input symbols that constitute the content. Traditional error
correcting codes, such as Reed-Solomon or other LDPC codes,
generate a fixed number of output symbols for a fixed length
content. For example, for K input symbols, N output symbols might
be generated. These N output symbols may comprise the K original
input symbols and N-K redundant symbols. If storage permits, then
the sender can compute the set of output symbols for each piece of
data only once and transmit the output symbols using a carousel
protocol.
[0013] One problem with some FEC codes is that they require
excessive computing power or memory to operate. Another problem is
that the number of output symbols must be determined in advance of
the coding process. This can lead to inefficiencies if the error
rate of the symbols is overestimated, and can lead to failure if
the error rate is underestimated. Moreover, traditional FEC schemes
often require a mechanism to estimate the reliability of the
communications channel on which they operate. For example, in
wireless transmission the sender and the receiver are in need of
probing the communications channel so as to obtain an estimate of
the noise and hence of the reliability of the channel. In such a
case, this probing has to be repeated quite often, since the actual
noise is a moving target due to rapid and transient changes in the
quality of the communications channel.
[0014] For traditional FEC codes, the number of valid output
symbols that can be generated is of the same order of magnitude as
the number of input symbols the content is partitioned into and is
often a fixed ratio called the "code rate." Typically, but not
exclusively, most or all of these output symbols are generated in a
preprocessing step before the sending step. These output symbols
often have the property that all the input symbols can be
regenerated from any subset of the output symbols equal in length
to the original content or slightly longer in length than the
original content. Typically, a code rate is selected to match an
expected error rate.
[0015] Fountain decoding is a form of forward error-correction that
addresses the above issues in cases where a transmission error
constitutes an erasure. For fountain codes, the pool of possible
output symbols that can be generated is orders of magnitude larger
than the number of the input symbols typically limited only by a
resolution of a counter on the encoder rather than by a code rate,
and a random output symbol from the pool of possibilities can be
generated very quickly. For fountain codes, the output symbols can
be generated on the fly on an as needed basis concurrent with the
sending step. Fountain codes have a property that all input symbols
of the content can be regenerated from any subset of a set of
randomly generated and/or selected output symbols slightly longer
in length than the original content.
[0016] Other descriptions of various fountain coding systems can be
found in documents such as U.S. Pat. No. 6,486,803, entitled "On
Demand Encoding with a Window" and U.S. Pat. No. 6,411,223 entitled
"Generating High Weight Output Symbols using a Basis," assigned to
the assignee of the present application.
[0017] Some embodiments of a fountain coding system comprise an
encoder and a decoder. Data may be presented to the encoder in the
form of a block, or a stream, and the encoder may generate output
symbols from the block or the stream on the fly. In some
embodiments, for example those described in Shokrollahi I and
Luby05, data may be pre-encoded off-line, or concurrently with the
process of transmission, using a static encoder, and the output
symbols may be generated from the static input symbols, defined as
the plurality of the original data symbols, and the output symbols
of the static encoder.
[0018] In general, the block or stream of symbols from which the
dynamic output symbols are generated is referred to herein as
"source symbols." Thus, in the case of the codes described in
Shokrollahi I and Luby05, the source symbols are the static input
symbols, while for codes described in Luby I, the source symbols
are the input symbols. The term "input symbols" herein refers to
the original symbols presented to the encoder for encoding. Thus,
for chain reaction codes described in Luby I, the source symbols
are identical with input symbols. Sometimes, to distinguish between
an MS code, as for example one of the codes described in
Shokrollahi I or Luby05, and the codes described in Luby I, we will
refer herein to the output symbols generated by a coding system
employing an MS code as the "dynamic output symbols."
[0019] In certain applications, it may be preferable to transmit
the input symbols first, and then continue transmission by sending
generated output symbols. Such a coding system is called a
systematic coding system and systematic coding systems for codes
such as those described in Luby I and Shokrollahi I are disclosed
in U.S. Pat. No. 6,909,383 entitled, "Systematic Encoding and
Decoding of Chain Reaction Codes," assigned to the assignee of the
present application (hereinafter "Systematic MS"), whereas Luby05
is itself a systematic MS code that is designed using the elements
described in "Systematic MS".
[0020] Various methods for generating source symbols from the input
symbols are described in Shokrollahi I and Luby05. Generally, but
not exclusively, the source symbols are preferably generated
efficiently on a data processing device, and, at the same time, a
good erasure correcting capability is required of the multi-stage
code. One of the teachings in Shokrollahi I and Luby05 is to use a
combination of codes to produce the source symbols. In one
particular embodiment described in Luby05, this combination
comprises using an LDPC code to produce a second set of source
symbols from the input symbols (the input symbols are the first set
of source symbols) and then using a Half-Weight encoder to produce
a third set of source symbols from the first two sets of source
symbols, and then the dynamic output symbols are calculated based
on all three sets of source symbols.
[0021] Other methods and processes for both the generation of
source symbols and dynamic output symbols, and the decoding of
these output symbols, have been described in U.S. Pat. No.
6,856,263 entitled "Systems and Processes for Decoding a Chain
Reaction Code Through Inactivation," assigned to the assignee of
the present application (hereinafter "Inactivation Decoding"). One
advantage of a decoder according teachings Inactivation Decoding
over a multi-stage chain reaction decoder described in Shokrollahi
I is that an inactivation decoding decoder has typically a lower
probability of error.
[0022] The encoding for an MS encoder in some embodiments can be
partitioned into two stages. The first stage computes redundant
symbols from the original input symbols, and the second stage
generates output symbols from combinations of the original input
symbols and redundant symbols. In some embodiments of an MS
encoder, the first stage can be further partitioned into two or
more steps, where some of these steps compute redundant symbols
based on Low Density Parity Check (LDPC) codes or other codes, and
where other steps compute redundant symbols based on other codes.
To lower the probability of error of the decoder, both multi-stage
fountain decoding and some embodiments of decoding described in
Inactivation Decoding may make use of a Half-Weight code in these
other steps, and thus a Half-Weight code is used in these
embodiments in one of the primary stages of static encoding.
[0023] The techniques of multi-stage encoding, multi-stage fountain
decoding and Inactivation Decoding may also be applied to other
forward error correction codes including LDPC ("low density parity
check") codes, IRA ("Irregular Repeat Accumulate") code, AIRA
("Accumulate Irregular Repeat Accumulate") codes, and LDGM ("low
density generator matrix") codes, and many other classes of codes
based on graphs, and thus the techniques described herein for
Half-Weight code generation may also be applied in those
settings.
[0024] As described in Luby05, a Half-Weight code generates, for a
given number k of first source symbols, a number h of redundant
symbols (referred to as the "Half-Weight symbols" hereinafter). The
number h is the smallest integer with the property illustrated in
Equation 1, where for positive integers i and j with i>j, the
function choose (i,j) is equal to i!/((i-j)!j!) and n!=12 . . .
(n-1)n, and where for real-valued x the function ceil(x) is the
smallest integer greater than or equal to x.
choose(h,ceil(h/2)).gtoreq.k (Equ. 1)
[0025] As described in Luby05, Half-Weight symbols can be
calculated in a specific way from the k source symbols. Using the
straightforward method for the computation of these symbols, each
Half-Weight symbol requires, on average, around k/2 XORs of source
symbols, and thus, in total, the h Half-Weight symbols require
around (k/2)h XORs of source symbols. Since h is of the order
log(k), this amounts to roughly klog(k)/2 XORs of input symbols for
the calculation of the Half-Weight symbols. Taking into account
that other redundant symbols calculated via, for example LDPC
encoding, require much less computational time, the calculation of
the Half-Weight symbols using the straightforward approach would
constitute a computational bottleneck for the design of some
embodiments of reliable multi-stage encoders.
[0026] What is therefore needed is an apparatus or process for
calculating the Half-Weight symbols that is much more efficient
than the straightforward one, and can be implemented easily on
various computing devices.
BRIEF SUMMARY OF THE INVENTION
[0027] In embodiments of encoders and/or decoders according to the
present invention, efficient methods for encoding and decoding
Half-Weight codes and similar high density codes are disclosed.
Such efficient methods can operate with at most 3(k-1)+h/2+1 XORs
of symbols to calculate h Half-Weight symbols from k source
symbols, where h is of the order of log(k).
