Apparatus and method for generating polarization code
阅读说明：本技术 生成极化码的设备和方法 (Apparatus and method for generating polarization code ) 是由 米哈伊尔·谢尔盖维奇·加米涅夫 尤利娅·鲍里索文·卡梅涅娃 金杰 奥列格·菲特维奇·库尔马耶夫 于 2018-02-22 设计创作，主要内容包括：本发明涉及一种用于基于由码序列S定义的原始极化码生成长度为N和维度为K的极化码的设备(202b、204b),其中所述码序列S具有从最不可靠到最可靠子信道排序的N<Sub>max</Sub>个比特索引。所述设备(202b、204b)包括处理单元(202c、204c),所述处理单元(202c、204c)用于：(a)通过从所述码序列S中删除大于或等于N<Sub>max</Sub>/2个比特索引来生成具有N<Sub>max</Sub>/2个比特索引的辅助码序列；(b)从所述辅助码序列中删除后N<Sub>R</Sub>个比特索引以生成修改后的辅助码序列,其中N<Sub>R</Sub>表示删除的比特数,并且所述处理单元(202c、204c)用于基于N<Sub>max</Sub>、N和预定义码率R确定所述删除的比特数N<Sub>R</Sub>；(c)通过基于由所述修改后的辅助码序列的后p＝N<Sub>max</Sub>–N个比特索引定义的打孔集合对所述原始极化码进行打孔,生成所述长度为N和维度为K的极化码。此外,本发明涉及一种用于生成极化码的相应方法。(The invention relates to a device (202b, 204b) for generating a polarization code of length N and dimension K based on an original polarization code defined by a code sequence S having N ordered from least reliable to most reliable sub-channels max A bit index. The device (202b, 204b) comprises a processing unit (202c, 204c), the processing unit (202c, 204c) being configured to: (a) by deleting N or more from the code sequence S max Generating a bit index having N max An auxiliary code sequence of/2 bit indices; (b) deleting post-N from the auxiliary code sequence R Bit index to generate a modified auxiliary code sequence, where N R Represents the number of bits deleted, and the processing unit (202c, 204c) is configured to base the N on max N and a predefined code rate R determining the number N of said deleted bits R (ii) a (c) By post-p-N based on the modified auxiliary code sequence max -N bit index definitionsThe original polarization code is punctured by the puncturing set to generate the polarization code with the length of N and the dimensionality of K. Furthermore, the invention relates to a corresponding method for generating a polarization code.)
1. Based on having NmaxDevice (202b, 204b) for generating a polarization code of length N and dimension K from an original polarization code defined by a code sequence S of individual bit indices, characterized in that the device (202b, 204b) comprises a processing unit (202c, 204c), the processing unit (202c, 204c) being configured to:
(a) by deleting N or more from the code sequence SmaxGenerating a bit index having NmaxAn auxiliary code sequence of/2 bit indices;
(b) deleting post-N from the auxiliary code sequenceRBit index to generate a modified auxiliary code sequence, where NRRepresents the number of bits deleted, and the processing unit (202c, 204c) is configured to base the N onmaxN and a predefined code rate R determining the number N of said deleted bitsR；
(c) By means of a code sequence based on the auxiliary code modified by the modificationAfter row p ═ Nmax-a puncturing set defined by N bit indices punctures the original polar code, generating the polar code of length N and dimension K.
2. The apparatus (202b, 204b) of claim 1, wherein said processing unit (202c, 204c) is configured to determine the number of deleted bits N based on the following equationR：
NR＝round(ap+b)，
Where round (.) represents a rounding function, a and b represent parameters that depend on NmaxAnd the predefined code rate R.
3. The apparatus (202b, 204b) of claim 1, wherein said processing unit (202c, 204c) is configured to determine the number of deleted bits N based on the following equationR：
NR＝max(0，round(ap+b))，
Where round (..) denotes a rounding function, max (..) denotes a maximum function, and a and b denote parameters that depend on NmaxAnd the predefined code rate R.
4. The device (202b, 204b) according to claim 2 or 3, wherein the device (202b, 204b) further comprises a memory unit, the processing unit (202c, 204c) being configured to determine the parameters a and b based on a look-up table stored in the memory unit.