[0028] According to one embodiment of the invention, a method of
encoding data operates on an ordered set of input symbols and
includes generating intermediate symbols from the set of input
symbols and possibly a set of a plurality of redundant symbols from
the intermediate symbols. The method also includes generating a
plurality of output symbols from a combined set of symbols
including the input symbols, the intermediate symbols and possibly
the redundant symbols. The intermediate symbols might include a
plurality of Half-Weight symbols wherein their number is a function
of the number of input symbols.
[0029] According to still another embodiment of the invention, a
system for receiving data transmitted from a source over a
communications channel is provided using similar techniques. The
system comprises a receive module coupled to a communications
channel for receiving output symbols transmitted over the
communications channel, including a decoder for Half-Weight
symbols.
[0030] According to yet another embodiment of the invention, a
computer data signal embodied in a carrier wave is provided,
wherein Half-Weight symbols are encoded.
[0031] Numerous benefits are achieved by way of the present
invention. For example, in a specific embodiment, the computational
expense of encoding data for transmission over a channel is
reduced. In another specific embodiment, the computational expense
of decoding such data is reduced. Depending upon the embodiment,
one or more of these benefits may be achieved. These and other
benefits are provided in more detail throughout the present
specification and more particularly below.
[0032] A further understanding of the nature and the advantages of
the inventions disclosed herein may be realized by reference to the
remaining portions of the specification and the attached
drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
[0033] FIG. 1 is a block diagram of a communications system
according to one embodiment wherein the present invention might be
used.
[0034] FIG. 2 is a block diagram of an encoder usable in the
communication system of FIG. 1.
[0035] FIG. 3 is a simplified block diagram of a method of
generating redundant symbols usable in the communication system of
FIG. 1.
[0036] FIG. 4 is a simplified block diagram of the basic operation
of a static encoder usable in the communication system of FIG.
1.
[0037] FIG. 5 is a simplified block diagram of a dynamic encoder
usable in the communication system of FIG. 1.
[0038] FIG. 6 is a simplified block diagram of a basic operation of
a dynamic encoder usable in the communication system of FIG. 1.
[0039] FIG. 7 is a simplified block diagram of a static encoder
usable in the communication system of FIG. 1.
[0040] FIG. 8 is a simplified block diagram of the basic operation
of a static encoder usable in the communication system of FIG.
1.
[0041] FIG. 9 is a simplified block diagram of a decoder usable in
the communication system of FIG. 1.
[0042] FIG. 10 is an illustration of an example of the first few
elements of a Gray code sequence.
[0043] FIG. 11 is an illustration of an example of a Half-Weight
matrix H.
[0044] FIG. 12 is a flowchart of an overall efficient method for
encoding a Half-Weight code according to embodiments of the present
invention.
[0045] FIG. 13 is a flowchart of a method for implementing the
Initialize Computation phase of FIG. 12.
[0046] FIG. 14 is a flowchart of a method for implementing the
Perform Core Computation phase of FIG. 12.
[0047] FIG. 15 is a flowchart of a method for implementing the
Clean Up phase of FIG. 12.
DETAILED DESCRIPTION OF THE INVENTION
[0048] In the specific embodiments described herein, a coding
scheme denoted as "multi-stage coding" is described, embodiments of
which are provided in Shokrollahi I.
[0049] Multi-stage encoding encodes the data in a plurality of
stages. Typically, but not always, a first stage adds a
predetermined amount of redundancy to the data. A second stage then
uses a fountain code, or the like, to produce output symbols from
the original data and the redundant symbols computed by the first
stage of the encoding. The first stage's redundancy might be in the
form of LDPC symbols, Half-Weight symbols, source symbols, etc.
[0050] As used herein, input symbols are data that an encoder seeks
to encode such that a decoder can recover those input symbols. In
some multi-stage encoders described herein, a pre-coder generates a
set of intermediate symbols from the input symbols using some
control inputs known to the encoder and to the decoder (at least
when the decoder is tasked with decoding information that depends
on knowing those control inputs). A next stage might generate a set
of redundant symbols from the intermediate symbols, which might
include Half-Weight symbols. The input symbols and the redundant
symbols might then collectively form a set of source symbols that
form input to a dynamic encoder such as a fountain encoder. For
systematic coding, the source symbols include the input symbols,
whereas for nonsystematic coding, the source symbols contain less
than all or none of the input symbols and rely more on redundant
symbols. In the latter case, the redundant symbols might not be
referred to as redundant, but just as being source symbols.
Furthermore, in some embodiments the labels "intermediate symbol"
and "redundant symbol" are interchangeable and the source symbols
comprise input symbols and redundant/intermediate symbols.
[0051] In some embodiments, the output symbols can be generated
from source symbols as needed or based on a maximum anticipated
error rate. In embodiments in which the second stage comprises
fountain encoding, each output symbol can be generated without
regard to how other output symbols are generated, in part because
they can be generated statistically with high likelihood of
information additivity between independently generated output
symbols due to the large key alphabet. Once generated, these output
symbols can then be placed into packets and transmitted to their
destination, with each packet containing one or more output
symbols. Non-packetized transmission techniques can be used instead
or as well.
[0052] As used herein, the term "file" refers to any data that is
stored at one or more sources and is to be delivered as a unit to
one or more destinations. Thus, a document, an image, and a file
from a file server or computer storage device, are all examples of
"files" that can be delivered. Files can be of known size (such as
a one megabyte image stored on a hard disk) or can be of unknown
size (such as a file taken from the output of a streaming source).
Either way, the file is a sequence of input symbols, where each
input symbol has a position in the file and a value.
[0053] As used herein, the term "stream" refers to any data that is
stored or generated at one or more sources and is delivered at a
specified rate at each point in time in the order it is generated
to one or more destinations. Streams can be fixed rate or variable
rate. Thus, an MPEG video stream, AMR audio stream, and a data
stream used to control a remote device, are all examples of
"streams" that can be delivered. The rate of the stream at each
point in time can be known (such as 4 megabits per second) or
unknown (such as a variable rate stream where the rate at each
point in time is not known in advance). Either way, the stream is a
sequence of input symbols, where each input symbol has a position
in the stream and a value.
[0054] Transmission is the process of transmitting data from one or
more senders to one or more recipients through a channel in order
to deliver a file or stream. A sender is also sometimes referred to
as the encoder. If one sender is connected to any number of
recipients by a perfect channel, the received data can be an exact
copy of the input file or stream, as all the data will be received
correctly. Here, we assume that the channel is not perfect, which
is the case for most real-world channels. Of the many channel
imperfections, two imperfections of interest are data erasure and
data incompleteness (which can be treated as a special case of data
erasure). Data erasure occurs when the channel loses or drops data.
Data incompleteness occurs when a recipient does not start
receiving data until some of the data has already passed it by, the
recipient stops receiving data before transmission ends, the
recipient chooses to only receive a portion of the transmitted
data, and/or the recipient intermittently stops and starts again
receiving data. As an example of data incompleteness, a moving
satellite sender might be transmitting data representing an input
file or stream and start the transmission before a recipient is in
range. Once the recipient is in range, data can be received until
the satellite moves out of range, at which point the recipient can
redirect its satellite dish (during which time it is not receiving
data) to start receiving the data about the same input file or
stream being transmitted by another satellite that has moved into
range. As should be apparent from reading this description, data
incompleteness is a special case of data erasure, since the
recipient can treat the data incompleteness (and the recipient has
the same problems) as if the recipient was in range the entire
time, but the channel lost all the data up to the point where the
recipient started receiving data. Also, as is well known in
communication systems design, detectable errors can be considered
equivalent to erasures by simply dropping all data blocks or
symbols that have detectable errors.
[0055] In some communication systems, a recipient receives data
generated by multiple senders, or by one sender using multiple
connections. For example, to speed up a download, a recipient might
simultaneously connect to more than one sender to transmit data
concerning the same file. As another example, in a multicast
transmission, multiple multicast data streams might be transmitted
to allow recipients to connect to one or more of these streams to
match the aggregate transmission rate with the bandwidth of the
channel connecting them to the sender. In all such cases, a concern
is to ensure that all transmitted data is of independent use to a
recipient, i.e., that the multiple source data is not redundant
among the streams, even when the transmission rates are vastly
different for the different streams, and when there are arbitrary
patterns of loss.