5. The device (202b, 204b) of claim 4, wherein the processing unit (202c, 204c) is configured to determine the parameters a and b based on auxiliary parameters a 'and b' provided in the look-up table and the following equation:
a＝f(R)a′，
b＝f(R)b′，
wherein f (R) represents a function of the predefined code rate R, the auxiliary parameters a 'and b' depending on Nmax。
6. The device (202b, 204b) according to claim 5, wherein the processing unit (202c, 204c) is configured to determine the auxiliary parameters a 'and b' based on the following look-up table.
7. The device (202b, 204b) according to claim 5 or 6, wherein the processing unit (202c, 204c) is configured to determine the parameters a and b based on the auxiliary parameters a 'and b' provided in the look-up table and the following equation:
a＝R2a′，
b＝R2b′。
8. a communication apparatus (202), characterized in that the communication apparatus (202) comprises a channel encoder (202a), the channel encoder (202a) comprising a device (202b) for generating a polarization code according to any of the preceding claims.
9. The communication apparatus (202) of claim 8, wherein the channel encoder (202a) is configured to encode an information message of size K into a codeword of size N based on the polarization code generated by the processing unit (202c) of the device (202 b).
10. The communication apparatus (202) of claim 9, wherein the channel encoder (202a) is configured to encode an information message of size N into a codeword of size N based on the polar code generated by the processing unit (202c) of the device (202b) using a set of frozen bits comprising the N of the code sequence S defined by the set of puncturesmax-bits defined by N bit indices and bits defined by the first N-K bit indices of the code sequence S, excluding the puncturing set.
11. A communication apparatus (204), characterized in that the communication apparatus (204) comprises a channel decoder (204a), the channel decoder (204a) comprising a device (204b) for generating a polarization code according to any one of claims 1 to 7.
12. The communication apparatus (204) of claim 11, wherein the channel decoder (204a) is configured to decode log likelihood ratios of N codeword bits into an information message of size K based on the polar code generated by the processing unit (204c) of the device (204 b).
13. The communication apparatus (204) of claim 12, wherein the channel decoder (204a) is configured to decode log-likelihood ratios of N codeword bits into an information message of size K based on the polar code generated by the processing unit (204c) of the device (204b) using a set of frozen bits comprising the N of the code sequence S defined by the set of puncturesmax-bits defined by N bit indices and bits defined by the first N-K bit indices of the code sequence S, excluding the puncturing set.
14. Based on having NmaxMethod (900) of generating a polar code of length N and dimension K from an original polar code defined by a code sequence S of individual bit indices, characterized in that said method (900) comprises:
(a) by deleting N or more from the code sequence SmaxGenerating (901) a bit index having NmaxAn auxiliary code sequence of/2 bit indices;
(b) deleting (903) the post-N from the auxiliary code sequenceRBit index to generate a modified auxiliary code sequence, where NRIndicating the number of bits of said erasure, and the number of bits N of said erasureRIs based on NmaxN and a predefined code rate R;
(c) by the post-p-N based on the modified auxiliary code sequencemax-puncturing the original polar code by a puncturing set defined by N bit indices, generating (905) a polar code of length N and dimension K.
15. A computer program product, characterized in that the computer program product comprises program code for performing the method of claim 14 (900) when executed on a computer.
Technical Field
The present invention relates generally to data encoding in a communication system. More particularly, the present invention relates to an apparatus and method for generating polarization codes and encoding data using the polarization codes.
Background
It has been proved that the shannon capacity of many channels can be achieved by using polar codes, and the length of the traditional polar code is generally equal to N2m. However, different application scenarios require polarization codes with different dimensions and finer code length granularity. For this reason, different methods have been proposed for constructing lengths different from N-2mThe polarization code of (1). These methods include reliability recalculation, bit reversal (brv) shortening, packet puncturing, etc., some of which are described further below.
The future 3GGP 5G standard adopts a long code sequence to construct a length N-2mThe polarization code of (1). Although this sequence allows the construction of a length N-2 with good performancemBut requires additional rate matching techniques in order to construct a polar code of any length. For example, the BRMIAA rate matching scheme uses block puncturing/shortening and some additional computations to evaluate the number of information (not frozen) bits in the N/2 high and N/2 low bits of the polar code. According to the rate matching scheme, an online calculation is performed according to the following equation:
block punching:
block shortening:
wherein the content of the first and second substances,
BRV shortening allows the construction of arbitrary length polarization codes by setting predefined values for certain information bits. The predefined set of bits is selected in such a way that after the encoding process the value of the respective encoded bit can be determined based on the set only. It is assumed that the shortened set of bits and their values are already known at the receiver side and therefore it is useless to transmit them over noisy channels. Prior to the decoding process, the decoder sets the LLR (log likelihood ratio) of the untransmitted bits to some larger value according to its value. For simplicity, it is generally assumed that all shortened bits are equal to zero.
To construct a length of N ≠ 2mThe polarization code of (2) needs to be shortened by S bits. The shortened set of bits of size S is typically determined by: (i) for the sequence [0, 1., N-1 ]]:B＝BRV([0，1，...，N-1]) Carrying out bit reversal arrangement; (ii) the last S elements are selected from the sequence B and the bit with the corresponding index is set as the freeze bit. This process is illustrated in figure 1. The benefit of BRV shortening is simplicity, but performance is worse than puncturing at low code rates.
Another approach adopted in the future 3GGP 5G standard is packet puncturing. In packet puncturing, codewords are divided into groups, for example, 32 groups of size N/32, and then the groups are arranged. The punctured/shortened bit set is determined from the arranged sequence by acquiring the necessary number of bits from the start/end. The disadvantage of this approach is that the constructed code is punctured severely and therefore performs poorly, especially for code rates from 1/4 to 2/5. This problem can be partially solved using repetition techniques, but does not work for all cases.
Accordingly, there is a need for improved apparatus and methods for generating polar codes of any given length and dimension with good performance using puncturing techniques and for encoding data using these polar codes.
Disclosure of Invention
It is an object of the present invention to provide improved apparatus and methods for generating polar codes of any given length and dimension using puncturing techniques and for encoding data using these polar codes.
The above object and other objects are achieved by the subject matter claimed in the independent claims. Further implementations are apparent in the dependent claims, the detailed description and the drawings.
In general, embodiments of the present invention are based on the idea proposed to puncture bits in a code sequence in a way that preserves the order of reliability of the bits. This allows use with a length of 2mThe punctured polarization code is constructed by the same sequence of the polarization code without recalculating the bit reliabilities.
According to a first aspect, the invention relates to a device for generating a polar code of length N and dimension K based on an original polar code defined by a code sequence S having N ordered from least reliable to most reliable sub-channelsmaxA bit index. The apparatus comprises a processing unit to: (a) by deleting N or more from the code sequence SmaxGenerating a bit index having NmaxAn auxiliary code sequence of/2 bit indices; (b) deleting post-N from the auxiliary code sequenceRBit index to generate a modified auxiliary code sequence, where NRRepresenting the number of bits deleted, and the processing unit being adapted to base N onmaxN and a predefined code rate R determining the number N of said deleted bitsR(ii) a (c) By post-p-N based on the modified auxiliary code sequencemax-puncturing set defined by N bit indices for said originalAnd punching the polarization code to generate the polarization code with the length of N and the dimensionality of K.
An advantage of generating a punctured set in a device according to the first aspect of the invention is to allow a punctured polar code to be obtained from one long code sequence for small code rates and various code dimensions without degrading performance.
In another possible implementation manner of the first aspect, the processing unit is configured to determine the number N of deleted bits based on the following equationR：
NR＝round(ap+b)，
Where round (.) represents a rounding function, a and b represent parameters, which depend on NmaxAnd the predefined code rate R. In another possible implementation manner of the first aspect, the processing unit is configured to determine the number N of deleted bits based on the following equationR：
NR＝max(0，round(ap+b))，
Wherein max (. -) represents the maximum function.
In another possible implementation manner of the first aspect, the apparatus further comprises a memory unit, wherein the processing unit is configured to determine the parameters a and b based on a look-up table stored in the memory unit.
In another possible implementation manner of the first aspect, the processing unit is configured to determine the parameters a and b based on the auxiliary parameters a 'and b' provided in the look-up table and the following equation:
a＝f(R)a′，
b＝f(R)b′，
wherein f (R) represents a function of the predefined code rate R, the auxiliary parameters a 'and b' depending on Nmax。
In another possible implementation manner of the first aspect, the processing unit is configured to determine the auxiliary parameters a 'and b' based on the following look-up table:
Nmax
b′
a′
64
32.455
-0.87273
128
77.319
-1.2332
256
138.56
-1.1028
512
275.89
-1.1111
1024
572.83
-1.1649
in another possible implementation manner of the first aspect, the processing unit is configured to determine the parameters a and b based on the auxiliary parameters a 'and b' provided in the lookup table and the following equation:
a＝R2a′，
b＝R2b′。
according to a second aspect, the invention relates to a communication device comprising a channel encoder comprising an apparatus for generating a polarization code according to the first aspect of the invention.