[0056] In general, a communication channel is that which connects
the sender and the recipient for data transmission. The
communication channel could be a real-time channel, where the
channel moves data from the sender to the recipient as the channel
gets the data, or the communication channel might be a storage
channel that stores some or all of the data in its transit from the
sender to the recipient. An example of the latter is disk storage
or other storage device. In that example, a program or device that
generates data can be thought of as the sender, transmitting the
data to a storage device. The recipient is the program or device
that reads the data from the storage device. The mechanisms that
the sender uses to get the data onto the storage device, the
storage device itself and the mechanisms that the recipient uses to
get the data from the storage device collectively form the channel.
If there is a chance that those mechanisms or the storage device
can lose data, then that would be treated as data erasure in the
communication channel.
[0057] When the sender and recipient are separated by a
communication channel in which symbols can be erased, it is
preferable not to transmit an exact copy of an input file or
stream, but instead to transmit data generated from the input file
or stream (which could include all or parts of the input file or
stream itself) that assists with recovery of erasures. An encoder
is a circuit, device, module or code segment that handles that
task. One way of viewing the operation of the encoder is that the
encoder generates output symbols from input symbols, where a
sequence of input symbol values represents the input file or a
block of the stream. Each input symbol would thus have a position,
in the input file or block of the stream, and a value. A decoder is
a circuit, device, module or code segment that reconstructs the
input symbols from the output symbols received by the recipient. In
multi-stage coding, the encoder and the decoder are further divided
into sub-modules each performing a different task.
[0058] In embodiments of multi-stage coding systems, the encoder
and the decoder can be further divided into sub-modules, each
performing a different task. For instance, in some embodiments, the
encoder comprises what is referred to herein as a static encoder
and a dynamic encoder. As used herein, a "static encoder" is an
encoder that generates a number of redundant symbols from a set of
input symbols, wherein the number of redundant symbols is
determined prior to encoding. Examples of static encoding codes
include Reed-Solomon codes, Tornado codes, Hamming codes, Low
Density Parity Check (LDPC) codes, etc. The term "static decoder"
is used herein to refer to a decoder that can decode data that was
encoded by a static encoder. For some encoders, the input symbols
are used to generate intermediate symbols, which are used to
generate the source symbols and the intermediate symbols are not
transmitted.
[0059] As used herein, a "dynamic encoder" is an encoder that
generates output symbols from a set of input symbols, where the
number of possible output symbols is orders of magnitude larger
than the number of input symbols, and where the number of output
symbols to be generated need not be fixed. One example of a dynamic
encoder is a chain reaction encoder, such as the encoders described
in Luby I and Luby05. The term "dynamic decoder" is used herein to
refer to a decoder that can decode data that was encoded by a
dynamic encoder.
[0060] Embodiments of multi-stage coding need not be limited to any
particular type of input symbol. Typically, the values for the
input symbols are selected from an alphabet of 2.sup.M symbols for
some positive integer M. In such cases, an input symbol can be
represented by a sequence of M bits of data from the input file or
stream. The value of M is often determined based on, for example,
the uses of the application, the communication channel, and/or the
size of the output symbols. Additionally, the size of an output
symbol is often determined based on the application, the channel,
and/or the size of the input symbols. In some cases, the coding
process might be simplified if the output symbol values and the
input symbol values were the same size (i.e., representable by the
same number of bits or selected from the same alphabet). If that is
the case, then the input symbol value size is limited when the
output symbol value size is limited. For example, it may be desired
to put output symbols in packets of limited size. If some data
about a key associated with the output symbols were to be
transmitted in order to recover the key at the receiver, the output
symbol would preferably be small enough to accommodate, in one
packet, the output symbol value and the data about the key.
[0061] As an example, if an input file is a multiple megabyte file,
the input file might be broken into thousands, tens of thousands,
or hundreds of thousands of input symbols with each input symbol
encoding thousands, hundreds, or only few bytes. As another
example, for a packet-based Internet channel, a packet with a
payload of size of 1024 bytes might be appropriate (a byte is 8
bits). In this example, assuming each packet contains one output
symbol and 8 bytes of auxiliary information, an output symbol size
of 8128 bits ((1024-8)*8) would be appropriate. Thus, the input
symbol size could be chosen as M=(1024-8)*8, or 8128 bits. As
another example, some satellite systems use the MPEG packet
standard, where the payload of each packet comprises 188 bytes. In
that example, assuming each packet contains one output symbol and 4
bytes of auxiliary information, an output symbol size of 1472 bits
((188-4)*8), would be appropriate. Thus, the input symbol size
could be chosen as M=(188-4)*8, or 1472 bits. In a general-purpose
communication system using multi-stage coding, the
application-specific parameters, such as the input symbol size
(i.e., M, the number of bits encoded by an input symbol), might be
variables set by the application.
[0062] As another example, for a stream that is sent using variable
size source packets, the symbol size might be chosen to be rather
small so that each source packet can be covered with an integral
number of input symbols that have aggregate size at most slightly
larger than the source packet.
[0063] Each output symbol has a value. In one preferred embodiment,
which we consider below, each output symbol also has associated
therewith an identifier called its "key." Preferably, the key of
each output symbol can be easily determined by the recipient to
allow the recipient to distinguish one output symbol from other
output symbols. Preferably, the key of an output symbol is distinct
from the keys of all other output symbols. There are various forms
of keying discussed in previous art. For example, Luby I describes
various forms of keying that can be employed in embodiments of the
present invention.
[0064] Multi-stage coding is particularly useful where there is an
expectation of data erasure or where the recipient does not begin
and end reception exactly when a transmission begins and ends. The
latter condition is referred to herein as "data
incompleteness."Regarding erasure events, multi-stage coding shares
many of the benefits of fountain coding described in Luby I. In
particular, multi-stage output symbols are information additive, so
any suitable number of packets can be used to recover an input file
or stream to a desired degree of accuracy.
[0065] These conditions do not adversely affect the communication
process when multi-stage coding is used, because the output symbols
generated with multi-stage coding are information additive. For
example, if a hundred packets are lost due to a burst of noise
causing data erasure, an extra hundred packets can be picked up
after the burst to replace the loss of the erased packets. If
thousands of packets are lost because a receiver did not tune into
a transmitter when it began transmitting, the receiver could just
pickup those thousands of packets from any other period of
transmission, or even from another transmitter.
[0066] With multi-stage coding, a receiver is not constrained to
pickup any particular set of packets, so it can receive some
packets from one transmitter, switch to another transmitter, lose
some packets, miss the beginning or end of a given transmission and
still recover an input file or block of a stream. The ability to
join and leave a transmission without receiver-transmitter
coordination helps to simplify the communication process. With
suitable selection of the stages, it might be that a coding scheme
will result in output symbols that can be more quickly decoded, or
with less computational effort.
[0067] In some embodiments, transmitting a file or stream using
multi-stage coding can include generating, forming or extracting
input symbols from an input file or block of a stream, computing
redundant symbols, encoding input and redundant symbols into one or
more output symbols, where each output symbol is generated based on
its key independently of all other output symbols, and transmitting
the output symbols to one or more recipients over a channel.
Additionally, in some embodiments, receiving (and reconstructing) a
copy of the input file or block of a stream using multi-stage
coding can include receiving some set or subset of output symbols
from one of more data streams, and decoding the input symbols from
the values and keys of the received output symbols.
[0068] Suitable FEC erasure codes as described herein can be used
to overcome the above-cited difficulties and would find use in a
number of fields including multimedia broadcasting and multicasting
systems and services. An FEC erasure code hereafter referred to as
"a multi-stage fountain code" has properties that meet many of the
current and future requirements of such systems and services.
[0069] Some basic properties of multi-stage fountain codes are
that, for any packet loss conditions and for delivery of source
files of any relevant size or streams of any relevant rate: (a)
reception overhead of each individual receiver device ("RD") is
minimized; (b) the total transmission time needed to deliver source
files to any number of RDs can be minimized (c) the quality of the
delivered stream to any number of RDs can be maximized for the
number of output symbols sent relative to the number of input
symbols, with suitable selection of transmission schedules. The RDs
might be handheld devices, embedded into a vehicle, portable (i.e.,
movable but not typically in motion when in use) or fixed to a
location.