In another possible implementation manner of the second aspect, the channel encoder is configured to encode an information message of size K into a codeword of size N based on the polarization code generated by the processing unit of the device.
In another possible implementation of the second aspect, the channel encoder is configured to encode an information message of size K into a codeword of size N based on the polar code generated by the processing unit of the device using a set of frozen bits comprising the N of the code sequence S defined by the set of puncturesmax-bits defined by N bit indices and bits defined by the first N-K bit indices of the code sequence S, excluding the puncturing set.
According to a third aspect, the invention relates to a communication device comprising a channel decoder comprising an apparatus for generating a polarization code according to the first aspect of the invention.
In another possible implementation manner of the third aspect, the channel decoder is configured to decode log likelihood ratios of N codeword bits into an information message with a size K based on the polarization code generated by the processing unit of the device.
In another possible implementation form of the third aspect, the channel decoder is configured to decode log-likelihood ratios of N codeword bits into an information message of size K based on the polar code generated by the processing unit of the device using a set of frozen bits comprising the N of the code sequence S defined by the punctured setmax-bits defined by N bit indices and bits defined by the first N-K bit indices of the code sequence S without the puncturing set, excluding the puncturing set.
According to a fourth aspect, the invention relates to a method for generating a polar code of length N and dimension K based on an original polar code defined by a code sequence S having N ordered from least reliable to most reliable sub-channelsmaxA bit index. The method comprises the following steps: (a) by means of a code derived from said codeDeletions in the sequence S being greater than or equal to NmaxGenerating a bit index having NmaxAn auxiliary code sequence of/2 bit indices; (b) deleting the post-N from the auxiliary code sequenceRBit index to generate a modified auxiliary code sequence, where NRIndicating the number of bits of said erasure, and the number of bits N of said erasureRIs based on NmaxN and a predefined code rate R; (c) by post-p-N based on the modified auxiliary code sequencemax-a puncturing set defined by N bit indices punctures the original polar code, generating the polar code of length N and dimension K.
The method of the fourth aspect of the invention may be performed by the apparatus of the first aspect of the invention. Further features of the method of the fourth aspect of the invention may be implemented directly by the functionality of the device of the first aspect of the invention and the different implementations described above and below.
According to a fifth aspect, the invention relates to a computer program comprising program code for performing the method of the fourth aspect when executed on a computer.
The present invention may be implemented in hardware and/or software.
Drawings
Specific implementations of the present invention will be described with reference to the following drawings, in which:
FIG. 1 shows a schematic diagram of conventional BRV shortening;
fig. 2 shows a schematic diagram of a communication system with a communication apparatus including a device for generating a polarization code according to an embodiment;
FIG. 3 shows a flowchart of various processing steps implemented in an apparatus for generating polarization codes, provided by an embodiment;
fig. 4 is a diagram illustrating an example of a puncturing scheme implemented in an apparatus for generating a polarization code, according to an embodiment;
FIG. 5 shows a flowchart of various processing steps implemented in an apparatus for generating polarization codes, provided by an embodiment;
fig. 6 shows a schematic diagram of a communication system including an apparatus for generating a polarization code provided by an embodiment;
FIG. 7 is a diagram illustrating performance of a polar code generated by an apparatus for generating a polar code, provided by an embodiment;
FIG. 8 is a diagram illustrating performance of a polar code generated by an apparatus for generating a polar code, provided by an embodiment;
fig. 9 shows a flowchart of a method for generating a polarization code according to an embodiment.
In the following figures, identical or at least functionally equivalent features are denoted by the same reference numerals.
Detailed Description
Reference is now made to the accompanying drawings, which form a part hereof, and in which is shown by way of illustration specific aspects in which the invention may be practiced. It is to be understood that other aspects may be utilized and structural or logical changes may be made without departing from the scope of the present invention. The following detailed description is, therefore, not to be taken in a limiting sense, and the scope of the present invention is defined by the appended claims.
It will be appreciated that the same applies to apparatus or systems corresponding to the method for performing the method, and vice versa, in connection with the method described. For example, if a specific method step is described, the corresponding apparatus may comprise means for performing the described method step, even if such means are not elaborated or illustrated in the figures. Further, it is to be understood that features of the various exemplary aspects described herein may be combined with each other, unless specifically noted otherwise.
Fig. 2 shows a schematic diagram of a communication system 200, a
In general, the polar code is formed by a matrixK rows j ofi∈ { 0., N-1} \ F, 0 ≦ i < K, (N ═ 2 ≦ KmK) linear block codes, whereinRepresenting the m-times kronecker product of the matrix S with itself. The encoding algorithm is as follows:whereinAnd the remaining elements of the vector u are set by the payload symbols to be encoded. For example, F is a set of frozen bits constructed as the index l of the low capacity or high error rate composite bit subchannel.
The
The second
As can be seen from fig. 2, the apparatus 202b of the
As described in further detail below, the processing unit 202c of the apparatus 202b of the
The bits or bits are shown in fig. 3The construction of the puncture set is indexed. Having a length NmaxCode sequence S (stage 301) of length NmaxIn the code sequence S (stage 303) of (a) deleting all the sequences S greater than or equal to NmaxIndex of/2. In a further stage 305, from the length NmaxN after deletion in code sequence of/2RAn index. If it is not
Fig. 5 shows how the construction of the set of bit index punctures shown in fig. 3 is implemented in said
In one embodiment, said processing unit 202c of said apparatus 202b of said
NR＝round(ap+b)，
Where round (.) represents a rounding function, a and b represent parameters, which depend on NmaxAnd the predefined code rate R. In one embodiment, the rounding function may be a ceiling function or a floor function. In one embodiment, said processing unit 202c of said apparatus 202b of said
NR＝max(0，round(ap+b))，
Wherein max (. -) represents the maximum function.
In one embodiment, the apparatus 202b of the first communication device 202 (and likewise the
In one embodiment, the processing unit 202c of the apparatus 202b of the
a＝f(R)a′，
b＝f(R)b′，
wherein f (R) represents a function of the predefined code rate R, the auxiliary parameters a 'and b' depending on Nmax。
In one embodiment, the processing unit 202c of the apparatus 202b of the
Nmax
b′
a′
64
32.455
-0.87273
128
77.319
-1.2332
256
138.56
-1.1028
512
275.89
-1.1111
1024
572.83
-1.1649
in one embodiment, the processing unit 202c of the apparatus 202b of the
a＝R2a′，
b＝R2b′。
in one embodiment, the
Also, in one embodiment, the
Some of the embodiments described above are summarized algorithmically below.
According to one embodiment, the scheme implemented in the
inputting:
length NmaxA code sequence S;
the number of punctured bits p;
a look-up table LT (as described above);
the code rate R.
And (3) outputting:
the set of bits P to be punctured.
1. Will S2Is set to be N in sizemaxA vector of/2;
2.j＝0；
i from 0 to Nmax-1：
a. If S [ i ]]＜NmaxAnd/2, then:
i.S2[j]＝S[i]；
ii.j＝j+1。
4. according to NmaxAnd a look-up table LT, setting a ', b';
5.a＝a′R2，b＝b′R2；
6.NRround (ap + b) or NR＝max(0，round(ap+b))；
7.P＝S2[Nmax-NR-p，...，Nmax-NR-1]；
According to one embodiment, the use as implemented in said
inputting:
a bit sequence I of length K;
required codeword length N，
Length NmaxA code sequence S;
a look-up table LT (as described above).
And (3) outputting:
a codeword C of length N.
1. By algorithm 1, the size is determined as p ═ Nmax-N punctured set of bits P;
2.S＝S\P；
3. a set of frozen bits F ═ S [ 0., N-K-1], [ u ] P;
4. use (N)maxK) the polar code encodes the message I into a codeword C, wherein a frozen channel is defined according to the set F;
5.C＝C[{0，...，Nmax-1}\P]。
according to one embodiment, the scheme of decoding the obtained codeword using algorithm 2 as implemented in the
inputting:
a vector L of Log Likelihood Ratios (LLRs) of codeword bits of length N, where
The required information message length K;
length NmaxA code sequence S;
a look-up table LT (as described above).
And (3) outputting:
information message I of length K.