[0070] The amount of working memory needed for decoding is low and
can still provide the above properties, and the amount of
computation needed to encode and decode is minimal. In this
document, we provide a simple and easy to implement description of
some variations of multi-stage fountain codes.
[0071] Multi-stage chain reaction codes are fountain codes, i.e.,
as many encoding packets as needed can be generated on-the-fly,
each containing unique encoding symbols that are equally useful for
recovering a source file or block of a stream, while decoding often
occurs in a chain reaction fashion, i.e., as one symbol is
recovered, it might be the last remaining symbol needed to recover
other symbols, which in turn might by the last remaining symbols to
recover other symbols, etc.
[0072] There are many advantages to using fountain codes versus
other types of FEC codes. One advantage is that, regardless of
packet loss conditions and RD availability, fountain codes minimize
the number of encoding packets each RD needs to receive to
reconstruct a source file or block of a stream. This is true even
under harsh packet loss conditions and when, for example, mobile
RDs are only intermittently turned-on or available over a long file
download session.
[0073] Another advantage is the ability to generate exactly as many
encoding packets as needed, making the decision on how many
encoding packets to generate on-the-fly while the transmission is
in progress. This can be useful if for example there is feedback
from RDs indicating whether or not they received enough encoding
packets to recover a source file or block of a stream. When packet
loss conditions are less severe than expected the transmission can
be terminated early. When packet loss conditions are more severe
than expected or RDs are unavailable more often than expected the
transmission can be seamlessly extended.
[0074] Another advantage is the ability to inverse multiplex.
Inverse multiplexing is when a RD is able to combine received
encoding packets generated at independent senders to reconstruct a
source file or block of a stream. One practical use of inverse
multiplexing is described in below in reference to receiving
encoding packets from different senders.
[0075] Where future packet loss, RD availability and application
conditions are hard to predict, it is important to choose an FEC
solution that is as flexible as possible to work well under
unpredictable conditions. Multi-stage fountain codes provide a
degree of flexibility unmatched by other types of FEC codes.
[0076] Aspects of the invention will now be described with
reference to the figures.
System Overview
[0077] FIG. 1 is a block diagram of a communications system 100
that uses multi-stage coding. In communications system 100, an
input file 101, or an input stream 105, is provided to an input
symbol generator 110. Input symbol generator 110 generates a
sequence of one or more input symbols (IS(0), IS(1), IS(2), . . . )
from the input file or stream, with each input symbol having a
value and a position (denoted in FIG. 1 as a parenthesized
integer). As explained above, the possible values for input
symbols, i.e., its alphabet, is typically an alphabet of 2.sup.M
symbols, so that each input symbol codes for M bits of the input
file or stream. The value of M is generally determined by the use
of communication system 100, but a general purpose system might
include a symbol size input for input symbol generator 110 so that
M can be varied from use to use. The output of input symbol
generator 110 is provided to an encoder 115.
[0078] Static key generator 130 produces a stream of static keys
S.sub.0, S.sub.1, . . . . The number of the static keys generated
is generally limited and depends on the specific embodiment of
encoder 115. The generation of static keys will be subsequently
described in more detail. Dynamic key generator 120 generates a
dynamic key for each output symbol to be generated by the encoder
115. Each dynamic key is generated so that a large fraction of the
dynamic keys for the same input file or block of a stream are
unique. For example, Luby I describes embodiments of key generators
that can be used. The outputs of dynamic key generator 120 and the
static key generator 130 are provided to encoder 115.
[0079] From each key I provided by dynamic key generator 120,
encoder 115 generates an output symbol, with a value B(I), from the
input symbols provided by the input symbol generator. The operation
of encoder 115 will be described in more detail below. The value of
each output symbol is generated based on its key, on some function
of one or more of the input symbols, and possibly on or more
redundant symbols that had been computed from the input symbols.
The collection of input symbols and redundant symbols that give
rise to a specific output symbol is referred to herein as the
output symbol's "associated symbols" or just its "associates". The
selection of the function (the "value function") and the associates
is done according to a process described in more detail below.
Typically, but not always, M is the same for input symbols and
output symbols, i.e., they both code for the same number of
bits.
[0080] In some embodiments, the number K of input symbols is used
by the encoder 115 to select the associates. If K is not known in
advance, such as where the input is a streaming file, K can be just
an estimate. The value K might also be used by encoder 115 to
allocate storage for input symbols and any intermediate symbols
generated by encoder 115.
[0081] Encoder 115 provides output symbols to a transmit module
140. Transmit module 140 is also provided the key of each such
output symbol from the dynamic key generator 120. Transmit module
140 transmits the output symbols, and depending on the keying
method used, transmit module 140 might also transmit some data
about the keys of the transmitted output symbols, over a channel
145 to a receive module 150. Channel 145 is assumed to be an
erasure channel, but that is not a requirement for proper operation
of communication system 100. Modules 140, 145 and 150 can be any
suitable hardware components, software components, physical media,
or any combination thereof, so long as transmit module 140 is
adapted to transmit output symbols and any needed data about their
keys to channel 145 and receive module 150 is adapted to receive
symbols and potentially some data about their keys from channel
145. The value of K, if used to determine the associates, can be
sent over channel 145, or it may be set ahead of time by agreement
of encoder 115 and decoder 155.
[0082] As explained above, channel 145 can be a real-time channel,
such as a path through the Internet or a broadcast link from a
television transmitter to a television recipient or a telephone
connection from one point to another, or channel 145 can be a
storage channel, such as a CD-ROM, disk drive, Web site, or the
like. Channel 145 might even be a combination of a real-time
channel and a storage channel, such as a channel formed when one
person transmits an input file from a personal computer to an
Internet Service Provider (ISP) over a telephone line, the input
file is stored on a Web server and is subsequently transmitted to a
recipient over the Internet.
[0083] Because channel 145 is assumed to be an erasure channel,
communications system 100 does not assume a one-to-one
correspondence between the output symbols that exit receive module
150 and the output symbols that go into transmit module 140. In
fact, where channel 145 comprises a packet network, communications
system 100 might not even be able to assume that the relative order
of any two or more packets is preserved in transit through channel
145. Therefore, the key of the output symbols is determined using
one or more of the keying schemes described above, and not
necessarily determined by the order in which the output symbols
exit receive module 150.
[0084] Receive module 150 provides the output symbols to a decoder
155, and any data receive module 150 receives about the keys of
these output symbols is provided to a dynamic key regenerator 160.
Dynamic key regenerator 160 regenerates the dynamic keys for the
received output symbols and provides these dynamic keys to decoder
155. Static key generator 163 regenerates the static keys S.sub.0,
S.sub.1, . . . and provides them to decoder 155. The static key
generator has access to random number generator 135 used both
during the encoding and the decoding process. This can be in the
form of access to the same physical device if the random numbers
are generated on such device, or in the form of access to the same
algorithm for the generation of random numbers to achieve identical
behavior. Decoder 155 uses the keys provided by dynamic key
regenerator 160 and static key generator 163 together with the
corresponding output symbols, to recover the input symbols (again
IS(0), IS(1), IS(2), . . . ). Decoder 155 provides the recovered
input symbols to an input file reassembler 165, which generates a
copy 170 of input file 101 or input stream 105.
An Encoder
[0085] FIG. 2 is a block diagram of one specific embodiment of
encoder 115 shown in FIG. 1. Encoder 115 comprises a static encoder
210, a dynamic encoder 220, and a redundancy calculator 230. Static
encoder 210 receives the following inputs: a) original input
symbols IS(0), IS(1), . . . , IS(K-1) provided by the input symbol
generator 110 and stored in an input symbol buffer 205; b) the
number K of original input symbols; c) static keys S.sub.0,
S.sub.1, . . . provided by the static key generator 130; and d) a
number R of redundant symbols. Upon receiving these inputs static
encoder 205 computes R redundant symbols RE(0), RE(1), . . . ,
RE(R-1) as will be described below. Typically, but not always, the
redundant symbols have the same size as the input symbols. In one
specific embodiment, the redundant symbols generated by static
encoder 210 are stored in input symbol buffer 205. Input symbol
buffer 205 may be only logical, i.e., the file or block of the
stream may be physically stored in one place and the positions of
the input symbols within symbol buffer 205 could only be renamings
of the positions of these symbols within the original file or block
of the stream. Some of the redundant symbols RE( ) might be
Half-Weight symbols calculated as describe below.
[0086] Dynamic encoder receives the input symbols and the redundant
symbols, and generates output symbols as will be described in
further detail below. In one embodiment in which the redundant
symbols are stored in the input symbol buffer 205, dynamic encoder
220 receives the input symbols and redundant symbols from input
symbol buffer 205.
[0087] Redundancy calculator 230 computes the number R of redundant
symbols from the number K of input symbols. This computation is
described in further detail below.
Overview of Static Encoder
[0088] The general operation of static encoder 210 is shown with
reference to FIGS. 3 and 4. FIG. 3 is a simplified flow diagram
illustrating one embodiment 300 of a method of statically encoding.
In a step 305, a variable j, which keeps track of how many
redundant symbols have been generated, is set to zero. Then, in a
step 310, a first redundant symbol RE(0) is computed as a function
F.sub.0 of at least some of the input symbols IS(0), . . . ,
IS(K-1). Then, in a step 315, the variable j is incremented. Next,
in a step 320, it is tested whether all of the redundant symbols
have been generated (i.e., is j greater than R-1?). If yes, then
the flow ends. Otherwise, the flow proceeds to step 325. In step
325, RE(j)) is computed as a function F.sub.j of the input symbols
IS(0), . . . , IS(K-1) and of the previously generated redundant
symbols RE(0), . . . , RE(j-1), where F.sub.j need not be a
function that depends on every one of the input symbols or every
one of the redundant symbols. Steps 315, 320, and 325 are repeated
until R redundant symbols have been computed. In some cases F.sub.j
might be such that Half-Weight symbols are generated.
[0089] Referring again to FIGS. 1 and 2, in some embodiments,
static encoder 210 receives one or more static keys S.sub.0,
S.sub.1, . . . from static key generator 130. In these embodiments,
the static encoder 210 uses the static keys to determine some or
all of functions F.sub.0, F.sub.1, . . . , F.sub.j-1. For example,
static key S.sub.0 can be used to determine function F.sub.0,
static key S.sub.1 can be used to determine function F.sub.1, etc.
Or, one or more of static keys S.sub.0, S.sub.1, . . . can be used
to determine function F.sub.0, one or more of static keys S.sub.0,
S.sub.1, . . . can be used to determine function F.sub.1, etc. In
other embodiments, no static keys are needed, and thus static key
generator 130 is not needed.
[0090] Referring now to FIGS. 2 and 3, in some embodiments, the
redundant symbols generated by static encoder 210 can be stored in
input symbol buffer 205. FIG. 4 is a simplified illustration of the
operation of one embodiment of static encoder 210. Particularly,
static encoder 210 generates redundant symbol RE(j) as a function
Fj of input symbols IS(0), . . . , IS(K-1), RE(0), . . . , RE(j-1),
received from input symbol buffer 205, and stores it back into
input symbol buffer 205. The exact form of the functions F.sub.0,
F.sub.1, . . . , F.sub.R-1 depends on the particular application.
Typically, but not always, functions F.sub.0, F.sub.1, . . . ,
F.sub.R-1 include an exclusive OR of some or all of their
corresponding arguments. As described above, these functions may or
may not actually employ static keys generated by static key
generator 130 of FIG. 1. For example, in one specific embodiment
described below, the first few functions implement a Hamming code
and do not make any use of the static keys S.sub.0, S.sub.1, . . .
, whereas the remaining functions implement a Low-Density
Parity-Check code and make explicit use of the static keys.
Overview of Multi-Stage Encoder
[0091] Referring again to FIG. 2, dynamic encoder 220 receives
input symbols IS(0), . . . , IS(K-1) and the redundant symbols
RE(0), . . . , RE(R-1) and a key I for each output symbol it is to
generate. The collection comprising the original input symbols and
the redundant symbols will be referred to as the collection of
"dynamic input symbols" hereafter.
[0092] FIG. 5 is a simplified block diagram of one embodiment of a
dynamic encoder, including a weight selector 510, an associator
515, a value function selector 520 and a calculator 525. As shown
in FIG. 5, the K+R dynamic input symbols are stored in a dynamic
symbol buffer 505. In effect, dynamic encoder 500 performs the
action illustrated in FIG. 6, namely, to generate an output symbol
value B(I) as some value function of selected input symbols.
[0093] FIG. 6 illustrates the action of a function used by
calculator 525.
[0094] FIG. 7 is a simplified block diagram of one specific
embodiment of a static encoder according to the present invention.
Static encoder 600 comprises a parameter calculator 605, a Hamming
encoder 610, and a low-density-parity-check (LDPC) encoder 620.
Hamming encoder 610 is coupled to receive the input symbols IS(0),
. . . , IS(K-1) from an input symbol buffer 625, the number K of
input symbols, and the parameter D. In response, Hamming encoder
610 generates D+1 redundant symbols HA(0), HA(1), . . . , HA(D)
according to a Hamming code.
[0095] FIG. 8 illustrates the operation of one embodiment of the
present invention that employs the static encoder shown in FIG.
7.
[0096] FIG. 9 is a simplified block diagram illustrating one
embodiment of a decoder according to the present invention. Decoder
900 can be used, for example, to implement decoder 155 of FIG. 1.
Decoder 900 comprises a dynamic decoder 905 and a static decoder
910. Input symbols and redundant symbols recovered by dynamic
decoder 905 are stored in a reconstruction buffer 915. Upon
completion of dynamic decoding, static decoder 910 attempts to
recover any input symbols not recovered by dynamic decoder 905, if
any. In particular, static decoder 910 receives input symbols and
redundant symbols from reconstruction buffer 915.
Variations
[0097] Many variations of LDPC decoders and Hamming decoders are
well known to those skilled in the art, and can be employed in
various embodiments according to the present invention. In one
specific embodiment, a Hamming decoder is implemented using a
Gaussian elimination algorithm. Many variations of Gaussian
elimination algorithms are well known to those skilled in the art,
and can be employed in various embodiments according to the present
invention.
[0098] Multi-stage fountain codes as described above are not
systematic codes, i.e., all of the original source symbols of a
source block are not necessarily among the encoding symbols that
are sent. However, systematic FEC codes are useful for a file
download system or service, and very important for a streaming
system or service. As shown in the implementation below, a modified
code can be made to be systematic and still maintain the fountain
code and other described properties.
[0099] One reason why it is easy to architect a variety of
supplemental services using multi-stage codes is that it can
combine received encoding symbols from multiple senders to
reconstruct a source file or stream without coordination among the
senders. The only requirement is that the senders use differing
sets of keys to generate the encoding symbols that they send in
encoding packets to the code. Ways to achieve this include
designating different ranges of the key space to be used by each
such sender, or generating keys randomly at each sender.
[0100] As an example of the use of this capability, consider
providing a supplemental service to a file download service that
allows multi-stage fountain codes that did not receive enough
encoding packets to reconstruct a source file from the file
download session to request additional encoding packets to be sent
from a make-up sender, e.g., via a HTTP session. The make-up sender
generates encoding symbols from the source file and sends them, for
example using HTTP, and all these encoding symbols can be combined
with those received from the file download session to recover the
source file. Using this approach allows different senders to
provide incremental source file delivery services without
coordination between the senders, and ensuring that each individual
receiver need receive only a minimal number of encoding packets to
recover each source file.
Implementations of a MS Code Intermediate Symbol Generator Using
Half-Weight Symbols
[0101] For a given set of K input symbols, a static encoder may
generate L intermediate symbols wherein L>K, wherein the first K
of these are the input symbols (in the case of a systematic
encoder) and the remaining L-K intermediate symbols are redundant
symbols including Half-Weight symbols. The intermediate symbols are
such that the input symbols can be recovered from the intermediate
symbols. The source symbols used as input to a dynamic encoder can
comprise some, all or none of the intermediate symbols (and/or the
input symbols if these are not part of the intermediate
symbols).
[0102] A dynamic encoder generates output symbols from the source
symbols. Where the dynamic encoder is a fountain or fountain
encoder, the code rate can be "rateless" and as many output symbols
can be generated as needed without having to resort to repeating
information duplicative symbols.