1. By algorithm 1, the size is determined as p ═ Nmax-N punctured set of bits P;
2. mixing L withFIs set to be N in sizemaxAll-zero vector of (2);
3.j＝0；
i from 0 to Nmax-1：
a. If i belongs to P, continuing;
b.LF[i]＝L[j]；
c.j＝j+1；
5.S＝S\P；
6. a set of frozen bits F ═ S [ 0., N-K-1], [ u ] P;
7. use (N)maxK) polarization code to LLR vector LFThe defined codeword is decoded into the information message I, wherein the freeze channel is defined according to the set F.
Fig. 6 shows another embodiment of the communication system 200 of fig. 2. In the embodiment shown in fig. 6, the
The puncturing scheme implemented in embodiments of the present invention allows for the construction of arbitrary length polar codes that exhibit better or similar performance when compared to polar codes generated by rate matching schemes. The corresponding simulation results for code rates 1/3, 1/4 … … 1/8 are given in fig. 7. Fig. 8 shows the results of the case related to ultra-reliable low-latency channel (URLLC) communication.
In fig. 7, each curve corresponds to a certain fixed code rate. The solid curve corresponds to the scheme adopted by the future 5G eMBB control channel. The dashed line corresponds to the puncturing scheme implemented in the embodiments of the present invention. Drawing dimension along X-axis, drawing implementation 10 along Y-axis-3Frame Error Rate (FER). The results are obtained by a CA-SCL decoder with a list size of 8 and a CRC of 11.
In fig. 8, each curve corresponds to a fixed dimension. The solid curve corresponds to the scheme adopted by the future 5G eMBB control channel. The dashed line corresponds to the puncturing scheme implemented in the embodiments of the present invention. Plotting code length along the X-axis while plotting implementation 10 along the Y-axis-5Required for frame error rateSNR of (d). The results are obtained by a CA-SCL decoder with a list size of 8 and a CRC of 24.
Fig. 9 shows a corresponding method 900 for generating a polarization code of length N and dimension K based on an original polarization code defined by a code sequence S having N ordered from least reliable to most reliable sub-channelsmaxA bit index. The method 900 includes the steps of: (a) by deleting N or more from the code sequence SmaxGenerating 901 bit indices with NmaxAn auxiliary code sequence of/2 bit indices; (b) deleting 903 the post-N from the auxiliary code sequenceRBit index to generate a modified auxiliary code sequence, where NRIndicating the number of bits of said erasure, and the number of bits N of said erasureRIs based on NmxxN and a predefined code rate R; (c) by the post-p-N based on the modified auxiliary code sequencemax-puncturing said original polar code by a puncturing set defined by N bit indices, generating 905 a polar code of length N and dimension K.
While a particular feature or aspect of the invention may have been disclosed with respect to only one of several implementations or embodiments, such feature or aspect may be combined with one or more other features or aspects of the other implementations or embodiments as may be desired and advantageous for any given or particular application. Furthermore, to the extent that the terms "includes," "has," "having," or any other variation thereof, are used in either the detailed description or the claims, such terms are intended to be inclusive in a manner similar to the term "comprising" as "comprising" is interpreted. Also, the terms "exemplary," "e.g.," are merely meant as examples, and not the best or optimal. The terms "coupled" and "connected," along with their derivatives, may be used. It will be understood that these terms may be used to indicate that two elements co-operate or interact with each other, whether or not they are in direct physical or electrical contact, or they are not in direct contact with each other.
Although specific aspects have been illustrated and described herein, it will be appreciated by those of ordinary skill in the art that a variety of alternate and/or equivalent implementations may be substituted for the specific aspects shown and described without departing from the scope of the present invention. This application is intended to cover any adaptations or variations of the specific aspects discussed herein.
Although the elements of the above claims below are enumerated in a particular order using corresponding labels, the elements are not necessarily limited to being executed in the particular order described unless the recitation of the claims otherwise implies a particular order for executing some or all of the elements.
Many alternatives, modifications, and variations will be apparent to those skilled in the art in light of the foregoing teachings. Of course, those skilled in the art will readily recognize that there are numerous other applications of the present invention beyond those described herein. While the present invention has been described with reference to one or more particular embodiments, those skilled in the art will recognize that many changes may be made thereto without departing from the scope of the present invention. It is therefore to be understood that within the scope of the appended claims and their equivalents, the invention may be practiced otherwise than as specifically described herein.
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