[0103] Intermediate symbols are generated using an inverse encoding
process. The output symbols do not need to include the intermediate
symbols, i.e., intermediate symbols are not included in data
packets. A source block is sometime broken into sub-blocks, each of
which is sufficiently small to be decoded in working memory. For a
source block comprising K source symbols, each sub-block comprises
K sub-symbols, wherein each symbol of the source block comprises
one sub-symbol from each sub-block.
Generating Intermediate Symbols Including Half-Weight Symbols
[0104] A method and apparatus for generating intermediate symbols
including Half-Weight symbols will now be described. A Half-Weight
symbol is a symbol that depends on approximately half of the source
symbols, such as being an XOR of approximately half of the source
symbols and wherein each symbol lends a dependency to about half of
the Half-Weight symbols. Where the number of source symbols and the
number of Half-Weight symbols are both divisible by two, it might
be such that exactly half of the Half-Weight symbols depend from
any one source symbol and each of the Half-Weight symbols depend on
exactly half of the source symbols. The graph mapping source
symbols on one side to Half-Weight symbols on the other side might
be generated using a Gray sequence.
[0105] Values for a set of intermediate symbols are derived from
the original input symbols such that knowledge of the intermediate
symbols is sufficient to reconstruct the input symbols. With
multiple stages, an encoder might produce redundant symbols that
are each a function (such as the exclusive OR, or "XOR") of a
number of the intermediate symbols. The redundant symbols are
produced in such a way that the intermediate symbols, and therefore
also the source symbols, can be recovered from any sufficiently
large set of output symbols, typically just the amount of data
representing the original data plus some small extra amount. The
intermediate symbols might comprise source symbols, Half-Weight
symbols, or the like.
[0106] Luby-Prov teaches the use of a systematic code encoder
wherein a number of decoding algorithms are possible, along with
some decoding algorithms. As taught there, construction of
intermediate and repair symbols is based in part on a pseudo-random
number generator based on a fixed set of random numbers that are
available to both sender and receiver. The construction of the
intermediate symbols from the source symbols might be governed by a
systematic index.
[0107] A pre-coder might be programmed or constructed to generate L
intermediate symbols from K input symbols (L>K where the symbols
are the same size). In this example, the first K intermediate
symbols are the K input symbols being pre-coded for and the last
L-K intermediate symbols are generated in terms of the first K
intermediate symbols. The first K intermediate symbols might be
generated in some other manner.
[0108] The last L-K intermediate symbols C[K], . . . , C[L-1] might
comprise S LDPC symbols and H Half-Weight symbols. The values of S
and H are determined from K as described below, wherein L=K+S+H.
Let X be the smallest positive integer such that X(X-1)=2K and let
S be the smallest prime integer such that S.gtoreq.ceil(0.01K)+X,
where the function ceil(x) denotes the smallest positive integer
which is greater than or equal to x. Then, determine H such that it
is the smallest integer such that choose(H, ceil(H/2)).gtoreq.K+S,
where the function choose(i,j) denotes the number of ways j objects
can be chosen from among i objects without repetition.
[0109] Next, determine the Half-Weight symbols using a process
wherein H'=ceil(H/2). Let: [0110] C[0], . . . , C[K-1] denote the
first K intermediate symbols [0111] C[K], . . . , C[K+S-1] denote
the S LDPC symbols, initialized to zero. [0112] C[K+S], . . . ,
C[L-1] denote the H Half-Weight symbols, initialized to zero.
[0113] The S LDPC symbols are defined to be the values of C[K], . .
. , C[K+S-1] at the end of the following process, wherein floor(x)
denotes the largest positive integer which is less than or equal to
x, "i % j" denotes i modulo j, and X Y denotes, for equal-length
bit strings X and Y, the bitwise exclusive-or of X and Y: [0114]
For i=0, . . . , K-1 do [0115] a=1+(floor (i/S)%(S-1)) [0116] b=i %
S [0117] C[K+b]=C[K+b] C[i] [0118] b=(b+a)% S [0119] C[K+b]=C[K+b]
C[i] [0120] b=(b+a)% S [0121] C[K+b]=C[K+b] C[i]
[0122] The H Half-Weight symbols are defined as follows:
[0123] Let g[i]=i (floor(i/2)) for all positive integers i, wherein
g[i] is a Gray sequence in which each element differs from the
previous one in a single bit position. Let g[, k] denote the
j.sup.th element, j=0, 1, 2, . . . , etc. of the subsequence of
g[i] whose elements have exactly k non-zero bits in their binary
representation. Then, the Half-Weight symbols are defined as the
values of C[K+S], . . . , C[L-1] after the following process:
[0124] For h=0, . . . , H-1 do [0125] For j=0, . . . , K+S-1 do
[0126] If bit h of g[j,H'] is equal to 1, then C[h+K+S]=C[h+K+S]
C[j].
[0127] A more efficient method and apparatus will now be described
for quickly generating the Half-Weight symbols.
Efficient Half-Weight Code Encoders/Decoders
[0128] An encoder for Half-Weight codes computes, for a sequence of
k source symbols, a plurality of h Half-Weight symbols. Each of
these Half-Weight symbols is the XOR of some of the source symbols,
where generally the lengths of all the symbols are the same. These
Half-Weight symbols can be used as intermediate symbols that
provide input to a dynamic encoder. We use matrix notation to
describe which source symbols are XORed to yield a given
Half-Weight symbol.
[0129] A Gray code sequence g[.cndot.] is used to describe the
relationship between the source symbols and the Half-Weight symbols
generated from the source symbols. To illustrate the concept, we
use the following particular Gray code. For all positive integers
i, let g[i] be defined as follows. Let b[i] be the highest order
bit that is different in the binary representation of i-1 and i.
Then, the binary representation of g[i] is obtained by flipping bit
b[i] of g[i-1]. FIG. 10 shows the first few elements of a Gray code
sequence generated according to this rule. Note that g[] has the
property that each pair of consecutive elements in the sequence
differ in exactly one bit position. Note also that there are other
Gray code sequences besides the ones described here that would be
equally suitable as the basis for a Half-Weight code.
[0130] We also use the function cnt[i], where cnt[i] returns the
number of bits that are set to one in the binary representation of
i. For example, if i=25 in decimal, then its binary representation
is 11001, and thus cnt[25]=3.
[0131] For any fixed positive integer j, let g[j] be the
subsequence of g[] where for each element in the sequence exactly j
bits are set to 1. Thus, g[j] can be defined based on g[] as
follows:
Initialize c=0, i=0 Do forever
[0132] While (cnt(g[c]).noteq.j)c=c+1
[0133] g[i,j]=g[c]
[0134] c=c+1
[0135] i=i+1
[0136] Thus, for example, for j=3, g[0,3]=g[5]=7, g[1,3]=g[9]=13,
g[2,3]=g[11]=14, g[3,3]=g[13]=11, g[4,3]=g[17]=25, etc. Note that
g[,j] has the property that each pair of consecutive elements in
the sequence differ in exactly two bit positions in their binary
representation. For all i.gtoreq.1, let Pos1[i,j] be one of the two
bit positions in which g[i,j] and g[i-1,j] differ and let Pos2[i,j]
be the other bit position in which they differ. Thus, for example,
Pos1[4,3]=1 and Pos2[4,3]=4.
[0137] Let (x.sub.0, x.sub.1, . . . , x.sub.k-1) denote the
sequence of k source symbols, and let (y.sub.0, y.sub.1, . . . ,
y.sub.h-1) denote the sequence of Half-Weight symbols, where h
satisfies Equation 1. The relationship between the source symbols
and the h Half-Weight symbols can be represented as shown in FIG.
11 by a matrix H with k rows and h columns with each entry being 0
or 1 such that the Half-Weight symbols can be calculated from the
source symbols using Equation 2.
(X.sub.0, X.sub.1, . . . , X.sub.k-1)H=(y.sub.0, y.sub.1, . . . ,
y.sub.h-1) (Equ. 2)
[0138] In Equation 2, if H[l,k] denotes the entry of the matrix H
at position (l, k), then y.sub.j is equal to
H[0,j]X.sub.0.sym.H[1,j]x.sub.1.sym. . . . .sym.H[k-1,j]X.sub.k-1,
wherein the symbol ".sym.+" is used to denote the XOR operation.
Herein, H[l,j]x.sub.1 is defined as 0 if H[l,j]=0, and as X.sub.1
if H[l,j]=1.
[0139] The matrix H has a special form that can be described as
follows. Let h'=ceil(h/2), i.e., h' is equal to h/2 rounded up to
the nearest integer. The matrix H is the k by h matrix in which the
rows are indexed from 0 to k-1, where row i is equal to the binary
representation of g[i,h']. For example, when k=7, we have h=5 (from
Equ. 1) and h'=3, the corresponding matrix H is as shown in FIG.
111. In this example, the seven rows of the matrix are indexed 0
through 6 and the entries in the rows correspond to the binary
representations of the integers 7, 13, 14, 11, 25, 26, and 28.
[0140] In a straightforward calculation of the product in Equation
2, the number of XORs of source symbols performed for calculating
each Half-Weight symbol y.sub.i for i=0, 1, . . . , h-1 is one less
than the number of ones in the column corresponding to i in the
matrix H. Since H has exactly kh' ones, the number of XORs of
source symbols performed by the straightforward method is k(h'-1),
where h' is of the order of log(k).
An Embodiment of an Efficient Method for Computing Half-Weight
Symbols
[0141] We present a method by which the Half-Weight symbols can be
calculated with at most 3(k-1)+h'+1 XORs of symbols.
[0142] An exemplary embodiment of an efficient method for
calculation of Half-Weight symbols is shown in FIG. 12. The method
comprises three phases: an Initialize Computation phase shown in
step 1210, a Perform Core Computation phase shown in step 1215, and
a Clean Up phase shown in step 1220. Other variations are possible.
The method starts by receiving the source symbols x.sub.0, x.sub.1,
. . . , x.sub.k-1 in step 1205, performs the phases shown in steps
1210, 1215, and 1220 on the source symbols, and outputs the
Half-Weight symbols y.sub.0, y.sub.1, y.sub.h-1 in step 1225.
[0143] An exemplary embodiment of the Initialize Computation phase
1210 of FIG. 12 is shown in FIG. 13. The Initialize Computation
phase 1210 shown in FIG. 13 comprises two steps. Step 1305
initializes the values of each of the Half-Weight symbols to zero.
The second Step 1310 initializes the value of a symbol variable SUM
to x.sub.0. During the Perform Core Computation phase 1215 of FIG.
12, the value of SUM will be the XOR of all the source symbols
visited during the execution of the process.
[0144] FIG. 14 shows one exemplary embodiment of the Perform Core
Computation phase 1215 of FIG. 12. The overall structure of this
phase is a loop over the source symbols x.sub.1, x.sub.2, . . . ,
x.sub.k-1. A variable i keeps track of the indices of these
symbols, and is, accordingly, initialized to one in 1405. In Step
1410, the value of SUM is XORed into the two Half-Weight symbols
indexed by Pos1[i,h'] and Pos2[i,h'].
[0145] The functions Pos1[i,h'] and Pos2[i,h'] can be calculated in
a variety of ways, as is well known to those of skill in the art.
For example, one way to do this is to explicitly calculate for
consecutive indices of i the g[i,h'] sequence, and then calculate
Pos1[i,h'] and Pos2[i,h'] based on the two positions where g[i-1,h]
and g[i,h'] differ in their binary representation. As another
example, J. R. Bitner, G. Ehrlich, E. M. Reingold, "Efficient
generation of the binary reflected Gray code", Communications of
the ACM (1976) describes an efficient process that requires only a
constant number of operations on standard central processing units
to calculate each consecutive value of Pos1[i,h'] and Pos2[i,h']
for a fixed value of h'. As another example, a lookup table might
be used to hold pre-computed values of Pos1[i,h'] and Pos2[i,h']
for indices i=1, . . . , K-1 and h'=1, . . . , H', where K' is an
upper-bound on values of k to be used and H' is an upper-bound on
values of h' to be used.
[0146] In step 1420, the value of x.sub.i is XORed into SUM to
update the value of SUM. This update ensures that SUM remains the
XOR of the values of all of the x.sub.i visited so far.
[0147] The counter i is incremented by one in step 1425. In step
1430, the value of i is compared to k, and if i<k, then steps
1410 through 1430 are repeated, whereas if i is equal to k, then
the loop terminates at step 1435, and the process continues with
the Clean Up phase 1220.
[0148] An exemplary embodiment of the Clean Up phase 1220 is shown
in FIG. 15. This phase comprises an initialization step 1505 and a
computation that loops through values of a variable j ranging from
0 to h-1. The computation loop encompasses steps 1510 through 1550
in FIG. 15. In the initialization step 1505 the value of j is
initialized to zero and the value of MASK is initialized to
g[k-1,h'].
[0149] In Step 1510 of the computation loop, it is checked whether
the least significant bit of the binary representation of MASK is
equal to 1, and if the test is true, then the next is step 1520. In
step 1520, the value of y.sub.j is updated by XORing it with the
value of SUM, and the process continues with step 1540.
[0150] If the result of the test in Step 1510 is false, then the
process jumps directly to step 1540. In step 1540, the value of j
is incremented by one and the least significant bit of MASK is
shifted out of MASK, where this shift is implemented by the DIV
operator comprising dividing MASK by two and dropping the
remainder. This second part of step 1540 could be implemented in a
variety of ways, including for example applying a shift operator to
MASK. Step 1550 checks to see if j<h, and if j<h then
processing returns to step 1510, but if this condition is not true
then processing stops at step 1560. Each of these steps can be
performed by digital logic, software or other physical computation
device.
[0151] One skilled in the art will recognize after reading this
disclosure that there are other alternative methods to those shown
in FIGS. 12, 13, 14 and 15 for efficient computation of the
Half-Weight symbols.
Calculating Several Sets of Half-Weight Symbols
[0152] As was mentioned above, one of the reasons for using the
Half-Weight code is the excellent erasure correction capability of
this code. In certain applications, however, it is important to
protect the data with more redundant symbols than a Half-Weight
code is capable of producing. This is particularly important when
the redundant symbols are used to decrease the error in a
probabilistic decoding algorithm, such as those described in
Shokrollahi I. For this reason, a variation of using Half-Weight
codes is described herein that is capable of producing several sets
of redundant symbols, wherein each set of redundant symbols has the
same number of redundant symbols as the Half-Weight code. This
encoder is referred to as the "parallel Half-Weight encoder"
herein.
[0153] Denoting again by k the number of source symbols to the
parallel Half-Weight encoder, and by s the number of sets of
independent Half-Weight symbols that are to be generated, the
method starts by calculating s random or pseudorandom permutations
of the integers 0, 1, . . . , k-1. There are a number of ways to
calculate these permutations, as is well known to those of skill in
the art.
[0154] In some embodiments, one property of these permutations is
that they are easy to calculate, i.e., it is easy to determine for
any given j in among the integers 0, 1, . . . , k-1, what the image
of j under the permutation is. For example, where k is a prime
number, one of the many possibilities for this permutation can be
to independently choose two random integers a and b, where a is
chosen uniformly at random in the range 1, 2, . . . , k-1, and b is
independently chosen at random in the range 0, 1, . . . , k-1.
Then, the image of j under the permutation defined by a and b is
defined as aj+b modulo the integer k. In other words, this image is
the remainder of the division of aj+b by k.
[0155] If k is not a prime number, then the numbers a and b can be
chosen by choosing a uniformly at random from the set of positive
integers less than k which do not have a common divisor with k,
while b can be chosen uniformly at random in the range 0, 1, . . .
, k-1 as before. Various other methods for generating permutations
can be envisioned by those of skill in the art upon studying this
application, and the method described below does not depend on the
specific choice of the permutations, except that in preferred
applications it is desirable to have s different permutations.
[0156] Returning to the description of the parallel Half-Weight
encoder, the encoder generates s random or pseudorandom
permutations, denoted .pi..sub.1, .pi..sub.2, . . . , .pi..sub.s.
There will be s sets of redundant Half-Weight symbols denoted
y.sub.0,0, y.sub.0,1, . . . , y.sub.0,h-1; y.sub.1,0, y.sub.1,1, .
. . , y.sub.1,h-1; . . . ; y.sub.s-1,0, y.sub.s-1,1, . . . ,
y.sub.s-1,h-1. As before, h is the smallest positive integer that
satisfies Equation 1. The generation of the redundant symbols of
the parallel Half-Weight encoder is done one set at a time.
[0157] To generate the jth set, where j=0, 1, . . . , s-1, the
encoder proceeds in a similar manner as the original Half-Weight
encoder described above. The encoder might comprise three phases,
similar to those shown in FIG. 12, with a modification for parallel
operation of step 1420, when updating the value of SUM. In the
parallel encoder, the value of SUM is updated as SUM=SUM
.sym.x(.pi..sub.j(i)) for the jth set.
Using the Fast Encoder for Recovering from Erasures
[0158] A decoder described in Shokrollahi I might be used to
recover the original source symbols from the received dynamic
symbols. If a Half-Weight code is used by the encoder, one of the
steps in the decoding process in some embodiments is to recover
some of the symbols involved in a Half-Weight code based on the
recovery of the remainder of the symbols involved in the
Half-Weight code. The Half-Weight code has the property that with
high probability any combination of r.ltoreq.h missing symbols can
be recovered if the remaining symbols among the k+h symbols
y.sub.0, y.sub.1, . . . , y.sub.h-1 and x.sub.0, x.sub.1, . . . ,
x.sub.k-1 have already been recovered. When used in conjunction
with an Inactivation Decoder or other decoder wherein a Gaussian
Elimination of the decoding matrix is performed, then the
Half-Weight code has the property that addition of the Half-Weight
rows to the decoding matrix formed from the other decoding stages
in a multi-stage decoder will result in a matrix of full rank, with
high probability, and hence reduce the error probability of the
decoder.
[0159] The relationship between a missing set of r symbols z.sub.0
can be efficiently calculated using at most 3(k-1)+h/2+1 XORs of
symbols given the values of the remaining t=k+h-r symbols, using a
variant of the efficient method for calculating the Half-Weight
code disclosed above. This can dramatically speed up the overall
decoding time, since this avoids using klog(k) XORs of symbols to
calculate the relationships between the r missing symbols based on
a variant of the straightforward method of calculating the
Half-Weight code.
[0160] Such a variant comprises making the following changes to the
Half-Weight encoding method described in FIGS. 12, 13, 14 and 15.
In step 1205 of FIG. 12, the symbols received are the t symbols
among the k+h source and Half-Weight symbols. In step 1305 of FIG.
13, the missing Half-Weight symbols are initialized to zero and the
remaining received Half-Weight symbols retain their received
values. In step 1310 of FIG. 13, if source symbol x.sub.0 is among
the received source symbols, then SUM is set to x.sub.0, and if
x.sub.0 is among the missing source symbols, then SUM is
initialized to zero. In step 1420 of FIG. 14, if x.sub.i is among
the received source symbols, then SUM is updated as described in
step 1420, and if x.sub.i is among the missing source symbols, then
SUM is left unchanged. Suppose r' of the source symbols are missing
and r'' of the Half-Weight symbols are missing, and thus r'+r''=r.
Then, in step 1225 of FIG. 12, the dependency on the k-r' received
source symbol of the h-r'' received Half-Weight symbols has been
removed, and thus the h-r'' received Half-Weight symbol values
depend only on the missing r' source symbols. Thereafter, a process
that can solve the small system of equations involving the h-r''
received Half-Weight symbols and the missing r' source symbols can
be used to recover the missing r' source symbol values. Examples of
such systems of equations should be apparent after review of this
disclosure.
[0161] As one skilled in the art will recognize, there are many
other variants of the above efficient decoding method. For example,
in step 1410 of FIG. 14, one variant is to skip updates of missing
Half-Symbol values, thus making the overall computation somewhat
more efficient. As another variant, the missing Half-Symbol updates
are not skipped in step 1410 of FIG. 14, and then in step 1225 of
FIG. 12, the Half-Symbol values corresponding to missing
Half-Symbols only depend on the missing source symbol values and
their own missing value, and these equations may be useful for
other part of an overall decoding process, e.g., if there are other
equations that involve the missing Half-Symbols and missing source
symbols from other parts of an overall code. This is the case, for
example, when using an Inactivation Decoder with a multi-stage
code.
[0162] In most of the examples described above, the input and
output symbols encode for the same number of bits and each output
symbol is placed in one packet (a packet being a unit of transport
that is either received in its entirety or lost in its entirety).
In some embodiments, the communications system is modified so that
each packet contains several output symbols. The size of an output
symbol value is then set to a size determined by the size of the
input symbol values in the initial splitting of the file or blocks
of the stream into input symbols, based on a number of factors. The
decoding process remains essentially unchanged, except that output
symbols arrive in bunches as each packet is received.
[0163] The setting of input symbol and output symbol sizes is
usually dictated by the size of the file or block of the stream and
the communication system over which the output symbols are to be
transmitted. For example, if a communication system groups bits of
data into packets of a defined size or groups bits in other ways,
the design of symbol sizes begins with the packet or grouping size.
From there, a designer would determine how many output symbols will
be carried in one packet or group and that determines the output
symbol size. For simplicity, the designer would likely set the
input symbol size equal to the output symbol size, but if the input
data makes a different input symbol size more convenient, it can be
used.
[0164] The above described encoding process produces a stream of
packets containing output symbols based on the original file or
block of the stream. Each output symbol in the stream is generated
independently of all other output symbols, and there is no lower or
upper bound on the number of output symbols that can be created. A
key is associated with each output symbol. That key, and some
contents of the input file or block of the stream, determines the
value of the output symbol. Consecutively generated output symbols
need not have consecutive keys, and in some applications it would
be preferable to randomly generate the sequence of keys, or
pseudorandomly generate the sequence.
[0165] Multi-stage decoding has a property that if the original
file or block of the stream can be split into K equal sized input
symbols and each output symbol value is the same length as an input
symbol value, then the file or block can be recovered from K+A
output symbols on average, with very high probability, where A is
small compared to K. For example, for the weight distributions
introduced above, the probability that the value of A exceeds
.alpha.*K is at most 10.sup.-12 if K is larger than 19,681, and it
is at most 10.sup.-10 for any value of K. Since the particular
output symbols are generated in a random or pseudorandom order, and
the loss of particular output symbols in transit is assumed random,
some small variance exists in the actual number of output symbols
needed to recover the input file or block. In some cases, where a
particular collection of K+A packets are not enough to decode the
entire input file or block, the input file or block is still
recoverable if the receiver can gather more packets from one or
more sources of output packets.
[0166] Because the number of output symbols is only limited by the
resolution of I, well more than K+A output symbols can be
generated. For example, if I is a 32 bit number, 4 billion
different output symbols could be generated, whereas the file or
block of the stream could include K=50,000 input symbols. In some
applications, only a small number of those 4 billion output symbols
may be generated and transmitted and it is a near certainty that an
input file or block of a stream can be recovered with a very small
fraction of the possible output symbols and an excellent
probability that the input file or block can be recovered with
slightly more than K output symbols (assuming that the input symbol
size is the same as the output symbol size).
[0167] In some applications, it may be acceptable to not be able to
decode all of the input symbols, or to be able to decode all of
input symbols, but with a relatively low probability. In such
applications, a receiver can stop attempting to decode all of the
input symbols after receiving K+A output symbols. Or, the receiver
can stop receiving output symbols after receiving less than K+A
output symbols. In some applications, the receiver may even only
receive K or less output symbols. Thus, it is to be understood that
in some embodiments of the present invention, the desired degree of
accuracy need not be complete recovery of all the input
symbols.
[0168] Further, in some applications where incomplete recovery is
acceptable, the data can be encoded such that all of the input
symbols cannot be recovered, or such that complete recovery of the
input symbols would require reception of many more output symbols
than the number of input symbols. Such an encoding would generally
require less computational expense, and may thus be an acceptable
way to decrease the computational expense of encoding.
[0169] It is to be understood that the various functional blocks in
the above described figures may be implemented by a combination of
hardware and/or software, and that in specific implementations some
or all of the functionality of some of the blocks may be combined.
Similarly, it is also to be understood that the various methods
described herein may be implemented by a combination of hardware
and/or software.
[0170] The above description is illustrative and not restrictive.
Many variations of the invention will become apparent to those of
skill in the art upon review of this disclosure. The scope of the
invention should, therefore, be determined not with reference to
the above description, but instead should be determined with
reference to the appended claims along with their full scope of
equivalents.
* * * * *